Entangled State Representations in Quantum Optics 9819923328, 9789819923328

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Xiang-Guo Meng · Ji-Suo Wang · Bao-Long Liang

Entangled State Representations in Quantum Optics

Entangled State Representations in Quantum Optics

Xiang-Guo Meng · Ji-Suo Wang · Bao-Long Liang

Entangled State Representations in Quantum Optics

Xiang-Guo Meng Weifang Medical University Weifang, China

Ji-Suo Wang Qufu Normal University Qufu, China

Bao-Long Liang Liaocheng University Liaocheng, China

ISBN 978-981-99-2332-8 ISBN 978-981-99-2333-5 (eBook) https://doi.org/10.1007/978-981-99-2333-5 Jointly published with Science Press The print edition is not for sale in China mainland. Customers from China mainland please order the print book from: Science Press. Translation from the Chinese Simplified language edition: “Liang Zi Guang Xue Zhong Jiu Chan Tai Biao Xiang De Ying Yong” by Xiang-Guo Meng et al., © Science Press 2020. Published by Science Press. All Rights Reserved. © Science Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

In quantum mechanics, when several particles interact, the properties of each particle have been integrated into the overall nature of a system. So, it is no longer possible to describe the properties of each particle separately, but only the properties of the overall system. This phenomenon is called quantum entanglement. In history, Einstein, Podolsky, and Rosen first proposed the idea of quantum entanglement, while Schrödinger further defined the term “quantum entanglement” and pointed out that it is a characteristic property of quantum mechanics. Quantum entangled state refers to a superposition state of multi-particle (or multidegree of freedom) systems that cannot be expressed as a direct product state. When one particle in the entangled two-particle system is measured, the other particle collapses automatically to a specific state, so new quantum states of light field can be prepared by using quantum entanglement. If multi-particle entangled states (such as graph state and cluster state) can be prepared and manipulated with a series of measurements, thus the manufacture of single-channel quantum computer becomes possible. Therefore, in the existing architecture of quantum computers, quantum entanglement plays a very important role. In addition, the quantum nonlocality reflected by entangled states has become the theoretical basis of quantum information and also makes quantum remote communication possible. However, there is unavoidable environmental noise in the quantum communication channel. This causes the fact that the quality of quantum entangled state continuously decreases with the increase of transmission distance, which means that quantum communication means only stay on short-distance applications. In short, the entanglement of quantum states is very helpful for understanding the fundamental problems of quantum physics, and also provides key physical resources to complete quantum information processing tasks (e.g., quantum key distribution, dense coding, and quantum teleportation). Therefore, the preparation, exploration, and final manipulation of quantum entangled states has become one of the important research tasks in the field of quantum optics. This book has focused on the applications of the continuous-variable entangled state representations in a series of important topics in quantum optics via the integration method within an ordered product of operators (IWOP). As one way to v

vi

Preface

develop Dirac’s symbolic method, the IWOP method made the integration of noncommutative operators possible by arranging non-commutable operators within an ordered product symbol. The continuous-variable entangled state representations originated from the IWOP method can not only deal with many problems in quantum optics but also explore new research topics (e.g., solving quantum master equation, deriving new integral formula, and finding mother wavelet). Based on the continuous-variable entangled state representations and the IWOP method, the main research contents of this book are arranged as follows: Introducing several continuous-variable entangled state representations, integration method within an ordered product of operators and operator ordering method; dealing with dynamics of two-body Hamiltonian systems, quantum theory of mesoscopic circuit systems and master equations of density operators; establishing entangled state representation theory of quantum decoherence in open systems, Wigner operator and Tomography; and proposing several new entangled states, generalized binomial theorems, and multivariable special polynomials. This book establishes a theoretical system for treating important problems of quantum optics by fully using the entangled state representations, in which its methods are novel and original, and its contents are abundance and completion. So, this book is suitable for teachers and students who are interested in quantum mechanics, quantum optics, and quantum information science, and provides reference for researchers in quantum physics. The completion of this book has received long-term support from the National Natural Science Foundation of China (Project’s numbers: 10574060, 11147009, 11244005, 11347026) and the Natural Science Foundation of Shandong Province (Project’s numbers: ZR2010AQ027, ZR2012AM004, ZR2013AM012, ZR2016AM03, ZR2017MA011, ZR2020MA085, ZR2020MF113). We would especially like to thank Professor Wang Jisuo of Qufu Normal University, who is the leader of our research on quantum theory, and Professor Fan Hongyi of the University of Science and Technology of China, for coming to Weifang from thousands of miles to personally review this book. At the same time, we would like to express our gratitude to Prof. Yang Zhenshan and Associate Professor Zhang Zhentao of Liaocheng University, Profs. Hu Liyun and Xu Xuexiang of Jiangxi Normal University, Prof. Yuan Hongchun of Changzhou Insitute of Technology, Associate Professor Wang Zhen of Changzhou Vocational Institute of Mechatronic Technology, Associate Professor Wang Shuai of Jiangsu University of Technology, and Profs. Xu Xinglei and Xu Shimin of Heze University for their encouragement and assistance. We have benefited greatly from our interaction with many of our colleagues, friends, and students in the course of writing this book. Finally, we are grateful to our family members for their support and understanding. Weifang, China Qufu, China Liaocheng, China

Xiang-Guo Meng Ji-Suo Wang Bao-Long Liang

Contents

1 Integration Method Within an Ordered Product of Operators and Continuous-Variable Entangled State Representations . . . . . . . . . 1.1 Integration Method Within an Ordered Product of Operators . . . . . . 1.1.1 Normal Ordering Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Anti-normal Ordering Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Weyl Ordering Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Two-Particle Entangled State Representation . . . . . . . . . . . . . . . . . . . 1.3 Thermal Entangled State Representation . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Bosonic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fermionic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dynamics of Two-Body Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . 2.1 Energy-Level Distribution and Wave Function of Two-Body Hamiltonian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Two Moving Charged Particles with Elastic Coupling . . . . . 2.1.2 Two Moving Charged Particles with Coulomb Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Path Integral Theory in Entangled State Representation . . . . . . . . . . 2.3 Evolution of Atomic Coherent State Governed by the Hamiltonian f (t)J+ + f ∗ (t)J− + g(t)Jz . . . . . . . . . . . . . . . . 2.4 Atomic Coherent States as Energy Eigenstates of the Hamiltonian for Two-Dimensional Anisotropic Harmonic Potential in a Uniform Magnetic Field . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 New Bipartite Entangled States in Two-Mode Fock Space . . . . . . . . . . 3.1 Coherent-Entangled States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Entangled States Describing Parametric Down Conversion . . . . . . . 3.3 Parameterized Entangled States Induced by the Common Eigenstates of a † + ib and a− ib† . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 6 8 14 18 18 20 23 25 26 27 32 36 46

54 61 63 64 72 78

vii

viii

Contents

3.4 Parameterized Entangled States as the Common Eigenstates of AQa + BPb and CQb + DPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86 92

4 Solutions of Density Operator Master Equations . . . . . . . . . . . . . . . . . . 4.1 Solutions of Several Boson Master Equations . . . . . . . . . . . . . . . . . . . 4.1.1 Diffusion Master Equation Under Linear Resonance Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Master Equation for Single-Mode Cavity Driven by Oscillating External Field in a Heat Reservoir . . . . . . . . . 4.1.3 Master Equation for Damped Harmonic Oscillator Acted by Linear Resonance Force in a Squeezed Heat Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Master Equation Describing a Diffusive Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Solutions of Several Fermi Quantum Master Equations . . . . . . . . . . 4.2.1 Fermi Master Equations for Amplitude Damping and Phase Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Master Equation for Fermi Heat Reservoir . . . . . . . . . . . . . . . 4.3 Generation of Displaced Thermal State . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 122 126 129

5 Wigner Distribution Function and Quantum Tomogram via Entangled State Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Wigner Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Wigner Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Wigner Distribution Functions for Quantum States . . . . . . . . 5.2 Quantum-state Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Quantum Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Quantum Tomograms of Quantum States . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 132 132 137 152 152 156 164

6 Evolution and Decoherence of Quantum States in Open Systems . . . . 6.1 Evolution of Quantum States in the Amplitude Damping Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Displaced Thermal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Thermal-State Superpositions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Two-Mode Squeezed Vacuum States . . . . . . . . . . . . . . . . . . . . 6.2 Evolution of Quantum States in the Laser Process . . . . . . . . . . . . . . . 6.2.1 Squeezed Number States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Squeezed Thermal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Multi-photon Subtracted Squeezed Vacuum States . . . . . . . . 6.2.4 Multi-photon Added Two-Mode Squeezed Thermal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 96 96 100

105 108 117

167 167 169 175 180 188 192 200 205 211 224

Contents

7 Generalized Binomial Theorems and Multi-variable Special Polynomials Involving Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . 7.1 Generalized Binomial Theorems Involving Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Multi-variable Special Polynomials and Their Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Three-variable Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Six-variable Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 New Operator Identity and Integral Formula . . . . . . . . . . . . . 7.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Quantum Theory of Mesoscopic Circuit Systems . . . . . . . . . . . . . . . . . . 8.1 Quantum Theory of Mesoscopic LC Circuits . . . . . . . . . . . . . . . . . . . 8.1.1 Single Mesoscopic LC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Mutual Inductance Coupling Mesoscopic LC Circuit . . . . . . 8.2 Quantum Theory of Mesoscopic Circuits Containing Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Mutual-Inductance Coupling Mesoscopic Circuit Including Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Mesoscopic LC Circuit with Josephson Junction Coupled by Mutual Inductance . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

225 226 237 237 239 241 243 253 255 256 256 260 272 272 276 282

Chapter 1

Integration Method Within an Ordered Product of Operators and Continuous-Variable Entangled State Representations

In this chapter, we first review the basic properties of normal ordering, anti-normal ordering, and Weyl ordering of Boson operators and the integration method within an ordered product of operators (IWOP), and then we expound the basic theory of two-body entangled state representations with continuous variables.

1.1 Integration Method Within an Ordered Product of Operators 1.1.1 Normal Ordering Case For an operator function F(a, a † ) related to Boson operators a and a † F(a, a † ) =

 i

···



a †i a j a †k · · · a n F(i, j, k, . . . , n),

(1.1)

n

where i, j, k, . . . , n are zero or positive integers, we can always move all the creation operators a † to the left of all the annihilation operators a by using the commutation relation a, a † = 1, and thus F(a, a † ) is arranged in the normal ordering product form. The main properties of normal ordering product are [1]. (I) The Boson operators in the normal ordering product symbol : : are commutative with each other. (II) c-numbers can freely enter and exit the normal ordering product symbol : : . (III) Normal product symbol : : in normal ordering product can be canceled, for example,   : H(a, a † ) : F(a, a † ) : : = : H(a, a † )F(a, a † ) : , a †m : F(a, a † ) : a n = : a †m F(a, a † )a n : . © Science Press 2023 X.-G. Meng et al., Entangled State Representations in Quantum Optics, https://doi.org/10.1007/978-981-99-2333-5_1

(1.2) 1

2

1 Integration Method Within an Ordered Product …

(IV) Hermite conjugate operation can freely enter and exit the symbol : : , i.e., : (V · · · W) :

†

= : (V · · · W)† : .

(1.3)

(V) The sum or difference of operator functions in normal ordering can be splitted, i.e., (1.4) : H(a, a † ) ± F(a, a † ) : = : H(a, a † ) : ± : F(a, a † ) : . However, the product of normal ordering operators is no longer a normal ordering form. (VI) Within the normal ordering of the Boson operator function F(a, a † ), it satisfies   ∂ F(a, a † ) : , a, : F(a, a † ) : = : ∂a †



 ∂ F(a, a † ) : . (1.5) : F(a, a † ) : , a † = : ∂a

The above relations are also applicable to the multimode cases, i.e., :

   ∂ ∂ F(ai , a j , ai† , a †j ) : = : F(ai , a j , ai† , a †j ) : , a †j , ai† . ∂ai ∂a j

(1.6)

(VII) The normal ordering of projection operator of vacuum is   |0 0| = : exp −a † a : .

(1.7)

A strict proof of the formula (1.7) is given as below. Letting |0 0| = : f (a † , a) : ,

(1.8)

and operating the coherent state |z on both sides of Eq. (1.8), thus we obtain   f (z ∗ , z) = |0 |z|2 = exp −zz ∗ .

(1.9)

By comparing Eqs. (1.8) and (1.9), we easily have Eq. (1.7). In addition, according to the completeness relation of number states |n, we obtain 1=

∞ 

|n n|

n=0 ∞ 

 1 d n ∗ n 

|n n  √ (z ) ∗ z =0 n!n  ! dz ∗ n,n  =0 

∂ = exp a † ∗ |0 0| exp(z ∗ a) z ∗ =0 . ∂z

=

(1.10)

1.1 Integration Method Within an Ordered Product of Operators

3

Supposing that the normal ordering of |0 0| is : G : , thus from Eq. (1.10) we have 

† ∂ ∗

: G : exp(z a) . 1 = exp a z ∗ =0 ∂z ∗

(1.11)

Because the left side of : G : is all the creation operator a † , while its right side is all annihilation operator a, we can move the parts on the left and right sides of : G : to the normal ordering product symbol : : , and then using the properties (I) and (III), we have 

∂ 1 = : exp a † ∗ G exp(z ∗ a) : z ∗ =0 ∂z = : exp(a † a)G : = : exp(a † a) : G : : ,

(1.12)

which also leads to the identity   : G : = : exp −a † a : = |0 0| .

(1.13)

(VIII) When the integration is convergent, the c number in normal ordering can be integrated or differentiated. The properties

(VII) and (VIII) show that if the asymmetric ket-bra operator integration (e.g., dx | f (x) x|) can be converted into its normal ordering product, thus we can integrate the real integral parameters since Boson operators are commutative in the normal ordering symbol : : . Obviously, there are the symbol : : in the integration and in the result after the integration. If we want to remove the symbol : : from the final integral result, we arrange it in normal ordering form. This is the normal ordering case of the IWOP method. In history, Dirac used the symbols of his own invent and delta function to derive the

completeness relation of the eigenstates |q of the coordinate operator Q, that is, dq |q q| = 1, while Professor Fan Hong-Yi (University of Science and Technology of China) used the expansion of |q in Fock space |q = π

−1/4

 2 √ q a †2 † |0 exp − + 2qa − 2 2

(1.14)

to obtain the

normal ordering product of the density operator |q q|, thus the completeness dq |q q| = 1 can be rewritten as the pure Gaussian integral form, i.e., 

∞ −∞

  dq √ : exp −(q − Q)2 : = 1. π

(1.15)

Further, using the IWOP method, the integration for the following asymmetric ket-bra operator

4

1 Integration Method Within an Ordered Product …



∞

−∞

 dq

q √ q| μ μ

(1.16)

is completed, where μ > 0. The detailed derivation is as follows: using Eqs. (1.7) and (1.14), thus Eq. (1.16) becomes 

∞ −∞

  √

  ∞ q2 dq

q dq 2q † a †2 a − √ q| = √ exp − 2 + μ μ πμ 2μ μ 2 −∞  2 √  †  q a2 . × : exp −a a : exp − + 2qa − 2 2

(1.17)

  Noting that the left side of : exp −a † a : is all the creation operator a † , while its right side is all annihilation operator a, we can move the parts on the left and   right sides of : exp −a † a : to the normal ordering product symbol : : , thus the integral operator function can be arranged into its normal product form. Because Boson operators are commutative in the symbol : : , three ex p exponent functions in Eq. (1.17) can be rewritten as a ex p exponential function, that is, 

∞

−∞

  2  ∞ 1 dq

q dq q 1+ 2 √ q| = √ : exp − μ μ πμ 2 μ −∞   † √ 1 a + 2q + a − (a + a † )2 : . μ 2

(1.18)

Further, the Boson operators a † , a are regarded as two parameters in normal ordering product, so using the property (VIII) to complete the integration in Eq. (1.18), we have

  ∞ dq

q √ q| μ μ −∞   †2 √ a a2 tanh r + (sech r − 1)a † a + tanh r : , (1.19) = sech r : exp − 2 2 where er = μ, sech r =

μ2 − 1 2μ , tanh r = . μ2 + 1 μ2 + 1

(1.20)

To get rid of the normal ordering symbol : : in Eq. (1.19), we first use the properties (I)–(V) and (VIII) to derive the operator identity that is exp(λa † a) =

∞ 

exp(λn) |n n| =

n=0

=

∞  n=0

:

∞ 

an a †n exp(λn) √ |0 0| √ n! n! n=0

1 λ † n (e a a) exp(−a † a) : = : exp[(eλ − 1)a † a] : . n!

(1.21)

1.1 Integration Method Within an Ordered Product of Operators

5

So, based on Eqs. (1.21), (1.19) can be rewritten as

 dq

q √ q| μ μ −∞    2  †2 a a 1 † tanh r exp a a + ln sech r exp tanh r , = exp − 2 2 2



∞

(1.22)

which is just the standard single-mode squeezing operator, denoted as S1 (r ). If the left and right sides of Eq. (1.22) are differentiated at the same time, it is obtained that ∂ ∂r



∞ −∞

dq √ k



q  r   ∞ dq

q 

2 †2 a −a √ q| .

q| = k 2 k k −∞

(1.23)

And using the following boundary conditions 

∞

−∞

we therefore have 

∞

−∞

dq √ k

dq √ k

q 



= 1,

q|

k k=1

q  r  

a 2 − a †2 ≡ S1 (r ).

q| = exp k 2

(1.24)

(1.25)

So, we can say that Eq. (1.22) is just the expansion of single-mode operator

squeezing  S1 (r ). Using the nature of δ function in the inner product q q  = δ(q − q  ), we easily prove that S1 (r ) is a unitary operator, i.e.,  S1 (r )S1† (r ) =  =

∞

dqdq  μ −∞

∞

−∞

 

q q





μ μ δ(q − q )

dq |q q| = 1 = S1† (r )S1 (r ).

(1.26)

Further, using the Baker-Hausdorff operator formula e A Be−A = B + [A, B] +

1 1 [A, [A, B]] + [A, [A, [A, B]]] + · · · , 2! 3!

(1.27)

we also derive the famous Bogoliubov transform (or squeezed transformation), i.e., S1 (r )aS1† (r ) = a cosh r + a † sinh r, S1 (r )a † S1† (r ) = a † cosh r + a sinh r.

(1.28)

The above results show that, starting from the asymmetric ket-bra operator constructed by Dirac’s coordinate eigenstates, we give the unitary operator S1 (r ) (i.e.,

6

1 Integration Method Within an Ordered Product …

squeezing operator) that leads to the Bogoliubov transformation and its normal ordering product by using the IWOP method. Besides, combining Eqs. (1.16) and (1.28) leads to the fact that quantum unitary transformations S1 (r )Q S1† (r ) = μQ, S1 (r )P S1† (r ) = P/μ can be mapped by classical scale transformation q → q/μ in phase space. That is, a new way of direct transition from classical canonical transformation to quantum unitary transformation is found by using the IWOP method and some useful quantum mechanical transformations are revealed from Dirac’s basic representation.

1.1.2 Anti-normal Ordering Case Different from the rule of normal ordering of operators, anti-normal ordering product requires that all the annihilation operators a to the left of all the creation operators .. a † , marked with the symbol .. ... However, the basic properties of the Boson operators .. in the symbol .. .. are very similar to those in the normal product symbol : : , i.e., (I) The Boson operators are commutative in the normal ordering product .. symbol .. ... .. (II) The normal ordering product symbol .. .. is allowed free access of c numbers. . . (III) Anti-normal product symbol .. .. in anti-normal ordering product can be canceled. (IV) c numbers in anti-normal ordering product can be integrated if the integration is convergent. (V) Anti-normal ordering product of the projection operator |0 0| is  |0 0| = πδ(a)δ(a ) = †

d2 ξ iξa iξ ∗ a † e e . π

(1.29)

The detailed proof of Eq. (1.29) is as follows. By using the normal ordering of projection operator of vacuum in Eq. (1.7) and the mathematics integral formula 

   ξη d2 z 1 exp ζ |z|2 + ξz + ηz ∗ = − exp − , π ζ ζ

which holds for Re ζ < 0, we give

(1.30)

1.1 Integration Method Within an Ordered Product of Operators

7



∗

d2 ξ −iξ(z−a) −iξ ∗ (z ∗ −a † ) e e π  2   d ξ : exp − |ξ|2 − iξ(z − a) − iξ ∗ (z ∗ − a † ) : = π   = : exp − |z|2 + z ∗ a + za † − aa † : = |z z| , (1.31)

πδ(z − a)δ(z − a ) = †

where |z is the coherent state. When z = 0, Eq. (1.31) reduces to Eq. (1.29). For a given density operator ρ, its diagonal coherent state representation is expressed as (Glauber-Sudarshan P-representation) [2],  ρ=

d2 z P(z) |z z| . π

(1.32)

Noting that the eigenstate equations of the coherent state |z are a |z = z |z and z| a † = z| z ∗ , so inserting the completeness relation of coherent states |z into the anti-normal ordering product of the density operator ρ leads to its P-representation. If substituting Eq. (1.31) into Eq. (1.32), we obtain the P-representation of the density operator, which has the following normal ordering form:  ρ=

    d2 z P(z) : exp − z ∗ − a † (z − a) : . π

(1.33)

Besides, Mahta once gave an integral formula of P-representation the density operator ρ [3], that is, |z|2

P(z) = e



  d2 α −α| ρ |α exp |α|2 + α∗ z − αz ∗ , π

(1.34)

where |α is also a coherent state. Noting that α∗ z − αz ∗ is a pure imaginary number, so Eq. (1.34) can be regarded as Fourier transform. Thus, using Eqs. (1.34), (1.33) can be rewritten as   2   . d2 ξ d α |α|2 . −α| ρ |α e . exp iξa + iξ ∗ a † ρ= π π  2       . d z × exp − |z|2 + z a † − iξ + α∗ exp z ∗ a − iξ ∗ − α ... (1.35) π Further, using the IWOP method to integrate the variables z and ξ, respectively, we derive the new formula that converts the density operator ρ into its anti-normal ordering product, that is,  ρ=

.  . d2 α −α| ρ |α .. exp |α|2 + α∗ a − αa † + a † a ... π

(1.36)

8

1 Integration Method Within an Ordered Product …

Equation (1.36) shows that, if the normal ordering of the density operator ρ is known, we can easily calculate the matrix element −α| ρ |α and then obtain the antinormal ordering product of ρ via performing integration over the variable α in the .. symbol .. ... This gives a new way to transform the normal ordering product of the operator ρ into its anti-normal ordering product. † For example, substituting the normal ordering of the exponential operator eλa a into Eq. (1.36) and using the IWOP method, the anti-normal ordering product of the † operator eλa a reads eλa

†



a

.  . d2 α −α| : exp[(eλ − 1)a † a] : |α .. exp |α|2 + α∗ a − αa † + a † a .. π  2 .  d α .. = . exp −eλ |α|2 + α∗ a − αa † + a † a .. π .  . = e−λ .. exp (1 − e−λ )a † a ... (1.37) =

In particular, for ρ = 1, Eq. (1.36) becomes  1=

.  d2 α .. . exp − |α|2 + α∗ a − αa † + a † a ... π

(1.38)

1.1.3 Weyl Ordering Case Noting that the coordinate operator Q and the momentum operator P are not commutative, so the quantum mechanics operator corresponding to the classical function h( p, q) is always uncertain. For this reason, physicists must give a corresponding rule whose correctness needs to be verified by experiments. Next, we introduce a correspondence rule widely used in path integral theory and present its main properties. The Weyl correspondence rule can also be said to be a Weyl ordering, so we can obtain a new formula for calculating the Weyl ordering product of the density operator ρ. Noting that

 ∂ q| P q  = −i δ(q − q  ) = ∂q



∞

−∞

 q + q q + q q| Q q  = δ(q − q  ) = 2 2 so Weyl gives a corresponding rule, that is,

d p i p(q−q  ) pe , 2π 

∞

−∞

d p i p(q−q  ) e , 2π

(1.39)

(1.40)

1.1 Integration Method Within an Ordered Product of Operators

 q| H(P, Q) q  =



∞

−∞

dp h 2π

 p,

q + q 2

9





ei p(q−q ) .

(1.41)

The specific correspondence the operator function H(P, Q) and  between    relation the classical function h p, q + q  /2 is derived as follows. For this, inserting the completeness relation of the coordinate eigenstates |q into Eq. (1.41) and letting q − q  = t, we have  dp q + q  i p(q−q  ) H(P, Q) = dq h p, e 2 −∞ −∞ −∞ 2π



   ∞

t t

1 (1.42) q −

, = dtd pdqh ( p, q) ei pt

q + 2π 2 2 −∞ 

∞





∞



dq |q q 



∞

where the states |q ± t/2 are eigenstates of the coordinate operator Q. Further, inserting the single-mode Wigner operator in the coordinate representation, that is, 

∞

−∞

 t dt

q+ q− 2π

2

t

it p e =  ( p, q) = † ( p, q) 2

(1.43)

into Eq. (1.42), thus the Weyl correspondence rule can be expressed as H(P, Q) =

1 2π



∞ −∞

d pdqh ( p, q)  ( p, q) ,

(1.44)

which shows that the operator function H(P, Q) and its corresponding classical function h ( p, q) are related to each other with the single-mode Wigner operator  ( p, q). Therefore, for a given operator H(P, Q), its corresponding classical function reads h ( p, q) = 2πtr[H(P, Q) ( p, q)].

(1.45)

The Weyl rule implies a sort of operator ordering, called Weyl ordering. The Weyl corresponding operator of the classical function q m p n is q m pn →

m  1  m Q m−l P n Q l , 2m l=0 l

(1.46)

where the item on the right is the Weyl ordering that is different from other operator ordering cases, such as coordinate Q-momentum P ordering q m p n → Q m P n , momentum P-coordinate Q ordering q m p n → P n Q m . Naturally, such a problem arises: “what is the Weyl ordering expansion of the density operator ρ?”. If the symbol :: :: is used to mark the Weyl ordering of operators, thus Eq. (1.44) can be rewritten as  ∞ : : d pdqh ( p, q)  ( p, q) , (1.47) h(P, Q) = : : −∞

10

1 Integration Method Within an Ordered Product …

which shows that the classical correspondence h ( p, q) of a Weyl ordering operator :: h(P, Q) :: can be obtained directly by making the substitutions Q → q, P → p in the operator :: h(P, Q) :: . For example, Eq. (1.46) represents the following Weyl classical correspondence: m  m  : : 1  m 1  m m−l n l P Q = Q Q m−l P n Q l m m 2 l=0 l : : 2 l=0 l   ∞ m 1  m d pdq m = q m p n  ( p, q) 2 l=0 l −∞  ∞ = d pdqq m p n  ( p, q) . (1.48) −∞

Besides, using the relations between the operators Q, P and Boson operators a, a † , i.e., a + a† a − a† q + ip Q= √ , P= √ , α= √ , (1.49) 2 i 2 2 Equations (1.44) and (1.47) can be rewritten as : : f (a, a † ) = 2 : :



    d2 α f α, α∗  α, α∗ = F(a, a † ),

(1.50)

where the classical corresponding function for the operator F(a, a † ) is     f α, α∗ = 2πtr[F(a, a † ) α, α∗ ],

(1.51)

where  (α, α∗ ) is the Wigner operator in the coherent representation, its normal ordering form reads       1  α, α∗ = : exp −2 a † − α∗ (a − α) : . π

(1.52)

The main properties of Boson operators in Weyl ordering are listed below: (I) The Boson operators are commutative within the symbol :: :: . (II) The symbol :: :: can be removed in the symbol :: :: . (III) If the integration converges, c numbers in the symbol :: :: can be integrated. (IV) c numbers can freely enter and exit the symbol :: :: . (V) The Weyl ordering form of the projection operator |0 0| is |0 0| =

: −2a † a : . 2e : :

(1.53)

1.1 Integration Method Within an Ordered Product of Operators

11

From the properties (I)-(V), we obtain the Weyl ordering of the Wigner operator  ( p, q) as : :  ( p, q) = δ( p − P)δ(q − Q) (1.54) : : or

 1:  : δ(α − a)δ(α∗ − a † ) .  α, α∗ = 2: :

(1.55)

So, substituting Eqs. (1.54) and (1.55) into Eqs. (1.47) and (1.50), respectively, we have  ∞ : : : : d pdqh ( p, q) δ( p − P)δ(q − Q) , (1.56) h(P, Q) = : : : : −∞ or

: : f (a, a † ) = : :



 : : d2 α f α, α∗ δ(α − a)δ(α∗ − a † ) . : :

(1.57)

For example, the Weyl ordering form of the operator corresponding to the classical function q m p n is 

∞

d pdqq m p n

−∞

To remove the symbol

: : : :

in

: :

: : : : δ( p − P)δ(q − Q) = Q m P n . : : : :

(1.58)

Q m P n :: , it must be rearranged as follows:

m  : 1  m : Q m−l P n Q l . : 2m l=0 l :

(1.59)

Based on Eqs. (1.51) and (1.52), the classical correspondence function for the coherent state projection operator |z z| is       2πtr[|z z|  α, α∗ ] = 2 z| : exp −2 a † − α∗ (a − α) : |z     = 2 exp −2 z ∗ − α∗ (z − α) .

(1.60)

Inserting Eq. (1.60) into Eq. (1.57) leads to the Weyl ordering of |z z|, that is, 

 :   : d2 α exp −2 z ∗ − α∗ (z − α) δ(α − a)δ(α∗ − a † ) : :  :   ∗ : † . (1.61) = 2 exp −2 z − a (z − a) : :

|z z| = 2

When z = 0, Eq. (1.61) becomes the Weyl ordering of the projection operator |0 0|. Thus, using Eq. (1.61) and the IWOP method, we can obtain the completeness relation of coherent states |z, i.e.,

12

1 Integration Method Within an Ordered Product …



d2 z |z z| = 2 π



 :   d2 z : exp −2 z ∗ − a † (z − a) = 1. π : :

(1.62)

Further, using Eq. (1.61), the Weyl ordering of P-representation of the density operator ρ reads 

d2 z P(z) |z z| π  2    : : d z P(z) exp −2 z ∗ − a † (z − a) . =2 π : :

ρ=

(1.63)

It can be seen that if the P-representation of the density operator ρ is known, the Weyl ordering of the operator ρ can be obtained via the formula (1.63) and the IWOP method. For example, substituting the P-representation of the exponential † operator eλa a into Eq. (1.63), we easily obtain λa † a

e



  d2 z exp (1 − e−λ ) |z|2 |z z| π  2  :   d z: exp − 1 + e−λ |z|2 + 2z ∗ a + 2za † − 2a † a = 2e−λ π : :   λ : 2(e − 1) † 2 : exp a a = λ . (1.64) e +1 : eλ + 1 : −λ

=e

So, inserting Eq. (1.34) into Eq. (1.63), we obtain the formula of calculating the Weyl ordering of the operator ρ as 



 d2 α −α| ρ |α exp |α|2 π  :  ∗ ∗ ∗ +α z − αz − 2 z − a † (z − a) :  2 : : d α ∗ −α| ρ |α exp[2(α a − αa † + a † a)] . = 2 π : :

ρ=2

d2 z |z|2 : e π :

(1.65)

As can be seen from Eq. (1.65), if the normal ordering of the density operator ρ is known, that is, the matrix elements of coherent states (−α| ρ |α) can be known, thus the integration can be performed within the symbol :: :: and the Weyl ordering of the operator ρ can be obtained. The invariance of Weyl ordering operators under similar transformation is introduced as follows. Introducing an operator S that causes the following similarity transformations SaS −1 = μa + νa † ,

Sa † S −1 = σa + τ a † ,

(1.66)

1.1 Integration Method Within an Ordered Product of Operators

13

  where μτ − σν = 1 and μa + νa † , σa + τ a † = 1, thus for the coherent state representation of Wigner operator     1  α, α∗ = : exp[−2 a † − α∗ (a − α)] : π  1 = d2 z exp[z(a † − α∗ ) − z ∗ (a − α)], 2π 2

(1.67)

its form under the similar transformation is        1 1 2 |z|2 + z σa + τ a † − α∗ S α, α∗ S −1 = z : exp − σν + d 2 2π 2      1 στ z 2 + μνz ∗2 : −z ∗ μa + νa † − α + 2  †   1 = : exp[−2 a − μα∗ + σα a − τ α + να∗ ] : π  =  τ α − να∗ , μα∗ − σα . (1.68) Further, making a similarity transformation on Eq. (1.50), and using Eqs. (1.55) and (1.68), we have   S F a, a † S −1 = 2



    d2 α f α, α∗ S α, α∗ S −1

      2 d2 α f α, α∗ : exp[−2 a † − μα∗ + σα a − τ α + να∗ ] : π       2 = d2 α f μα + να∗ , σα + τ α∗ : exp[−2 a † − α∗ a − α ] : π    : : = d2 α f μα + να∗ , σα + τ α∗ δ(α − a)δ(α∗ − a † ) : :  :  † † : . (1.69) = f μa + νa , σa + τ a : :

=

On Comparing Eqs. (1.50) and (1.69), we have : : :  :  S f a, a † S −1 = f SaS −1 , Sa † S −1 . : : : :

(1.70)

Equation (1.50) shows that the similarity transformation operator S can freely enter and exit the Weyl ordering symbol :: :: , which is just the invariance of the Weyl ordering operator under the similarity transformation. To sum up, the essence of the IWOP method is to popularize the Newton Leibniz’s integration for ordinary functions and realize the integration of operator functions (e.g., asymmetric ket-bra operators). It not only reveals the internal beauty of the mathematical structure of quantum mechanics, but also develops the representation and transformation theory of quantum mechanics. More importantly, the continuous-

14

1 Integration Method Within an Ordered Product …

variable entangled state representation can be established in Fock space by using the IWOP method, so that the quantum entanglement phenomenon of the system can be revealed more clearly.

1.2 Two-Particle Entangled State Representation In history, Einstein, Podolsky, and Rosen (EPR) first introduced the concept of quantum entanglement. It is a unique phenomenon of quantum mechanics that reflects the quantum correlation between the parts of two or more bodies [4]. For any twoparticle entangled systems, since their relative coordinate Q a − Q b (the coordinate of the center of mass is q0 ) and the total momentum Pa + Pb (the eigenvalue is p0 ) are commutative, they have the following common eigenstates, i.e., φ(qa , pa ; qb , pb ) = δ(qa − qb + q0 )δ( pa + pb ).

(1.71)

Later, Professor Fan found the common eigenstates of the operators Q a − Q b and Pa + Pb in two-mode Fock space [5–9], denoted as the continuous-variable entangled state |η, that is,  1 |η = exp − |η|2 + ηa † − η ∗ b† + a † b† |00 . 2

(1.72)

where η = η1 +iη2 , and a † , b† are Boson creation operators of two particles. In fact, the correctness of the state |η can be verified via the IWOP method. First, assuming that the state |η can satisfy the following eigenstate equations: (Q a − Q b ) |η =

√

2η1 |η , (Pa + Pb ) |η =

√ 2η2 |η .

(1.73)

Inspired by the above eigenstate equations and the IWOP method, the pure Gaussian integration in normal ordering product is constructed as follows:  1=

    d2 η Qa − Qb 2 Pa + Pb 2 : exp − η1 − − η2 − √ :. √ π 2 2

(1.74)

 √  √   Owing to Q i = i + i † / 2, Pi = i − i † /(i 2) (i = a, b), thus Eq. (1.74) is further decomposed into  1 d2 η exp − |η|2 + ηa † − η ∗ b† + a † b† π 2   †  1 × : exp −a a − b† b : exp − |η|2 + η ∗ a − ηb + ab . 2 

1=

(1.75)

1.2 Two-Particle Entangled State Representation

15

Using the normal ordering of the projector operator |00 00|, i.e.,   |00 00| = : exp −a † a − b† b :

(1.76)

to express Eq. (1.75) as the completeness relation of the states |η, that is, 

d2 η |η η| . π

1=

(1.77)

So, the states |η are capable of forming a useful representation for describing continuous entangled systems. Operating the annihilation operators a and b on the states |η, respectively, we have   a |η = η + b† |η ,

  b |η = −η ∗ + a † |η ,

(1.78)

which leads to √    1  a + a † − b + b† |η = 2η1 |η = (Q a − Q b ) |η , √ 2 √    1  a − a † + b − b† |η = 2η2 |η = (Pa + Pb ) |η . √ i 2

(1.79)

It is clearly seen that the states |η are indeed √ the common eigenstates of the √ operators Q b and Pa + Pb , and the real part 2η1 and the imaginary part 2η2 of the Qa − √ plural 2η are, respectively, the eigenvalues of the operators Q a − Q b and Pa + Pb . Using Eq. (1.78), we obtain η| (a † − b) = η ∗ η| ,

η| (b† − a) = −η η| .

(1.80)

Combining Eqs. (1.78) and (1.80) leads to 

η (a − b† ) |η = η η  |η = η  η  |η , 

η (b − a † ) |η = −η ∗ η  |η = −η ∗ η  |η .

(1.81)

So it is not difficult to prove the orthogonality of the state |η, that is,  η |η = πδ(η − η  )δ(η ∗ − η ∗ ) ≡ πδ (2) (η  − η).

(1.82)

The entanglement property of the state |η can be further explained via its Schmidt decomposition in the coordinate representation or momentum representation, that is, −iη1 η2 /2

|η = e



∞

−∞

dq |qa ⊗ |q − η1 b e−iqη2 ,

(1.83)

16

1 Integration Method Within an Ordered Product …

or |η = e−iη1 η2 /2



∞

−∞

d p | p + η2 a ⊗ |− pb e−i pη1 .

(1.84)

From Eqs. (1.83) and (1.84), we find that when particle a is measured in the coordinate eigenstate |qa (or momentum eigenstate | p + η2 a ), then particle b automatically collapses to the coordinate eigenstate |q − η1 b (or momentum eigenstate |− pb ), this is the entanglement property of two particles. In terms of the definition of the generation function of two-variable Hermite polynomials Hm,n (x, y), i.e., ∞    t m t n Hm,n (x, y) = exp −tt  + t x + t  y , m!n! m,n=0

(1.85)

we can expand the state |η as |η = e−|η|

2

/2

∞  (−)n Hm,n (η, η ∗ ) |m, n , √ m!n! m,n=0

(1.86)

√ where |m, n = a †m b†n / m!n! |0, 0 is a two-mode Fock state, and Hm,n (η, η ∗ ) yields     (1.87) H∗m,n η, η ∗ = Hn,m η, η ∗ . Besides, for the asymmetric ket-bra operator integration IWOP method to integrate over all values of η leads to

d2 η μπ

|η/μ η|, using the

 d2 η

η η| μπ μ  †     2 |η|2 a b† 1 d η : exp − 1+ 2 +η − b + η∗ a − = μπ 2 μ μ μ  † † † † +a b + ab − a a − b b :   †     †  2μ μ2 a b† † = 2 : exp 2 −b a− − a−b a −b : μ +1 μ +1 μ μ   = sech r exp(a † b† tanh r ) : exp[ a † a + b† b (sech r − 1)] : exp(−ab tanh r )   = exp(a † b† tanh r ) exp[ a † a + b† b − 1 ln sech r ] exp(−ab tanh r ), (1.88)



where the relations between the parameters μ and r can be found in Eq. (1.20). So, it is just a two-mode squeezed operator, denoted as S2 (r ), with the following compact form:  2  d η

η η| = exp[r (a † b† − ab)]. (1.89) S2 (r ) = μπ μ

1.2 Two-Particle Entangled State Representation

Clearly, (μπ)−1

17

 d2 η |η/μ η| is the natural expression of two-mode squeezing

operator in entangled state |η representation. This also theoretically explains the fact that the two-mode squeezed state itself is an entangled state. Therefore, the operator S2 (r ) can naturally squeeze the entangled state |η as

 d2 η 

η  

η η S2 (r ) |η = μπ μ

 2   d η

η (2)  = δ (η − η) μ μ

 1 η =

. μ μ 

(1.90)

As the conjugate state of the entangled state |η, the common eigenstates |ζ describing the mass-center coordinate Q a + Q b and the relative momentum Pa − Pb of two particles have the following form in two-mode Fock space [8]:  1 2 † ∗ † † † |ζ = exp − |ζ| + ζa + ζ b − a b |00 . 2

(1.91)

It yields the eigenstate equations (Q a + Q b ) |ζ = or

√ √ 2ζ1 |ζ , (Pa − Pb ) |ζ = 2ζ2 |ζ

    a + b† |ζ = ζ |ζ , a † + b |ζ = ζ ∗ |ζ ,

(1.92)

(1.93)

and the complete orthogonal relations 

d2 ζ |ζ ζ| = 1, ζ  |ζ = πδ (2) (ζ  − ζ), π

(1.94)

where ζ is complex, ζ = ζ1 +iζ2 . Also, the inner product of the conjugate entangled states η| and |ζ is   1 1 ∗ ∗ η |ζ = exp (ζη − ζ η) (1.95) 2 2 and the Fourier transform relation reads  2 d η ∗ ∗ |ζ = |η eζη −ζ η . π

(1.96)

The introduction of these two continuous-variable entangled state representations not only enriches the mathematical basic theory of quantum mechanics, but also promotes the development of quantum optics, information optics, Fourier optics,

18

1 Integration Method Within an Ordered Product …

condensed matter physics, etc. For example, in order to clearly reflect the properties of entangled physical systems, the entangled state representation theory of Wigner operator and tomography describing the system is established. Experimentally, when two single-mode squeezed vacuum states are input into the two input ports of a 50:50 symmetric optical beam splitter, the output state is just a continuous-variable entangled state.

1.3 Thermal Entangled State Representation 1.3.1 Bosonic Case In order to equivalently convert the average value of the statistical ensemble under T = 0 into the expected value in a pure state, Takahashi and umezawa first introduced the theory of thermal field dynamics [10]. For the free Boson system H = ωa † a, Takahashi and umezawa found that the thermal vacuum state |0(θ) is expressed as



 



|0(θ) = S(θ) 0, 0˜ = sech θ exp(a † a˜ † tanh θ) 0, 0˜ .

(1.97)

Clearly, the state |0(θ) can be obtained by operating

the  thermal squeezing opera

tor S(θ) = exp[θ(a † a˜ † − a a)] ˜ on a vacuum state 0, 0˜ at T = 0. a˜ † is the creation operator of a fictitious mode accompanied with the photon creation operator a † of



˜ a † ] = 0. The parameter θ is a real field and a˜ can annihilate 0˜ , [a, ˜ a˜ † ] = 1 and [a, related to the temperature T , i.e., tanh θ = exp(−ω/2kT ). At ultra-high temperature, ω 0, the Airy function becomes the second kind of Bessel function. Thus, Eq. (2.26) becomes

2.1 Energy-Level Distribution and Wave Function of Two-Body Hamiltonian System

  −1/3 

1 8επ 1 1 η | p, E n  = C −K p − η2 δ λ F 2μ 2

   √  2 3/2 × exp iη1 (μa − μb ) η2 + λξ0 J1/3 ε 3   −1/3 1 6 1 1 ε→∞ −→ Cπ √ √ −K ε λ F 2μ   √  × exp iη1 (μa − μb ) η2 + λξ0 

 

1 2 3/2 . p − η2 exp − ε ×δ 2 3

31



(2.28)

For ε < 0, the Airy function becomes the first kind of Bessel function, thus we have

 −1/3   √  1 8 |ε| π 1 η | p, E n  = C exp iη1 (μa − μb ) η2 + λξ0 −K λ F 2μ 





 2 3/2 2 3/2 1 |ε| |ε| + J−1/3 . (2.29) p − η2 J1/3 ×δ 2 3 3 For the reason that the motion of two particles is governed by the condition Q r  0, then the energy-level distribution is determined by η = 0 | p, E n  = 0, that is, E n is given by the equation

J1/3



 2 3/2 2 3/2 + J−1/3 = 0, g g 3 3

(2.30)

whose solutions can be obtained from the Bessel function table, i.e., g = 2.3381, 4.0880, 5.5206, 6.7867, 7.9441, . . . .

(2.31)

From Eq. (2.27), the energy of the ground state can be obtained as

E 1 = 2.3381F

2/3

1 −K 2μ

1/3 + T.

(2.32)

Further, using Eq. (2.21) and recovering the reduced Planck constant , thus Eq. (2.32) becomes

  1 − 2μK 1/3 1 − μM K 2 p 2 . + E 1 = 2.3381 2 F 2 2μ 1 − 2μK 2M

(2.33)

32

2 Dynamics of Two-Body Hamiltonian Systems

2.1.2 Two Moving Charged Particles with Coulomb Coupling For two moving charged particles with Coulomb coupling, its Hamiltonian includes not only the kinetic coupling K Pa Pb but also the Coulomb potential F/Q r , so we have P2 F P2 . (2.34) H = a + b + K Pa Pb + 2m a 2m b Qr Because the total momentum P of the system is also conserved, we introduce the eigenstate equations that are similar to the equations in Eq. (2.11). To obtain the energy levels of a system with Hamiltonian H , we change the Hamiltonian H to the form  

1 F 1 + K μa μb P 2 + − K Pr2 + K (μb − μa ) P Pr + . (2.35) H= 2M 2μ Qr From Eqs. (2.6) and (2.11), we find that, in the entangled state η| representation, the energy eigenstate equation has the following form: 

 1 1 2 η| H | p, E n  = + 2K μa μb η2 η | p, E n  + − K η| Pr2 | p, E n  M 2μ  √ F η | p, E n  . (2.36) +K (μb − μa ) 2η2 η| Pr | p, E n  + √ 2η1 

Using Eqs. (2.4)–(2.6), we obtain the entangled state η| representation of the operator Pr , i.e., η| Pr = η|

  ∂ 1 d2 ξ √ i 2ξ2 |ξ ξ| = − + (μa − μb ) η2 η| . π 2 ∂η1

(2.37)

Substituting Eq. (2.37) into Eq. (2.36), we derive a new differential equation for η | p, E n , i.e.,   2 1 1 ∂ K− E n η | p, E n  = − i (μa − μb ) η2 − iη2 K (μb − μa ) 2 2μ ∂η1    1 ∂ F η | p, E n  . + 2K μa μb η22 + √ × − i (μa − μb ) η2 + ∂η1 M 2η1

(2.38) Clearly, Eq. (2.38) is the differential equation about the variable η1 , while the variable η2 appears as a constant. Making an ansatz η | p, E n  = n exp[i (μa − μb ) η1 η2 ]

(2.39)

2.1 Energy-Level Distribution and Wave Function of Two-Body Hamiltonian System

33

and using the transformation  ∂ ∂ − i (μa − μb ) η2 exp[i (μa − μb ) η1 η2 ] = , exp[−i (μa − μb ) η1 η2 ] ∂η1 ∂η1 (2.40) thus we obtain the equation for wave function n , i.e., 



 2 1 1 ∂ ∂ − −K − iη2 K (μb − μa ) 2 2μ ∂η1 ∂η12 

1 F 2 n = E n n . + 2K μa μb η2 + √ + M 2η1

(2.41)

Assuming that the wave function n is  n = exp

 i2η1 η2 K (μa − μb ) μ ψn = eiη1 ρ ψn , 1 − 2μk

where ρ=

2η2 K (μa − μb ) μ , 1 − 2μk

(2.42)

(2.43)

thus the former two terms on left-hand side of Eq. (2.41) becomes    ∂ 1 ∂ 4μ −K iη2 K (μb − μa ) eiη1 ρ ψn + 2μ ∂η1 ∂η1 1 − 2μk 

  ∂ ∂ 1 1 −K =− − i2ρ eiη1 ρ ψn . 2 2μ ∂η1 ∂η1

1 − 2

(2.44)

Further, using the transformation e−iη1 ρ

∂ iη1 ρ ∂ e = + iρ, ∂η1 ∂η1

(2.45)

Equation (2.44) becomes    ∂ 1 iη1 ρ −iη1 ρ ∂ iη1 ρ −iη1 ρ −K e e e e − i2ρ eiη1 ρ ψn 2μ ∂η1 ∂η1

   ∂ 1 1 ∂ iη1 ρ −K e =− + iρ − iρ ψn 2 2μ ∂η1 ∂η1

2 

 ∂ 1 1 2 ψn . − K eiη1 ρ =− + ρ 2 2μ ∂η12

1 − 2

(2.46)

34

2 Dynamics of Two-Body Hamiltonian Systems

Substituting Eq. (2.46) into Eq. (2.41) leads to a new differential equation for wave function ψn , that is, 

 2 1 1 ∂ 1 − K 2 μM 2 1 − ψn = E n ψn . −K η + F√ + 2 2μ M(1 − 2μK ) 2 ∂η12 2η1

(2.47)

For the reason that, in the entangled state η| representation there exist the corresponding relations

  √ ∂ 1 i Pr → − + (μa − μb ) η2 , Q r → 2η1 , 2 ∂η1

P→

√ 2η2

(2.48)

and the transformation in Eq. (2.40), thus Eq. (2.48) becomes √ 1 ∂ , Q r → 2η1 , Pr → −i √ 2 ∂η1

P→

√ 2η2 .

(2.49)

So, the Hamiltonian H in Eq. (2.34) can be rewritten as

H=

1 −K 2μ

 Pr2 +

1 − K 2 μM F P2 + , 2M(1 − 2μK ) Qr

(2.50)

which is indeed the entangled state η| representation of the Hamiltonian H . To obtain the energy levels of H , we rewrite H as the sum of two terms, that is, H=

1 − K 2 μM P 2 + Hr , 2M(1 − 2μK )

where Hr =

1 −K 2μ

 Pr2 +

F . Qr

(2.51)

(2.52)

Next, we use the entangled state ξ| representation to derive the energy levels of Hr . Acting 1/Q r on the state ξ|, we have ξ|

2  2 1 1 d ξ       d η |η η| ξ ξ = ξ| Qr π Qr π 2  2 λ d ξd η = √ exp {i (μa − μb ) (η1 η2 − ξ1 ξ2 ) π 2 η1 4 2 √          + i λ η1 ξ2 − ξ2 − η2 ξ1 − ξ1 ξ1 , ξ2  .

Integrating over the parameters η2 and ξ1 in Eq. (2.53), we obtain

(2.53)

2.1 Energy-Level Distribution and Wave Function of Two-Body Hamiltonian System

ξ|

∞     1 dξ2 λ = √ exp i (μa − μb ) ξ1 ξ2 − ξ2 Qr 2 2 −∞ π ∞ √     dη1 exp i λη1 ξ2 − ξ2 ξ1 , ξ2  . × −∞ η1

35

(2.54)

Using the definition of the function θ (τ ), that is,

θ (τ ) = lim+ ε→0

1 − i2π



∞

−iτ ω

e −∞

dω = ω + iε



1, τ > 0 0, τ < 0

(2.55)

to perform the integration over the variable η1 leads to

∞ √ √     dη1 dη1  exp i λη1 ξ2 − ξ2 exp i λη1 ξ2 − ξ2 = lim+ ε→0 η1 −∞ η1 + iε  1, ξ2 > ξ2 . (2.56) = i2π 0, ξ2 < ξ2

∞

−∞

Thus, Eq. (2.54) becomes

     1 λ ξ2  ξ| =i dξ2 exp i (μa − μb ) ξ1 ξ2 − ξ2 ξ1 , ξ2  . Qr 2 −∞

(2.57)

√ Noticing that the eigenvalue of Q cm is λ/2ξ1 in the entangled state |ξ and there is no operator Q cm in Hr , so we take ξ| as ξ1 = 0, ξ2 |, thus ξ1 = 0, ξ2 |

 1 λ ξ2   =i dξ2 ξ1 = 0, ξ2  . Qr 2 −∞

(2.58)

In the entangled state ξ1 = 0, ξ2 | representation, the Schrödinger equation for Hr reads



 1 λ 2 λ ξ2  ξ1 = 0, ξ2 | Hr |ψ = −K ξ2 ψ(0, ξ2 ) + iF dξ ψ(0, ξ2 ) 2μ 2 2 −∞ 2 (2.59) = Eψ(0, ξ2 ). Owing to the binding energy E < 0 and taking f 2 = 4μE/[(2μK − 1)λ], we have ψ(0, ξ2 )

iFμ =−  ξ2   (1 − 2μK )(ξ22 + f 2 ) −∞ dξ2 ψ(0, ξ2 )



8 , λ

(2.60)

its solution is

ξ2

−∞



dξ2 ψ(0, ξ2 )



Fμ ξ2 8 arctan = exp −i λ (1 − 2μK ) f f

 + C,

(2.61)

36

2 Dynamics of Two-Body Hamiltonian Systems

where C is the integral constant. From Eq. (2.61), we can see that the function arctan(ξ2 / f ) has multiple values for a fixed ξ2 /f√. To ensure  the uniqueness of the 2/λFμ / [(1 − 2μK )n], n is an wave function ψ(0, ξ2 ), here we only take f = integer. Thus, the energy levels of Hr read E=

μF 2 (2μK − 1)λ 2 f = 2 , 4μ 2n (2μK − 1)

(2.62)

so the energy levels of H are En = E +

1 − K 2 μM 2 η . M(1 − 2μK ) 2

(2.63)

  Obviously, the term 1 − K 2 μM η22 / [M(1 − 2μK )] is the kinetic energy of the √ center of mass because 2η2 is the eigenvalue of the momentum P. Thus, Eq. (2.63) clearly shows the impact of the kinetic potential and Coulomb coupling potential on the total energy E n .

2.2 Path Integral Theory in Entangled State Representation By using the Fourier transform of |η, we obtain a new entangled state as    i  ∗ d2 η ∗ |ς = |η exp ςη + ς η 2π 2 

1 2 † ∗ † † † = exp − |ς| + iςa − iς b − a b |00 , 2

(2.64)

where ς is complex. By using the IWOP method, we find that the state |ς obeys the eigenvalue equations     a + b† |ς = iς |ς , a † + b |ς = −iς ∗ |ς

(2.65)

and the set of |ς is complete and orthonormal

d2 ς |ς ς| = 1, π

 ς| ς  = πδ(ς − ς  )δ(ς ∗ − ς ∗ ).

(2.66)

And the inner product of the entangled states ς| and |η reads    1 i  ∗ ∗ ς| η = exp − ςη + ς η . 2 2

(2.67)

2.2 Path Integral Theory in Entangled State Representation

37

Based on the complete and orthonormal entangled state |ς and |η representations, we have the ability to establish a path integral theory. Here, the transition amplitude is   as the matrix element of the evolved operator between the two states  represented  ς → ς  with all the paths that go from ς  to ς  .   For a Hamiltonian operator H a † , a; b† , b , by using the completeness relations in Eqs. (2.8) and (2.66), we have

d2 η  d2 ς  d2 η  d2 ς                        ς ς η η H η η ς ς π4 2  2  2  2  1 d η d ς d η d ς             ς ς η H η = 4 π4    i   ∗ ∗  ∗   ∗ × exp − ς η + ς η − ς η − ς η . (2.68) 2

  H a † , a; b† , b =

Further, making the following transformation of integral variables: η = η +

σ σ λ , η  = η − , ς  = ς + , 2 2 2

such that

ς  = ς −

λ , 2

d2 ς  d2 ς  = d2 ςd2 λ, d2 η  d2 η  = d2 ηd2 σ,

(2.69)

(2.70)

so we obtain   ! λ λ  d2 ςd2 ηd2 λd2 σ  H= ς + 2 ς − 2  4π 4   " # σ −i(λη∗ +λ∗ η+ς ∗ σ+ςσ∗ )/2 σ  × η +  H η − e . 2 2

(2.71)

Introducing the entangled state |ς representation of two-mode Wigner operator

  ! λ d2 λ  λ  −i(λη∗ +λ∗ η)/2 ς + e ≡  (ς, η) ς − π  2 2

(2.72)

and the classical correspondence function of H

σ # − i (ς ∗ σ+ςσ∗ ) σ   d2 σ " η +  H η − e 2 ≡ h (ς, η) , π 2 2

(2.73)

we can express Eq. (2.71) as the compact form H=

d2 ςd2 η h (ς, η)  (ς, η) , 4π 2

(2.74)

38

2 Dynamics of Two-Body Hamiltonian Systems

which is indeed the Weyl correspondence relation between h (ς, η) and H in the entangled state representation. As a result of Eqs. (2.66) and (2.71), the matrix element of the operator H in the entangled state ς| representation is obtained as 2 d λ −i(λη∗ +λ∗ η)/2 d2 ςd2 η h (ς, η) e 4 π

  λ λ (2) ς − − ς  δ × δ (2) ς  − ς − 2 2 2 2  $ %  d ςd η = h (ς, η) δ (2) ς  + ς  − 2ς exp −i[(ς  − ς)η ∗ + (ς  − ς)∗ η] π  2     % $   d η ς + ς = (2.75) h , 2η exp −i η ∗ ς  − ς  + ς ∗ − ς ∗ η . π 2

      ς  H ς =



In terms of the Feynman’s path integral theory, the transition amplitude reads 

         ς  , t  ς  , t  = ς   exp −iH t  − t  ς  & n n+1  d2 ςi &   −iH  ς j−1 , = ςj e π j=1 i=1

(2.76)

  where ς  = ςn+1 , ς  = ς0 and  =  t  − t /(n + 1). When  → 0, using Eqs. (2.66) and (2.75), the monomial ς j e−iH ς j−1 in Eq. (2.76) can be calculated as          −iH  ς j−1 = πδ (2) ς j − ς j−1 − i ς j  H ς j−1 ςj e 2 $      d ηj exp −i η ∗j ς j − ς j−1 + η j ς ∗j − ς ∗j−1 = π  

ς j + ς j−1 × 1 − ih , 2η j 2 2 $      d ηj exp −i η ∗j ς j − ς j−1 + η j ς ∗j − ς ∗j−1 = π

 ς j + ς j−1 −ih . (2.77) , 2η j 2 Substituting Eq. (2.77) into Eq. (2.76), we can obtain the path integral in the entangled state ς| representation, that is,   ς , t ς  , t  =







& 2 d ς (t) d2 η (t) t

π2

 exp i t

t 

 dtL ,

(2.78)

where L is the Lagrangian function L = −ς˙∗ (t) η (t) − ς˙ (t) η ∗ (t) − h (ς (t) , 2η (t)) ,

(2.79)

2.2 Path Integral Theory in Entangled State Representation

39

    t and exp i t  dtL represents a certain kind of action. Thus, we have defined “paths” in the phase space by virtue of the entangled state |ς and |η representations. Further, we calculate the classical correspondence function h(ς, η) for the Hamiltonian of the generalized parametric amplifier. A general Hamiltonian for the twophoton process is expressed as H = ω1 a † a + ω2 b† b + g(ab + a † b† ) + f (a 2 + a †2 ) + k(b2 + b†2 ),

(2.80)

here ω1 , ω2 , g, f , and k are real in order to ensure that the Hamiltonian H is Hermite operator. For f = k = 0, Eq. (2.80) becomes the Hamiltonian of optical parametric amplifier. Conveniently, letting H1 = (ω1 a † a + ω2 b† b), H3 = f (a 2 + a †2 ), we have H=

H2 = g(ab + a † b† ), H4 = k(b2 + b†2 ),

4 '

(2.81)

Hi .

(2.82)

    " σ# i ∗ d2 σ σ   ∗ exp − ς σ + ςσ . η +  H1 η − π 2 2 2

(2.83)

i=1

For this purpose, we first calculate h1 ≡

Letting α = η − σ/2, we can rewrite Eq. (2.83) as h 1 = 4e

−i(ης ∗ +ςη ∗ )



d2 α i(ας ∗ +ςα∗ ) 2η − α| H1 |α . e π

(2.84)

Owing to |α belonging to the entangled state |η and using Eq. (2.7), we get 

 α∗ ∂ ∂ α |α , a |α = − |α , + − ∂α 2 ∂α∗ 2 



∂ α∗ ∂ α |α |α |α . , b = − b† |α = − + ∂α∗ 2 ∂α 2

a † |α =

(2.85)

Thus, we obtain

2   

|α| ω1 − ω2 1 ∂2 ∂ ∗ ∂ |α + − − α −α H1 |α = (ω1 + ω2 ) 4 2 ∂α∂α∗ 2 ∂α ∂α∗ = H1 |α . (2.86)

40

2 Dynamics of Two-Body Hamiltonian Systems

Further, letting H1 = H11 + H12 , where



∂ ∗ ∂ H11 , H12 . α −α ∂α ∂α∗ (2.87) Substituting the completeness relation of the states |ς |ς=ξ in Eq. (2.66) into Eq. (2.84) and using the inner product ξ| α = 21 exp[−i(ξα∗ + αξ ∗ )/2], thus we obtain h 1 = h 11 + h 12 , where h 11 and h 12 , respectively, read |α|2 1 ∂2 = (ω1 + ω2 ) − − 4 2 ∂α∂α∗



ω1 − ω2 = 2



d2 α d2 ξ i(ας ∗ +ςα∗ ) 2η − α| ξ ξ| H11 |α e π π 2 2 d α d ξ i(ας ∗ +ςα∗ ) i [ξ(2η−α)∗ +ξ ∗ (2η−α)] ∗ ∗ = e−i(ης +ςη ) e2 e π π   i |α|2 1 ∂2 ∗ ∗ × (ω1 + ω2 ) e− 2 (αξ +ξα ) − − 4 2 ∂α∂α∗   2 2 d αd ξ i(ας ∗ +ςα∗ )+i[ξ(η∗ −α∗ )+ξ ∗ (η−α)] |α|2 + |ξ|2 1 ∗ ∗ = e−i(ης +ςη ) e − (ω1 + ω2 ) π2 4 2   |α|2 ω1 + ω2 1 ∂2 −i(ης ∗ +ςη ∗ ) 2 i(ας ∗ +ςα∗ ) =e + d αe − δ (2) (η − α) (ω1 + ω2 ) − 4 ∂α∂α∗ 4 2    |η|2 ∂2 1 1 ∗ ∗ ∗ ∗  = − (ω1 + ω2 ) e−i(ης +ςη ) ei(ας +ςα )  + (ω1 + ω2 ) − ∗ 4 ∂α∂α 4 2 α=η,α∗ =η ∗     1 1 |ς|2 + |η|2 − 1 , (2.88) = (ω1 + ω2 ) 2 2

h 11 = 4e−i(ης

∗ +ςη ∗

)

and

d2 α d2 ξ i(ας ∗ +ςα∗ ) 2η − α| ξ ξ| H12 |α e π π 2 2 d α d ξ i(ας ∗ +ςα∗ ) i [ξ(2η−α)∗ +ξ ∗ (2η−α)] ∗ ∗ = e−i(ης +ςη ) e2 e π π 

i ∂ ω1 − ω2 ∂ ∗ ∗ × α − α∗ ∗ e− 2 (αξ +ξα ) 2 ∂α ∂α 2 2 d αd ξ i(ας ∗ +ςα∗ )+i[ξ(η∗ −α∗ )+ξ ∗ (η−α)] − i (αξ ∗ −α∗ ξ ) ω1 − ω2 −i(ης ∗ +ςη∗ ) = e e 2 e 2 π2

 ∗ ∗ α ∂ α∗ ∂ (ω1 − ω2 ) e−i(ης +ςη ) ∗ ∗ d2 αei(ας +ςα ) = − δ (2) (η − α) ∗ 4 2 ∂α 2 ∂α     ∗ ∗ ∂  ∗ i(ας ∗ +ςα∗ )  ∂  i(ας ∗ +ςα∗ )  (ω1 − ω2 ) e−i(ης +ςη ) α αe =− − e   ∗ ∗ 4 ∂α ∂α∗ α=η α =η

h 12 = 4e−i(ης

∗ +ςη ∗

)

  i = − (ω1 − ω2 ) ης ∗ − η ∗ ς . 4

(2.89)

2.2 Path Integral Theory in Entangled State Representation

41

So we have h 1 (ς, η) =

     1 2 i 1 |ς| + |η|2 − 1 − (ω1 − ω2 ) ης ∗ − η ∗ ς . (2.90) (ω1 + ω2 ) 2 2 4

Further, we calculate h 2 as     " σ# i  ∗ d2 σ σ   ∗ exp − ς σ + ςσ h2 = η +  H2 η − π 2 2 2 2   α d ∗ ∗ ∗ ∗ ei(ας +ςα ) 2η − α| g ab + a † b† |α . = 4e−i(ης +ςη ) π

(2.91)

Using Eq. (2.85), we see

  † † ab + a b |α = −2

|α|2 ∂2 − ∂α∂α∗ 2

 |α .

(2.92)

Substituting the completeness relations of |ς |ς=ξ in Eq. (2.66) into Eq. (2.91) and using Eq. (2.92), we arrive at   |α|2 d2 α d2 ξ i(ας ∗ +ςα∗ ) ∂2 |α 2η − α| ξ ξ| −2 e − π π ∂α∂α∗ 2   2 2 ∗ +ςη ∗ |α|2 d α d ξ i(ας ∗ +ςα∗ )+i[ξ(η ∗ −α∗ )+ξ ∗ (η−α)] |ξ|2 −i ης ( ) e − = ge π π 2 2   ∗ ∗ ∗ ∗ |α|2 1 ∂2 = ge−i(ης +ςη ) d2 αei(ας +ςα ) − − δ (2) (η − α) ∗ 2 ∂α∂α 2 ⎡ ⎤   2 2 ∗ ∗ ∗ ∗ |η| 1 ∂  ⎦ ei(ας +ςα )  − = g ⎣− e−i(ης +ςη )  2 ∂α∂α∗ 2 ∗ ∗ α=η,α =η  g 2 2 |ς| − |η| . (2.93) = 2

∗ ∗ h 2 = 4ge−i(ης +ςη )



Now we calculate H3 . Using Eq. (2.85), we have

α2 α∗2 ∂2 ∂2 ∂ ∗ ∂ + + + − α + α ∂α∗2 ∂α2 ∂α∗ ∂α 4 4 = H3 |α ,

(a 2 + a †2 ) |α =

here we can divide H3 into the following three parts:

 |α (2.94)

42

2 Dynamics of Two-Body Hamiltonian Systems

 ∂2 ∂2 , = f + ∂α∗2 ∂α2 

∂ ∂ , = f −α ∗ + α∗ ∂α ∂α

2  α α∗2 + = f . 4 4

H31 H32 H33

(2.95)

Noticing the similarity between H12 and H32 and using Eq. (2.88), we find −i(ης ∗ +ςη ∗ )



h 32 = 4 f e

 d2 αd2 ξ i(ας ∗ +ςα∗ ) ∂ ∗ ∂ 2η − α| ξ ξ| −α ∗ + α |α e π2 ∂α ∂α

 i  = − f η ∗ ς ∗ − ης . 2

(2.96)

Similarly, we can calculate h 31 and h 33 as h 31

2  ∂ ∂2 d2 α d2 ξ i(ας ∗ +ςα∗ ) 2η − α| ξ ξ| |α e = 4fe + π π ∂α∗2 ∂α2 2 2  1 d α d ξ i(ας ∗ +ςα∗ )+i[ξ(η∗ −α∗ )+ξ ∗ (η−α)]  2 ∗ ∗ e = − f e−i(ης +ςη ) ξ + ξ ∗2 4 π π

2  ∂ f −i(ης ∗ +ςη∗ ) ∂2 2 i(ας ∗ +ςα∗ ) δ (2) (η − α) = e + d αe 4 ∂α∗2 ∂α2     ∂ 2 i(ας ∗ +ςα∗ )  f −i(ης ∗ +ςη∗ ) ∂ 2 i(ας ∗ +ςα∗ )  e + e = e   ∗ ∗ 4 ∂α∗2 ∂α2 α=η α =η −i(ης ∗ +ςη ∗ )

=−



 f  2 ς + ς ∗2 4

(2.97)

and 2 2  1 −i(ης ∗ +ςη∗ ) d α d ξ i(ας ∗ +ςα∗ )+i[ξ(η∗ −α∗ )+ξ ∗ (η−α)]  2 fe e α + α∗2 4 π π  f −i(ης ∗ +ςη∗ ) ∗ ∗  2 = e d αei(ας +ςα ) α2 + α∗2 δ (2) (η − α) 4  f  2 η + η ∗2 . = (2.98) 4

h 33 =

So we have h3 = −

 i    f  2 f  2 ς + ς ∗2 − f η ∗ ς ∗ − ης + η + η ∗2 . 4 2 4

(2.99)

2.2 Path Integral Theory in Entangled State Representation

43

Finally, we calculate H4 . From Eq. (2.85), we obtain

∂2 ∂2 ∂ α2 α∗2 ∗ ∂ + + α − α (b + b ) |α = + + ∂α2 ∂α∗2 ∂α∗ ∂α 4 4 = H4 |α , 2

†2

 |α (2.100)

thus we divide H4 into the following parts, i.e.,  ∂2 ∂2 = H31 , =k + ∂α∗2 ∂α2 

∂ ∂ = −H32 , = k α ∗ − α∗ ∂α ∂α 

2 α∗2 α + = H33 . =k 4 4

H41 H42 H43

(2.101)

Similar to deriving h 3 , here we can obtain h 4 as h4 = −

   k 2 k k 2 ς + ς ∗2 + i η ∗ ς ∗ − ης + η + η ∗2 . 4 2 4

(2.102)

,4 Using Eqs. (2.90), (2.93), (2.99), and (2.102), we finally have h(ς, η) = i=1 hi , i.e.,     1 2 g 2 1 2 |ς| + |η| − 1 + |ς| − |η|2 h(ς, η) = (ω1 + ω2 ) 2 2 2    − ω i ( f + k)  2 (ω ) 1 2 ς + ς ∗2 − ης ∗ − η ∗ ς − 4 4   ( f + k)  2  ∗ ∗ i η + η ∗2 . − ( f − k) η ς − ης + (2.103) 2 4 Besides, we investigatethe Lagrange  function in path integral theory. Using ς +ς Eq. (2.103) we can give h j 2 j−1 , 2η j , and then inserting it into Eq. (2.77), we obtain    −iH  ς j−1 ςj e   2  d2 η j  exp −i (ω1 + ω2 − 2g) η j  =  ς j , ς j−1 π

   ∗  ςj ς j + ς j−1 ∗ − η j i ς j − ς j−1 + (ω1 − ω2 ) −  ( f − k) 2 2

     ςj ς j + ς j−1 ∗ −  ( f − k) + η j −i ς j − ς j−1 + (ω1 − ω2 ) 2 2 % , −i ( f + k) η 2j − i ( f + k) η ∗2 j

 + ς j−1 2   + ς j−1 ∗ 2 (2.104)

44

2 Dynamics of Two-Body Hamiltonian Systems

  where the term  ς j , ς j−1 reads 

 ς j , ς j−1







   ς j + ς j−1 2 1 1  + ( f + k)  = exp −i − (ω1 + ω2 − 2g)   4 2 4     

ς j + ς j−1 2 ς j + ς j−1 ∗2 1 + . (2.105) + (ω1 + ω2 ) × 2 2 2

By using the integral formula

  d2 z exp ς |z|2 + ξz + ηz ∗ + gz 2 + hz ∗2 π

 −ςξη + ξ 2 h + η 2 g 1 exp =, ς 2 − 4gh ς2 − 4 f g

(2.106)

  2 −4gh < 0, which holds for the convergence conditions Re(ξ ± g ± h) < 0 and Re ςξ±g±h to integrate over η j , we further obtain   −iH   ς j−1 ςj e

   ς j , ς j−1

 −i =  exp − + ω − 2g)2 + 4 ( f + k)2 (ω 2 2 1 2 − (ω1 + ω2 − 2g) + 4 ( f + k) ⎡ 2  

 (ω − ω ) ς + ς  ς j − ς j−1  ς j + ς j−1 ∗ 2 j j−1  1 ⎣ × (ω1 + ω2 − 2g)  −i − ( f − k)    4 2     ∗

 

i ς j − ς j−1 ς j + ς j−1 (ω1 − ω2 ) ς j + ς j−1 ∗ − ( f + k) − ( f − k) +  2 2 2   

  

−i ς j − ς j−1 ς j + ς j−1 ∗ (ω1 − ω2 ) ς j + ς j−1 + ( f + k) . + − ( f − k)  2 2 2

(2.107) By letting A = (ω1 + ω2 − 2g) ,

B = − (ω1 + ω2 − 2g)2 + 4 ( f + k)2

and calculating the matrix element of Feynman transition

(2.108)

2.2 Path Integral Theory in Entangled State Representation

45

  2   −iH   ς j−1 = √1 exp i − ω1 + ω2 + 2g + A (ω1 − ω2 ) + A ( f − k)2 ςje 4 4B B B ⎤    2 2  2 (ω1 − ω2 ) f − k  ς + ς j−1 2  − − f + k + A (ω1 − ω2 ) ( f − k) ⎦ j +   B 2 4 2B     

ς j + ς j−1 2 ς j + ς j−1 ∗2 f 2 − k2 ( f + k) (ω1 − ω2 ) + + + B 4B 2 2      ς j + ς j−1 ∗ 2 A ( f + k) (ω1 − ω2 ) + −i f 2 − k2 ( f − k) + B B B 2 

  ς j − ς j−1 ∗ 2  2 A ( f + k) (ω1 − ω2 ) + f − k2 × +i (ω1 − ω2 ) +  2B B B  

 

ς j − ς j−1 ∗ ς ς + ς ς j + ς j−1 i j j−1 j − ς j−1 + ( f − k) × 2  B 2 

  ω1 + ω2 i (ω1 − ω2 ) ς j + ς j−1 ∗ ς j − ς j−1 + 2B 2  2  2 

∗2   A ς j − ς j−1  2 ( f + k) ς j − ς j−1 +  +  , B  B  −

(2.109)

we obtain the transition amplitude as 

  ς  , t  ς  , t  =

   & 2 t d ς (t) exp i dtL , π t t

(2.110)

where the Lagrange function L is 

  2 (ω1 − ω2 ) f 2 − k 2 ω1 + ω2 + 2g A (ω1 − ω2 )2 A |ς (t)|2 + + ( f − k)2 + 4 4B B B    f +k A (ω1 − ω2 ) ( f − k) f 2 − k2 ( f + k) (ω1 − ω2 )  ς (t)2 + ς (t)∗2 − − + + + 4 2B B 4B   A 2  2 ( f + k) 2 −i f −k ς (t)∗ ς˙ (t)∗ ( f − k) + (ω1 − ω2 ) + B B B   A 2  2 ( f + k) f − k 2 ς (t) ς˙ (t)∗ +i (ω1 − ω2 ) + (ω1 − ω2 ) + 2B B B i ( f − k) 2 ( f + k) i (ω1 − ω2 ) ∗ + ς (t) ς˙ (t) − ς (t) ς˙ (t) + ς˙ (t)∗2 B 2B B A ω1 + ω2 + |˙ς (t)|2 + (2.111) . B 2

L= −

As a specific application of the above theory, we discuss the path integral problem of the nondegenerate parametric amplifier using the entangled state |ς representation. Generally, the Hamiltonian of the nondegenerate parametric amplifier reads

46

2 Dynamics of Two-Body Hamiltonian Systems

  H = ω a † a + b† b + g(ab + a † b† ).

(2.112)

By comparing Eqs. (2.80) and (2.112), we have ω1 = ω2 = ω and f = k = 0, thus the Lagrange function for the nondegenerate parametric amplifier is L=ω−

1 1 ς˙ (t) ς˙ (t)∗ (ω + g) ς (t) ς (t)∗ − 2 2 (g − ω)

(2.113)

and the classical Euler–Lagrange equations are   ς¨∗ (t) − ω 2 − g 2 ς ∗ (t) = 0,

  ς¨ (t) − ω 2 − g 2 ς (t) = 0.

(2.114)

To verify the correctness of the above method, here we use the Heisenberg equation and Eq. (2.80) to obtain the identities ( = 1) i

     d  † a + b = a † + b, H = (g − ω) a † − b dt

(2.115)

and      d2  † d  † a − b = ω2 − g2 a † + b . a + b = i (g − ω) 2 dt dt

(2.116)

Clearly, Eq. (2.116) just corresponds to Eq. (2.114) because that the entangled state |ς is the eigenstate of the operator a † + b. Also, this correspondence proves the feasibility of the path integral method in the entangled state representations.

2.3 Evolution of Atomic Coherent State Governed by the Hamiltonian f (t) J+ + f ∗ (t) J− + g (t) Jz For the coherent state with the amplitude α that is expressed as |α = exp(αa † − α∗ a) |0, its form always remains unchanged when it is driven by Hamiltonian H0 = ωa † a + f (t) a † + f ∗ (t) a.

(2.117)

That is to say, if the coherent state |α is regarded as an initial state, it will still evolve into a coherent state under the action of Hamiltonian H0 . Next, we will find out which kind of Hamiltonian can keep the atomic coherent state unchanged. The definition of atomic coherent state is |ϑ = exp(μJ+ − μ∗ J− ) | j, − j ,

(2.118)

2.3 Evolution of Atomic Coherent State Governed by the Hamiltonian …

47

where J+ , J− are, respectively, the raising and lowering operators of angular  momentum, J+ , J− = 2Jz , J± , Jz = ∓J± , and the operator Jz has the following eigenequation: (2.119) Jz | j, m = m | j, m and | j, − j is the lowest weight state annihilated by the lowering operator J− , that is, J− | j, − j = 0. Using the disentangling formula, the exponential term exp(μJ+ − μ∗ J− ) can be decomposed into   ∗ exp(μJ+ − μ∗ J− ) = eϑJ+ exp Jz ln(1 + |ϑ|2 ) e−ϑ J− ,

(2.120)

where μ = 2θ e−iϕ and ϑ = e−iϕ tan 2θ , thus Eq. (2.118) becomes 1 ϑJ |ϑ =   j e + | j, − j . 1 + |ϑ|2

(2.121)

The state |ϑ has the non-orthogonal property 

ϑ |ϑ =

(1 + ϑ ϑ∗ )2 j (1 + |ϑ|2 ) j (1 + |ϑ |2 ) j

(2.122)

and its set makes up the completeness relation in the j-subspace

j ' d | j, m  j, m| = 1, d = sin θdθdϕ. |ϑ ϑ| = 4π m=− j

(2.123)

Using the Schwinger Boson realization of   angular momentum operators, i.e., J+ = a † b, J− = ab† and Jz = 21 a † a − b† b , and the eigenstate of the operator Jz | j, m = √

a † j+m b† j−m |00 , ( j + m)! ( j − m)!

(2.124)

where |00 is two-mode vacuum, a |00 = 0, b |00 = 0, and b†2 j | j, − j = √ |00 = |0 ⊗ |2 j . (2 j)! Thus, we can rewrite the atomic coherent state |ϑ as

(2.125)

48

2 Dynamics of Two-Body Hamiltonian Systems †2 j 1 ϑa † b b |00 |ϑ =  √ j e (2 j)! 1 + |ϑ|2  †  1 † 2j =√  j b + a ϑ |00  (2 j)! 1 + |ϑ|2  2j '  2 j−l 1 2j |00 =√ b†l a † ϑ j  (2 j)! 1 + |ϑ|2 l=0 l 1/2 2j ' 1 2j = ϑ2 j−l |2 j − l ⊗ |l , j l 2 1 + |ϑ| l=0

(2.126)

which shows quantum entanglement between two modes. For the unnormalized continuous-variable entangled state |ζ   |ζ = exp a † ζ + ζ ∗ b† − a † b† |00 ,

ζ = ζ1 + iζ2 ,

(2.127)

which is indeed the common eigenvector of a + b† and a † + b, its set is complete, that is, 2 d ζ −|ζ|2 |ζ ζ| = 1. e (2.128) π In two-mode Fock space, the state |ζ is expanded as |ζ =

∞ ' a †m b†n Hm,n (ζ, ζ ∗ ) |00 , m!n! m,n=0

(2.129)

where two-variable Hermite polynomial Hm,n (ξ, ξ ∗ ) can be defined by ∞ '   sm t n Hm,n (ξ, ξ ∗ ) = exp −st + sξ + tξ ∗ m!n! m,n=0

(2.130)

or   ∂ m+n exp −st + sξ + tξ ∗ |s=t=0 m n ∂s ∂t min(m,n) ' m!n!(−1)l ξ m−l ξ ∗n−l . = l!(m − l)!(n − l)! l=0

Hm,n (ξ, ξ ∗ ) =

(2.131)

From Eq. (2.129), we have ζ |m, n = √

1 m!n!

Hm,n (ζ, ζ ∗ ).

(2.132)

2.3 Evolution of Atomic Coherent State Governed by the Hamiltonian …

49

Using Eqs. (2.126) and (2.129), we obtain the entangled state ζ| representation of the wave function for atomic coherent state, that is, √ 2j (2 j)! ' ϑ2 j−l ζ |ϑ j =  H2 j−l,l (ζ, ζ ∗ ). j 2 l! j − l)! (2 1 + |ϑ| l=0

(2.133)

To give the summation involving H2 j−l,l in Eq. (2.133), we first use the new identity 2j  ' . 2 j .. .H2 j−l,l (a † , a)..ϑ2 j−l l l=0

(2.134)

. . to derive a generalized binomial formula, where ..H2 j−l,l (a † , a).. is in anti-normal ordering form. Noting that Boson operators are commutative (permuted) within the .. anti-normal ordering symbol .. .., so using Eq. (2.130) we have ∞ '  . . . s m t n .. .Hm,n (a † , a).. = .. exp −st + sa † + ta .. m!n! m,n=0

= e−st eta esa = esa eta †

=

†

∞ ' s m t n †m n a a . m!n! m,n=0

(2.135)

By comparing s m t n on both sides of Eq. (2.135), we obtain the operator identity as . .. .Hm,n (a † , a).. = a †m a n = : a †m a n : = : a n a †m : .

(2.136)

2j  2j  ' ' . 2 j .. 2 j 2 j−l †2 j−l l ϑ a a .H2 j−l,l (a † , a)..ϑ2 j−l = l l l=0 l=0 2j  ' 2 j 2 j−l †2 j−l l ϑ = :a a : l l=0  2 j = : ϑa † + a :.

(2.137)

Thus, we have

Further, introducing the operator identity

50

2 Dynamics of Two-Body Hamiltonian Systems ∞ ' λn n=0

n!

 n    : ϑa † + a : = : exp λ ϑa † + a : . † 2 . = ..eλa eλϑa e−λ ϑ .. √ n ∞ ϑλ . ϑa † + a  . ' .. ..H = √ n n! 2 ϑ n=0

(2.138)

and comparing λn on both sides of Eq. (2.138), we obtain √ n . n  ϑ ..Hn : ϑa † + a : =

ϑa † + a √ 2 ϑ



.. ..

(2.139)

So Eq. (2.134) becomes 

† 2j  ' . . 2 j .. ϑa + a .. .. .H2 j−l,l (a † , a)..ϑ2 j−l = ϑ j ..H2 j √ l 2 ϑ l=0

(2.140)

Noting that both sides of Eq. (2.140) are in anti-normal ordering, so we obtain 2j  ' 2j l=0

l

∗

H2 j−l,l (ζ, ζ )ϑ

2 j−l

= ϑ H2 j j

ϑζ + ζ ∗ √ 2 ϑ

 ,

(2.141)

which is related to a single-variable Hermite polynomial H2 j (·). Further, letting χ=

ϑζ + ζ ∗ √ , 2 ϑ

(2.142)

then Eq. (2.133) reads ζ |ϑ = √

ϑj  j H2 j (χ) ,  (2 j)! 1 + |ϑ|2

(2.143)

which shows that the inner product ζ |ϑ is proportional to a single-variable Hermite polynomial H2 j (·) of order 2 j. Taking the state ϑ (0)| as the initial state of the system controlled by Hamiltonian H = f (t) J+ + f ∗ (t) J− + g (t) Jz

(2.144)

and supposing that it still evolves into an atomic coherent state, which means H |ϑ (t) = E (t) |ϑ (t), thus the Schrödinger equation is given by

2.3 Evolution of Atomic Coherent State Governed by the Hamiltonian …

i

∂ |ϑ (t) = H |ϑ (t) = E (t) |ϑ (t) . ∂t

51

(2.145)

Substituting the completeness relation of |ζ into Eq. (2.145) leads to ∂ i |ϑ (t) = ∂t



d2 ζ −|ζ|2 |ζ ζ| H |ϑ (t) e π

(2.146)

and using the relations 

∂ ∂ ζ| J+ = ζ| a † b = ζ − ∗ ζ| , ∂ζ ∂ζ ∗ 

∂ ∂ ζ| J− = ζ| b† a = ζ ∗ − ζ| , ∂ζ ∂ζ 

 †  1 ∂ 1 † ∗ ∂ ζ| Jz = ζ| a a − b b = ζ| , ζ −ζ 2 2 ∂ζ ∂ζ ∗

(2.147)

we obtain   

∂ ∂ ∂ ∂ ∗ ∗ f ζ− ∗ ζ + f − ∗ ∂ζ ∂ζ ∂ζ ∂ζ

 ∂ g ∂ ζ |ϑ (t) ζ − ζ∗ ∗ + 2 ∂ζ ∂ζ = E ζ |ϑ (t) .

ζ| H |ϑ (t) =

(2.148)

Supposing that |ϑ (t) is still an atomic coherent state and using Eq. (2.143), we can change Eq. (2.148) to  

 ∂ ∂ ∂ ∂ ∗ ∗ f ζ− ∗ + f ζ − ∗ ∂ζ ∂ζ ∂ζ ∂ζ 

∂ ∂ g H2 j (χ) ζ − ζ∗ ∗ + 2 ∂ζ ∂ζ = EH2 j (χ) .

(2.149)

From Eq. (2.142), the partial differential with respect to the variable ζ is transformed into √ ∂χ ∂ ∂ ϑ ∂ 1 ∂ ∂ = = , (2.150) = √ ∂ζ ∂ζ ∂χ 2 ∂χ ∂ζ ∗ 2 ϑ ∂χ and using the differential identity of Hn (χ) ,that is, 2χHn (χ) − Hn (χ) = 2nHn (χ),

(2.151)

52

2 Dynamics of Two-Body Hamiltonian Systems

we can rewrite the terms on the left side of Eq. (2.149) as  ∂ ∂ H2 j (χ) I1 ≡ ζ − ∗ ∂ζ ∂ζ ∗ 

1 ∂2 1 ∂ H2 j (χ) − = ζ √ 2 ϑ ∂χ 4ϑ ∂χ2 1 1 = ζ √ H2 j (χ) − [2χH2 j (χ) − 4 jH2 j (χ)] 4ϑ 2 ϑ 

1 χ j H2 j (χ) + H2 j (χ) = ζ √ − 2ϑ ϑ 2 ϑ

(2.152)

and 

∂ ∂ ∗ H2 j (χ) I2 ≡ ζ − ∂ζ ∂ζ   √ ϑ ∂2 ∗ ϑ ∂ − H2 j (χ) = ζ 2 ∂χ 4 ∂χ2 √ ϑ ∗ ϑ  H2 j (χ) − [2χH2 j (χ) − 4 jH2 j (χ)] =ζ 2  √ 4 ϑ ϑχ − = ζ∗ H2 j (χ) + ϑ jH2 j (χ). 2 2

(2.153)

 √  Further, using the inverse relation of χ = (ζ ∗ + ϑζ) / 2 ϑ , that is, 4 |ϑ| ζ= |ϑ|2 − 1

√

 χ∗ ϑ∗ χ− √ , 2 2 ϑ

(2.154)

we obtain 

χ 1 j H2 j (χ) + H2 j (χ) I1 = ζ √ − ϑ 2 ϑ 2ϑ   2 ∗ |ϑ| + 1 χ − 2χ |ϑ|  j   = H2 j (χ) + H2 j (χ), ϑ 2ϑ |ϑ|2 − 1  1  ∗√ ζ ϑ − ϑχ H2 j (χ) + ϑ jH2 j (χ) 2   ϑ 2χ∗ |ϑ| − χ |ϑ|2 + 1   = H2 j (χ) + jϑH2 j (χ) 2 |ϑ|2 − 1

(2.155)

I2 =

(2.156)

2.3 Evolution of Atomic Coherent State Governed by the Hamiltonian …

53

and   2 

|ϑ| + 1 χ − 2 |ϑ| χ∗  ∂ 1 ∗ ∂   H2 j (χ) = ζ −ζ I3 = H2 j (χ) . 2 ∂ζ ∂ζ ∗ 2 |ϑ|2 − 1

(2.157)

So Eq. (2.149) becomes EH2 j (χ) = f I1 + f ∗ I2 + g I3   f − ϑ2 f ∗ + gϑ  2 |ϑ| + 1 χ − 2χ∗ |ϑ|  2  = 2 |ϑ| − 1 × H2 j (χ) + f Noting that

ϑ2 + 1 jH2 j (χ) . ϑ

H2 j (χ) = 4 jH2 j−1 (χ)

(2.158)

(2.159)

and using the property that the Hermite polynomials of different orders are mutually orthogonal, we have f − ϑ2 f ∗ + gϑ |ϑ|2 − 1



 |ϑ|2 + 1 ∗ χ − χ |ϑ| = 0. 2

(2.160)

By comparing H2 j (χ) on both sides of Eq. (2.158), we give E=

 f  2 ϑ + 1 j. ϑ

(2.161)

Since the eigenvalue equation is irrelevant to the entangled state parameter ζ (or χ), from Eq. (2.160), we obtain the value of ϑ, that is, f − ϑ2 f ∗ + gϑ = 0,

ϑ=

g±

-

g 2 + 4 | f |2 2f∗

(2.162)

and the eigenvalues E, i.e.,

E=

f + f ∗ϑ ϑ



 j = ± j g 2 + 4 | f |2 .

(2.163)

From above, the parameter ϑ and the eigenvalues E fully depend on the coefficients f (t), f ∗ (t) and g(t) in the Hamiltonian H = f (t) J+ + f ∗ (t) J− + g (t) Jz .

54

2 Dynamics of Two-Body Hamiltonian Systems

2.4 Atomic Coherent States as Energy Eigenstates of the Hamiltonian for Two-Dimensional Anisotropic Harmonic Potential in a Uniform Magnetic Field Now we consider the Hamiltonian of two-dimensional anisotropic harmonic potential in a uniform magnetic field H=

1 1 2 2 2 2 x + ω0y y ), (P − eA)2 + m(ω0x 2m 2

(2.164)

where P is the electron canonical momentum, A is the vector potential of magnetic field. Taking the vector potential as the symmetric gauge A = 21 (−By, Bx, 0), then we can rewrite the Hamiltonian H as H=

 1     1  2 P + Py2 + m ωx2 x 2 + ω 2y y 2 − ω L x Py − y Px , 2m x 2

where ωL =

eB 2 2 , ωa2 = ω0x + ω 2L , ωb2 = ω0y + ω 2L . 2m

(2.165)

(2.166)

For convenience, letting  mω 1/2

x + i (2mωa )−1/2 Px , 2  mω 1/2 b b= y + i (2mωb )−1/2 Py , 2

a=

a

(2.167)

thus the Hamiltonian H in Eq. (2.165) becomes the following Boson operator form:

    1 1 + ωb b† b + + iω L a † b − ab† . H = ωa a † a + 2 2

(2.168)

In order to determine whether the atomic coherent states are the eigenstates of two-dimensional anisotropic harmonic oscillator potential in a uniform magnetic field, we directly operate the √Hamiltonian H in Eq. (2.168) √ on the state |ϑ, and using the identities a † |n = n + 1 |n + 1 and a |n = n |n − 1, thus we obtain

2.4 Atomic Coherent States as Energy Eigenstates of the Hamiltonian …

55

 2j    ' 2 j 1/2 1  2 j−l ω 2 j − l + H |ϑ = ϑ a 2 (1 + |ϑ|2 ) j l=0 l

1/2  2j ' 1 2j |2 j − l ⊗ |l + iω L +ωb l + l −1 2 l=1 × ϑ2 j−l (2 j − l + 1) |2 j − l + 1 ⊗ |l − 1  1/2 2j ' 2j 2 j−l −iω L ϑ (l + 1) |2 j − l − 1 ⊗ |l + 1 . l +1

(2.169)

l=0

Taking l  = l ∓ 1 in the last two terms on the right-hand side of Eq. (2.169), we arrive at    1 ωb i2ω L j  |ϑ − + + H |ϑ =  ωa 2 j + 2 2 ϑ (1 + |ϑ|2 ) j



  2j 1/2 ' 1 2j 2 j−l (ωa − ωb ) + iω L ϑ + l |2 j − l ⊗ |l . (2.170) × ϑ l ϑ l=0

From Eq. (2.170) we find that, when the equation iω L ϑ2 + (ωa − ωb )ϑ + iω L = 0

(2.171)

holds, the states |ϑ±  with ϑ± =

(ωb − ωa ) ±



(ωa − ωb )2 + 4ω 2L

i2ω L

are the eigenstates of H with the eigenvalues    1 ωb i2ω L j . + + E ± =  ωa 2 j + 2 2 ϑ±

(2.172)

(2.173)

From above, we see that the eigenstates of H are classified as the atomic coherent states |ϑ±  according to the spin values j, where ϑ± are governed by the dynamic parameters m, A, ω0x , and ω0y in H . Particularly, when ωa = ωb = ω, the Hamiltonian H in Eq. (2.168) describes the two-dimensional isotropic harmonic oscillator potential in a uniform magnetic field, thus we use Eqs. (2.172) and (2.173) to obtain ϑ± = ∓i,

E ± = [ω + 2(ω ∓ ω L ) j].

(2.174)

56

2 Dynamics of Two-Body Hamiltonian Systems

Clearly, for the spin value j = 1/2, the eigenstates of H read 1 |ϑ±  = √ (|0, 1 ± i |1, 0). 2

(2.175)

In terms of Eq. (2.126), we find that all values of ϑ are permitted because the set of |ϑ makes up a complete quantum mechanics representation. However, from Eqs. (2.172) and (2.173), we know that, for a given Hamiltonian H , there exist only two atomic coherent states determined by the specific values ϑ± . Indeed, using the wave function of |ϑ in the entangled state η| representation, we can also solve the energy eigenstates of the Hamiltonian H . For this, introducing the unnormalized entangled state η = 00| exp (η ∗ a − ηb + ab), which yields the eigenequations     η a − b† = η η, η a † − b = η η ∗

(2.176)

and the partial differential identities −

∂ η = η b, ∂η

∂ η = η a. ∂η ∗

(2.177)

So, in the entangled state η basis, there exist the following corresponding relations: a→

∂ ∂ ∂ ∂ , b → − , a† → − − η, + η ∗ , b† → ∂η ∗ ∂η ∂η ∂η ∗

(2.178)

which help one to deal with the eigenvalue problems of Hamiltonian in quantum mechanics. Using the generating functions for Hermite polynomials Hm,n (x, y) in Eq. (2.131), in two-mode Fock space the entangled state η is expanded as η =

∞ '

  (−1)n m, n| √ Hn,m η, η ∗ , m!n! m,n=0

(2.179)

it then follows that the inner product η m, n reads   (−1)n η m, n = √ Hn,m η, η ∗ . m!n!

(2.180)

Using Eqs. (2.179) and (2.180), we obtain the inner product of η and |ϑ as √ 2j (2 j)! ' (−1)l ϑ2 j−l Hl,2 j−l (η, η ∗ ) η ϑ = . l! (2 j − l)! (1 + |ϑ|2 ) j l=0

(2.181)

2.4 Atomic Coherent States as Energy Eigenstates of the Hamiltonian …

57

Using the integral expansion of Hm,n (ζ, ξ) , that is, Hm,n (ζ, ξ) = (−1)n eζξ



d2 z n ∗m z z exp(− |z|2 + ζz − ξz ∗ ), π

(2.182)

Equation (2.181) becomes √ 2 2j (−1)2 j (2 j)!e|η| ' (1 + |ϑ|2 ) j l=0 2 2 j |η|2 d z (−1) e = 2 j√ π (1 + |ϑ| ) (2 j)!

d2 z (ϑz)2 j−l z ∗l exp(− |z|2 + ηz − η ∗ z ∗ ) π l! (2 j − l)!

η ϑ =

 2 j ϑz + z ∗ exp(− |z|2 + ηz − η ∗ z ∗ ). (2.183)

For convenience, making the following transformation of integral variables z = z ∗ − ϑ∗ z  ,

z ∗ = z  − ϑz ∗ ,

ϑz + z ∗ = (1 − |ϑ|2 )z  , and letting

d2 z = −(1 − |ϑ|2 )d2 z 

(2.184)

κ = η ∗ + ηϑ∗ ,

(2.185)

2 j 2  2 ∂ (−1)e|η| d z 2 2 j+1 |ϑ| ) (1 − √ ∂κ π (1 + |ϑ|2 ) j (2 j)!     2 exp −(1 + |ϑ|2 ) z   + ϑ∗ z 2 + ϑz ∗2 − κz  + κ∗ z ∗ .

(2.186)

thus Eq. (2.183) becomes η ϑ =

Using the integral formula (2.106), we have (−1)e|η| (1 − |ϑ|2 )2 j √ (1 + |ϑ|2 ) j (2 j)!  

2 j −(1 + |ϑ|2 ) |κ|2 + ϑκ2 + ϑ∗ κ∗2 ∂ exp × . ∂κ (1 − |ϑ|2 )2 2

η ϑ =

(2.187)

Further, using the differential expression of Hm (x)

 d m −x 2 − Hm (x) = e e dx x2

(2.188)

58

2 Dynamics of Two-Body Hamiltonian Systems

and letting

√ 1 + |ϑ|2 κ∗ ϑ η − ϑη ∗ − ≡ ζ, κ = √ √ 1 − |ϑ|2 2 ϑ 1 − |ϑ|2 2 ϑ

(2.189)

we obtain (−1) j+1 ϑ j H2 j (iζ)e|η| +|ζ| η ϑ = √ (1 + |ϑ|2 ) j (2 j)! ⎡  2 ⎤ 2 ∗ ∗2 ∗ ϑ κ 1 + |ϑ| κ ⎦. − × exp ⎣ √ (1 − |ϑ|2 )2 1 − |ϑ|2 2 ϑ 2

2

(2.190)

Inserting Eq. (2.189) into Eq. (2.190), we finally obtain the wave function of |ϑ in the entangled state η| representation that is

 (−1) j+1 ϑ j η − ϑη ∗ η ϑ = , H2 j i √ √ (1 + |ϑ|2 ) j (2 j)! 2 ϑ

(2.191)

which is related to a single-variable Hermite polynomial H2 j (·) of order 2 j. Supposing that the states |ϑ are the eigenstates of H with the eigenvalues E, thus we have η H |ϑ = E η ϑ . (2.192) Using Eqs. (2.85) and (2.178), we get     ∂ ∂ 1 ∗ η H |ϑ = ωa − +η + ∂η ∂η ∗ 2    ∂ 1 ∂ + + ωb − −η ∂η ∗ ∂η 2

2  ∂ ∂ ∂2 ∗ ∂ η ϑ . +η ∗ + iω L − ∗2 − η ∂η 2 ∂η ∂η ∂η

(2.193)

Inserting Eq. (2.191) into the right side of Eq. (2.193), we find 

 η (ωa + ωb )H2 j (ζ) + i ωb √ 2 2 ϑ   √ η∗ ϑ C   − iω L B H2 j (ζ) − H (ζ) , −ωa 2 4 2j

η H |ϑ = D

where

(2.194)

2.4 Atomic Coherent States as Energy Eigenstates of the Hamiltonian …

√ η∗ η ϑ B= √ + , 2 2 ϑ C = (ωa + ωb ) + iω L D=

59

 1 −ϑ , ϑ

(−1) j+1 ϑ j . √ (1 + |ϑ|2 ) j (2 j)!

(2.195)

Further, using Eq. (2.192) and the differential identity of Hm (x), that is, 

2λHm (x) − 2mHm (x) = Hm (x),

(2.196)

we obtain

η  (ωa + ωb + 2C j)H2 j (ζ) + i ωb √ 2 2 ϑ  √ ∗ η ϑ 1 − iω L B + i Cζ H2 j (ζ). −ωa 2 2

EH2 j (ζ) =

(2.197)

Owing to the differential identity H2 j (x) = 4 jH2 j−1 (x)

(2.198)

and two Hermite polynomials of different orders are orthogonal, so the coefficient of H2 j (ζ) shall be zero, thus we have √ η η∗ ϑ 1 − iω L B + i Cζ = 0. ωb √ − ωa 2 2 2 ϑ

(2.199)

For any values of η, η ∗ , Eq. (2.199) is always satisfied, so we obtain iω L ϑ2 + (ωa − ωb )ϑ + iω L = 0,

(2.200)

thus its solutions are ϑ± =

(ωb − ωa ) ±

 (ωa − ωb )2 + 4ω 2L i2ω L

,

(2.201)

which shows that the values of ϑ± coincide quite well with the results in Eq. (2.172). On the other hand, when the uniform magnetic field B changes with time t, thus the Hamiltonian describing the two-dimensional anisotropic harmonic potential in a uniform magnetic field is also time-dependent, thus its Boson form is

60

2 Dynamics of Two-Body Hamiltonian Systems

 

  1 1 + ωb (t) b† b + + iω L (t) a † b − ab† . H (t) = ωa (t) a † a + 2 2 (2.202) To derive the energy eigenstates and eigenvalues of H (t), we must solve the Schrödinger equation ∂ |φ(t) = H (t) |φ(t) . (2.203) i ∂t Taking |φ(t) = f (t) |ϑ ,

(2.204)

and substituting Eq. (2.204) into Eq. (2.203) lead to the differential equation i

d f (t) d |ϑ + i f (t) |ϑ = H (t) f (t) |ϑ . dt dt

(2.205)

Using the explicit expression of the state |ϑ in Eq. (2.126), we arrive at the time evolution of the state |ϑ , that is, d 2 j dϑ j 1 d(|ϑ|2 ) |ϑ = |ϑ − |ϑ − 2 dt ϑ dt dt 1 + |ϑ| (1 + |ϑ|2 ) j 1/2 2j ' dϑ 2j × lϑ2 j−l−1 |2 j − l ⊗ |l . l dt

(2.206)

l=0

Using a similar method to deriving Eq. (2.169), we have  

ωb (t) i2ω L (t) j 1 + + f (t) |ϑ H (t) f (t) |ϑ =  ωa (t) 2 j + 2 2 ϑ  2j  f (t) ' 2 j 1/2 − [(ωa (t) − ωb (t)) (1 + |ϑ|2 ) j l=0 l 

1 +iω L (t) ϑ + (2.207) lϑ2 j−l |2 j − l ⊗ |l . ϑ 

Substituting Eqs. (2.206) and (2.207) into Eq. (2.205) and using the linear independence of Fock states, we arrive at i and

dϑ = [ωa (t) − ωb (t)]ϑ + iω L (t)(ϑ2 + 1) dt

(2.208)

References

61

i

 

ωb (t) i2ω L (t) j d f (t) 1 = f (t) ωa (t) 2 j + + + dt 2 2 ϑ

 ∗ dϑ i2 j dϑ dϑ ij ϑ∗ +ϑ + . − dt dt ϑ dt 1 + |ϑ|2

(2.209)

If letting dϑ/dt = 0 in Eq. (2.209), the Hamiltonian H (t) reduces to H in Eq. (2.168). Substituting Eq. (2.208) into Eq. (2.209) leads to i

 

   ωa (t) d f (t) 1 = f (t) ωb (t) 2 j + + + i jω L (t) ϑ∗ − ϑ . dt 2 2

(2.210)

If taking f (t) = exp[iϕ(t)],

(2.211)

then the phase ϕ(t) is represented as t





 ∗  ωa (ε) + + i jω L (ε) ϑ (ε) − ϑ(ε) dε. ϕ(t) = − 2 0 (2.212) From above, we conclude that the states |ϑ with the exponential phase factor exp[iϕ(t)] are the eigenstates of the two-dimensional time-dependent anisotropic harmonic oscillator potential in a uniform magnetic field.

1 f (ε) ωb (ε) 2 j + 2

References 1. Dirac PAM (1958) Principles of quantum mechanics. Oxford University Press, New York 2. Feynman RP, Hibbs AR (1965) Quantum mechanics and path integrals. McGraw-Hill, New York 3. Klauder JR, Skargerstam B-S (1985) Coherent states: applications in physics and mathematical physics. World Scientific Publishing, Singapore 4. Klauder JR, Sudarshan ECG (1968) Fundamentals of quantum optics. W. A. Benjamin, New York 5. Hillery M, Zubairy MS (1982) Path-integral approach to problems in quantum optics. Phys Rev A 26(1):451–460 6. Arecchi FT, Courtens E, Gilmore R, Thomas H (1972) Atomic coherent states in quantum optics. Phys Rev A 6(6):2211–2237 7. Fan HY, Klauder JR (1994) Eigenvectors of two particles’ relative position and total momentum. Phys Rev A 49(2):704–707 8. Fan HY (2004) Entangled states, squeezed states gained via the route of developing Dirac’s symbolic method and their applications. Int J Mod Phys B 18(10–11):1387–1455 9. Wang JS, Meng XG, Fan HY (2008) Energy-level and wave functions of two moving charged particles with elastic coupling derived by virtue of the entangled state representations. Phys A: Stat Mech Appl 387(16–17):4453–4458 10. Meng XG, Wang JS, Liang BL (2010) Energy level formula for two moving charged particles with Coulomb coupling derived via the entangled state representations. Chin Phys B 19(4):044202

62

2 Dynamics of Two-Body Hamiltonian Systems

11. Wang JS, Meng XG, Feng J, Gao YF (2007) Establishing path integral in the entangled state representation for Hamiltonians in quantum optics. Chin Phys B 16(1):23–31 12. Wang JS, Meng XG, Fan HY (2019) Time evolution of angular momentum coherent state derived by virtue of entangled state representation and a new binomial theorem. Chin Phys B 28(10):100301 13. Meng XG, Wang JS, Liang BL (2010) Atomic coherent states as energy eigenstates of a Hamiltonian describing a two-dimensional anisotropic harmonic potential in a uniform magnetic field. Chin Phys B 19(12):124205 14. Meng XG, Wang JS, Fan HY (2009) Atomic coherent states as the eigenstates of a twodimensional anisotropic harmonic oscillator in a uniform magnetic field. Mod Phys Lett A 24(38):3129–3136

Chapter 3

New Bipartite Entangled States in Two-Mode Fock Space

In quantum optics, in order to clearly describe the unique nonclassical properties of quantum light fields, physicists usually introduce some quantum states with different characteristics. At present, in addition to the commonly used number states, coherent states, thermal states and squeezed states, are there other useful quantum states in physics? In particular, since quantum entanglement has been found to be widely used in quantum information, it is natural to think whether there is an entangled state with definite wave function? Historically, based on the relative coordinate Q a − Q b of two particles and their total momentum Pa + Pb , EPR first introduced the concepts of quantum entanglement and proposed the specific wave function describing two particles in an entangled state [1]. Later, Fan constructed the entangled states |η as the common eigenstates of the operators Q a − Q b and Pa + Pb in the twomode Fock space, whose specific form is shown in Sect. 1.2, and the complete set of the orthogonal normalized states |η can form an entangled state representation [2]. Entangled state |η representation is a natural representation of EPR entanglement thought, so it plays an important role in quantum mechanics and quantum information. For examples, the entangled state |η is an important bridge between classical optical transformation and quantum optical unitary operator, and it is helpful to study optical fractional Fourier transform, fractional Hankel transform, and optical Wigner transform [3]. In this chapter, using the IWOP methods, several new bipartite entangled states are constructed in two-mode Fock space, such as coherent-entangled states with both entanglement and coherence properties [4], entangled states describing parametric down-conversion process [5], and parameterized entangled states [6]. Also, the properties and generation schemes of these entangled states, and their applications (e.g., new squeezing operators and squeezed states, new entangled Wigner operators, entangled Fresnel transform, quantum correspondence of classical circular harmonic correlator, and tomography theory of two-mode correlated states) are discussed in detail.

© Science Press 2023 X.-G. Meng et al., Entangled State Representations in Quantum Optics, https://doi.org/10.1007/978-981-99-2333-5_3

63

64

3 New Bipartite Entangled States in Two-Mode Fock Space

3.1 Coherent-Entangled States For the bipartite entangled systems, noting that the operators μQ a + ν Pb and νa+iμb are commutative, i.e., [μQ a + ν Pb , νa+iμb] = 0, so their common eigenstates can be found in two-mode Fock space. For this, constructing the following Gaussian operator in normal ordering [4], that is,    μQ a + ν Pb 2 1 − O (α, x) ≡ : exp − x − λ 2    √ √ 2 2 × α∗ − (νa † − iμb† ) α − (νa + iμb) :, λ λ

(3.1)

where λ, μ, and ν are real. Using the IWOP method, we obtain the following two marginal integrations: 

    μQ a + ν Pb 2 d2 α O (α, x) = : exp − x − : 2π λ

(3.2)

and 

∞

dx √ O (α, x) π −∞     √ √ 2 2 1 ∗ † † (νa − iμb ) α − (νa + iμb) :. = : exp − α − 2 λ λ

(3.3)

Thus we have     dx μQ a + ν Pb 2 : = 1. √ : exp − x − λ π −∞ −∞ (3.4) Further, using the normal ordering of two-mode vacuum in Eq. (2.7) [3] and letting λ = μ2 + ν 2 , thus the operator O (α, x) can be decomposed as the form of the following projection operator: 

∞

dx √ π



d2 α O (α, x) = 2π



∞

O (α, x) = |α, x α, x| .

(3.5)

Based on Eqs. (3.1) and (3.5), the state |α, x is defined as

|α|2 x2 1 − + √ (2μx + να)a † 4 2 2λ  i 1 † † † 2 + √ (2νx − μα)b − 2 (μa + iνb ) |00 , 2λ 2λ

|α, x = exp −

(3.6)

3.1 Coherent-Entangled States

65

and its completeness relation is 

d2 α 2π



∞

−∞

dx √ |α, x α, x| = 1, π

(3.7)

which shows that the complete set of the states |α, x can form a quantum mechanical representation. Using the Boson commutation relation [i, f (i, i † )] =

∂ f (i, i † ), (i = a, b), ∂i †

(3.8)

we obtain

and

 1 μ † † a |α, x = √ (2μx + να) − 2 (μa + iνb ) |α, x , λ 2λ

(3.9)

 i iν b |α, x = √ (2νx − μα) − 2 (μa † + iνb† ) |α, x . λ 2λ

(3.10)

Combining Eqs. (3.9) and (3.10) leads to the following eigenvalue equations: λα (μQ a + ν Pb ) |α, x = λx |α, x , (νa + iμb) |α, x = √ |α, x . 2

(3.11)

Equation (3.11) shows that the state |α, x has some characteristics of not only coherent states but also entangled states, but it is completely different from the existed coherent states and entangled states in structure. So, the state |α, x is called as the bipartite coherent-entangled state. Especially, when μ = ν, the state |α, x becomes another coherent-entangled state, i.e., |α, xμ=ν

|α|2 x2 1 − + (2x + α)a † = exp − 4 2 2  i 1 † † † 2 + (2x − α)b − (a + ib ) |00 . 2 4

(3.12)

It can be proved that the states |α, xμ=ν obey the completeness relation 

d2 α 2π

and the eigenvalue equations



∞

−∞

dx √ |α, xμ=ν μ=ν α, x| = 1 π

(3.13)

66

3 New Bipartite Entangled States in Two-Mode Fock Space

(Q a + Pb ) |α, xμ=ν =

√ 2x |α, xμ=ν ,

(a + ib) |α, xμ=ν = α |α, xμ=ν . (3.14) The mutual orthogonality of the states |α, x is discussed as follows. Performing the partial differential ∂/∂α in Eq. (3.6), we give √



α∗ ∂ + 2λ ∂α 4

 |α, x = (νa † − iμb† ) |α, x .

(3.15)

Thus, it follows from Eq. (3.15) that

λα α , x  (νa + iμb) |α, x = √ α , x  |α, x 2   √ ∂ α   = 2λ α , x |α, x . + ∂α∗ 4

(3.16)

Taking the complex conjugate of Eq. (3.16) and exchanging the parameters α , x  and α, x, we have   ∗ α∗   α ∂   α , x |α, x = − α , x |α, x . (3.17) ∂α 2 4 On the other hand, carrying out the partial differential ∂/∂α∗ in Eq. (3.6) leads to α ∂ |α, x = − |α, x . ∂α∗ 4

(3.18)

∂   α∗   α , x |α, x . α , x |α, x = −  ∂α 4

(3.19)

Similarly, we obtain

Solving the partial differential equations (3.17) and (3.19), and using the completeness relation in Eq. (3.7) and the identity

α , x  (μQ a + ν Pb ) |α, x = λx α , x  |α, x = λx  α , x  |α, x ,

(3.20)

thus we have   √ −1 α , x |α, x = πe 4 





|α|2 +|α | −2αα∗ 2



δ(x − x  ),

(3.21)

2 − |α|2 +|α | −2αα∗ /4 where the term e reflects the non-orthogonal characteristics like ordinary coherent states, but δ(x − x  ) represents the orthogonality of entangled states. This also confirms that the state |α, x is indeed a coherent-entangled state. Particularly, when α = α , the inner product α , x  |α, x becomes

3.1 Coherent-Entangled States

67



α , x  |α, x =

√ πδ(x − x  ).

(3.22)

On the other hand, by performing Fourier transform on the state |α = α1 + iα2 , x, we obtain  1 dα2 |α = α1 + iα2 , x eiuα2 2π  

1 να1  1 2 μx − uν + = exp − (2u − α1 ) y = 8 λ 2 a

  

1 μα 1 νx + μu − ⊗

p = , (3.23) λ 2 b where the states |ya , | pb are, respectively, the coordinate eigenstate of the particle a and the momentum eigenstate of the particle b in Fock space, i.e.,

and

  √ 1 1 |ya = π −1/4 exp − y 2 + 2ya † − a †2 |0 2 2

(3.24)

  √ 1 1 | pb = π −1/4 exp − p 2 + i 2 pb† + b†2 |0 . 2 2

(3.25)

Further, performing the inverse Fourier transform of Eq. (3.23), we have 

1 du exp − (2u − α1 )2 − iuα2 8

 

1 να1  μx − uν + ×

y = λ 2 a

 

1 μα 1 νx + μu − ⊗

p = , λ 2 b 

|α = α1 + iα2 , x =

(3.26)

which is the Schmidt decomposition of the state |α = α1 + iα2 , x. According to the standard Schmidt decomposition theory [7], the state |α = α1 + iα2 , x is an entangled state. Optical beam splitter plays an important role in the generation of entanglement, so the asymmetrical beam splitter can be used as an experimental device of generating some quantum states. Experimentally, the role of an asymmetrical beam splitter with free phase can be represented by the operator [8, 9] 

θ † † B(θ) = exp −i (a b + ab ) . 2

(3.27)

68

3 New Bipartite Entangled States in Two-Mode Fock Space

  Inputting the ideal single-mode coordinate eigenstate |x = 0a ≡ exp −a †2 /2 |0a and the vacuum state |0b into the two input ports of an asymmetrical beam splitter and making them overlapped, and taking θ = 2 cos−1 μλ , thus the output state reads  1 † † 2 B(θ) |x = 0a ⊗ |0b = exp − 2 (μa + iνb ) |00 . 2λ

(3.28)

Further, operating two local oscillator laser beams represented by the displaced operators   μx + να = exp(ε1 a † − ε∗1 a) D a ε1 = √ (3.29) 2λ   νx − μα = exp(ε2 b† − ε∗2 b) D b ε2 = i √ (3.30) 2λ   on the output state exp − 2λ1 2 (μa † + iνb† )2 |00, thus we achieve the generation of |α, x, that is, and



1 Da Db exp − 2 (μa † + iνb† )2 |00 2λ  |ε1 |2 + |ε2 |2 1 (με∗1 + iνε∗2 )2 − 2 (μa † + iνb† )2 = exp − − 2 2λ2 2λ   



μ(με∗1 + iνε∗2 ) iν(με∗1 + iνε∗2 ) † † |00 a + + ε + + ε 1 2 b λ2 λ2 = |α, x .

(3.31)

As applications of |α, x, we present the following four aspects: A coherent-entangled state conjugate to the state |α, x Performing the Fourier transform on the state |α, x leads to 1

  1 ∗ dx i px ∞ d2 α ∗ |α, x exp (α β − αβ ) √ e 4 2π −∞ −∞ 4π  2 |β| 1 p2 − + √ (νβ + i2μ p)a † = exp − 2 4 2λ  1 1 † † † 2 − √ (iμβ + 2ν p)b + 2 (μa + iνb ) |00 , 2λ 2λ 

|β, p =

∞

(3.32)

which is the common eigenstate of the commutation operators μPa − ν Q b and νa+iμb since it obeys the eigenvalue equations λβ (μPa − ν Q b ) |β, p = λ p |β, p , (νa + iμb) |β, p = √ |β, p . 2

(3.33)

3.1 Coherent-Entangled States

69

So, the states |β, p and |α, x are mutually conjugated. Similarly, using the IWOP method, we find that the states |β, p yield the completeness relation 

d2 β 2π



∞ −∞

dp √ |β, p β, p| = 1, π

(3.34)

which shows that the set of the states |β, p can form a new coherent-entangled state representation. 2 Superpositions of the states |α, x In quantum optics, superpositions of quantum states can produce some new quantum states with significant nonclassical properties. For example, quantum superpositions of coherent states possess second-order squeezing, higher order squeezing, and subPoisson photon number statistical distribution [10–12]. Integrating over α1 in the state |α, x, we have  1 dα1 |α = α1 + iα2 , x √ 2 π   x2 1  α2 +√ = exp − 2 − (2μx + iνα2 )a † + (μα2 + i2νx)b† 4 2 2λ    1 + 2 (ν 2 − μ2 )(a †2 + b†2 ) − i4μνa † b† |00 2λ = |α2 , x . (3.35) It is proved that the states |α2 , x are the common eigenstates of the mutually commutative operators μX b + ν Pa and μQ a + ν Pb , i.e., (μX b + ν Pa ) |α2 , x =

λα2 |α2 , x , 2

(μQ a + ν Pb ) |α2 , x = λx |α2 , x . (3.36) Therefore, the superposition of the state |α = α1 + iα2 , x along the α1 -axis can lead to the newly derived EPR entangled state [13]. 3 One- and two-mode combinatorial squeezing operator Integrating over α1 in the density operator |α = α1 + iα2 , x α = α1 + iα2 , x| and acting the integral result on vacuum, we have 1 √ 2π



dα1 |α = α1 + iα2 , x α = α1 + iα2 , x| 00

  1 †2 α2 3 a − b†2 − 2 (μa † + iνb† )2 = exp −x 2 − 2 + 2 4 2λ  1 i + √ (2μx + iνα2 )a † + √ (2νx − iμα2 )b† |00 ≡ |s . 2λ 2λ

Especially, when α2 = x = 0,

(3.37)

70

3 New Bipartite Entangled States in Two-Mode Fock Space

|s = exp

 

1 †2 3 a − b†2 − 2 (μa † + iνb† )2 |00 . 4 2λ

(3.38)

Since there exist the operators a †2 , b†2 and a † b† in Eq. (3.38), the state |s is a one- and two-mode combinatorial squeezed state. Thus, the superposition of |α = α1 + iα2 , x α = α1 + iα2 , x|α2 =x=0 along the α1 -axis is indeed a one- and two-mode combinatorial squeezing operator. Further, letting μ = ν, Eq. (3.38) becomes

 1 †2 3 † † †2 |s = exp (3.39) (a − b ) − i a b |00 . 16 8 Further, constructing the integration for the asymmetric ket-bra operator  Sc (r ) =

∞

−∞

dx √ kπ



d2 α

x 

α, α, x| , 2π k

(3.40)

where k = er is the squeezing parameter, using the IWOP method and the integral formula in Eq. (1.30), as well as the operator identity in Eq. (1.21), we directly have       tanh r †2 tanh r 2 exp R † R ln sech r exp R R , sech r exp − 2 2 (3.41)   where R † = μa † + iνb† /λ. Noting that R †2 , R 2 and R † R can form a SU(1,1) Lie algebra because 

2   R R †2 1 , = R† R + , (3.42) R, R † = 1, 2 2 2 Sc (r ) =

√

so Sc (r ) is another one- and two-mode combinatorial squeezing operator. In terms of Eqs. (3.21) and (3.40), the operator Sc (r ) can naturally squeeze the state |α, x, that is, 1 (3.43) Sc (r ) |α, x = √ |α, x/k . k It then follows that 

√ tanh r † † 2 Sc (r ) |00 = sech r exp − (μa + iνb ) |00 , 2λ2

(3.44)

which is also a new one- and two-mode combinatorial squeezed state. Besides, the state |α, x can be used to derive some new operator identities. For example, for the operator O = exp[ f (νa + iμb)2 ] exp[g(μQ a + ν Pb )2 ],

(3.45)

3.1 Coherent-Entangled States

71

inserting the completeness relation of the states |α, p into Eq. (3.45) and using the IWOP method, we obtain 

d2 α 2π



∞

dx √ exp[ f (νa + iμb)2 ] |α, x α, x| exp[g(μQ a + ν Pb )2 ] π −∞   2  2  ∞ dx d α λ 2 f α |α, x α, x| exp(λ2 gx 2 ) = √ exp 2π −∞ π 2

g 1 (μa † + iνb† )2 : exp = 2 2(1 − λ2 g) 1−λ g g + (μa † + iνb† )(μa − iνb) 1 − λ2 g  g 2 2 :, (3.46) (μa − iνb) + + f (νa + iμb) 2(1 − λ2 g)

O=

which refers to the normal ordering product of the operator O. Further, using Eq. (1.21), thus Eq. (3.46) becomes O=

1

exp

g(μa † + iνb† )2 2(1 − λ2 g)



1 − λ2 g 

(μa † + iνb† )(μa − iνb) 1 × exp ln λ2 1 − λ2 g 

2 g(μa − iνb) 2 . + f (νa + iμb) × exp 2(1 − λ2 g)

(3.47)

Particularly, letting f = 0, we have a one- and two-mode combinatorial squeezing operator, i.e.,  g(μa † + iνb† )2 exp[g(μQ a + ν Pb ) ] = exp 2(1 − λ2 g) 1 − λ2 g 

(μa † + iνb† )(μa − iνb) 1 × exp ln λ2 1 − λ2 g 

2 g(μa − iνb) . (3.48) × exp 2(1 − λ2 g) 2

1



Operating the operator exp[g(μQ a + ν Pb )2 ] on vacuum leads to  g(μa † + iνb† )2 |00 . exp exp[g(μQ a + ν Pb ) ] |00 = 2(1 − λ2 g) 1 − λ2 g 2

1



(3.49)

72

3 New Bipartite Entangled States in Two-Mode Fock Space

4 Generalized P-representation Noting that the state |α, x has the same completeness and non-orthogonality as the coherent state, so by using the states |α, x as a basis vector set, the expansion of the density operator ρ(νa † − iμb† , νa+ iμb) leads to the so-called generalized P-representation, i.e., [14]

 ρ(νa − iμb , νa + iμb) = †

†

dx √ π



d2 α P(α, α∗ ) |α, x α, x| . 2π

(3.50)

 Operating the states −α , −x  and α , x  on left and right side of Eq. (3.50), respectively, and using Eq. (3.22), we obtain

 −α , −x  ρ(νa † − iμb† , νa + iμb) α , x   2   d α dx P(α, α∗ ) −α , −x  |α, x α, x| α , x  = √ 2π π  2  d α dx   P(α, α∗ ) = √ δ(x − x )δ(x + x ) 2 π  

 2

α |α|2 α∗ α − αα∗ × exp − − + 2 2 2  

 2  2

α |α|2 α∗ α − αα∗ d α  ∗ − + = δ(2x ) . √ P(α, α ) exp − 2 2 2 2 π



(3.51)

Since the term (α∗ α − αα∗ )/2 is a pure plural, the inverse Fourier transform of Eq. (3.51) reads  2  

α  |α|2 αα∗ − α∗ α 1 2  + P(α, α ) = e 2 d α exp 2π 3/2 2 2 

 × dx  −α , −x  ρ(νa † − iμb† , νa + iμb) α , x  , ∗

(3.52)

which is exactly the generalization of single-mode P-representation given by Mehta [15].

3.2 Entangled States Describing Parametric Down Conversion In the process of light fields interacting with some nonlinear crystals, the parametric down-conversion process with both squeezing and entanglement can be realized [16, 17]. The signal and idle lights in the process of parametric down conversion are

3.2 Entangled States Describing Parametric Down Conversion

73

entangled with each other in the frequency domain, so the Hamiltonian describing this process is written as [5, 18, 19]   H = ωs a † a + ωi b† b +  a † b† e−i2ω p t + abei2ω p t ,

(3.53)

where 2ω p = ωs +ωi , the subscripts p, s, and i are, respectively, pump, signal, and idle lights,  is the coupling constant depending on the amplitude of pump light and the second-order magnetic susceptibility of nonlinear crystal. On the other hand, using the Heisenberg equation of motion in interaction representation, we have da = −ib† , dt

db† = −ia, dt

(3.54)

which leads to a(t) = a(0) cosh t − ib† (0) sinh t,

b(t) = b(0) cosh t − ia † (0) sinh t, (3.55) that is, a squeezing transformation is involved in the parametric down-conversion process. So, we plan to study the parametric down-conversion process from the view of squeezing mechanism and quantum entanglement. In two-mode Fock space, the entangled state |τ  is structured as  |τ |2 † ∗ † † † |τ  = exp − + τ a − iτ b + ia b |00 , 2 

(3.56)

where τ = τ1 +iτ2 . Operating the operators √ a and b on the state |τ  , respectively, we find that the real and imaginary parts of 2τ are just the eigenvalues of the operators (Q a − Pb ) and (Pa − Q b ), i.e., (Q a − Pb ) |τ  =

√

2τ1 |τ  , (Pa − Q b ) |τ  =

√

2τ2 |τ  .

(3.57)

Using the normal ordering of the projection operator of two-mode vacuum in Eq. (2.7), we find that the states |τ  possess the completeness relation 

d2 τ |τ  τ | = π



d2 τ −(τ ∗ −a † −ib)(τ −a+ib† ) :e : π  2 2  2  P −Q b d τ − τ1 − Qa√−P − τ2 − a√2 b 2 :e = : =1 π

(3.58)

and the orthogonality

τ  |τ  = πδ(τ − τ  )δ(τ ∗ − τ ∗ ).

(3.59)

Further, performing the Fourier transform on the state |τ = τ1 + iτ2 , we have

74

3 New Bipartite Entangled States in Two-Mode Fock Space

  √

τ1 τ1 dτ2 √ |τ = τ1 + iτ2  ei 2τ2 x =

y = √ − x ⊗

p = − √ − x , 2π 2 2 −∞ a b (3.60) where the states |ya , | pb are, respectively, the eigenstates of the coordinate operator Q a and the momentum operator Pb . Further, performing the inverse Fourier transform of Eq. (3.60), we obtain the Schmidt decomposition of |τ  as 

∞

|τ  = eiτ1 τ2



∞ −∞

 √ √

dx y = 2τ1 − x ⊗ | p = −xb e−i 2τ2 x . a

(3.61)

Similar to preparing the coherent-entangled   state |α, x, inputting  the ideal momentum eigenstates | p = 0a = exp a †2 /2 |0a and | p = 0b = exp b†2 /2 |0b into the two input ports of a symmetric beam splitter and taking θ = −π/2, so we obtain the entangled state |τ = 0, i.e.,  π | p = 01 ⊗ | p = 02 = exp(ia † b† ) |00 = |τ = 0 , B − 2

(3.62)

thus, after acting the displaced operator D (τ ) on the state |τ = 0 via laser modulation, the ideal entangled states |τ  can naturally be prepared, that is, D (τ ) exp(ia † b† ) |00 = D (τ ) |τ = 0 = |τ  .

(3.63)

Making the classical scale transformation τ → τ /k in the state |τ , we can introduce the integration for the asymmetrical ket-bra operator, that is, 

d2 τ |τ /k τ | πk 

  †   2 1 |τ |2 a d τ : exp − 1 + 2 +τ + ib = πk k 2 k   † b + ia † b† − iab − a † a − b† b : +τ ∗ a − i k

= sech r : exp[ia † b† tanh r + (sech r − 1)(a † a + b† b) + iab tanh r ] : ,

(3.64)

where k = er . Using Eq. (1.21), we can rewrite Eq. (3.64) as 

d2 τ |τ /k τ | πk = exp(ia † b† tanh r ) exp[(a † a + b† b + 1) ln sech r ] exp(iab tanh r )   = exp ir (a † b† + ab) ≡ S p (r ). (3.65)

3.2 Entangled States Describing Parametric Down Conversion

75

  Comparing with the ordinary squeezing operator S2 (r ) = exp r (a † b† − ab) , we find that S p (r ) is indeed a new two-mode squeezing operator. Using Eqs. (3.59) and (3.65), we have 

S p (r )S †p (r )

d2 τ |τ /k τ | = πk 2 = 1 = S †p (r )S p (r ),



d2 τ 

  

τ τ /k π (3.66)

that is, the new squeezing operator S p (r ) is unitary. Besides, S p (r ) can lead to the following squeezed transformations: † S p (r )aS −1 p (r ) = a cosh r − ib sinh r,

† S p (r )bS −1 p (r ) = b cosh r − ia sinh r, (3.67) which is just the squeezed transformation involved in the parametric down conversion process as shown in Eq. (3.55). Therefore, the squeezing in this process can be described by the squeezing operator S p (r ), whose concise entangled state form is S p (r ) = (πk)−1 d2 τ |τ /k τ |. This shows that the parametric down-conversion process is not only a squeezing but also an entanglement process. Using Eq. (3.67), we naturally obtain r S p (r )(Q a − Pb )S −1 p (r ) = e (Q a − Pb ), −r S p (r )(Pa − Q b )S −1 p (r ) = e (Pa − Q b ).

(3.68)

Acting the squeezing operator S p (r ) on the state |00 leads to a new squeezed state, i.e., S p (r ) |00 = sech r exp(ia † b† tanh r ) |00 = sech r

∞ 

(−1)n/2 tanhn r |n, n .

(3.69)

n=0

Besides, using the √ state |τ  to construct another new squeezing operator that can squeeze |τ1 , τ2  as 1/ k |τ1 , τ2 /k. Using Eq. (3.61), we have  (Q a + Pb ) |τ  =

∞

dx −∞

√  √  √

2τ1 − 2x 2τ1 − x ⊗ | p = −xb e−iτ2 ( 2x−τ1 ) a

√ ∂ |τ  . = −i 2 ∂τ2

(3.70)

Combining Eqs. (3.57) and (3.70) and letting τ2 = er , thus we have

1 ∂ ∂ τ | (Q a + Pb )(Pa − Q b ) = iτ2 τ | = i τ1 , τ2 = er . 2 ∂τ2 ∂r

(3.71)

76

3 New Bipartite Entangled States in Two-Mode Fock Space ∂

Owing to e−r ∂λ f (λ) = f (λ − r ), thus

∂ τ | eir (Q a +Pb )(Pa −Q b )/2 = e−r ∂λ τ1 , τ2 = eλ



= τ1 , τ2 = eλ−r = τ1 , e−r τ2 ,

(3.72)

so the operator e[ir (Q a +Pb )(Pa −Q b )−r ]/2 ≡ S p (r ) can squeeze the state τ | as

τ | S p (r ) = e−r/2 τ1 , e−r τ2 ,

(3.73)

which shows that when making the dilation transformation τ2 −→ e−r τ2 in the state |τ1 , τ2 , we naturally obtain a new operator S p (r ) that is called as one-side squeezing operator. Using Eqs. (3.58) and (3.73) to obtain  r   d2 τ

|τ  τ1 , e−r τ2 S p (r ) = exp − 2 π  tanh r † [(a + ib† )2 − (a − ib)2 ] = sech1/2 r : exp − 4  1 † † + (a + ib )(a − ib)(sech r − 1) : 2 

tanh r † 1/2 † 2 (a + ib ) = sech r exp − 4 



1 † tanh r † 2 (a − ib) , × exp (a + ib )(a − ib) ln sech r exp 2 4

(3.74)

which leads to a two-mode one-side squeezed state 

tanh r † (a + ib† )2 |00 . S p (r ) |00 = sech1/2 r exp − 4

(3.75)

Similarly, making the dilation transformation τ1 −→ e−r τ1 in the state τ1 , τ2 | also leads to another one-side squeezing operator S p (r ) that squeezes the state |τ1 , τ2  as √ 1/ k |τ1 /k, τ2 , i.e.,  r   d2 τ

|τ  e−r τ1 , τ2 S p (r ) = exp − 2 π 

ir r = exp − (Pa + Q b )(Q a − Pa ) − 2 2 

tanh r (a † − ib† )2 = sech1/2 r exp − 4 



1 † tanh r † 2 (a + ib) . × exp (a − ib )(a + ib) ln sech r exp 2 4

(3.76)

3.2 Entangled States Describing Parametric Down Conversion

77

For a two-particle system with entanglement, it is necessary to discuss its Wigner operator from the view of quantum entanglement because the entangled state representation of the Wigner operator and its marginal distributions can reveal the entangled properties of the system. In terms of the state |τ , the Wigner operator can be expressed as the form  (ρ, σ) =

d2 τ |ρ − τ  ρ + τ | exp(τ σ ∗ − τ ∗ σ), π3

(3.77)

where ρ and σ are complex, ρ = ρ1 +iρ2 , σ = σ1 +iσ2 . Substituting Eq. (3.56) into Eq. (3.77) and using the IWOP method to complete it, thus the normal ordering product of the Wigner operator reads (ρ, σ) = π −2 : exp[−(ρ∗ − a † − ib)(ρ − a + ib† ) − (σ ∗ − a † + ib)(σ − a − ib† )] : ,

(3.78)

which is just the entangled state |τ  representation of two-mode Wigner operator. This is because, when taking ρ = α − iβ ∗ , σ = α + iβ ∗ .

(3.79)

Equation (3.78) can be rewritten as the direct product of single-mode Wigner operators a (α, α∗ ) and b (β, β ∗ ) [20], that is, (ρ, σ) = π −2 : exp[−2(α∗ − a † )(α − a) − 2(β ∗ − b† )(β − b)] : = a (α, α∗ ) ⊗ b (β, β ∗ ).

(3.80)

Further, integrating over the variables σ and ρ in the Wigner operator (ρ, σ), two marginal distributions are 

1 1 : exp[−(ρ∗ − a † − ib)(ρ − a + ib† )] : = |τ  τ |τ =ρ , π π (3.81)  1 1 2 ∗ † † d ρ(ρ, σ) = : exp[−(σ − a + ib)(σ − a − ib )] : = |ς ς|ς=σ . π π (3.82) As the conjugate states of |τ , the entangled states |ς are the common eigenstates of the operators Q a + Pb and Pa + Q b , i.e., d2 σ(ρ, σ) =

√

2ς1 |ς , (Pa + Q b ) |ς =

√

2ς2 |ς ,

(3.83)

  |ς|2 |ς = exp − + ςa † + iς ∗ b† − ia † b† |00 , ς = ς1 + iς2 , 2

(3.84)

(Q a + Pb ) |ς =

where the states |ς have the form

78

3 New Bipartite Entangled States in Two-Mode Fock Space

and yield the complete orthogonal relations 

d2 ς |ς ς| = 1, π



ς  |ς = πδ(ς − ς  )δ(ς ∗ − ς ∗ ).

(3.85)

So, for the bipartite entangled state |φ, the marginal distributions of its Wigner distribution function are, respectively, 

1 |τ |φ|2τ =ρ , π  1 d2 ρ φ| (ρ, σ) |φ = |ς |φ|2ς=σ , π d2 σ φ| (ρ, σ) |φ =

(3.86)

where |τ |φ|2τ =ρ and |ς |φ|2ς=σ refer to the measurement probabilities of joint observables of two particles. Specifically, |τ |φ|2τ =ρ shows that the probabilities of measuring√the joint √ observables Q a − Pb and Pa − Q b of two particles are, respectively, 2ρ1 and 2ρ2 in the entangled state |φ, while |ς |φ|2ς=σ means that the probabilities of measuring the joint observables Q a + Pb and Pa + Q b are, √ √ respectively, 2σ1 and 2σ2 . Therefore, the marginal distributions derived from the Wigner operator (ρ, σ) in the entangled state |τ  representation can give the probabilities of finding two particles in an entangled way, which is a new physical meaning endowed with the marginal distributions of Wigner distribution functions for entangled particles. Based on it, we can name (ρ, σ) the entangled Wigner operator.

3.3 Parameterized Entangled States Induced by the Common Eigenstates of a† + ib and a− ib† Based on the basic idea of quantum entanglement [1], and noting that the commutative  relation a † + ib, a − ib† = 0, constructing a Gaussian operator whose integration in normal ordering is equal to one [6], that is, 

d2 τ : exp[−(τ ∗ − a † − ib)(τ − a + ib† )] : ≡ 1. π

(3.87)

Using Eq. (2.7) to decompose the above Gaussian operator into : exp[−(τ ∗ − a † − ib)(τ − a + ib† )] : = |τ  τ | ,

(3.88)

where the form of the state |τ  can be seen in Eq. (3.56). Similarly, noting that the operators a † − ib and a+ ib† are commutative, we also find their common eigenstates |ς as shown in Eq. (3.84). Using Eqs. (3.56) and (3.84), the inner product of the states

3.3 Parameterized Entangled States Induced by the Common Eigenstates …

τ | and |ς reads τ |ς =



1 1 exp (τ ∗ ς − τ ς ∗ ) , 2 2

79

(3.89)

which is just a complex Fourier transform kernel. So, the states |ς and |τ  obey the following Fourier transform relation, that is, 

 1 d2 ς |ς exp (τ ς ∗ − τ ∗ ς) = |τ  . 2π 2

(3.90)

Making the substitutions a → s ∗ a − s  b† and b → s ∗ b − s  a † in the complete 2 −1 d τ τ | = 1, where s, s  are complex and yield the unimodularity ness relation π ∗  ∗ condition ss − s s = 1, we still obtain the resolvent of unity, that is, 

d2 τ : exp{−[τ ∗ − (sa † − s ∗ b) − i(s ∗ b − s  a † )] π × [τ − (s ∗ a − s  b† ) + i(sb† − s ∗ a)]} :    ∗ 1 d2 τ τ − (s − is  )a † : exp − = 2 2   |s − is | π |s − is |   +(s ∗ − is ∗ )b × [τ − (s ∗ + is ∗ )a + (s  + is)b† ] : ,

1=

(3.91)

2 where the appearance of the factor 1/ s − is  is to ensure that thenormal ordering  product of the vacuum projection operator, i.e., |00 00| = : exp −a † a − b† b : can be obtained in the process of exponential expansion. By placing all the creation operators to the left of the annihilation operators and using the identity 1 s ∗ − is ∗ s + is  = + , 2 2(s ∗ + is ∗ ) 2(s − is  ) |s − is  |

(3.92)

we can decompose the integral kernel in Eq. (3.91) into 

d2 τ |τ s,s  s,s  τ | = 1, π

(3.93)

where the state |τ s,s  is expressed as

1 s ∗ − is ∗ τ a† |τ |2 + ∗ exp − ∗ ∗ ∗ ∗ s + is 2(s + is ) s + is ∗  τ ∗ b† s  + is † † − ∗ + ∗ a b |00 ∗ s − is s + is ∗

|τ s,s  =

and obeys the eigenstate equations

(3.94)

80

3 New Bipartite Entangled States in Two-Mode Fock Space

[(s ∗ + is ∗ )a − (s  + is)b† ] |τ s,s  = τ |τ s,s  , [(s − is  )a † − (s ∗ − is ∗ )b] |τ s,s  = τ ∗ |τ s,s  .

(3.95)

Further, introducing the identities related to the real parameters A, B, C, D, i.e., s=

1 1 [A + D − i(B − C)], s  = [B + C + i(A − D)], 2 2

(3.96)

where ss ∗ − s  s ∗ = 1 guarantees AD − BC = 1, thus the state |τ s,s  becomes

D − iC 1 τ a† |τ |2 + exp − A + iB 2(A + iB) A + iB  ∗ † B + iA † † τ b + a b |00 , − B − iA A + iB

|τ s,s  =

(3.97)

thus the eigenstate equations in Eq. (3.95) can be rewritten as √ 2τ1 |τ s,s  , √ = 2τ2 |τ s,s  ,

[A(X a − Pb ) − B(Pa + X b )] |τ s,s  = [A(Pa − X b ) + B(X a + Pb )] |τ s,s 

(3.98)

and the orthogonality of the state |τ s,s  reads s,s 

 τ τ s,s  = πδ(τ − τ  )δ(τ ∗ − τ ∗ ).

(3.99)

On the other hand, according to the expression of the state |ς in Eq. (3.84), the parameterized entangled state |ςs,s  conjugate to the state |τ s,s  is obtained as |ςs,s 

1 s ∗ + is ∗ ςa † 2 |ς| = ∗ exp − + s − is ∗ 2(s ∗ − is ∗ ) s ∗ − is ∗  ∗ †  iς b s + is + ∗ − ∗ ia † b† |00 , ∗ s − is s − is ∗

(3.100)

which yields the eigenequations [(s ∗ − is ∗ )a − (s  − is)b† ] |ςs,s  = ς |ςs,s  [(s − is  )a † + i(s ∗ + is ∗ )b] |ςs,s  = ς ∗ |ςs,s 

(3.101)

and the complete orthogonal relations s,s 







∗

∗

ς |ςs,s  = πδ(ς − ς )δ(ς − ς ),



d2 ς |ςs,s  s,s  ς| = 1. π

(3.102)

3.3 Parameterized Entangled States Induced by the Common Eigenstates …

81

New parameterized entangled state |τ s,s  representation can provide convenience for finding some new unitary operators that can be widely used in establishing some relationship between the entangled state |τ s,s  and some known states. Next, we want to find a kind of unitary operator that converts the state |τ  to the state |τ s,s  . For this, constructing the integration for the following ket-bra operator: 

U (s, s ) ≡



d2 τ |τ s,s  τ | , π

(3.103)

which leads to the identity 

d2 τ 

  τ s,s  τ  τ  π 

   = d2 τ  τ  s,s  δ (2) τ  − τ = |τ s,s  .

U (s, s  ) |τ  =

(3.104)

Using the IWOP method to complete the integration in Eq. (3.103), we have 



d2 τ |τ s,s  τ | π

 2 s ∗ − is ∗ τ a† d τ 1 2 |τ | exp − + = ∗ s + is ∗ π 2(s ∗ + is ∗ ) s ∗ + is ∗ 

 ∗ †  |τ |2 iτ b s − is † † ∗ |00 00| exp − + τ − ∗ + ia b a + iτ b − iab s + is ∗ s ∗ + is ∗ 2

 2 2 ∗ † s |τ | [a + i(s ∗ + is ∗ )b]τ d τ 1 : exp − ∗ + = ∗ ∗ ∗ s + is π s + is s ∗ + is ∗  [(s ∗ + is ∗ )a − ib† ]τ ∗ s − is  † † † † + + ia b − iab − a a − b b s ∗ + is ∗ s ∗ + is ∗ 

  ∗    s 1 s = exp ∗ a † b† exp (a † a + b† b + 1) ln ∗ exp − ∗ ab . (3.105) s s s

U (s, s ) =

to the classical optical Fresnel transform and U (s, s  ) Since U (s, s  ) corresponds   † † ∗ |00 = exp s a b /s |00 is an entangled state, the operator U (s, s  ) is called as entangled Fresnel operator. Actually, using Eqs. (3.56) and and the complete (3.97), −2 2 2 |α, β α, β| = 1, |α, d αd β ness relation of two-mode coherent states β, i.e., π we can give the integral kernel τ  U (s, s  ) |τ  as  2 2   d αd β 



τ α, β α, β |τ s,s  τ U (s, s ) |τ  = τ |τ s,s  = π2  2 2 d αd β 1 = ∗ exp[−(|α|2 + |β|2 ) ∗ s + is π2

82

3 New Bipartite Entangled States in Two-Mode Fock Space



 2

τ (s ∗ − is ∗ ) |τ |2 − + τ α + iτ β − iαβ] exp − ∗ ∗ 2(s + is ) 2  ∗ ∗ ∗  τα iτ β s − is + ∗ − ∗ + ∗ iα∗ β ∗ ∗ ∗ s + is s + is s + is ∗ 

 2 i 1 2 ∗  ∗

= exp (D |τ | + A τ − τ τ − τ τ ) i2Bπ 2B ∗





= κ2(s,s ) (ρ, τ ),

(3.106)

 where the state τ  belongs to the set of |τ . Further, using the Schmidt decomposition of the state |τ  in Eq. (3.61), we have the following inner products, i.e., √ √ τ | y, p = eiτ1 τ2 e−i 2τ2 y δ( 2τ1 + p − y), √   √      y , p τ = e−iτ1 τ2 ei 2τ2 y δ( 2τ1 + p  − y  ),

(3.107)

|y, p ≡ |ya ⊗ | pb . Thus, in terms of Eq. (3.106), the matrix element where y  , p  U (s, s  ) |y, p reads

y  , p  U (s, s  ) |y, p  2  2  d τ d τ  

  y , p τ τ |y, p κ2(s,s ) (τ  , τ ) = π

  D 1 i A  2  2   (y − p) (y = exp + − p ) − (y − p)(y − p ) i4Bπ 2 2B 2 2   

     i  2 y +p y+p − iτ2 + Dτ2 + Aτ22 − 2τ2 τ2 × dτ2 dτ2 exp iτ2 √ √ 2B 2 2

  i 1 (Dζ 2 + Aζ 2 − 2ζζ  ) exp = √ 2B i2Bπ 

 i 1 (Dξ 2 + Aξ 2 − 2ξξ  ) , (3.108) exp × √ −2C −i2Cπ



where ζ, ζ  , ξ and ξ  are, respectively, y−p y  − p y  + p y+p ζ = √ , ζ  = √ , ξ = √ , ξ = √ . (3.109) 2 2 2 2

From Eq. (3.108), we can find that y  , p  U (s, s  ) |y, p is completely different from the direct product of two one-dimensional Fresnel transformations in space domain because variables are the joint forms of coordinate and momentum, so   the transform y , p U (s, s  ) |y, p is called as the entangled Fresnel transformation [21, 22].

3.3 Parameterized Entangled States Induced by the Common Eigenstates …

83

 On the other hand, via the entangled state τ = ρeiψ representation, the quantum correspondence of the classical circular harmonic correlator is found and its essence in quantum optics is given. Operating two-mode number-difference operator N =

 a † a − b† b on the state τ = ρeiψ s,s  leads to

 (a † a − b† b) τ = ρeiψ s,s  =



τ a† τ ∗ b† − ∗ ∗ ∗ s + is s − is ∗  ∂

τ = ρeiψ s,s  , = −i ∂ψ





τ = ρeiψ  s,s (3.110)

which indicates that the operator N = a † a −b† b is equivalent to the differential ∂ in the entangled state τ = ρeiψ s,s  representation. Using Eq. (3.110), operation −i ∂ψ we obtain eiβ(a

†

a−b† b)



  

τ = ρeiψ  = eβ ∂ψ∂ τ = ρeiψ  = τ = ρei(ψ+β)  , s,s s,s s,s †

(3.111)

†

which means that the operator eiβ(a a−b b) is considered as a rotation operator in  iψ

the entangled τ = ρe s,s  representation. On the other hand, based on the entan  gled state τ = ρeiψ s,s  and introducing an integer q, we construct the following integration, that is, 1 2π

 0

2π

 dψe−iqψ τ = ρeiψ s,s  ≡ |q, ρs,s  ,

(3.112)

which is called as the deduced entangled state with the following complete orthogonality: ∞  

∞

dρ2 |q, ρs,s  s,s  q, ρ| = 1,

q=−∞ 0 s,s 

 1 q, ρ q  , ρ s,s  = δq,q  δ(ρ − ρ ). 2ρ

(3.113)

Acting the operator N and the correlated-amplitude operator M = [(s ∗ + is ∗ )a − (s  + is)b† ][(s − is  )a † − (s ∗ − is ∗ )b]

(3.114)

on the states |q, ρs,s  , respectively, we have N |q, ρs,s  = q |q, ρs,s  , M |q, ρs,s  = ρ2 |q, ρs,s  ,

(3.115)

84

3 New Bipartite Entangled States in Two-Mode Fock Space

thus the states |q, ρs,s  are just the common eigenstates of the operators N and M because of [N , M] = 0. On the other hand, the reciprocal relation of Eq. (3.112) is just the circular harmonic expansion, i.e., ∞ 



τ = ρeiψ  = |q, ρs,s  eiqψ , s,s

(3.116)

q=−∞

so, for an arbitrary two-dimensional function f (ρ, ψ) in polar coordinates, the function f (ρ, ψ) can be expressed as f (ρ, ψ) =

s,s 

τ = ρeiψ f 



(3.117)

in the entangled state s,s  τ = ρeiψ representation. Using Eqs. (3.93), (3.111), and † † (3.117), the inner product of the states  f | and the rotated state eiβ(a a−b b) | f  is calculated as †

†

 f | eiβ(a a−b b) | f   2π 

 1 ∞ † † = ρdρ dψ  f | τ = ρeiψ s,s  s,s  τ = ρeiψ eiβ(a a−b b) | f  π 0 0  2π 

 1 ∞ = ρdρ dψ  f | τ = ρeiψ s,s  s,s  τ  = ρei(ψ−β) f  π 0 0  2π  1 ∞ = ρdρ dψ f ∗ (ρ, ψ) f (ρ, ψ − β) ≡ Rβ , (3.118) π 0 0 which is the cross correlation denoted as Rβ between the function f ∗ (ρ, ψ) and its rotation function f (ρ, ψ − β). Using Eq. (3.116), we give f (ρ, ψ) =

∞  s,s  q=−∞

q, ρ| f  e−iqψ =

∞ 

f q (ρ)e−iqψ ,

(3.119)

q=−∞

where f q (ρ) = s,s  q, ρ| f , that is, the entangled state s,s  q, ρ| representation of the state | f . Acting the completeness relation in Eq. (3.113) on the state | f  leads to |f =

∞   q=−∞ 0

∞

dρ2 |q, ρs,s  s,s  q, ρ| f  =

∞  

∞

dρ2 |q, ρs,s  f q (ρ).

q=−∞ 0

(3.120) Further, substituting Eq. (3.120) into Eq. (3.118) leads to another form of Rβ , i.e.,

3.3 Parameterized Entangled States Induced by the Common Eigenstates …

Rβ =

∞  

∞

q  =−∞ 0

 ∞ 

= =2

q=−∞

∞

dρ2

0

q  ,q=−∞ ∞ 

∞  

† † dρ2 s,s  q  , ρ f q∗ (ρ)eiβ(a a−b b)

 iqβ

e

∞

1 ∗ f  (ρ)eiqβ 2ρ q

q=−∞ 0



∞

∞

85

 dρ2 q, ρ s,s  f q (ρ )

dρ2 f q (ρ )δq,q  δ(ρ − ρ )

0

2 ρ f q (ρ) dρ.

(3.121)

0

Obviously, the cross-correlation factor Rβ is not rotation invariant because each of the circular harmonic components of Rβ depends on a different phase shift qβ. Based on this, the cross-correlation factor Rβ is endowed with a definite physical meaning, † † that is, the inner product of the states  f | and eiβ(a a−b b) | f , in the entangled state |q, ρs,s  representation. Based on the entangled state |τ s,s  representation, tomogram function for two

2 mode correlated state |φ is obtained as φ |τ s,s  . For this, using the Weyl correspondence rule, the relation between the projection operator |τ s,s  s,s  τ | and its classical correspondence function h(ρ, σ; τ ) reads  |τ s,s  s,s  τ | =

d2 ρd2 σh(ρ, σ; τ )(ρ, σ),

(3.122)

it then from Eq. (1.45) that the function h(ρ, σ; τ ) is given by h(ρ, σ; τ ) = 4π 2 tr[(ρ, σ) |τ s,s  s,s  τ |]  2 

 d τ   τ  σ ∗ −τ ∗ σ

=4 . s,s  τ ρ − τ ρ + τ τ s,s  e π

(3.123)

Substituting Eq. (3.56) into Eq. (3.123) and using the integral formula in Eq. (1.30), we have  2 

 d τ    ∗ ∗

h(ρ, σ; τ ) = 4 s,s  τ ρ − τ ρ + τ τ s,s  exp(τ σ − τ σ) π   2 

2 d τ 1  −D |τ |2 − A ρ + τ  = exp 2 πB i2B  ∗  +τ (ρ + τ ) + τ (ρ∗ + τ ∗ ) exp(τ  σ ∗ − τ ∗ σ)  

2 1  −D |τ |2 − A ρ − τ  + τ (ρ∗ − τ ∗ ) + τ ∗ (ρ − τ  ) × exp − i2B    2   1  d τ ∗  ∗ (−iAρ + iτ + Bσ) exp τ − (−iAρ + iτ + Bσ)τ = π B2 B = πδ(Aρ1 − Bσ2 − τ1 )δ(Aρ2 + Bσ1 − τ2 ). (3.124)

86

3 New Bipartite Entangled States in Two-Mode Fock Space

Combining Eqs. (3.122) and (3.124) leads to the result  |τ s,s  s,s  τ | = π

d2 ρd2 σδ(Aρ1 − Bσ2 − τ1 )δ(Aρ2 + Bσ1 − τ2 )(ρ, σ),

(3.125) which shows that the pure projection operator |τ s,s  s,s  τ | is seen as the Radon transformation of the Wigner operator (ρ, σ) via the standard definition of Radon transformation [23]. For any bipartite entangled states |φ, we obtain

φ |τ s,s  2 = π

 d2 ρd2 σWφ (ρ, σ)δ(Aρ1 − Bσ2 − τ1 )δ(Aρ2 + Bσ1 − τ2 ),

(3.126) that is, the Radon transformation of Wφ (ρ, σ) can be expressed as the module square of the wave function |φ in the entangled state |τ s,s  representation. For example, using Eq. (3.126), we can directly obtain the tomogram function for two-mode squeezed state S2 (r ) |00 because the amplitude of its tomogram function is just the inner product of the states 00| S2† (r ) and |τ s,s  , i.e., 00| S2† (r ) |τ s,s   2 2 d αd β 00| exp(ab tanh r ) |α, β α, β |τ s,s  = π2   2 2

 1 s ∗ − is ∗ d αd β 2 |τ | = ∗ exp − exp − |α|2 ∗ ∗ ∗ 2 s + is 2(s + is ) π  ∗ ∗β∗ τ α τ s  + is ∗ ∗ 2 − |β| + αβ tanh r + ∗ − ∗ + ∗ α β s + is ∗ s − is ∗ s + is ∗   ⎧ ⎫ ⎨ s ∗2 − s ∗2 + |s + is  |2 − i2 tanh r ⎬ 1       |τ |2 . = ∗ exp ⎩ 2[s ∗ + is ∗ − s + is  tanh r ] s ∗ + is ∗ ⎭ s + is ∗ − s + is  tanh r

(3.127)

3.4 Parameterized Entangled States as the Common Eigenstates of AQa + BPb and CQb + DPa In quantum optics, the operator μQ + ν P (μ and ν being arbitrary and real) can be seen as all possible linear combinations of the coordinate operator Q and the momentum operator P. For a bipartite entangled system, two Hermite operators AQ a + B Pb and C Q b + D Pa can be introduced as orthogonal components, where A, B, C, and D are arbitrary real numbers. Obviously, when AD − BC = 0, the operators AQ a + B Pb and C Q b + D Pa are commutative, that is,

3.4 Parameterized Entangled States as the Common Eigenstates …

[AQ a + B Pb , C Q b + D Pa ] = 0.

87

(3.128)

So, there may be the common eigenstates of the operators AQ a + B Pb and C Q b + D Pa in two-mode Fock space, denoted as |η1 , η2 . Since the common eigenstate |η1 , η2  completely depends on the linear parameters A, B, C, and D, it is called as a parameterized entangled state. To obtain the specific expression of the state |η1 , η2  in two-mode Fock space, we shall make full use of the Weyl correspondence rule of the operators AQ a + B Pb and C Q b + D Pa and the Radon transformation of two-mode Wigner operator. Here, the entangled state |η representation of two-mode Wigner operator   (σ, γ) =

d2 η ∗ ∗ |σ − η σ + η| eηγ −η γ 3 π

(3.129)

∗ is used, where |σ ± η belong√to the entangled states |η √ in Eq. (1.72), γ = α − β , ∗ σ = α + β , α = (qa +i pa )/ 2 and β = (qb +i pb )/ 2. In terms of Weyl correspondence rule, the Weyl correspondence functions of the operators AQ a + B Pb and C Q b + D Pa are, respectively, Aqa + Bpb and Cqb + Dpa . So, the Radon transformation of Wigner operator  (σ, γ) is



∞

   d pa dqa d pb dqb δ η1 − Cqb + Dpa −∞    × δ η2 − Aqa + Bpb  (σ, γ) . (3.130)

|η1 , η2  η1 , η2 | =

On the other hand, using the IWOP method to complete the integration in Eq. (3.129), we obtain the normal ordering form of  (σ, γ), that is,  (σ, γ) =

1 & −(qi −Q i )2 −( pi −Pi )2 : e :. π 2 i=a,b

(3.131)

Thus, Eq. (3.130) can be rewritten as 

∞

  d pa dqa d pb dqb  δ η1 − Cqb + Dpa 2 π −∞  & −(q  −Q  )2 −( p −P  )2   I i e i i :. × δ η2 − Aqa + Bpb :

|η1 , η2  η1 , η2 | =

(3.132)

i=a,b

Further, using the nature of the delta function to remove the integral symbol and decomposing the right side of Eq. (3.132) into the form of h(i † ) |00 00| h(i), thus the state |η1 , η2  can be expressed as

88

3 New Bipartite Entangled States in Two-Mode Fock Space

  2  η2 2 −1/4 |η1 , η2  = [π A + B C + D ] exp −  2 1 2  2 A +B √   η2 2 (Cη1 + iBη2 )a † + (Aη2 + iDη1 )b† −  2 2 2 + AC + B D 2 C +D  AD + BC † † B D − AC †2 †2 (3.133) (a + b ) − i a b |00 + 2 (AC + B D) AC + B D 2



2

2

in the Fock space. Using the commutative relation in Eq. (3.8), we have √

2(Cη1 + iBη2 ) + (B D − AC) a † − i (AD + BC) b† |η1 , η2  , AC + B D √ 2(Aη2 + iDη1 ) + (B D − AC) b† − i (AD + BC) a † |η1 , η2  , b |η1 , η2  = AC + B D (3.134)

a |η1 , η2  =

which leads to the eigenstate equations (AQ a + B Pb ) |η1 , η2  = η1 |η1 , η2  ,

(C Q b + D Pa ) |η1 , η2  = η2 |η1 , η2  , (3.135) that is, the parameterized entangled states |η1 , η2  are the common eigenstates of the operators AQ a + B Pb and C Q b + D Pa . Indeed, using Eq. (3.133), it can prove that the states |η1 , η2  possess the completeness relation 

∞

−∞

dη1 dη2 |η1 , η2  η1 , η2 | = 1

(3.136)

and the orthogonality

    η1 , η2 |η1 , η2  = δ η1 − η1 δ η2 − η2 .

(3.137)

Constructing the integration for the asymmetric ket-bra operator 1 Se (r ) = √ k



∞ −∞

dη1 dη2 |η1 , η2 /k η1 , η2 | ,

(3.138)

and substituting Eq. (3.133) into Eq. (3.138) lead to 'σ

tanh r [(Ab − iBa)2 − (Ab† + iBa † )2 ] 2  +σ(sech r − 1)(Ab† + iBa † )(Ab − iBa) : , (3.139)

Se (r ) = sech1/2 r : exp

where

3.4 Parameterized Entangled States as the Common Eigenstates …

σ=

C 2 + D2 2k k2 − 1 . , sech r = , tanh r = 2 1 + k2 k2 + 1 (AC + B D)

89

(3.140)

Using the operator identity   exp( f i † i) = : exp (e f − 1)i † i : , (i = a, b),

(3.141)

Equation (3.139) can be rewritten as  σ  Se (r ) = exp − (Ab† + iBa † )2 tanh r 2 σ    × exp σ(Ab† + iBa † )(Ab − iBa) ln sech r exp (Ab − iBa)2 tanh r . 2 (3.142) It is then from Eq. (3.137) that the operator Se (r ) can naturally squeeze the state |η1 , η2  as |η1 , η2 /k, that is, 1 Se (r ) |η1 , η2  = √ |η1 , η2 /k , k

(3.143)

and can lead to the following new two-mode squeezed state: Se (r ) |00 = sech1/2 r exp

σ 2

 (iBa † + Ab† )2 tanh r |00 .

(3.144)

Using Eq. (3.142) and the Baker-Hausdorff formula in Eq. (1.27), we obtain the squeezing transformations as iB(cosh r − 1) (Ab − iBa) + iBσ(iBa † + Ab† ) sinh r, A2 + B 2 A(cosh r − 1) Se (r )bSe−1 (r ) = b + (Ab − iBa) + Aσ(iBa † + Ab† ) sinh r, A2 + B 2 (3.145)

Se (r )aSe−1 (r ) = a +

it then follows that Se (r )(AQ a + B Pb )Se−1 (r ) = (AQ a + B Pb ) +

  2 er − 1 AB

(B Q a + A Pb ), A2 + B 2   r e − 1 (AC + B D) Se (r )(C Q b + D Pa )Se−1 (r ) = (C Q b + D Pa ) + (AQ b + B Pa ). A2 + B 2

(3.146)

Similar to Eq. (3.122), the operator |η1 , η2  η1 , η2 | and its classical correspondence function h(σ, γ; η1 , η2 ) yield the identity

90

3 New Bipartite Entangled States in Two-Mode Fock Space

 |η1 , η2  η1 , η2 | =

d2 σd2 γh(σ, γ; η1 , η2 ) (σ, γ) ,

(3.147)

which leads to the classical correspondence function h(σ, γ; η1 , η2 ) h(σ, γ; η1 , η2 ) = 4π 2 tr [ (σ, γ) |η, η2  η1 , η2 |]  2 d η ∗ ∗ η1 , η2 | σ − η σ + η| η1 , η2  eηγ −η γ . =4 π

(3.148)

Substituting the completeness relation of coherent states |α, β into the inner product η |η1 , η2  and using Eqs. (1.72) and (3.133), we obtain  1 η |η1 , η2  = [π 2 X 2 (A2 + B 2 )(C 2 + D 2 )]−1/4 exp − |η|2 X  S + iS M + 2N T ∗ N  2 T + iM T + 2S N η+ η + η + η ∗2 − X X X  η22 ST (1 + iM) + N (S 2 + T 2 ) η12 − + , − 2(A2 + B 2 ) 2(C 2 + D 2 ) X (3.149) where the parameters X, M, N , S, and T are, respectively, AD + BC B D − AC i4 AD , M= , N= , AC + B D AC + B D 2 (AC + B D) √ √ 2(Cη1 + iBη2 ) 2(Aη2 + iDη1 ) S= , T = . AC + B D AC + B D

X=

(3.150)

Substituting Eq. (3.149) into the integration in Eq. (3.148) and completing it, we have √ √ h(σ, γ; η1 , η2 ) = 4 ADδ(2B Dσ2 − 2Dη1 − 2Bη2 + 2 ADγ1 ) √ √ × δ(−2 ACσ1 + 2Cη1 − 2 Aη2 + 2 ADγ2 ),

(3.151)

thus Eq. (3.147) becomes 

√ √ d2 γd2 σδ(2B Dσ2 − 2Dη1 − 2Bη2 + 2 ADγ1 ) √ √ × δ(−2 ACσ1 + 2Cη1 − 2 Aη2 + 2 ADγ2 ) (σ, γ) , (3.152)

|η1 , η2  η1 , η2 | = 4 AD

which means that the Radon transformation of Wigner operator  (σ, γ) is just the projection operator |η1 , η2  η1 , η2 |. Therefore, for a two-mode correlated state |(or ρ = | |), its tomogram function is

3.4 Parameterized Entangled States as the Common Eigenstates …

91



√ √ d2 γd2 σδ(2B Dσ2 − 2Dη1 − 2Bη2 + 2 ADγ1 ) √ √ × δ(−2 ACσ1 + 2Cη1 − 2 Aη2 + 2 ADγ2 )W (σ, γ) , (3.153)

|η1 , η2 ||2 = 4 AD

which is just the Radon transformation of W (ρ, σ), i.e., the module square |η1 , η2 ||2 . In terms of Eq. (3.152), the Fourier transformation of the projection operator |η1 , η2  η1 , η2 | is 

   dη1 dη2 η1 , η2 η1 , η2 exp −iζ1 η1 − iζ2 η2 −∞ 

  A(σ1 + γ1 ) B(σ2 − γ2 ) 2 2 =π d γd σ (σ, γ) exp −iζ1 + √ √ 2 2 

D(σ2 + γ2 ) C(γ1 − σ1 ) −iζ2 , (3.154) + √ √ 2 2 ∞

which is indeed a special Fourier transformation of Wigner operator  (σ, γ). Using its inverse Fourier transformation, thus the inverse Radon transformation of Eq. (3.152) can be obtained as  ∞    ∞

dη1 dη2 1 dζ1 ζ1  (σ, γ) = 4 π (2π) −∞ −∞  π  ∞  π



   × dζ2 ζ2 dθ1 dθ2 η1 , η2 η1 , η2 −∞ 0 0

  η1 γ1 + σ1 σ2 − γ2  × exp −iζ1 √ − √ cos θ1 − √ sin θ1 2 2 A2 + B 2    η2 σ2 + γ2 γ1 − σ1  (3.155) −iζ2 √ − √ cos θ2 − √ sin θ2 , 2 2 C 2 + D2 where ζ1 = ζ1 A2 + B 2 ,

ζ2 = ζ2 C 2 + D 2 , cos θ1 = √

A , A2 + B 2 B D C sin θ1 = √ , cos θ2 = √ , sin θ2 = √ . (3.156) A2 + B 2 C 2 + D2 C 2 + D2 From Eq. (3.155), we easily obtain  ∞    ∞

dη1 dη2 1 dζ1 ζ1 W (σ, γ) = 4 (2π) π −∞ −∞  π  ∞  π

 2 × dζ2 ζ2 dθ1 dθ2  η1 , η2 −∞

0

0

92

3 New Bipartite Entangled States in Two-Mode Fock Space

 

η γ1 + σ1 σ2 − γ2 × exp −iζ1 √ 1 − √ cos θ1 − √ sin θ1 2 2 A2 + B 2    η σ2 + γ2 γ1 − σ1 −iζ2 √ 2 (3.157) − √ cos θ2 − √ sin θ2 , 2 2 C 2 + D2 which shows that, when the parameters A, B, C, and D in | |η1 , η2 |2 are given, we may reconstruct the Wigner distribution function W (σ, γ) for the state | via the inverse Radon transformation of the measurable probability distribution | |η1 , η2 |2 .

References 1. Einstein A, Podolsky B, Rosen N (1935) Can quantum-mechanical description of physical reality be considered complete? Phys Rev 47(10):777–780 2. Fan HY, Klauder JR (1994) Eigenvectors of two particles’ relative position and total momentum. Phys Rev A 49(2):704–707 3. Fan HY (2004) Entangled states, squeezed states gained via the route of developing Dirac’s symbolic method and their applications. Int J Mod Phys B 18(10–11):1387–1455 4. Meng XG, Wang JS, Fan HY (2011) New bipartite coherent-entangled state in two-mode Fock space and its applications. Opt Commun 284(7):2070–2074 5. Meng XG, Wang JS, Zhang XY (2012) A new bipartite entangled state describing the parametric down-conversion process and its applications in quantum optics. Chin Phys B 21(10):100305 6. Meng XG, Wang JS, Zhang XY, Liang BL (2011) New parameterized entangled state representation and its applications. J Phys B: At Mol Opt Phys 44(16):165506 7. Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press, New York 8. Campos RA, Saleh BEA, Teich MC (1989) Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics. Phys Rev A 40(3):1371–1384 9. Sudarshan ECG (1963) Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys Rev Lett 10(7):277–279 10. Dodonov VV, Malkin IA, Man’ko VI (1974) Even and odd coherent states and excitations of a singular oscillator. Physica 72(3):597–615 11. Bužek V, Vidiella-Barranco A, Knight PL (1992) Superpositions of coherent states: squeezing and dissipation. Phys Rev A 45(9):6570–6585 12. Dodonov VV (2002) ‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years. J Opt B: Quantum Semiclass Opt 4(1):R1–R33 13. Xu SM, Xu XL, Li HQ, Wang JS (2009) New two-mode intermediate momentum-coordinate representation with quantum entanglement and its application. Chin Phys B 18(6):2129–2136 14. Klauder JR, Sudarshan ECG (1968) Fundamentals of quantum optics. W. A. Benjamin Inc, New York 15. Mehta CL (1967) Diagonal coherent-state representation of quantum operators. Phys Rev Lett 18(18):752–754 16. Schumaker BL, Caves CM (1985) New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation. Phys Rev A 31(5):3093–3111 17. Wu LA, Kimble HJ, Hall JL, Wu HF (1986) Generation of squeezed states by parametric down conversion. Phys Rev Lett 57(20):2520–2523 18. Mandel L, Wolf E (1995) Optical coherence and quantum optics. Cambridge University Press, Cambridge

References

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19. Fan HY, Jiang NQ (2010) Entangled state representation for describing both squeezing and entanglement involved in the parametric down-conversion process. Phys Scr 82(5):055403 20. Wigner E (1932) On the quantum correction for thermodynamic equilibrium. Phys Rev 40(5):749–759 21. Goodman JW (1968) Introduction to Fourier optics. McGraw-Hill Book Company, New York 22. James DFV, Agarwal GS (1996) The generalized Fresnel transform and its application to optics. Opt Commun 126(4–6):207–212 23. Radon J (1986) On the determination of functions from their integral values along certain manifolds. IEEE Trans Med Imaging 5(4):170–176

Chapter 4

Solutions of Density Operator Master Equations

Quantum decoherence is a common essential problem in quantum information and quantum computing and has brought great trouble to the development and application of information technology. So, the decoherence effect of interaction system is of great concern. In the interaction representation, the quantum decoherence effect of an open system can be described by the quantum master equation of the reduced density operator of the system. In Refs. [1–5], the decoherence evolution law of the system is usually analyzed by solving the P-representation of the density operator or the Fock-Planck equation satisfied by the Wigner distribution function. Different from the previous methods, based on thermal field dynamics, Fan introduced a thermal field entangled state representation describing the interaction between the system and its surrounding environment (the system being the real mode and the environment being the fictitious mode), derived the operator identities about the real and fictitious modes and converted the quantum master equation of density operator into the equation of the corresponding state vector. Thus, a series of quantum master equations describing the decoherence evolution of the system are successfully solved, and their analytical solutions are given [6]. In this chapter, we use the thermal entangled state representation to solve a series of quantum master equations of Boson or Fermi systems, including diffusion master equation under linear resonance force [7], master equation describing single-mode cavity driven by an oscillating external field in a heat reservoir [8], master equation for damped harmonic oscillator acted by linear resonance force in a squeezed heat reservoir [9], master equation describing a diffusive anharmonic oscillator [10], and Fermi master equations for amplitude damping, phase damping, and thermal reservoir [11, 12], and obtain the Kraus operator-sum representation of the time-evolution density operator. Finally, we discuss the generation mechanism of displaced thermal states [13].

© Science Press 2023 X.-G. Meng et al., Entangled State Representations in Quantum Optics, https://doi.org/10.1007/978-981-99-2333-5_4

95

96

4 Solutions of Density Operator Master Equations

4.1 Solutions of Several Boson Master Equations 4.1.1 Diffusion Master Equation Under Linear Resonance Force In quantum optics, the quantum master equation describing the diffusion process under the action of linear resonance force is     dρ(t) = iλ a † + a, ρ(t) − k a † aρ(t) − a † ρ(t)a − aρ(t)a † + ρ(t)aa † , (4.1) dt where a, a † are the creation and annihilation operator of the system, λ, k are respectively the resonance force strength and the decay rate. Especially, when k = 0, the system is only affected by linear resonance force, while λ = 0, Eq. (4.1) describes the pure diffusion process of light field. To solve the master equation in Eq. (4.1), we make full use of the operator identities derived from the entangled state |I  in Eq. (1.104). Acting both sides of Eq. (4.1) on the state |I , and denoting |ρ(t) = ρ(t) |I , we have     d |ρ(t) = iλ a † + a, ρ(t) |I  − k a † aρ(t) − a † ρ(t)a − aρ(t)a † + ρ(t)aa † |I  dt     (4.2) = iλ(a † + a − a˜ − a˜ † ) − k a † − a˜ a − a˜ † |ρ(t) ,

so the formal solution of Eq. (4.2) is † † † † |ρ(t) = eiλ(a−a˜ )t eiλ(a −a˜ )t e−k (a −a˜ )(a−a˜ )t |ρ(0) ,

(4.3)

where ρ(0) is the initial density operator, |ρ(0) = ρ(0) |I . Substituting the completeness relation of the states |χ in Eq. (1.102) into the right of Eq. (4.3), and using the eigenstate equations in Eq. (1.100), we obtain 

d2 χ iλt(χ+χ∗ )−kt|χ|2 |χ χ| ρ(0) e π  2 d χ −(1+kt)|χ|2 +χ(a † −a+iλt)+χ ∗ ˜ (a−a˜ † +iλt )−(a−a˜ † )(a † −a˜ ) : |ρ(0) :e = π 1 1 † ˜ (a−a˜ † +iλt )−(a † −a˜ )(a−a˜ † ) : |ρ(0) : e 1+kt (a −a+iλt) = 1 + kt 1 −kt iλt iλt λ2 t 2 † † † † : e 1+kt (a −a˜ )(a−a˜ )+ 1+kt (a−a˜ )+ 1+kt (a −a˜ )− 1+kt : |ρ(0) = 1 + kt

a † a+a˜ † a+1 ˜ 2t2 1 kt iλt kt iλt ˜ − λ1+kt a†a˜ † + 1+kt a † −a˜ † ) ˜ ( 1+kt 1+kt (a−a) |ρ(0) , e e 1+kt aa+ =e 1 + kt (4.4)

|ρ(t) =

4.1 Solutions of Several Boson Master Equations

97

where we have used the normal ordering of |0 0| in Eq. (1.7) and the operator identity in Eq. (1.21). Further, using Eq. (1.104), thus Eq. (4.4) can be rewritten as

m

a † a ∞  1 1 kt 1 λ2 t 2 iλt † e− 1+kt a †m e 1+kt a 1 + kt m!n! 1 + kt 1 + kt m,n=0

n

a † a 1 kt iλt iλt iλt † × a n e 1+kt a ρ(0)e− 1+kt a a †n e− 1+kt a a m |I  . 1 + kt 1 + kt

|ρ(t) =

(4.5)

After getting rid of |I  from both sides of Eq. (4.5), the solution of Eq. (4.1) is obtained as

m+n ∞  kt 1 1 λ2 t 2 iλt † e− 1+kt a †m e 1+kt a 1 + kt m!n! 1 + kt m,n=0

a † a

a † a 1 1 iλt iλt iλt n 1+kt a − 1+kt a † †n × a e ρ(0)e a e− 1+kt a a m , 1 + kt 1 + kt

ρ(t) =

(4.6)

which is just the infinitive operator-sum representation of the time-dependent density operator ρ(t). Corresponding to the density operator ρ(t), the Kraus operator Mm,n is defined as

(m+n)/2 kt 1 + kt m!n!

a † a+1/2 1 iλt iλt † × a †m e 1+kt a a n e 1+kt a , 1 + kt

Mm,n = √

1

λ2 t 2

e− 2(1+kt)



(4.7)

thus ρ(t) can further represented as ρ(t) =

∞ 

† Mm,n ρ(0)Mm,n .

(4.8)

m,n=0

Next, we check the normalization condition of the Kraus operator Mm,n . Noting that ∞  m,n=0



m+n ∞  kt 1 1 λ2 t 2 iλt † e− 1+kt e− 1+kt a 1 + kt m!n! 1 + kt m,n=0

a † a

a † a 1 1 iλt iλt iλt † × a †n e− 1+kt a a m a †m e 1+kt a a n e 1+kt a 1 + kt 1 + kt

n

a † a ∞  1 kt 1 1 λ2 t 2 iλt † = e− 1+kt a a †n e− 1+kt 1 + kt n! 1 + kt 1 + kt n=0

† Mm,n Mm,n =

98

4 Solutions of Density Operator Master Equations

. kta† a . iλt † iλt × e− 1+kt a ..e 1+kt ..e 1+kt a



1 1 + kt

a † a

iλt

a n e 1+kt a ,

(4.9)

and using the operator identity in Eq. (1.37), thus Eq. (4.9) becomes ∞ 

† Mm,n Mm,n

m,n=0



n

a † a ∞  1 kt 1 iλt − 1+kt a † †n =e e a n! 1 + kt 1 + kt n=0

−a † a

a † a 1 1 iλt iλt iλt † × e− 1+kt a e 1+kt a a n e 1+kt a . 1 + kt 1 + kt 2 2

t − λ1+kt

(4.10)

Using the operator identity

† † eβa a f a, a † e−βa a = f (ae−β , a † eβ ),

(4.11)

thus we have ∞ 



n ∞  kt 1 iλt † e− 1+kt a n! 1 + kt n=0

a † a 1 iλt iλt † × a †n e− 1+kt a eiλta a n e 1+kt a . 1 + kt λ2 t 2

† Mm,n Mm,n = e− 1+kt

m,n=0

(4.12)

Further, using the operator formula e A e B = e B e A e[A,B] ,

(4.13)

which holds for [A, B] = 0, we obtain ∞ 

† Mm,n Mm,n

m,n=0



n

a † a ∞  1 kt 1 iλt † − 1+kt a † †n = e a eiλta a n . n! 1 + kt 1 + kt n=0

So, using Eqs. (1.21) and (4.11), such that ∞  m,n=0



n

a † a ∞  1 kt 1 iλt † e− 1+kt a a †n n! 1 + kt 1 + kt n=0 † †

−a a

a a 1 1 † × eiλta an 1 + kt 1 + kt

n

a † a ∞  kt 1 1 †n = a an n! 1 + kt 1 + kt n=0

† Mm,n Mm,n =

(4.14)

4.1 Solutions of Several Boson Master Equations

99



n ∞  kt 1 kt † = a †n : e− 1+kt a a : a n n! 1 + kt n=0 = : e 1+kt a a e− 1+kt a a : = 1, kt

†

kt

†

(4.15)

that is, the Kraus operator Mm,n is normalized, thus we have  trρ(t) = tr

∞ 

 † Mm,n ρ(0)Mm,n

= trρ(0) = 1,

(4.16)

m,n=0

which means that the Kraus operator Mm,n is a trace-preserving quantum operation, and the operator ρ(t) possesses an infinitive Kraus operator-sum representation. Using Eq. (4.6) and taking the coherent state ρ|α = |α α| as an initial state, thus the evolution of ρ|α in the diffusion process under the action of linear resonance force reads

m+n ∞ 2 +kt  |α|2n kt 1 iλt (α−α∗ )− λ1+kt 1+kt e ρ|α (t) = 1 + kt m!n! 1 + kt m,n=0

a † a

a † a 1 1 iλt iλt †m 1+kt a† |α α| ×a e e− 1+kt a a m . 1 + kt 1 + kt

(4.17)

Substituting the operator identity

1 1 + kt

a † a |α = ea

†

a ln

1 1+kt

|α|2 αa † |0 |α = exp − + 2 1 + kt

(4.18)

and the normal ordering of the projection operator of vacuum in Eq. (1.7) into Eq. (4.17), thus the state ρ|α (t) has the normal ordering product, that is ∞  |α|2n 1 iλt λ2 +kt 2 ∗ e−|α| + 1+kt (α−α )− 1+kt 1 + kt m!n! m,n=0

m+n kt iλt αa † α∗ a iλt † † × : a †m e 1+kt a e 1+kt + 1+kt −a a e− 1+kt a a m : 1 + kt 1 iλt λ2 +kt iλt+α † 1 −iλt+α∗ 2 ∗ † e−|α| + 1+kt (α−α )− 1+kt e 1+kt a : e− 1+kt a a : e 1+kt a . = 1 + kt

ρ|α (t) =

(4.19)

Further, using Eq. (1.21), we obtain ρ|α (t) =

1 iλt λ2 +kt iλt+α † kt −iλt+α∗ 2 ∗ † e−|α| + 1+kt (α−α )− 1+kt e 1+kt a ea a ln 1+kt e 1+kt a , 1 + kt

(4.20)

100

4 Solutions of Density Operator Master Equations

which is indeed a superposition of photon-added thermal states. So, when the pure coherent state passes through the diffusion process under the action of linear resonance force, it evolves into a mixed thermal superposition as a result of quantum noise.

4.1.2 Master Equation for Single-Mode Cavity Driven by Oscillating External Field in a Heat Reservoir For the single-mode cavity driven by an oscillating external field with a main frequency ωl and the strength λ in a heat reservoir, the Hamiltonian of the system is represented by [14, 15] H = ωc a † a − λ(a † e−iωl t − aeiωl t ),

(4.21)

where a † , a are the creation and annihilation operators of single-mode cavity field with the natural frequency ωc . When the interaction system is “immersed” in a heat reservoir, the evolution of density operator follows the quantum master equation   dρ(t) 1 = [H, ρ(t)] + κ(n¯ + 1) 2aρ(t)a † − a † aρ(t) − ρ(t)a † a dt i   (4.22) + κn¯ 2a † ρ(t)a − aa † ρ(t) − ρ(t)aa † , where κ, n¯ are respectively the dissipative coefficient and the mean thermal photon number of heat reservoir. To get rid of the exponential terms e±iωl t , using a unitary operator U(t) = exp(−iωl a † at) to transform the Hamiltonian H into the form H R = U † (t)H U(t) − iU † (t)

∂U(t) , ∂t

(4.23)

such that   dρU (t) 1 = [H R , ρU (t)] + κ(n¯ + 1) 2aρU (t)a † − a † aρU (t) − ρU (t)a † a dt i   (4.24) + κn¯ 2a † ρU (t)a − aa † ρU (t) − ρU (t)aa † , where ρU (t) = U † (t)ρ(t)U(t), H R = a † a − λ(a † − a),

(4.25)

 = ωc − ωl is the detuning for the effective frequency of the cavity field controlled by external field. Indeed, H R describes a damped harmonic oscillator system affected by time-independent linear resonance force. Since [a † a, ρU ] has no effect on decoherence effect, we have

4.1 Solutions of Several Boson Master Equations

101

dρU (t) = iλ[a † − a, ρU (t)] + κ(n¯ + 1)[2aρU (t)a † − a † aρU (t) − ρU (t)a † a] dt   (4.26) + κn¯ 2a † ρU (t)a − aa † ρU (t) − ρU (t)aa † . Operating both sides of Eq. (4.26) on the state |I , such that     d ρU (t) = [iλ(κ † − κ) + κ(a a˜ − a † a˜ † + 1) − κ(2n¯ + 1)κ † κ] ρU (t) , (4.27) dt whose analytical solution reads   ρ (t) = exp[iλt (κ † − κ) + κt (a a˜ − a † a˜ † + 1) − κt (2n¯ + 1)κ † κ] |ρ(0) , U (4.28) ˜ their common eigenstates are denoted as |χ and where κ = a − a˜ † , κ † = a † − a, obey the eigenstate equations κ |χ = χ |χ , κ † |χ = χ∗ |χ .

(4.29)

Noting that the operators in Eq. (4.28) yield the commutative relations [a a˜ − a † a˜ † , κ † − κ] = −(κ † − κ), [κ † κ, κ † + κ] = 0, [a a˜ − a † a˜ † , κ † κ] = −2κ † κ,

(4.30)

and using the disentangling formula eλ(A+σ B) = eλA e τ (1−e σ

−λτ

)B ,

(4.31)

which holds for [A, B] = τ B, thus we obtain

   

ρ (t) = exp[κt (a a˜ − a † a˜ † + 1)] exp n¯ + 1 1 − e2κt κ † κ U 2  

iλ 1 − eκt (κ † − κ) |ρ(0) . (4.32) × exp − κ Operating both sides of Eq. (4.32) on the state χ| and using the entangled state |χ representation of two-mode squeezing operator S(γ), that is 1 S(γ) = μ



  d2 χ  χ χ| , π μ

μ = eγ ,

(4.33)

102

4 Solutions of Density Operator Master Equations

which squeezes the state |χ into |χ/μ, i.e., S(γ) |χ = μ−1 |χ/μ, thus Eq. (4.32) becomes

  χ| ρU (t) = exp −(n¯ + 1/2) 1 − e−2κt |χ|2  

 iλ −κt ∗ 1−e (χ + χ ) χe−κt  ρ(0) . + κ

(4.34)

To obtain the explicit expression of ρU (t), using the completeness relation of the entangled states |χ to get rid of χ from χ| ρU (t) , that is   ρ (t) = U



 d2 χ |χ χ| ρU (t) π  2 

 d χ = : exp −[n¯ 1 − e−2κt + 1] |χ|2 + a † − ae ˜ −κt π 



  iλ 1 − e−κt iλ 1 − e−κt −κt † + χ + ae − a˜ + χ∗ κ κ

× exp(a † a˜ † + a a˜ − a † a − a˜ † a) ˜ : |ρ(0) = T1 exp(T1 T22 ) exp[(1 − T1 )a † a˜ † + T1 T2 a † − T1 T2 a˜ † ]

× : exp[ T1 e−κt − 1 (a † a + a˜ † a)] ˜ :

−2κt × exp[ 1 − T1 e a a˜ + T1 T2 ae−κt − T1 T2 ae ˜ −κt ] |ρ(0) ,

(4.35)

where in the last step we have used the integral formula in Eq. (1.30), and 1

, n¯ 1 − e−2κt + 1

T1 =

T2 =

iλ 1 − e−κt . κ

(4.36)

Further, using the operator identities in Eqs. (1.21) and (1.104), thus Eq. (4.35) can be rewritten as   ρ (t) = T1 eT1 T22 exp[(1 − T1 )a † a˜ † + T1 T2 a † − T1 T2 a˜ † ] U × exp[(a † a + a˜ † a) ˜ ln(T1 e−κt )] × exp[(1 − T1 e−2κt )a a˜ + T1 T2 ae−κt − T1 T2 ae ˜ −κt ] |ρ(0) 2

= T1 eT1 T2

∞  (1 − T1 e−2κt )l (1 − T1 )m †m T1 T2 a † a † a ln(T1 e−κt ) a e e l!m! l,m=0 −κt

× a l eT1 T2 ae ρ(0)e−T1 T2 a

† −κt

e

a †l ea

†

a ln(T1 e−κt ) −T1 T2 a m

e

a |I  ,

which leads to the infinitive operator-sum representation of ρU (t), that is

(4.37)

4.1 Solutions of Several Boson Master Equations

ρU (t) = T1 e

T1 T22

103

l ∞  1 − T1 e−2κt (1 − T1 )m †m T1 T2 a † a e l!m! l,m=0

a ln(T1 e−κt ) l T1 T2 ae−κt

ae ρ(0)e−T1 T2 a † −κt × a †l ea a ln(T1 e ) e−T1 T2 a a m . × ea

†

† −κt

e

(4.38)

Noting that ρ(t) = e−iωl a

†

ρU (t)eiωl a

†

,

(4.39)

† Ml,m ρ(0)Ml,m ,

(4.40)

at

at

thus the density operator ρ(t) can be expressed as ρ(t) =

∞  l,m=0

where Ml,m is the Kraus operator of ρ(t) describing the single-mode cavity driven by an oscillating external field in a heat reservoir, that is 

Ml,m



l 2 T1 1 − T1 e−2κt (1 − T1 )m eT1 T2 = l!m! −iωl a † at †m −T1 T2 a † a † a ln(T1 e−κt ) l −T1 T2 ae−κt ae ×e a e e .

(4.41)

In particular, when n¯ → 0 and κ being finite, T1 → 1, thus Eq. (4.39) becomes ρ(t) = e

T22

l ∞  1 − e−2κt −iωl a † at T2 a † a † a ln e−κt e e e l! l=0 −κt

× a l eT2 ae ρ(0)e−T2 a

† −κt

e

a †l ea

†

a ln e−κt −T2 a iωl a † at

e

e

,

(4.42)

which refers to the evolution of the initial state ρ(0) in the cavity acted by an external field for amplitude damping. Further, taking λ = 0, Eqs. (4.39) and (4.42) reduce to the evolution of the initial state ρ(0) for thermal noise and amplitude damping. For κ → 0 and n¯ → ∞, κn¯ is finite, T1 → (1 + 2e−2κt )−1 and T2 → iλte−κt , such that Eq. (4.39) represents the evolution of the state ρ(0) in the diffusion process at finite temperature. To certify whether or not the Kraus operator Ml,m is normalized, we must check ∞ 

?

† Ml,m Ml,m = 1.

(4.43)

l,m=0

Using the completeness relation of coherent states |α and the operator identity  1 2  † −2r e−ra a |α = e− 2 (1−e )|α| αe−r ,

(4.44)

104

4 Solutions of Density Operator Master Equations

we obtain 

∞ d2 α  † M |α α| Ml,m π l,m=0 l,m  2  ∞ (1 − T1 )m T1 T22 T1 (1−T1 e−2κt )|α|l −T1 T2 (α+α∗ )e−κt −iωl a † at d α = e e e e π m=0 m!   2 −2κt  † † × a †m e−T1 T2 a e−(1−T1 e ) αT1 e−κt αT1 e−κt  e−T1 T2 a a m eiωl a at .

(4.45)

Further, using the normal ordering of the coherent state |α in Eq. (1.31) and the integral formula in Eq. (1.30), we find that the Kraus operator Ml,m is normalized since ∞  † Ml,m Ml,m = 1. (4.46) l,m=0

Using the normalization condition in Eq. (4.46), we can prove that the Kraus operators † are trace-preserving quantum operations. Ml,m and Ml,m Substituting the coherent state ρ|β = |β β| as an initial state into Eq. (4.39), we have ρ|β (t) = T1 e

T1 T22

× e−iωl a

†

l ∞  1 − T1 e−2κt (1 − T1 )m l!m! l,m=0

at †m T1 T2 a † a † a ln(T1 e−κt ) l T1 T2 ae−κt

a e

ae

e

|β β|

−T1 T2 a † e−κt †l a † a ln(T1 e−κt ) −T1 T2 a m iωl a † at

×e

e a e a e 2 −κt ∗ −κt )|β| +T1 T2 βe −T1 T2 β e +T1 T22

= T1 e(1−T1 e

−2κt

×

∞  (1 − T1 )m −iωl a † at †m T1 T2 a † a † a ln(T1 e−κt ) e a e e m! m=0

× |β β| ea

†

a ln(T1 e−κt ) −T1 T2 a m iωl a † at

e

a e

.

(4.47)

Combining Eqs. (1.7) and (4.44) leads to the result ρ|β (t) = T1 e

−T1 (βe−κt +T2 )(β ∗ e−κt −T2 )

× a †m eT1 (βe

−κt

+T2 )a †

∞  (1 − T1 )m −iωl a † at e m! l,m=0

|0 0| eT1 (β

∗ −κt

e

−T2 )a m iωl a † at

a e

∞  (1 − T1 )m −κt ∗ −κt = T1 e−T1 (βe +T2 )(β e −T2 ) m! m=0 −iωl t

× : a †m eT1 e

(βe−κt +T2 )a † e−a † a eT1 eiωl t (β ∗ e−κt −T2 )a a m :

4.1 Solutions of Several Boson Master Equations

= T1 e−T1 (βe

−κt

× : e−T1 a a : e †

105

+T2 )(β ∗ e−κt −T2 ) T1 e−iωl t (βe−κt +T2 )a †

e

T1 eiωl t

(β

∗ −κt

e

−T2 )a

.

(4.48)

Further, using Eq. (1.21) and the completeness relation of number states |n, we give ρ|β (t) = T1 e−T1 (βe

−κt

+T2 )(βe−κt −T2 )

∞  (1 − T1 )n n=0

−iωl t

× eT1 e

(βe−κt +T2 )a † |n n| eT1 eiωl t (β ∗ e−κt −T2 )a .

(4.49)

Clearly, the initial state ρ|β loses its coherence and evolves into a new mixed state. Especially, for n¯ → 0 and κ being finite, T1 → 1, thus Eq. (4.48) refers to the evolution of ρ|β in the cavity acted by an external field for amplitude damping, that is 

 

 (4.50) ρ|β (t) = e−iωl t βe−κt + T2 e−iωl t βe−κt + T2  , which means that the coherence of ρ|β is partially preserved. For λ = 0, Eqs. (4.49) and (4.50) respectively become the analytical evolution of ρ|β for pure thermal noise and amplitude damping.

4.1.3 Master Equation for Damped Harmonic Oscillator Acted by Linear Resonance Force in a Squeezed Heat Reservoir Considering the damped harmonic oscillator system controlled by linear resonance force, the Hamiltonian operator of the system is H = ωa † a − λ(a † + a) [14, 15], where ω, λ are respectively the natural frequency of harmonic oscillator and the strength of the linear resonance force. When the system is “immersed” in a squeezed heat reservoir, the time-dependent density operator ρ(t) satisfies the following quantum master equation   dρ(t) = iλ[a † − a, ρ(t)] + κ(n¯ + 1) 2aρ(t)a † − a † aρ(t) − ρ(t)a † a dt    + κn¯ 2a † ρ(t)a − aa † ρ(t) − ρ(t)aa † + κM 2a † ρ(t)a † − a †2 ρ(t)    −ρ(t)a †2 + κM ∗ 2aρ(t)a − a 2 ρ(t) − ρ(t)a 2 , (4.51) where the parameter M is related to the squeezed heat reservoir. Operating both sides of Eq. (4.51) on the state |I  and using the operator identity in Eq. (1.104), thus we have

106

4 Solutions of Density Operator Master Equations

d |ρ(t)  = iλ(κ † − κ) + κ(a a˜ − a † a˜ † + 1) dt  −κ(2n¯ + 1)κ † κ − κMκ †2 − κM ∗ κ 2 |ρ(t) ,

(4.52)

its formal solution is  |ρ(t) = exp itλ(κ † − κ) + κt (a a˜ − a † a˜ † + 1)  −κt (2n¯ + 1)κ † κ − κt Mκ †2 − κt M ∗ κ 2 |ρ(0) .

(4.53)

Using the commutative relations [κ †2 , κ † + κ] = 0,

[a a˜ − a † a˜ † , κ †2 ] = −2κ †2 ,

(4.54)

and the disentangling formula in Eq. (4.31), thus Eq. (4.53) becomes |ρ(t) = exp[κt (a a˜ − a † a˜ † + 1)]  

  1 2κt † †2 ∗ 2 1−e (2n¯ + 1)κ κ + Mκ + M κ × exp 2  

† iλ κt 1 − e (κ + κ) |ρ(0) . × exp κ

(4.55)

Using the eigenstate equations in Eq. (4.29), thus the inner product χ| ρ(t) reads 

χ| ρ(t) = exp −(n¯ + 1/2) 1 − e−2κt |χ|2

1 1 − e−2κt (Mχ2 + M ∗ χ∗2 ) − 2   

iλ −κt ∗ 1−e (χ + χ ) χe−κt  ρ(0) . + κ

(4.56)

Similar to deriving Eq. (4.35), inserting the completeness relation of the states |χ into Eq. (4.56), such that 

d2 χ |χ χ| ρ(t) π   2

1 d χ : exp −[n¯ 1 − e−2κt + 1) |χ|2 − 1 − e−2κt = π 2 



iλ 1 − e−κt 2 ∗ ∗2 † −κt ˜ + × Mχ + M χ + a − ae χ κ 

  −κt iλ 1 − e + ae−κt − a˜ † + χ∗ κ

|ρ(t) =

× exp(a † a˜ † + a a˜ − a † a − a˜ † a) ˜ : |ρ(0) .

(4.57)

4.1 Solutions of Several Boson Master Equations

107

Further, using the integral formula in Eq. (2.106), we therefore obtain   T42 T22 2 2 |ρ(t) = T4 exp 2 Re(T4 T3 T2 ) + T1

  T42 † † 2 ∗ †2 2 †2 † ∗ † a a˜ + T4 T3 a + T4 T3 a˜ + T5 a + T5 a˜ × exp 1 − T1  2



T4 −κt × : exp e − 1 (a † a + a˜ † a) ˜ exp −2T42 T3∗ e−κt a † a˜ T1



T 2 e−2κt a a˜ + T42 T3 e−2κt a 2 × exp −2T42 T3 e−κt a a˜ † : exp 1 − 4 T1  +T42 T3∗ e−2κt a˜ 2 + T6 a + T6∗ a˜ |ρ(0) , (4.58) where the parameters T1 , T2 can be seen in Eq. (4.36), and other parameters are respectively

−1/2 1 2 T4 = − 4 |T3 | , T12

1 + 2T3 e−κt . T6 = T2 T42 T1

1 T3 = − 1 − e−2κt M, 2

1 2 ∗ T5 = T2 T4 2T3 + , T1

(4.59)

It then follows that 

l T42 e−2κt T42 T22  1− ρ(t) = T4 exp + T1 T1 l,m,n,r   r

m

n T2 −2T42 T3 e−κt 1 − T41 −2T42 T3∗ e−κt 2 ∗ †2 a †r eT4 T3 a × l!m!n!r !



2 Re(T42 T3 T22 )

†

a † a ln

× eT5 a a †n e ×e

T42 −κt −1 T1 e



T42 T3∗ e−2κt a †2 †l+n

a

a † a ln

e

2

−2κt 2

a l+m eT4 T3 e

a



T42 −κt −1 T1 e

∗ †

eT6 a ρ(0)eT6 a

∗

2

2

a m eT5 a eT4 T3 a a r ,

(4.60)

which is the infinite operator-sum representation of ρ(t) describing the damped harmonic oscillator acted by linear resonance force in a squeezed heat reservoir. Actually, Eq. (4.60) can also be expressed as ρ(t) =



† Ml,m,n,r ρ(0)Ml,m,n,r ,

l,m,n,r † where the operators Ml,m,n,r and Ml,m,n,r are respectively

(4.61)

108

4 Solutions of Density Operator Master Equations

Ml,m,n,r = Ne

T6∗ a † T42 T3∗ e−2κt a †2 †l+n

e

× am e † Ml,m,n,r

= Ne

T5∗ a

a r eiωl a

−iωl a at †r

a † a ln

×e

e

a

T42 T3 a 2

†



a e

at

T42 T3∗ a †2



T42 −κt −1 T1 e

†

e

a † a ln

e



T42 −κt −1 T1 e

, T5 a † †n

a

2

−2κt 2

a l+m eT4 T3 e

a

eT6 a

(4.62)

with ⎡  l T 2 e−2κt  2 2  T4 1 − 4 T1 T T ⎢ N = exp Re(T42 T3 T22 ) − 4 2 ⎣ 2T1 l!m!n!r !

r 1/2



T42 2 −κt m 2 ∗ −κt n −2T4 T3 e × −2T4 T3 e . 1− T1

(4.63)

† Clearly, the operators Ml,m,n,r and Ml,m,n,r are not conjugate to each other. However, it can still be proved that they meet the normalization condition, that is ! † l,m Ml,m,n,r Ml,m,n,r = 1. So, compared to the Kraus operator Ml,m , they are more general Kraus operators and trace-preserving quantum operations. Especially, when M = 0, owing to T3 = 0, T4 = T1 , T5 = T1 T2 and T6 = T1 T2 e−κt , Eq. (4.60) reduces to Eq. (4.38), as expected.

4.1.4 Master Equation Describing a Diffusive Anharmonic Oscillator As one of the most important nonlinear models, anharmonic oscillator model is defined via the relation between force and displacement, which is not linear but depends on the amplitude of displacement. In the past two decades, anharmonic oscillator model has been widely used in atomic and molecular physics and quantum field theory [16, 17], so many different effective methods for studying anharmonic oscillator model have been proposed, such as the state-dependent diagonalization method, Green’s function and equation of motion method, and effective operator method. Recently, diffusive anharmonic oscillator (i.e., the coupling system of anharmonic oscillator and heat reservoir in equilibrium) has attracted much attention [18–20]. In the case of small nonlinearity strength and weak coupling to the reservoir degrees of freedom, the master equation for the diffusive anharmonic oscillator in the interaction picture is

4.1 Solutions of Several Boson Master Equations

  d ρ(t) = −iκ (a † a)2 , ρ(t) + κ(n¯ + 1) dt   × 2aρ (t) a † − a † aρ (t) − ρ (t) a † a   + κn¯ 2a † ρ (t) a − aa † ρ (t) − ρ (t) aa † ,

109

(4.64)

where n¯ is the average thermal photon number in mode ω of the reservoir (ω is the natural frequency of oscillator), κ is the damping factor, and κ is a nonlinearity constant related to the nonlinear susceptibility χ(3) of the medium. For the case of the diffusive limit, i.e., κ → 0 and n¯ → ∞, thus k = κn¯ being finite, Eq. (4.64) reduces to    d ρ(t) = −iκ (a † a)2 , ρ(t) + 2k aρ (t) a † dt  +a † ρ (t) a − a † aρ (t) − ρ (t) a † a − ρ (t) .

(4.65)

Indeed, Peixoto de Faria solved Eq. (4.65) by using the Baker-Campbell-Hausdorff formula to expand the evolution operator, and obtained its classical and quantum mechanical standard solutions [18, 21]. However, when n¯ → 0 and κ being finite, Eq. (4.64) refers to the master equation for a dissipative Kerr medium, that is     d ρ(t) = −iκ (a † a)2 , ρ(t) + κ 2aρ (t) a † − a † aρ (t) − ρ (t) a † a . dt

(4.66)

Operating both sides of Eq. (4.64) on the state |I  leads to the result    d |ρ(t) = −iκ (a † a)2 , ρ(t) + κ(n¯ + 1)[2aρ (t) a † − a † aρ (t) dt   − ρ (t) a † a] + κn¯ 2a † ρ (t) a − aa † ρ (t) − ρ (t) aa † |I     = −iκ (a † a)2 − (a˜ † a) ˜ 2 + κ(n¯ + 1)[2a a˜ − a † a − a˜ † a] ˜  † † † † +κn[2a ¯ a˜ − aa − a˜ a˜ ] |ρ(t) , (4.67) such that    |ρ(t) = exp −iκt (a † a)2 − (a˜ † a) ˜ 2 + κ(n¯ + 1)t[2a a˜  † † −a † a − a˜ † a] ˜ + κnt[2a ¯ a˜ − aa † − a˜ a˜ † ] |ρ (0) .

(4.68)

Defining the operators K + = a † a˜ † , K − = a a, ˜ 1 † K z = (a a + a˜ † a˜ + 1), K 0 = a † a − a˜ † a, ˜ 2

(4.69)

110

4 Solutions of Density Operator Master Equations

  where K + , K − and K z obey SU(1,1) Lie algebra since K − , K + = 2K z   the operators and K z , K ± = ±K ± , and K 0 is the Casimir operator and commutative with the generators K ± and K z of SU(1,1) Lie algebra, thus Eq. (4.68) becomes    |ρ(t) = exp −iκt K 0 (2K z − 1) + κ(n¯ + 1)t ×(2K − − 2K z + 1) + κnt ¯ (2K + − 2K z + 1)} |ρ (0) = exp(iκt K 0 + κ(2n¯ + 1)t] exp (γ+ K + + γz K z + γ− K − ) |ρ (0) , (4.70) where γ+ = 2κnt, ¯ γ− = 2κ(n¯ + 1)t, γz = −2t[iκ K 0 + κ(2n¯ + 1)].

(4.71)

Using the disentangling formula for SU(1,1) Lie algebra [22], we obtain |ρ(t) = exp[iκt K 0 + κ(2n¯ + 1)t] exp(+ K + ) " × exp[(2 ln z )K z ] exp(− K − ) |ρ (0) ,

(4.72)

where 2γ± sinh φ , 2φ cosh φ − γz sinh φ " 2φ z = , 2φ cosh φ − γz sinh φ γ2 φ2 = z − γ+ γ− . 4 ± =

(4.73)

To get rid of the state |I  from Eq. (4.72), substituting the completeness relation ∞ 

|m, n ˜ m, n| ˜ =1

(4.74)

m,n=0 ˜

in the enlarged space into Eq. (4.70), and using a † j |n = |ρ(t) =

#

(n+ j)! n!

∞ j  − (m, n) exp[iκt K 0 + κ(2n¯ + 1)t] j! j,m,n=0

|n + j, we obtain

# ∞ i  + a †i a˜ †i |m, n ˜ m, n| ˜ a j ρ (0) a † j |I  zm+n+1 (m, n) i! i=0 $ ∞  (m + i)!(n + i)! m+n+1 z = (m, n) m!n! i, j,m,n=0

×

4.1 Solutions of Several Boson Master Equations

111

j

i + (m, n)− (m, n) exp[iκ(m − n)t + κ(2n¯ + 1)t] i! j! × |m + i, n˜ + i m, n| ˜ a j ρ (0) a † j |I ,

×

(4.75)

where ¯ n) sinh φ(m, n), + (m, n) = 2κnt(m, " z (m, n) = (m, n)φ(m, n), − (m, n) = 2κt (n¯ + 1)(m, n) sinh φ(m, n), (m, n)=[φ(m, n) cosh φ(m, n) + t[iκ(m − n) + κ(2n¯ + 1)] sinh φ(m, n)]−1 , φ(m, n) = t[−κ 2 (m − n)2 + i2κκ(2n¯ + 1)(m − n) + κ2 ]1/2 .

(4.76)

Using the relations n˜ |I  = |n,

n |I  = |n, ˜

(4.77)

we thus have m, n| ˜ a j ρ (0) a † j |I  = m| a j ρ (0) a † j n˜ |I  = m| a j ρ (0) a † j |n.

(4.78)

So, Eq. (4.75) becomes ∞ 

|ρ(t) =

$

i, j,m,n=0

(m + i)!(n + i)! m+n+1 z (m, n) m!n!

j i + (m, n)− (m, n)

exp[iκ(m − n)t + κ(2n¯ + 1)t] i! j! × |m + i, n˜ + i m| a j ρ (0) a † j |n .

×

(4.79)

Further, using Eq. (4.77) again, Eq. (4.79) can be rewritten as ∞ 

|ρ(t) =

i, j,m,n=0

$

j

i (m, n)− (m, n) (m + i)!(n + i)! + m!n! i! j!

# ×

zm+n+1 (m, n) exp[iκ(m − n)t + κ(2n¯ + 1)t]

× m| a j ρ (0) a † j |n |m + i n + i| I  ,

(4.80)

such that # j i + (m, n)− (m, n) ρ(t) = zm+n+1 (m, n) i! j! i, j,m,n=0 ∞ 

× exp[iκt (m − n) + κ(2n¯ + 1)t]a †i |m m| a j ρ (0) a † j |n n| a i

(4.81)

112

4 Solutions of Density Operator Master Equations

or

∞ 

ρ(t) =

Mm,n,i, j ρ (0) M†m,n,i, j ,

(4.82)

i, j,m,n=0

where the operators Mm,n,i, j and M†m,n,i, j are respectively 



 j 1/2 i 1 + (m, n)− (m, n)z (m, n) exp κ n¯ + t i! j! 2 " × exp[m(iκt + ln z (m, n))]a †i |m m| a j

(4.83)

⎧ 

 ⎨  i (m, n) j (m, n) 1/2 (m, n) 1 z − + exp κ n¯ + = t ⎩ i! j! 2 († " × exp[n(iκt + ln z (m, n))]a †i |n n| a j .

(4.84)

Mm,n,i, j =

and M†m,n,i, j

Clearly, the operators Mm,n,i, j and M†m,n,i, j are not Hermite conjugate to each other. However, they are still normalized since ∞ 

M†m,n,i, j Mm,n,i, j = 1,

(4.85)

i, j,m,n=0

which means that, in a general sense, Mm,n,i, j and M†m,n,i, j are trace-preserving, so the operators Mm,n,i, j and M†m,n,i, j are the generalized Kraus operators. In particular, when κ → 0 and n¯ → ∞, thus k = κn¯ is finite, via letting  = iκ(m − n) + 2k,

1/2 ,  = 2 − 4k 2

(4.86)

and (t) =

2k sinh t ,  cosh t +  sinh t

0 (t) =

 ,  cosh t +  sinh t

(4.87)

we have ρ(t) =

∞  i, j,m,n=0

√

(m + i)!(n + i)!(m + j)!(n + j)! m!n!i! j!

×  i+ j (t)0m+n+1 (t) exp[iκ(m − n)t + 2kt] × m + j| ρ (0) |n + j |m + i n + i| ,

(4.88)

4.1 Solutions of Several Boson Master Equations

113

which is just the evolution of the density operator of the diffusive anharmonic oscillator in the case of diffusion limit [18]. Besides, when n¯ = 0 and κ being finite, Eq. (4.82) refers to the density operator’s evolution in a dissipative cavity with Kerr medium, that is ρ(t) =

∞   j (t) exp[−iκ(m 2 − n 2 )t j! j,m,n=0

− κ(m + n)t] |m m| a j ρ (0) a † j |n n| , where

  κ 1 − e−2[iκ(m−n)+κ]t . (t) = iκ(m − n) + κ

(4.89)

(4.90)

Using the definition of the Wigner distribution function for the density operator ρ, that is W (α, α∗ ) = tr[ρ(α, α∗ )] and the concise form of single-mode Wigner † operator (α, α∗ ), i.e., (α, α∗ ) = π −1 D(2α)(−1)a a , thus we obtain the thermal entangled state representation of the Wigner distribution function as W (α, α∗ ) =

∞ 

n, n| ˜ (α, α∗ )ρ |m, m ˜

m,n=0

1 † I | D(2α)(−1)a a ρ |I  π 1 † = χ = −2α| (−1)a a |ρ π 1 =  = 2α |ρ , π

=

(4.91)

which is the new formula of calculating the Wigner distribution function in the context of thermal dynamics, that is, an inner product of the states  = 2α| and |ρ in an enlarged Fock space. Substituting the completeness relation in Eq. (1.108) into the inner product  = 2α| ρ(t), we give the Wigner distribution function evolution of the diffusive anharmonic oscillator as  2 d β K (α, β ∗ , t)W (β, β ∗ , 0), (4.92) W (α, α∗ , t) = 4 π where W (β, β ∗ , 0) is the initial Wigner distribution function W (β, β ∗ , 0) =

1  = 2β |ρ (0) , π

and K (α, β ∗ , t) =  = 2α| exp[iκt K 0 + κ(2n¯ + 1)t] exp(+ K + )

(4.93)

114

4 Solutions of Density Operator Master Equations

× exp[(2 ln

"

  z )K z ] exp(− K − )  = 2β .

(4.94)

Substituting the completeness relation in Eq. (4.74) into Eq. (4.94), and using Eq. (4.69), we can rewrite K (α, β ∗ , t) as K (α, β ∗ , t) =

∞ # 

zm+n+1 (m, n) exp [iκt (m − n) + κ(2n¯ + 1)t]

m,n=0

×  = 2α| exp[+ (m, n)a † a˜ † ]

  × |m, n ˜ m, n| ˜ exp[− (m, n)a a] ˜  = 2β .

(4.95)

Further, using the generating functions for two-variable Hermite polynomials in Eq. (1.85), the entangled state | in Eq. (1.107) can be expanded as | = e−||

2

/2

∞ 



1 Hm,n , ∗ |m, n ˜ √ m!n! m,n=0

(4.96)

a †m a˜ †n  ˜ ) in two-mode Fock space, where |m, n ˜ = √ 0, 0 is two-mode Fock state. m!n! Thus, the inner product | m, n ˜ reads | m, n ˜ =√

1 m!n!



2 Hm,n ∗ ,  e−|| /2 .

(4.97)

Then, using Eqs. (4.95) and (4.97), we obtain K (α, β ∗ , t) ∞ #  = zm+n+1 (m, n) exp [iκt (m − n) + κ(2n¯ + 1)t] ×

m,n=0 ∞ k l  + (m, n)− (m, n) k,l=0

=

k!l!

  ξ = 2α| a †k a˜ †k |m, n ˜ m, n| ˜ a l a˜ l ξ = 2β

∞ #  zm+n+1 (m, n) exp [iκt (m − n) + κ(2n¯ + 1)t]

m,n=0 ∞ k l  + (m, n)− (m, n) −2|α|2 −2|β|2 e Hm+k,n+k (2α∗ , 2α)Hm+l,n+l (2β, 2β ∗ ). × k!l!m!n! k,l=0

(4.98) Further, using a new sum formula related to two-variable Hermite polynomials, i.e.,

4.1 Solutions of Several Boson Master Equations

115

x yz

∞  y xl z e x+1 Hm+l,n+l (y, z) = ,√ , Hm,n √ l! (x + 1)(m+n+2)/2 x +1 x +1 l=0

(4.99)

thus Eq. (4.98) can be expressed as the form ∞ 

"

zm+n+1 (m, n) # (m+n+2) m,n=0 m!n! {[+ (m, n) + 1][− (m, n) + 1]}  + (m, n) − 1 2 |α| × exp[iκt (m − n) + κ(2n¯ + 1)t] exp 2 + (m, n) + 1   2α∗ − (m, n) − 1 2 |β| Hm,n √ +2 , − (m, n) + 1 + (m, n) + 1    2β 2β ∗ 2α Hm,n √ ,√ . (4.100) √ + (m, n) + 1 − (m, n) + 1 − (m, n) + 1 ∗

K (α, β , t) =

Finally, substituting the factor K (α, β ∗ , t) in Eq. (4.100) into Eq. (4.92) can lead to the time-dependent Wigner distribution function W (α, α∗ , t) for the diffusive anharmonic oscillator. Obviously, when κ → 0 and n¯ → ∞, Eq. (4.100) becomes ∞ 

40m+n+1 (t) exp [iκt (m − n) + 2kt m!n m,n=0    (t) − 1 2 2α∗ 2α +2 |α| Hm,n √ ,√ (t) + 1 (t) + 1 (t) + 1      2 2β (t) − 1 2 2β ∗ d β |β| Hm,n √ exp 2 ,√ W (β, β ∗ , 0). × π (t) + 1 (t) + 1 (t) + 1 (4.101)

W (α, α∗ , t) =

which is the evolution of the Wigner distribution function for the diffusive anharmonic oscillator for the diffusion limit case. When n¯ → 0 and κ is finite, Eq. (4.100) reduces to ∞ 

4Hm,n (2α∗ , 2α) exp[−iκt (m 2 − n 2 ) (m+n+2)/2 m!n![(t) + 1] m,n=0    2 (t) − 1 2 d β − κ(m + n)t − 2|α|2 ] exp 2 |β| π (t) + 1   2β 2β ∗ ,√ W (β, β ∗ , 0), × Hm,n √ (4.102) (t) + 1 (t) + 1

W (α, α∗ , t) =

which represents the Wigner distribution function evolution in a dissipative cavity filled with Kerr medium.

116

4 Solutions of Density Operator Master Equations

Taking the coherent state |z as an initial state, substituting its Wigner distribution function

1 (4.103) W (β, β ∗ ) = exp −2 |β − z|2 π into Eq. (4.92), and using Eqs. (1.30) and (4.100), we easily obtain the evolution of the Wigner distribution function for the state |z, i.e., " zm+n+1 (m, n) " W (α, α∗ , t) = m+n+2 m,n=0 πm!n! [+ (m, n) + 1]   + (m, n) − 1 2 × exp[iκt (m − n) + κ(2n¯ + 1)t] exp 2 |α| + (m, n) + 1   2α∗ 2α × Hm,n √ ,√ + (m, n) + 1 + (m, n) + 1   × z m z ∗n exp [− (m, n) − 1]|z|2 . ∞ 

(4.104)

Especially, for t = 0, z (m, n) = 1, + (m, n) = − (m, n) = 0, thus Eq. (4.104) becomes 2 2 ∞ e−2|α| −|z|  z m z ∗n Hm,n (2α∗ , 2α). W (α, α , 0) = π m!n! m,n=0

∗

(4.105)

Further, using the generating functions for two-variable Hermite polynomials in Eq. (1.85), we find that the function W (α, α∗ , 0) just reduces to the Wigner distribution function for the coherent state |z. Besides, using the evolution formula of Wigner distribution function in Eq. (4.92), we derive the evolution of photon number distribution of the diffusive anharmonic oscillator. In terms of thermal field dynamics theory, the photon number distribution p(s, t) = tr[ρ(t) |s s|] can be converted into the form p(s, t) =

∞ 

s, s˜ | ρ(t) |m, m ˜ = s, s˜ | ρ(t) ,

(4.106)

m=0

where |s is the number state. Substituting the completeness relation of | in Eq. (1.108) into Eq. (4.106) leads to  p(s, t) =

d2 ξ s, s˜ |  | ρ(t) . π

(4.107)

From Eq. (4.91), we find that π −1 | ρ(t) is the Wigner distribution function W (α, α∗ , t) for ρ(t), and π −1 s, s˜ |  is the Wigner distribution function for the number state |s, that is

4.2 Solutions of Several Fermi Quantum Master Equations

W|ss| (α, α∗ ) =

(−1)s −2|α|2 e Ls (4 |α|2 ). π

117

(4.108)

Therefore, for the state ρ(t), the relation between the photon number distribution and the Wigner distribution function yields  p(s, t) = 4

d2 α(−1)s e−2|α| Ls (4 |α|2 )W (α, α∗ , t). 2

(4.109)

Substituting Eq. (4.92) into Eq. (4.109) and completing this integration, we obtain the evolution of the photon number distribution of ρ(t) for the diffusive anharmonic oscillator, i.e., # ∞ 4(−1)(3m+n)/2 s!  m+n+1 (m, n) (2s−n−m) (m, n)  z + " p(s, t) = m!n!(s − m)! [− (m, n) + 1](m+n+2) m,n=0    − (m, n) − 1 2 |β| × exp[iκt (m − n) + κ(2n¯ + 1)t] d2 β exp 2 − (m, n) + 1   ∗ 2β 2β × Hm,n √ (4.110) ,√ W (β, β ∗ , 0), − (m, n) + 1 − (m, n) + 1 which indicates that, if the initial Wigner distribution function W (β, β ∗ , 0) is known, the evolution of the photon number distribution can be easily obtain from Eq. (4.110).

4.2 Solutions of Several Fermi Quantum Master Equations 4.2.1 Fermi Master Equations for Amplitude Damping and Phase Damping For the amplitude damping process, the time-dependent density operator ρ(t) of a two-level system obeys the following master equation [23]   dρ(t) = κ 2σ− ρ(t)σ+ − σ+ σ− ρ(t) − ρ(t)σ+ σ− , dt

(4.111)

where σ+ , σ− are respectively the raising and lowering operators of a two-level atom, κ is the decay rate. On the other hand, the corresponding master equation describing the phase damping process is [23]   dρ(t) = κ 2σ+ σ− ρ(t)σ+ σ− − σ+ σ− ρ(t) − ρ(t)σ+ σ− . dt

(4.112)

118

4 Solutions of Density Operator Master Equations

Usually, the density matrix method is used to investigate the evolution of two-level systems for the above two decoherence processes [23, 24]. If we consider the qubits of two-level atoms as fermions, thus decay and excitation processes correspond to the annihilation and creation operators f , f † , analogous to the lowering and raising operators σ− , σ+ . So, Eqs. (4.111) and (4.112) can be rewritten as the following Fermi operator master equations   dρa (t) = κ 2 f ρa (t) f † − f † f ρa (t) − ρa (t) f † f dt

(4.113)

and

  dρ p (t) = κ 2 f † f ρ p (t) f † f − ρ p (t) f † f − f † f ρ p (t) . (4.114) dt   Operating both sides of Eq. (4.113) on the state  I f in Eq. (1.118), and using Eq. (1.119), we have d |ρa (t) = −κ(2 f f˜ + f † f + f˜† f˜) |ρa (t) , dt   where |ρa (t) = ρa (t)  I f , thus its solution is given as |ρa (t) = exp[−κt (2 f f˜ + f † f + f˜† f˜)] |ρa (0) ,

(4.115)

(4.116)

  where ρa (0) is the initial density operator, |ρa (0) = ρa (0)  I f . Noting that the operators f f˜, f † f and f˜† f˜ obey the commutative relations [ f f˜, f † f ] = [ f f˜, f˜† f˜] = f f˜,



 f † f + f˜† f˜, f f˜ = −2 f f˜,

(4.117)

and using the operator identity in Eqs. (4.31), (4.116) becomes   |ρa (t) = exp −κt ( f † f + f˜† f˜) exp(−T f f˜) |ρa (0) ,

(4.118)

where T = 1− e−2κt . Considering that      exp(−T f f˜) |ρa (0) = (1 − T f f˜)ρa (0)  I f = ρa (0) + T f ρa (0) f †  I f , (4.119) thus we have     |ρa (t) = exp −κt ( f † f + f˜† f˜) ρa (0) + T f ρa (0) f †  I f     † † = e−κt f f ρa (0) + T f ρa (0) f † e−κt f f  I f =

1  Tn n=0

n!

e−κt f

†

f

f n ρa (0) f †n e−κt f

†

f

  I f .

(4.120)

4.2 Solutions of Several Fermi Quantum Master Equations

119

  Getting rid of the state  I f from both sides of Eq. (4.120) leads to the operator-sum representation of ρa (t), that is ρa (t) =

1  Tn n=0

n!

e

−κt f † f

†n −κt f † f

f ρa (0) f e n

=

1 

Mn ρa (0)Mn† ,

(4.121)

n=0

where Mn is the Kraus operator of the density operator ρa (t) in the amplitude damping process, i.e., √ † M0 = e−κt f f , M1 = T f. (4.122) Further, using the operator relations eλ f

†

f

= : exp[(eλ − 1) f † f ] : , eλ f

†

f

f e−λ f

†

f

= f e−λ ,

(4.123)

we can prove that the Kraus operator Mn yields the normalization condition, that is 1 

Mn† Mn

=

n=0

1  Tn

n!

n=0

= : e(e

2κt

†n −2κt f † f

f e

f = n

1  Tn n=0

−1) f † f

such that trρa (t) = tr

: e−2κt f

 1 

†

f

n!

e2nκt : f †n f n : e−2κt f

= 1,

†

f

(4.124) 

Mn ρa (0)Mn†

= trρa (0) = 1,

(4.125)

n=0

that is, Mn is a trace-preserving quantum operation. For a two-level atomic system, its ground (|0) and excited (|1) states have the following matrix forms



1 0 |0 = , |1 = , (4.126) 0 1 thus the matrix forms of Fermi operators f and f † are respectively f = |0 1| =

01 , 00

f = †

00 . 10

(4.127)

So we obtain M0 = e

−κt f † f

= 1 + f f (e †

and M1 =

−κt

− 1) =

√ T f =

1√ 0 0 1−T

√

0 T , 0 0

(4.128)

(4.129)

120

4 Solutions of Density Operator Master Equations

which are just the matrix forms of two Kraus operators in the amplitude damping † the evolution of the process [25, 26]. Here, the operator M0 = e−κt f f represents √ initial state when no quantum jump occurs, and M1 = T f refers to the decay (quantum jump) from the excited state |1 to the ground state |0 by emitting a photon.   Similarly, acting both sides of Eq. (4.114) on the state  I f , that is     d  ρ p (t) = κ 2 f † f ρ p (t) f † f − ρ p (t) f † f − f † f ρ p (t)  I f dt    = κ 2 f † f f˜† f˜ − f˜† f˜ − f † f ρ p (t) ,

(4.130)

using Eq. (1.119), we have      ρ p (t) = exp −κt ( f † f − 2 f † f f˜† f˜ + f˜† f˜) ρ p (0)

  = exp(−κt f † f ) exp(2κt f † f f˜† f˜)ρ p (0) exp(−κt f˜† f˜)  I f   = exp(−κt f † f ) exp(2κt f † f f˜† f˜)ρ p (0) exp(−κt f † f )  I f =

∞  (2κt)n n=0

n!

e−κt f f ( f † f )n ρ p (0)( f † f )n e−κt f †

†

f

  I f .

(4.131)

So, the formal solution of Eq. (4.131) reads ρ p (t) = exp(−κt f † f )ρ p (0) exp(−κt f † f ) + T f † f ρ p (0) f † f =

1 

Mn ρ p (0)Mn† ,

(4.132)

n=0

which is just the Kraus operator-sum representation of the density operator ρ p (t), and the corresponding Kraus operators are M0 = e−κt f f , †

M1 =

√ T f † f.

(4.133)

Since the Kraus operators M0 and M1 are only related to the operator f † f , no photon jump occurs in the phase damping process. After simple calculation, we find that the Kraus operator Mn is normalized because of 1 

Mn† Mn = 1.

(4.134)

n=0

Further, using Eq. (4.127), we can give the matrix forms of the Kraus operators M0 and M1 as

4.2 Solutions of Several Fermi Quantum Master Equations

M0 =

1√ 0 0 1−T

,

M1 =

121

√

T

00 . 01

(4.135)

Obviously, the analytical solutions of master equations in Eqs. (4.121) and (4.132) are very suitable for studying the evolutions of two-state systems (e.g., a two-level atom) because of two singlets. Now, we investigate a two-level atom (denoted by subscript A) that evolves from |1 A to |0 A via emitting a photon, thus photon field (denoted by subscript E) evolves from |0 E to |1 E . For the amplitude damping process, taking the state ρ A (0) = |0 A A 0| of the atom as the initial state and using Eq. (4.121), thus we have ρ A (t) =

1  Tn n=0

n!

e−κt f

†

f

f n |0 A A 0| f †n e−κt f

†

f

= |0 A A 0| ,

(4.136)

that is, no decay and no photon appear. Therefore, for the interaction system between two-level atom and the photon field, we find |0 A |0 E → |0 A |0 E .

(4.137)

However, when the initial state of an atom is an excited state ρ A (0) = |1 A A 1|, the atom system evolves into ρ A (t) =

1  Tn n=0

n!

e−κt f

†

f

f n |1 A A 1| f †n e−κt f

†

f

= (1 − T ) |1 A A 1| + T |0 A A 0| ,

(4.138)

where T refers to the transition probability of atomic system from excited state |1 A to ground state |0 A . It then follows that |1 A |0 E →

√ √ 1 − T |1 A |0 E + T |0 A |1 E .

(4.139)

Clearly, T is also the probability of the interaction system from the initial state |1 A |0 E to the state |0 A |1 E in the amplitude damping process. For t → ∞, the initial state |1 A |0 E becomes completely the final state |0 A |1 E , which shows that all atoms finally decay to the ground state after enough long time. Similarly, using Eq. (4.132), the evolution of the interaction system between two-level atom and photon field in the phase damping process read |0 A |0 E → |0 A |0 E ,

|1 A |0 E → |1 A |0 E ,

(4.140)

that is, the interaction system remains unchanged after undergoing the phase damping process.

122

4 Solutions of Density Operator Master Equations

4.2.2 Master Equation for Fermi Heat Reservoir A general Fermi master equation describing a heat reservoir reads [27]   dρt (t) = g 2 f † ρt (t) f − f f † ρt (t) − ρt (t) f f † dt   + κ 2 f ρt (t) f † − f † f ρt (t) − ρt (t) f † f .

(4.141)

  Operating both sides of Eq. (4.141) on the state  I f and using Eq. (1.119), we obtain the evolution equation of the state |ρ(t) as   d |ρt (t) = g(2 f † f˜† − f f † − f˜ f˜† ) + κ(2 f˜ f − f † f − f˜† f˜) |ρt (0) , dt (4.142) whose solution is   |ρt (t) = exp g(2 f † f˜† − f f † − f˜ f˜† )t + κ(2 f˜ f − f˜† f˜ − f † f )t |ρt (0) . (4.143) For convenience, letting H ≡ g(2 f † f˜† − f f † − f˜ f˜† )t + κ(2 f˜ f − f˜† f˜ − f † f )t,

(4.144)

so the operator exp H means quantum entanglement between the system and its surrounding environment. To disentangle the operator exp H, we rewrite it as exp H = e(κ+g)t exp



1 B B˜ , 2

(4.145)

where B is given as

B ≡ F † F ≡ ( f † f˜† f f˜), B˜ ≡



F˜ † F˜

,

and  is a 4 × 4 matrix, that is

2gt J2 (g − κ) t I2 ≡ , (κ − g) t I2 2κt J2T



10 0 1 I2 = , J2 = , J22 = −I2 . 01 −1 0

(4.146)

(4.147)

On the other hand, the fermionic coherent state representation of the operator exp H is

4.2 Solutions of Several Fermi Quantum Master Equations

exp H = √

1 det Q

123



 +   * 2  Q −L α α  , dα¯ i dαi  −N P α¯ α¯ 

(4.148)

i=1

where the two-mode fermionic coherent state reads  

 α α †  |00 = exp(F α − αF) ¯ , = (α1 , α2 , α¯ 1 , α¯ 2 ),  α¯ α¯

(4.149)

where (α, α) ¯ is a pair of Grassmann numbers, and

Q L N P



= exp(),  =

0 I2 I2 0

.

(4.150)

According to Eqs. (4.147) and (4.150), we obtain ge(g+κ)t + κe−(g+κ)t κe(g+κ)t + ge−(g+κ)t I2 , I2 , P= g+κ g+κ

(g+κ)t − e−(g+κ)t g e κ e(g+κ)t − e−(g+κ)t T J2 , N = J2 . L= g+κ g+κ

Q=

(4.151)

Thus, using the IWOP method to complete the integration in Eq. (4.148), we obtain the normal ordering of the operator exp H as √



1 † −1 ˜ † F F LP 2    1 −1 ˜ † −1 ˜ ˜ +F P − I F + F P N F : . 2

exp H =

det P : exp

(4.152)

Further, using the operator identity eλ f

†

f

= : exp[(eλ − 1) f † f ] : ,

(4.153)

we give  1 † −1 ˜ † F LP F exp H = det P exp 2     1 −1 ˜ † −1 ˜ ˜ F P N F . × exp F ln P F exp 2 √



(4.154)

Substituting Eqs. (4.146) and (4.151) into Eq. (4.154) leads to   exp H = T4 exp(T1 f † f˜† ) exp ( f˜† f˜ + f † f ) ln(T2 + 1) exp(T3 f˜ f ) = T4 (1 + T1 f † f˜† )(1 + T2 f † f )(1 + T2 f˜† f˜)(1 + T3 f˜ f )

124

4 Solutions of Density Operator Master Equations

 = T4 1 + T2 f˜† f˜ + T1 f † f˜† + T2 f † f

 +T22 f † f f˜† f˜ + T3 f˜ f + T1 T3 f † f˜† f˜ f ,

(4.155)

where T1 , T2 , T3 and T4 are respectively ge2(g+κ)t − g , κe2(g+κ)t + g κe2(g+κ)t − κ , T3 = 2(g+κ)t κe +g

T1 =

g+κ − 1, + κe(g+κ)t ge−2(g+κ)t + κ T4 = . g+κ

T2 =

ge−(g+κ)t

(4.156)

Using Eqs. (1.119) and (4.155), thus Eq. (4.143) can be expressed as   |ρt (t) = T4 ρt (0) + T3 f ρt (0) f † + T1 f † ρt (0) f + T2 ρt (0) f † f

  + f † f ρt (0) + T22 + T1 T3 f † f ρt (0) f † f |I  , (4.157) it then follows that the operator-sum representation of the density operator ρt (t) is



ρt (t) = T4 1 + T2 f † f ρt (0) 1 + T2 f † f + T3 T4 f ρt (0) f † + T1 T4 f † ρt (0) f + T1 T3 T4 f † f ρt (0) f † f =

4 

Mm ρt (0)Mm† ,

(4.158)

m=1

where Mm is the Kraus operator for the Fermi thermal reservoir, i.e., " T3 T4 f, M2 = T1 T4 f † , " "

M3 = T4 1 + T2 f † f , M4 = T1 T3 T4 f † f,

M1 =

"

(4.159)

so M1 refers to a quantum jump from the excited state |1 to the ground |0 of the atom by emitting a photon, M2 is the quantum jump contrary to M1 , however M3 , M4 means that no quantum jump occurs for the atom. Further, using the matrix representation of fermionic operators f , f † in Eq. (4.127), we can give the matrix form of the Kraus operator Mm as √

0 K , 0 0

" √ 0 0 † √ , M2 = T1 T4 f = 1 − P K0 √  P + (1 − P) (1 − K) # 0 "

† M3 = T4 1 + T2 f f = , 1−K 0 P+(1−P)(1−K) M1 =

"

T3 T4 f =

√ P

4.2 Solutions of Several Fermi Quantum Master Equations

M4 =

"

 T1 T3 T4 f f = †

0 # 0 P(1−P) 0 K P+(1−P)(1−K)

125

 ,

(4.160)

where P = κ/(g + κ), K = 1−e−2(g+κ)t . Similarly, the normalization for Mm is true, that is 4  Mm† Mm = 1. (4.161) m=1

By acting the unitary matrix ⎛

Unm

1 ⎜0 ⎜ =⎜ ⎜0 ⎝ 0

⎞ 0 0 0 ⎟ 1# 0 0 ⎟ # (1−P)(1−K) ⎟ P 0 P+(1−P)(1−P) − P+(1−P)(1−K) ⎟ ⎠ # # (1−P)(1−K) P 0 P+(1−P)(1−K) P+(1−P)(1−P)

(4.162)

on the Kraus operator Mm , we obtain the new Kraus operator Mn = Unm Mm , where Mn also yields the identities ρt (t) =

4 

4 

Mn ρ(0)M†n ,

n=1

M†n Mn = 1

(4.163)

n=1

with √



√ √ 0 0 0 K √ M1 = P , , M2 = 1 − P K0 0 0



√ √ √ 1√ 0 1−K 0 , M4 = 1 − P . M3 = P 0 1−K 0 1

(4.164)

Indeed, Mn is the Kraus operator of the generalized amplitude damping model [28]. In particular, replacing the parameters g and κ by κn¯ and κ(n¯ + 1), and n¯ → 0, κ being finite, P → 1, K → 1− e−2kt , so Eq. (4.164) becomes the matrix representation of the Kraus operator for the amplitude damping model, i.e., M1 =

√

0 K , 0 0

M2 = 0, M3 =

1√ 0 , 0 1−K

M4 = 0.

(4.165)

In this process, the atom decays from |1 to |0 (M1 ). When κ → 0 and n¯ → ∞, that is, κn¯ keeps finite, P → 0, K → 1− e−2k nt¯ ≡ T, thus Eq. (4.164) reduces to the matrix form of the Kraus operators that correspond to the Fermi diffusion process at finite temperature, i.e.,

126

4 Solutions of Density Operator Master Equations

M1 = 0,

M2 =

√



1−T 0 √0 0 , M3 = , 0 1 T0

M4 = 0.

(4.166)

Obviously, the quantum jump occurs from |0 to |1 (M2 ) for the atom in this process.

4.3 Generation of Displaced Thermal State In quantum optics, the light field described by thermal state is a typical “classical” light field, its density operator reads [29]

† ρth = 1 − e−ν/(k B T ) e−νa a/(k B T ) ,

(4.167)

where  is the reduced Planck constant, k B is the Boltzmann constant, T, ν are respectively the temperature and frequency of the thermal field. Using the integral formula in Eq. (1.30), the anti-normal ordering of ρth is

. . ν/(k B T ) −1)a † a . ., ρth = eν/(k B T ) − 1 ..e−(e

(4.168)

which leads to the P-representation of ρth in the coherent state representation, i.e.,

ρth = eν/(k B T ) − 1



d2 α −(eν/(k B T ) −1)|α|2 e |αα|. π

(4.169)

Comparing  ν/(k T ) with the2  pure coherent state |α, the exponential factor exp B − 1 |α| represents Gaussian noise. In terms of the P-representation − e of ρth , introducing a displaced thermal state ρd , its P-representation reads

ρd = eν/(k B T ) − 1



d2 α −(eν/(k B T ) −1)|α−z|2 e |αα|, π

(4.170)

where z is the displacement. Further, using the IWOP method to carry out the integration in Eq. (4.170), we give the normal ordering of ρd as



 ρd = 1 − e−ν/(k B T ) : exp − 1 − e−ν/(k B T ) (|z|2 − za † − z ∗ a + a † a) : . (4.171) Obviously, the state ρd can be viewed as an intermediate quantum state between the mixed thermal state ρth (z = 0) and the pure coherent state ρ|z = |zz| (T → 0). The generation mechanism of the state ρd is introduced as below, that is, when the pure coherent state as an initial state passes through the diffusion channel, its output state is just the state ρd . For this, introducing the diffusion equation of ρ(t) [30, 31]   dρ(t) = −k a † aρ(t) − aρ(t)a † − a † ρ(t)a + ρ(t)aa † . dt

(4.172)

4.3 Generation of Displaced Thermal State

127

Operating both sides of Eq. (4.172) on the state |I , and using Eq. (1.104), we obtain   d |ρ(t) = −k a † aρ(t) − aρ(t)a † − a † ρ(t)a + ρ(t)aa † |I  dt   ˜ − a † a˜ † ρ(t) + a˜ a˜ † ρ(t) |I  = −k a † aρ(t) − a aρ(t)

= −k a † − a˜ a − a˜ † |ρ(t) ,

(4.173)





|ρ(t) = exp −kt a † − a˜ a − a˜ † |ρ(0) .

(4.174)

such that

Projecting Eq. (4.174) onto the entangled state χ|, and using the eigenstate equations in Eq. (1.100), we obtain



 2 χ |ρ(t) = χ| exp −kt a † − a˜ a − a˜ † |ρ(0) = e−kt|χ| χ |ρ(0) . (4.175) 2 Further, multiplying both sides of Eq. (4.175) by π −1 d2 χ |χ, and using the completeness relation of |χ in Eq. (1.102) and the IWOP method, we have 

d2 χ −kt|χ|2 |χ χ |ρ(0) e π  2 

d χ : exp −(1 + kt) |χ|2 + χ a † − a˜ = π 

† ∗ +χ a − a˜ † + a † a˜ † + a a˜ − a † a − a˜ † a˜ : |ρ(0)  

kt † † 1 : exp a a˜ + a a˜ − a † a − a˜ † a˜ : |ρ(0) = 1 + kt 1 + kt

a † a+a˜ † a˜ 1 1 kt kt † † e 1+kt a a˜ e 1+kt a a˜ ρ(0) |I  , (4.176) = 1 + kt 1 + kt

|ρ(t) =

where we have used the operator identity 

a † a+a˜ † a˜ 

−kt † 1 † a a + a˜ a˜ : = . : exp 1 + kt 1 + kt

(4.177)

  Noting that a, ˜ ρ(0) = 0 and a˜ |I  = a † |I , we obtain e

kt 1+kt

a a˜



n ∞  kt 1 a ρ(0)a˜ n |I  ρ0 |I  = n! 1 + kt n=0

n ∞  kt 1 = a n ρ(0)a †n |I  . n! 1 + kt n=0

(4.178)

128

4 Solutions of Density Operator Master Equations

Since the operator a˜ † a˜ and all real field operators a n , a †n and a † a are commutative, we give

a † a 1 1 kt † † e 1+kt a a˜ 1 + kt 1 + kt

a˜ † a˜ ∞ n  1 1 kt |I  × a n ρ(0)a †n n! 1 + kt 1 + kt n=0

m

a † a ∞ 1 kt 1  1 †m = a 1 + kt m=0 m! 1 + kt 1 + kt

a † a ∞ n  1 1 kt × a n ρ(0)a †n a˜ †m |I  . n! 1 + kt 1 + kt n=0

|ρ(t) =

(4.179)

It then follows that ∞ 

1 (kt)m+n m!n! (kt + 1)m+n+1 m,n=0

a † a

a † a 1 1 †m n †n ×a a ρ(0)a a m |I  . 1 + kt 1 + kt

ρ (t) |I  =

(4.180)

So, the infinite Kraus operator-sum representation of ρ (t) reads ∞ 

ρ (t) =

† Mm,n ρ(0)Mm,n ,

(4.181)

m,n=0

where the Kraus operator Mm,n is  Mm,n =

1 (kt)m+n a †m m!n! (kt + 1)m+n+1



1 1 + kt

a † a an .

(4.182)

Substituting the initial coherent state ρ(0) ≡ |zz| into Eq. (4.181), and using a|z = z|z, we obtain the analytical evolution of coherent states in the diffusion process as ρ|z (t) =

∞ 

1 1 1 (kt)m+n † † a †m ea a ln 1+kt |z|2n |zz|ea a ln 1+kt a m . (4.183) m+n+1 m!n! (kt + 1) m,n=0

After summing over n, and using ea

†

a ln

1 1+kt

|z = e−

|z|2 2

†

za + 1+kt

|0 = e−

|z|2 2

3  3 z 3 3 1 + kt ,

(4.184)

References

129

3 z  is the unnormalized coherent state, we obtain ρ|z (t) as where 3 1+kt ∞  1 kt 1 1 (kt)m 2 † † e kt+1 |z| a †m ea a ln 1+kt |zz|ea a ln 1+kt a m m+1 m! (kt + 1) m 3 3 + ∞  1 3 z |z|2 z 3 (kt)m − kt+1 †m 3 3 am = e a 3 3 m+1 m! 1 + kt 1 + kt + 1) (kt m

ρ|z (t) =

∞  |z|2 1 a† z az ∗ (kt)m † e− kt+1 : a †m a m e 1+kt + 1+kt −a a : m+1 m! + 1) (kt m  

−1 2 1 |z| − a † z − az ∗ + a † a : . = : exp kt + 1 kt + 1

=

(4.185)

Using the IWOP method, we can prove that the state ρ|z (t) satisfies the normalization condition, that is 1 kt + 1 = 1.

trρ|z (t) =



 

−1 2 d2 β β| : exp |z| − a † z − az ∗ + a † a : |β π kt + 1 (4.186)

Now we compare the states ρ|z (t) with ρd in Eq. (4.171). Obviously, only when kt =

1 eν/(k B T )

−1

,

(4.187)

the pure coherent state ρ(0) evolves into the displaced thermal state ρd in the diffusion process.

References 1. Risken H (1996) The Fokker-Planck equation: methods of solutions and applications. Springer, New York 2. Gardiner CW (1983) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, Berlin 3. Haake F (1969) On a non-Markoffian master equation: II. Application to the damped oscillator. Zeitschrift für Physik A Hadrons and nuclei 223(4): 364–377 4. Agarwal GS, Wolf E (1970) Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators. Phys Rev D 2(10): 2161–2186 5. Schleich WP (2001) Quantum optics in phase space. Wiley-Vch, Berlin 6. Fan HY, Hu LY (2008) Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach. Mod Phys Lett B 22(25):2435–2468 7. Yao F, Wang JS, Xu TN (2015) Explicit solution of diffusion master equation under the action of linear resonance force via the thermal entangled state representation. Chin Phys B 24(7):070304

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4 Solutions of Density Operator Master Equations

8. Meng XG, Wang JS, Gao HC (2016) Kraus operator-sum solution to the master equation describing the single-mode cavity driven by an oscillating external field in the heat reservoir. Int J Theor Phys 55(8):3630–3636 9. Wu WF (2016) Infinitive operator-sum representation for damping in a squeezed heat reservoir via the thermo entangled state approach. Int J Theor Phys 55(12):5062–5068 10. Meng XG, Wang JS, Liang BL (2013) New approach for deriving the exact time evolution of the density operator for a disffusive anharmonic oscillator and its Wigner distribution function. Chin Phys B 22(3):030307 11. Chen XF, Hou LL (2015) Explicit Kraus operator-sum representations for time-evolution of Fermi systems in amplitude- and phase-decay process. Can J Phys 93(11):1356–1359 12. Meng XG, Wang JS, Fan HY, Xia CW (2016) Kraus operator solutions to a fermionic master equation describing a thermal bath and their matrix representation. Chin Phys B 25(4):040302 13. Meng XG, Fan HY, Wang JS (2018) Generation of a kind of displaced thermal states in the diffusion process and its statistical properties. Int J Theor Phys 57(4):1202–1209 14. Scully MO, Zubairy MS (1997) Quantum optics. Cambridge University Press, Cambridge 15. Restrepo J, Ciuti C, Favero I (2014) Single-polariton optomechanics. Phys Rev Lett 112(1):013601 16. Tana´s R, Miranowicz A, Kielich S (1991) Squeezing and its graphical representations in the anharmonic oscillator model. Phys Rev A 43(7):4014–4021 17. Chung NN, Chew LY (2007) Energy eigenvalues and squeezing properties of general systems of coupled quantum anharmonic oscillators. Phys Rev A 76(3):032113 18. Peixoto de Faria JG (2007) Time evolution of the classical and quantum mechanical versions of diffusive anharmonic oscillator: an example of Lie algebraic techniques. Eur Phys J D 42(1):153–162 19. Daniel DJ, Milburn GJ (1989) Destruction of quantum coherence in a nonlinear oscillator via attenuation and amplification. Phys Rev A 39(9):4628–4640 20. Oliveira AC, Peixoto de Faria JG, Nemes MC (2006) Quantum-classical transition of the open quartic oscillator: the role of the environment. Phys Rev E 73(4):046207 21. Chaturvedi S, Srinivasan V (1991) Class of exactly solvable master equations describing coupled nonlinear oscillators. Phys Rev A 43(7):4054–4057 22. Fan HY, Zaidi HR (1987) Application of IWOP technique to the generalized Weyl correspondence. Phys Lett A 124(6–7):303–307 23. Carvalho ARR, Mintert F, Palzer S, Buchleitner A (2007) Entanglement dynamics under decoherence: from qubits to qudits. Eur Phys J D 41(2):425–432 24. Ikram M, Li FL, Zubairy MS (2007) Disentanglement in a two-qubit system subjected to dissipation environments. Phys Rev A 75(6):062336 25. Ma J, Wang XG, Sun CP, Nori F (2011) Quantum spin squeezing. Phys Rep 509(2–3):89–165 26. Hellwig KE, Kraus K (1969) Pure operations and measurements. Commun Math Phys 11(3):214–220 27. Al-Qasimi A, James DFV (2008) Sudden death of entanglement at finite temperature. Phys Rev A 77(1):012117 28. Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press, Cambridge 29. Ouyang Y, Wang S, Zhang LJ (2016) Quantum optical interferometry via the photon-added two-mode squeezed vacuum states. J Opt Soc Am B 33(7):1373–1381 30. Carmichael HJ (1999) Statistical methods in quantum optics 1: master equations and FokkerPlanck equations. Springer, New York 31. Carmichael HJ (2008) Statistical methods in quantum optics 2: non-classical fields. Springer, Berlin

Chapter 5

Wigner Distribution Function and Quantum Tomogram via Entangled State Representations

The quasi-probability distribution functions (e.g., Wigner distribution function) in quantum mechanics have important applications in many fields of physics [1–3]. In particular, the partial negativity of Wigner distribution function clearly reflects the nonclassical properties of quantum states. At present, it has become an effective tool to study the nonclassical properties of quantum states in quantum optics. Moreover, there is a one-to-one correspondence between the quantum state and its Wigner distribution function. Therefore, the measurement of quantum states can be realized indirectly by reconstructing and measuring their Wigner distribution functions by virtue of some observable quantities in physics. So far, many methods of measuring quantum states with the reconstruction of Wigner distribution functions have been introduced in quantum optics, such as optical zero beat chromatography and quantum tomography. Experimentally, using the optical tomography technology, two-dimensional data of a three-dimensional object can be obtained, which can establish the slice image of the internal structure, so that the interior of the object can be peered noninvasively. Lately, several representative experiments with homodyne detection are completed in order to investigate the tomographic probabilities of single-mode quantum states and measure the precision of their quantum tomograms [4–6]. In quantum statistics, Vogel et al. pointed out that the probability distribution for the rotational quadrature phase X (θ) = Q cos θ + P sin θ (Q, P being respectively the coordinate and momentum operators) with the controlled angle θ as a parameter can be represented according to Wigner distribution function, and the reverse called Vogel-Risken relation is also true, that is, one can obtain the Wigner distribution function by tomographic inversion of a set of measured probability distributions, W (x, θ), of the quadrature amplitude [7]. Using the homodyne detection of single-mode quantum state |, the quantum tomogram M(x, θ) = |ϕ| x, θ|2 can be measured experimentally, where |x, θ is an eigenvector of the phase operator X (θ) with the exact eigenvalue x. And then, Smithey et al. stressed that, once the Wigner distribution function W (x, θ) is given, one can use the inverse Radon transformation in tomographic imaging to obtain

© Science Press 2023 X.-G. Meng et al., Entangled State Representations in Quantum Optics, https://doi.org/10.1007/978-981-99-2333-5_5

131

132

5 Wigner Distribution Function and Quantum Tomogram …

the Wigner distribution function of the quantum state and its density matrix. Thus, tomographic approach of quantum theory offers a description in terms of quantum tomographic probabilities [8]. In this chapter, via the IWOP method, the operator ordering method and several entangled state representations [9–11], we plan to review briefly the Wigner operator theory and quantum tomography of single-mode quantum states, and establish preliminarily the entangled state representation theory of two-mode Wigner operator and quantum tomography for two-mode cases. According to the entangled Wigner operator, Wigner distribution functions for several two-mode quantum states are investigated analytically and numerically [12–14]. Besides, using the Radon transformation between single- (two-) mode Wigner operator and the pure-state density operator |q, f, g q, f, g| (|η, κ1 , κ2  η, κ1 , κ2 |), quantum tomograms of several quantum states are discussed [15, 16].

5.1 Wigner Distribution Functions 5.1.1 Wigner Operator Theory In 1932, Wigner first proposed the concept of Wigner distribution function and used it for quantum correction of classical distribution function. The Wigner distribution function for quantum state is defined as a real function in phase space, which has the basic properties of quasi-probability distribution function. In terms of the definition of the Wigner operator [4]   ( p, q) =

∞

−∞

  t dt it p  e q + q− 2π 2

 t  2

(5.1)

in the coordinate representation, we can represent the Wigner distribution function for the single-mode quantum state |ϕ (or the corresponding density operator ρ = |ϕ ϕ|) as the average value of the operator  ( p, q) in the state ρ, i.e.,  W ( p, q) = tr [ρ ( p, q)] =

∞

−∞

 dt q− 2π

   t it p t   q + ρ e . 2  2

(5.2)

The basic properties of Wigner distribution function are (1) W ( p, q) is a real function in phase space, i.e., W ∗ ( p, q) = W ( p, q). (2) W ( p, q) has the meaning of probability distribution because 

∞ −∞

d pW ( p, q) = ψ ∗ (q)ψ(q),



∞ −∞

dqW ( p, q) = φ∗ ( p)φ( p),

(5.3)

5.1 Wigner Distribution Functions

133

where ψ ∗ (q)ψ(q), φ∗ ( p)φ( p) are respectively the probability distribution functions of particles in the coordinate space and momentum space. (3) W ( p, q) takes both positive and negative values, which means that it is a quasiprobability distribution function. Therefore, W ( p, q) cannot be simply regarded as the probability density of particles with coordinate q and momentum p at the same time, as in classical physics, because it completely violates the Heisenberg uncertainty relation. However, for the “quasi-classical” state, there always exists the relation W ( p, q) ≥ 0. For example, the Wigner distribution functions for coherent states always yield W ( p, q) ≥ 0 in phase space and present the standard Gaussian wave packets. Indeed, the partial negativity of Wigner distribution function W ( p, q) provides an effective tool to explore the nonclassicality of light field. In theory, the formula for calculating the negative volume of the function W ( p, q) is defined as [17]  ∞ 1 dqd p[|W ( p, q)| − W ( p, q)], (5.4) δ= 2 −∞ which means that, once the Wigner distribution function W ( p, q) for a quantum state is known, we easily obtain its negative volume δ by integrating over q and p in the whole phase space. Next, we use the IWOP method to derive several common expressions of Wigner operator. First, using the IWOP method to complete the integration (5.1) leads to the normal ordering product of the operator  ( p, q), that is   ( p, q) =

∞

−∞

dt − 1 (q+ t )2 +√2(q+ t )a † − a†2 2 2 2 e 2 2π 3/2 √

1 t 2 t a |0 0| e− 2 (q− 2 ) + 2(q− 2 )a− 2 ei pt 2  ∞ √ 2 (a+a † )2 e−q † − t ++ √t 2 (a † −a) = dt : e 4 e 2q(a+a )− 2 : ei pt 3/2 2π −∞ 1 −(q−Q)2 −( p−P)2 = :e :. π 2

(5.5)

† Further, using the relation between √ the operators Q, P and the Boson operators a, a and the identity α = (q + i p) / 2 in Eq. (1.49), as well as the integral formula in Eq. (1.30), we can rewrite Eq. (5.5) as

 2 d z − |z|2 +z ∗ (α−a)−z (α∗ −a † ) 1 −2(α∗ −a † )(α−a) : : = :e 2  ( p, q) = : e π 2π 2  2   d z z ∗ (α−a)−z (α∗ −a † ) = : ≡  α, α∗ :e 2π 2 or

(5.6)

134

5 Wigner Distribution Function and Quantum Tomogram …

  1 † ∗  α, α∗ = : e−2(a −α )(a−α) : π  2  d z : exp − |z|2 + (α + z)a † + (α∗ − z ∗ )a = 2 π −a † a + αz ∗ − α∗ z − |α|2 : .

(5.7)

Using the normal ordering form of the projection operator of vacuum in Eq. (1.7) 2 † and the exponential expansion of the coherent state |z, that is |z = e−|z| /2 eza |0, we obtain the coherent state representation of Wigner operator as 

 α, α or

∗





d2 z ∗ ∗ |α + z α − z| eαz −α z π2

=

  2  α, α∗ = e2|α|



d2 z ∗ ∗ |z −z| e−2(zα −z α) . π2

(5.8)

(5.9)

Taking  (α, α∗ ) as an integrand function and integrating over α, we have the completeness relation of  (α, α∗ ), i.e.,  2



d α α, α 2

∗



2 = π



d2 α : e−2(α

∗

−a † )(α−a)

: = 1.

(5.10)

  So, using the completeness relation in Eq. (5.10), any operator functions F a † , a can be expanded as   F a†, a = 2



    d2 α f α∗ , α  α, α∗ ,

(5.11)

  which is exactly the Weyl correspondence between the operator function F a † , a and its classical function f (α∗ , α). Further, letting α = 0 in Eq. (5.6), we have  (0, 0) =

1 −2a † a (−1) N :e , : = π π

(5.12)

where N = a † a is the particle-number operator. Thus, using the displaced operator † ∗ D(α) = eaa −α α , the Wigner operator  (α, α∗ ) reads   1  α, α∗ = D(α) (0, 0) D † (α) = D(α)(−1) N D † (α) π 1 1 2 † ∗ = e2αa (−1) N e2α α e−2|α| = D(2α)(−1) N . π π

(5.13)

If we take the Wigner operator  ( p, q) as an integrand function and using the IWOP method to integrate over q, p, we find that the Wigner operator  ( p, q) yields the

5.1 Wigner Distribution Functions

normalization condition



135 ∞ −∞

dqd p ( p, q) = 1

(5.14)

and has the following two marginal distributions 

∞

 1 dq ( p, q) = √ : exp − ( p − P)2 : = | p  p| , π −∞  ∞  1 d p ( p, q) = √ : exp − (q − Q)2 : = |q q| , π −∞

(5.15)

where the states |q, | p are respectively the eigenstates of the coordinate operator Q and momentum operator P. Thus, for any single-mode quantum state |ϕ, two marginal distributions of its Wigner distribution function W ( p, q) are, respectively, 

∞ −∞

 dqW ( p, q) = | p| ϕ|2 ,

∞

−∞

d pW ( p, q) = |q| ϕ|2 ,

(5.16)

which refer to the probability distributions of particles with the state |ϕ in the q − p phase space. On the other hand, based on the continuous-variable entangled state representations in Eqs. (1.72) and (1.91), we can establish the entangled Wigner operator theory for two-mode quantum states. Using the completeness relations in Eqs. (1.77) and (1.94), and the inner product in Eq. (1.95), thus any two-mode operators H can be represented as [10]   2 2 d η d ζ           d2 η d2 ζ η η ζ ζ H ζ ζ η η 2 π π2   2 2   2 2 1 d η d ζ    d ηd ζ η η = 2 4 π π2    1 ∗ ∗ ∗ ∗ × ζ  H ζ e 2 (ζ η −ζ η +ζ η −ζ η ) . (5.17) 

H=

Making the following variable substitutions η = σ − η, η = σ + η, σ = σ1 + iσ2 , ζ = γ − ζ, ζ = γ + ζ, γ = γ1 + iγ2 , d2 η d2 η = 4d2 σd2 η,

d 2 ζ d2 ζ = 4d2 γd2 ζ,

(5.18)

d2 ηd2 ζd2 σd2 γ |σ − η σ + η| π4 ∗ ∗ ∗ ∗ γ − ζ| H |γ + ζ e(ηγ −η γ)+(ζ σ−ζσ ) .

(5.19)

thus Eq. (5.17) can be simplified as 

H=4

136

5 Wigner Distribution Function and Quantum Tomogram …

Defining the following correspondences 

and

 4

d2 η ∗ ∗ |σ − η σ + η| eηγ −η γ ≡ (σ, γ) 3 π

(5.20)

d2 ζ ∗ ∗ γ − ζ| H |γ + ζ eζ σ−ζσ ≡ h(σ, γ), π

(5.21)

then the correspondence between the operator function H and its classical function h(σ, γ) in the entangled state representation can be represented as  H= d2 σd2 γ(σ, γ)h(σ, γ),

(5.22)

where (σ, γ) is named as the entangled state representation of two-mode Wigner operator, and h(σ, γ) = 4π 2 tr[H(σ, γ)]. (5.23) Using the IWOP method to calculate the integration in Eq. (5.20), and making the substitutions γ = α + β ∗ and σ = α − β ∗ , thus the entangled Wigner operator (σ, γ) can be rewritten as the direct product of two single-mode Wigner operators a (α, α∗ ) and b (β, β ∗ ), i.e., 1 : exp[−2(α∗ − a † )(α − a) − 2(β ∗ − b† )(β − b)] : π2 = a (α, α∗ )b (β, β ∗ ).

(σ, γ) =

(5.24)

So, the operator (σ, γ) can be viewed as the nature representation of two-mode Wigner operator in the entangled state |η representation. Moreover, the entangled Wigner operator (σ, γ) also has the general properties of Wigner operator. For instance, its marginal distributions give the probability density distributions of finding two particles in an entangled way. In terms of Eq. (1.74), the normal ordering product of the project operator |η η| is given by   

Qa − Qb 2 Pa + Pb 2 |η η| = : exp − η1 − − η2 − √ :. √ 2 2

(5.25)

Similarly, using Eq. (1.91), the normal ordering product of the project operator of the state |ζ conjugate with the entangled state |η is   

Qa + Qb 2 Pa − Pb 2 |ζ ζ| = : exp − ζ1 − − ζ2 − √ :. √ 2 2

(5.26)

5.1 Wigner Distribution Functions

137

So, similar to defining the Wigner operator  ( p, q) in Eq.  in terms of Eq. (5.15),  (5.5) the normal ordering product of the Wigner operator  σ, γ of two-mode entangled system can be constructed as  

1 Qa − Qb 2 Pa + Pb 2 − σ2 − √  σ, γ = 2 : exp − σ1 − √ π 2 2 2 2 

Qa + Qb Pa − Pb − γ1 − − γ2 − √ :. √ 2 2 



(5.27)

  Taking the operator  σ, γ as an integrand function and integrating over γ, we have 

  1 (5.28) d2 γ σ, γ = |η η|η=σ . π   Similarly, integrating over σ in the operator  σ, γ leads to another projection operator    1 (5.29) d2 σ σ, γ = |ζ ζ|ζ=γ . π For a two-mode entangled state|φ with the density   operator ρ (= |φ φ|), its Wigner distribution function reads W σ, γ = tr[ρ σ, γ ], thus its two marginal distributions are respectively  π

  d γW σ, γ = |η| φ|2η=σ , 2

 π

  d2 σW σ, γ = |ζ| φ|2ζ=γ ,

(5.30)

where |η| φ|2η=σ (or |ζ| φ|2ζ=γ ) represents the probability for finding the two par√ √ 2σ2 (or relative momentum ticles that have total momentum √ 2γ2 ) and simulta√ neously relative position 2σ1 (or center-of-mass position 2γ1 ) in the entangled state |φ. So, we can say that, for an entangled   particle system, the physical meaning of the Winger distribution function W σ, γ lies in that its marginal distributions give the probabilities of finding the entangled particles in the σ-γ phase space.

5.1.2 Wigner Distribution Functions for Quantum States 5.1.2.1

Multi-photon Subtracted Squeezed Vacuum States

Theoretically, the normalized squeezed vacuum state is defined as [12]

S1† (r ) |0 = sech1/2 r exp

 1 †2 a tanh r |0 , 2

(5.31)

138

5 Wigner Distribution Function and Quantum Tomogram …

where S1 (r ) = er (a −a )/2 is single-mode squeezed operator in Eq. (1.25). Repeatedly operating the annihilation operator a on a squeezed vacuum, we give the unnormalized multi-photon subtracted squeezed vacuum state as 2

†2

a m S1† (r ) |0 ≡ |r, m .

(5.32)

. . a †m a m = ..Hm,m (a † , a)..,

(5.33)

Note that the operator identity

where Hm,m (a † , a) is a two-variable Hermite polynomial related to the creation operator a † and annihilation operator a, thus the normalization factor of the state |r, m reads 



.. . 1 †2 1 2 † a tanh r Hm,m (a , a) exp a tanh r .. |0 . (5.34) Nm = sech r 0| . exp 2 2 Inserting the completeness relation of coherent states |z into Eq. (5.34), and using the generating functions for Hm,n (x, y) and the integral formula in Eq. (2.106), we therefore have    ∂ 2m 1 2 2 2 exp (τ + s ) sinh 2r + τ s sinh r  . (5.35) Nm = m m ∂τ ∂s 4 τ =s=0 Further, using the following differential result  ∂ 2m exp (−t 2 − τ 2 + 2xτ t)t=τ =0 m m ∂t ∂τ ∞   (−1)n+l ∂ 2m (2x)k m m τ 2n+k t 2l+k t=τ =0 = n!l!k! ∂t ∂τ n,l,k=0 [m/2]

= 2m m!



m!

n=0

22n (n!)2 (m

− 2n)!

x m−2n

(5.36)

and the new expression of Legendre polynomials [m/2]

x

m

 n=0

 1 n m! 1− 2 = Pm (x), 22n (n!)2 (m − 2n)! x

(5.37)

we thus arrive at the normalization factor of the state |r, m, which is of the form Nm = m!(−i sinh r )m Pm (i sinh r ).

(5.38)

5.1 Wigner Distribution Functions

139

Particularly, for m = 0, i.e., the case of no photon subtraction, N0 = 1, as expected; for m = 1, corresponding to single-photon subtracted squeezed vacuum, N1 = sinh2 r . It is noted that, the normalization factor Nm is related to the mth-order Legendre polynomial, so it can provide help for analytically studying the nonclassicality of the state |r, m. Using the coherent state representation of single-mode Wigner operator in Eq. (5.8), the Wigner distribution function for the state |r, m is obtained as   Wm (α, α∗ ) = Nm−1 0| S1 (r ) a †m  α, α∗ a m S1† (r ) |0 ,

(5.39)

where  (α, α∗ ) ≡  ( p, x) for α = (x+ i p). Using the Bogoliubov transformations in Eq. (1.28), we can rewrite the state |r, m as a m S1† (r ) |0 = S1† (r ) d m |0 ,

(5.40)

where d = a cosh r + a † sinh r . Thus, the Wigner operator  ( p, x) is directly sandwiched in between S (r ) · · · S1† (r ), and further using the following identities S1 (r ) Q S1† (r ) = er Q, S1 (r ) P S1† (r ) = e−r P,

(5.41)

where Q, P are respectively the coordinate and momentum operators, we obtain the squeezed Wigner operator as    er p, e−r x =



d2 z ∗ ∗ |β + z β − z| eβz −zβ , 2 π

(5.42)

where |β ± z are also coherent states, β = (e−r x+ ier p) = α cosh r − α∗ sinh r . Using Eqs. (5.40) and (5.42), Eq. (5.39) can be expressed as ∗

Wm (α, α ) =

Nm−1

 0| d

†m

d2 z ∗ ∗ |β + z β − z| eβz −zβ d m |0 . 2 π

(5.43)

Further, in order to obtain the function Wm (α, α∗ ), we need to calculate the normal ordering product of d m . For this reason, we introduce the state |q, f, g with the parameters f and g [18], i.e.,   |q, f, g = c exp c1 a † − c2 a †2 |0 ,

(5.44)

where the parameters c, c1 and c2 are respectively 1

c=    1/4 π f 2 + g2

 exp −

√  2q q2 , c1 = , 2( f 2 + g 2 ) f − ig

c2 =

f + ig . 2( f − ig) (5.45)

140

5 Wigner Distribution Function and Quantum Tomogram …

Using the relation between the coordinate operator Q (or momentum operator P) and bosonic operators a, a † in Eq. (1.49), we find that the state |q, f, g is just the eigenstate of the operator f Q + g P with the eigenvalue q, that is ( f Q + g P) |q, f, g = q |q, f, g .

(5.46)

Moreover, using the IWOP method, we can obtain the completeness relation of the states |q, f, g as 

∞

dq |q, f, g q, f, g|    ∞ 1 1 2 dq : exp − − ( f Q + g P)] =  : [q  1/2 f 2 + g2 −∞ π f 2 + g2 −∞

= 1,

(5.47)

which shows that the set of the states |q, f, g can form a complete quantum mechanics representation. Especially, when ( f, g) = (1, 0) (or ( f, g) = (0, 1)), the states |q, f, g become the eigenstates of the coordinate operator Q (or momentum operator P). So, we name the states |q, f, g as the intermediate coordinate-momentum states. Inserting the completeness relation of the state |q, f, g into the operator ( f Q + g P)n and using the IWOP method, we obtain the new operator identify as

 

 n    ρν ρ ν † † n (5.48) a+i a :, = −i : Hn i ρa + νa 2 2ν 2ρ which leads to the identity 



sinh 2r d |0 = −i 4



m

m

Hm

 tanh r † a |0 . i 2

(5.49)

After a direct calculation, we finally obtain Wm (α, α∗ ) =

2 m (m!)2 e−2|β| sinhm 2r  2n (− tanh r )n π4m Nm n! [(m − n)!]2 n=0

 2 √   Hm−n (−i 2 tanh r β) ,

(5.50) which is the analytical formula of the Wigner distribution function for the state √ |r, m, related to the module-square of Hm−n (−i 2 tanh r β). In particular, when m = 0, the function Wm (α, α∗ ) becomes Gaussian and positive, i.e., W0 (α, α∗ ) = 2 π −1 e−2|β| , corresponding to squeezed vacuum. For m = 1, corresponding to singlephoton subtracted squeezed vacuum, thus Eq. (5.50) reduces to W1 (α, α∗ ) =

 1 2 4|β|2 − 1 e−2|β| . π

(5.51)

5.1 Wigner Distribution Functions Wm ( , *)

141

Wm ( , *)

Wm ( , *) (b )

(a)

Re( )

Im( )

Wm ( , *)

(c )

Re( )

Im( )

Wm ( , *) ( e)

(d )

Re( )

Im( )

Wm ( , *)

(f)

Re( )

Im( )

Re( )

Im( )

Re( )

Im( )

Fig. 5.1 Wigner distribution functions for multi-photon subtracted squeezed vacuum states for a m = 1, r = 0.3; b m = 2, r = 0.3; c m = 5, r = 0.3; d m = 8, r = 0.3; e m = 2, r = 0.8; f m = 5, r = 0.8

Clearly, when |β|2 < 1/4, the function W1 (α, α∗ ) is always negative in the whole phase space [19]. In Fig. 5.1, we show the Wigner distribution functions for the states |r, m for several different values of the photon-subtraction number m and squeezing r . Clearly, squeezing in one of the quadratures can be always seen in these figures. For the large squeezing r , the peaks are not only further compressed along the Im(α) direction, but also can show the large negative region. For the large m, the multi-peak structure halfway between the two adjacent main peaks representing quantum interference is shown, and the interference frequency increases with the increase of m. This comes from mth-order photon-subtraction operation in the state |r, m since the function Wm (α, α∗ ) is completely different from the Gaussian distribution of W0 (α, α∗ ). Besides, the negativity is always shown in the whole phase space when m = 0, but the negativity cannot monotonically change with increasing m. This is because W0 (α, α∗ ) is always positive, while the negativity of the function Wm (α, α∗ ) fully depends on the successful photon-subtraction operations.

5.1.2.2

Two-variable Hermite Polynomial States

In theory, the single-variable Hermite polynomial state |m,  is defined as the solution of the following eigenvalue problem [20] (Y1 + iY2 ) |m,  = α |m,  ,

(5.52)

where the eigenvalue α and the parameters  and m yield the relation α = (2 − 1)1/2 (m + 1/2),   1, and m is a non-negative integer, Y1 = (a 2 + a †2 )/2 and Y2 = (a 2 − a †2 )/(2i) are the two components that describe the amplitude-squared squeezing and yield the uncertainty relation Y1 Y2 ≥ N + 1/2. It is proved that

142

5 Wigner Distribution Function and Quantum Tomogram …

state |m,  is just the minimum uncertainty state for the components Y1 and Y2 , and its explicit form reads |m,  = Cm ()S1† (r ) Hm (iξ()a † ) |0 ,

(5.53)

where the normalized factor Cm () is [Cm ()]

−2

[m/2]

=

 (m!)2 [4 |ξ()|2 ]m−2n , (m − 2n)! (n!)2 n=0

(5.54)

  Hm iξ()a † is the mth-order Hermite polynomial of creation operator a † , ξ() = [(2 − 1)1/2 /(2)]1/2 . Indeed, this state |m,  is great physical significance because it can be experimentally generated by a degenerate parametric amplifier. Especially, the state |1,  can be produced via acting a one-photon number state as the input for a degenerate parametric amplifier because it is a squeezed one-photon state in nature. As an important generalization of amplitude-squared squeezing, the concept of sum-frequency squeezing for the components Z 1 and Z 2 is introduced, where the components Z 1 and Z 2 read [21] Z1 =

1 † † i (a b + ab), Z 2 = (a † b† − ab), 2 2

(5.55)

which obey the uncertainty relation Z 1 Z 2 ≥

1 N1 + N2 + 1 . 4

(5.56)

If a certain quantum state obeys the condition (Z i )2
T 2 tanh2 r , such that e−2κt < 1, 1 − T 2 tanh2 r

(6.81)

which means that the squeezing amount of the initial squeezed vacuum states gradually decreases as the decay time κt goes on. Especially, suppose at κt = 0, thus T = 0, ß1 = sech2 r , ß2 = −1, and ß3 = tanh r , Eq. (6.77) becomes the normal ordering of the density operator of two-mode squeezed vacuum. For κt → ∞, note that T → 1, ß1 = −ß2 → 1, ß3 → 0, thus the state ρ (t) fully loses its entanglement and squeezing, and finally decays to vacuum, as expected. For a two-mode squeezed vacuum, its photon number distribution reads p (m, n, t) = tr[ρ (t) |m, n m, n|]. Using the normal ordering product of ρ (t) in Eq. (6.77) and the coherent state representation of two-mode number state |m, n, i.e., ∂ m+n 1 |α, β|m=n=0 , |m, n = √ (6.82) m!n! ∂αm ∂β n we thus have ß1 ∂ 2(m+n) exp {(ß2 + 1) m!n! ∂αm ∂β n ∂α∗m ∂β ∗n  ∗     × αα + ββ ∗ + ß3 αβ + α∗ β ∗ α,β,α ,β  =0 .

p (m, n, t) =

(6.83)

Performing the multiple-order differential operations and using the range of exponentials, such that

184

6 Evolution and Decoherence of Quantum States in Open Systems

∂ 2(m+n) ß1 n!m! ∂αm ∂β n ∂α∗m ∂β ∗n

p (m, n, t) =

(ß2 + 1) l!k! j! p!

l+k

×

= ß1

min(m,n)  p=0

∞ 

j+ p

ß3

l, j,k, p=0

  l+ p  ∗ k+ p   β αl+ j β j+k α∗ 

α,β,α ,β  =0

2p

m!n! (ß2 + 1)m+n−2 p ß3 . ( p!)2 (m − p)! (n − p)!

(6.84)

Without loss of generality, letting m  n, via comparing Eq. (6.84) with the standard (α,β) definition of Jacobi polynomials Pm (·), we can change Eq. (6.84) into the following compact expression m+n −2(m−1)κt e sech2 r p (m, n, t) = ßn+1 3 T

2  2 2m+n−1 (0,n−m) 1 + T tanh r r Pm × tanh , 1 − T 2 tanh2 r (α,β)

(6.85)

(β,α)

where the relation Pm (−x) = (−1)m Pm (x) can be used. Equation (6.85) shows that the analytical evolution of two-mode squeezing vacuum for amplitude damping is just related to the Jacobi polynomials. Next, using the power-series expansion in Eq. (6.84) to analyze several special cases as below. Suppose at some give decay time, say at κt = 0 (i.e., T = 0), thus Eq. (6.84) becomes the photon number distribution of two-mode squeezed vacuum, i.e., p (m, n, 0) = sech2 r lim

T →0

 =

min(m,n)  p=0

m!n! (tanh r )2m+2n−2 p T m+n−2 p ( p!)2 (m − p)! (n − p)!

sech r tanh r, m = n . 0, m = n 2

2n

(6.86)

Equation (6.86) shows that the photon number distribution is not zero only for m = n, and decreases with the increase of the squeezing r . For the limit case, i.e., κt → ∞ (T → 1), thus p (m, n, ∞) → 0, which means that the system no longer has any photons and becomes vacuum after a long time interaction with the amplitude damping, same as the result of the density-operator evolution for κt → ∞. In Fig. 6.3, the evolution of the photon number distribution of two-mode squeezed vacuum for amplitude damping is shown in the Fock space (m, n) for different values of r and κt. Obviously, for small squeezing r , the probabilities of finding the smaller (m, n) numbers of photons in the output mixed state are larger for small squeezing r , but slowly decreases as the squeezing r increases, that is, more photons appear the initial twoin larger number states. This is because the coefficient tanhl r of † † ∞ tanhl r |ll mode squeezed vacuum as shown in the expansion ea b tanh r |00 = l=0 possesses a larger weight to the lager-number photon component. Moreover, the

6.1 Evolution of Quantum States in the Amplitude Damping Channel

185

Fig. 6.3 Evolution law of photon number distribution of two-mode squeezed vacuum in the amplitude damping channel for a r = 0.5, κt = 0.1; b r = 0.9, κt = 0.1; c r = 0.9, κt = 3

probabilities of finding any photon numbers (m, n) excluding (0, 0) decreases with the parameter κt. Besides, for κt → ∞, the probability of finding (0, 0) is equal to one and the others become zero, which results from amplitude damping, same as the above analytical result. Using the normal ordering product of the density operator ρ (t) in Eq. (6.77) and the two-mode Wigner operator (α, β) in the coherent state representation, we obtain the Wigner distribution function for two-mode squeezing vacuum in the amplitude damping channel, that is

  d2 γd2 ζ exp ß3 ζ − 2α∗ γ − B |γ|2 W (α, β; t) = ß1 e 4 π     + ß3 ζ ∗ + 2α γ ∗ − B |ζ|2 − 2 ζβ ∗ − ζ ∗ β , (6.87) 2(|α|2 +|β|2 )

where γ, ζ are the complex amplitudes of two-mode coherent states |γ, ζ, and B=

1 + (1 − 2T ) T tanh2 r . 1 − T 2 tanh2 r

(6.88)

Further, using the integral formula in Eq. (1.30), we directly give

 ß1 4 |ß3 α − Bβ ∗ |2 2(2 − B) |α|2 − 2B |β|2  exp −  −  W (α, β, t) = 2  2 . B π B − ß23 B B 2 − ß23 (6.89) Clearly, the amplitude damping cannot change the Gaussianity of two-mode squeezed vacuum. Specially, when κt = 0 and κt → ∞, Eq. (6.89) becomes, respectively, W (α, β, 0) = and

1 −2(|α|2 +|β|2 ) cosh 2r +2(βα+β ∗ α∗ ) sinh 2r e π2

W (α, β, ∞) → π −2 e−2(|α|

2

+|β|2 )

,

(6.90)

(6.91)

186

6 Evolution and Decoherence of Quantum States in Open Systems

Fig. 6.4 The evolved Wigner distribution function W (α, β, t) for two-mode squeezed vacuum in the amplitude damping channel for a r = 0.5, κt = 0.1; b r = 0.9, κt = 0.1; c r = 0.9, κt = 3

which are, respectively, the Wigner distribution functions for two-mode squeezed vacuum and vacuum. In Fig. 6.4, the evolution of the Wigner distribution function for two-mode squeezed vacuum in the amplitude damping channel is plotted as a function Re α and Re β for different values of r and κt. Clearly, squeezing in one of the quadratures always occurs, as a feature of nonclassicality of this state. Also, the peak is further compressed along the diagonal direction as the squeezing r increases, which means that the larger squeezing can lead to the slower decay of nonclassicality and the longer decay time. However, squeezing can deteriorate rapidly as the decay time κt goes on. So, we say that squeezing can only suppress decoherence to a certain extent. For κt → ∞, the function W (α, β, t) finally becomes the standard Gaussian distribution belonging to vacuum as expected. Next, we plan to investigate how the marginal distributions of the function W (α, β, t) evolves in the amplitude damping channel. For this purpose, substituting Eq. (6.77) into the formula in Eq. (1.36), and using the inner product 2 2 −α, −β| α, β = e−2(|α| +|β| ) , we first derive the anti-normal ordering product of the density operator ρ(t), that is   .    . ρ(t) = h 1 .. exp h 2 a † a + b† b + h 3 ab + a † b† ..,

(6.92)

  h 1 = Cß1 , h 2 = C (ß2 + 1) ß2 − ß23 , −1  h 3 = Cß3 , C = (ß2 + 1)2 − ß23 .

(6.93)

where

Thus, using Eq. (6.92) and the completeness relation of coherent states |α, β, we thus obtain 2 2   d αd β 2 η| ρ (t) |η = h 1 e−|η| exp η ∗ α − ηβ + ηα∗ − η ∗ β ∗ 2 π      × exp (h 2 − 1) |α|2 + |β|2 + (h 3 + 1) αβ + α∗ β ∗ = κ(h 1 , h 2 , h 3 )e(h 1 ,h 2 ,h 3 )|η| , 2

(6.94)

6.1 Evolution of Quantum States in the Amplitude Damping Channel

187

where h2 − h3 . h3 − h2 + 2 (6.95) Further, using Eq. (5.28), the compact form of the evolved marginal distribution of the function W (σ, γ, t) in the “σ-direction” is therefore given by κ(h 1 , h 2 , h 3 ) =



h1 , (h 2 + h 3 ) (h 2 − h 3 − 2)

(h 1 , h 2 , h 3 ) =

d2 γW (σ, γ, t) = π −1 κ(h 1 , h 2 , h 3 )e(h 1 ,h 2 ,h 3 )|σ| . 2

(6.96)

Similarly, in terms of Eq. (5.29), the evolved marginal distribution of the function W (σ, γ, t) in the “γ-direction” reads 2 (6.97) d2 σW (σ, γ, t) = π −1 κ(h 1 , h 2 , −h 3 )e(h 1 ,h 2 ,−h 3 )|γ| . Finally, using Eqs. (5.112) and (5.121), the evolution of the optical tomogram of two-mode squeezed vacuum in the amplitude damping channel can be expressed as  . M (η, τ1 , τ2 ; t) = h 1 η, τ1 , τ2 | h 1 .. exp h 2 (a † a + b† b)   . +h 3 ab + a † b† .. |η, τ1 , τ2  .

(6.98)

Substituting the completeness relation of coherent states |α, β into Eq. (6.98) and using the integral formula in Eq. (2.106) to integrate α, β, respectively, we thus have  h 1 g2 (h 2 − 1) |g2 |2 M (η, τ1 , τ2 ; t) =   exp −  c1 c1 c22 − 4 |c4 |2  

c1 g1 − g22 g5∗ c2 |c3 |2 c2 c∗ , +2 Re + 2 3 4 2 − 2 c1 c2 − 4 |c4 | c2 − 4 |c4 |2

(6.99)

where the parameters c1 , c2 , c3 , and c4 are, respectively,   (h 2 − 1) c1 − |g4 + h 3 |2 c2 = , c1 = (h 2 − 1) − 4 |g5 | , c1    2  c1 g3∗ − g4∗ + h 3 g2 (h 2 − 1) + 2c1 g2∗ g5 c1 g ∗ + g5 g4∗ + h 3 c3 = , c4 = − 5 . c1 c1 (6.100) 2

2

Particularly, at κt = 0, owing to h 1 → − sinh−2 r, h 2 → 1, h 3 → − tanh−1 r , thus Eq. (6.99) becomes the optical tomogram for two-mode squeezed vacuum. However,

188

6 Evolution and Decoherence of Quantum States in Open Systems

for the limit case, κt → ∞, Eq. (6.99) reduces to the optical tomogram of vacuum, that is, M (η, τ1 , τ2 ; ∞) = g 2 e2g1 , a typical Gaussian distribution.

6.2 Evolution of Quantum States in the Laser Process In the field of optics, laser noise is also an important source of decoherence. When a quantum state passes through a laser channel, the evolution of its density operator under the lowest order approximation yields the following master equation [1, 8, 9], that is   dρ (t) = g 2a † ρ (t) a − aa † ρ (t) − ρ (t) aa † dt   + κ 2aρ (t) a † − a † aρ (t) − ρ (t) a † a ,

(6.101)

where the parameters g and κ are, respectively, the cavity gain and loss of the laser. Especially, when g = 0, Eq. (6.101) becomes the master equation for amplitude damping in Eq. (6.1). However, when g → κ¯n and κ → κ (¯n + 1), n¯ denoting the mean thermal photon number of the environment, thus Eq. (6.101) becomes the master equation describing the thermal environment, i.e.,   dρ (t) = κ¯n 2a † ρ (t) a − aa † ρ (t) − ρ (t) aa † dt   + κ (¯n + 1) 2aρ (t) a † − a † aρ (t) − ρ (t) a † a ,

(6.102)

Similar to solving the master equation in Eq. (6.1), operating simultaneously both sides of Eq. (6.102) on the entangled state |I  in Eq. (1.104), we have     d |ρ (t) = [g 2a † a˜ † − aa † − a˜ a˜ † + κ 2a a˜ − a † a − a˜ † a˜ ] |ρ (t) , dt

(6.103)

the formal solution of Eq. (6.103) is thus given by     |ρ (t) = exp[gt 2a † a˜ † − aa † − a˜ a˜ † + κt 2a a˜ − a † a − a˜ † a˜ ] |ρ (0)      = exp[t (κ + g) a˜ − a † a − a˜ † + t (κ − g) a a˜ − a˜ † a † + 1 ] |ρ (0) . (6.104) Using the following commutative relation,        a a˜ − a˜ † a † , a † − a˜ a − a˜ † = 2 a † − a˜ a − a˜ † and the operator formula (4.31), we then find

(6.105)

6.2 Evolution of Quantum States in the Laser Process

   κ+g |ρ (t) = exp a a˜ − a˜ † a † + 1 (κ − g) t + 2 (κ − g)     × 1 − e2(κ−g)t a † − a˜ a − a˜ † |ρ (0) .

189

(6.106)

So, in the entangled state χ| representation, the evolution law of the density operator ρ (t) is

    (κ + g) 1 − e−2(κ−g)t 2 |χ| χe−(κ−g)t |ρ (0) , χ |ρ (t) = exp − 2 (κ − g)

(6.107)

where we have used the eigenvalue equations in Eq. (1.100). It can be clearly seen −κt from Eq.(6.107) that and gain factor egt exist simultaneously in  the loss factor e −(κ−g)t  . Further, using the IWOP method and the completeness relation the state χe of the entangled state |χ representation, we therefore obtain

d2 χ |χ χ |ρ (t) π      = T3 exp gT1 a † a˜ † : exp (T2 − 1) a˜ † a˜ + a † a : exp (κT1 a a) ˜ |ρ (0) , (6.108)

|ρ (t) =

where T1 =

1 − e−2(κ−g)t (κ − g) e−(κ−g)t , T = , T3 = 1 − gT1 . 2 κ − ge−2(κ−g)t κ − ge−2(κ−g)t

(6.109)

Using the operator identities in Eqs. (1.21) and (1.103) to rewrite Eq. (6.108) as      |ρ (t) = T3 exp gT1 a † a˜ † exp a˜ † a˜ + a † a ln T2 exp (κT1 a a) ˜ ρ (0) |I  = T3

∞      κi g j i+ j † j T1 a exp a † a ln T2 a i ρ (0) a †i exp a † a ln T2 a j |I  . i! j! i, j=0

(6.110) Further, depriving of the state |I  from both sides of Eq. (6.110), thus the infinitive operator-sum representation of ρ(t) is the form ρ (t) = T3

∞ 

κi g j

2j i, j=0 i! j!T2

i+ j

T1

    exp a † a ln T2 a † j a i ρ (0) a †i a j exp a † a ln T2 (6.111)

or ρ (t) =

∞  i, j=0

Mi, j ρ (0) Mi,† j ,

(6.112)

190

6 Evolution and Decoherence of Quantum States in Open Systems

where Mi, j is the Kraus operator corresponding to the density operator ρ (t) in the laser channel, i.e., Mi, j =

T3

κi g j

i+ j

2j i! j!T2

T1

  exp a † a ln T2 a † j a i .

(6.113)

Now let’s prove the normalization of the Kraus operator Mi, j . Using the Glauber operator formula [A,B] [B,A] (6.114) e A+B = e A e B e− 2 = e B e A e− 2 , subject to the relation [[A, B], A] = [[A, B], B] = 0, we arrive at     2j exp 2a † a ln T2 a † j exp −2a † a ln T2 = T2 a † j .

(6.115)

On the other hand, a direct comparison of 

†

†







et a eta = eta et a ett = : et a+ta =

∞ 

†

+tt 

:

 n

(−it) (−it ) : Hm,n (ia † , ia) : m!n! m,n=0

(6.116)

∞  t m t n n †m a a m!n! m,n=0

(6.117)

m

and 

†

et a eta =

leads to the following operator identity:   a n a †m = (−i)m+n : Hm,n ia † , ia : ,

(6.118)

it then follows from Eqs. (6.113) and (6.118) that ∞  i, j=0

∞  κi g j i+ j T a †i (−1) j 2i 1 i! j!T 2 i, j=0  †  i   × : H j, j ia , ia : a exp 2a † a ln T2 .

Mi,† j Mi, j = T3

(6.119)

Further, using the relation between Laguerre polynomials Lm (x y) and two-variable Hermite polynomials Hm,m (x, y), which is of the form Hm,m (x, y) = (−1)m m!Lm (x y) , and the generating functions for Laguerre polynomials Lm (x), i.e.,

(6.120)

6.2 Evolution of Quantum States in the Laser Process

(1 − z)

−1

xz exp z−1

 =

191 ∞ 

Lm (x) z m ,

(6.121)

m=0

we therefore arrive at ∞  i, j=0

Mi,† j Mi, j

∞    κi g j i+ j †i i  †  = T3 T : a a L j −a a : exp 2a † a ln T2 2i 1 i!T 2 i, j=0 ∞ ∞       κi i †i i T : a a (gT1 ) j L j −a † a : exp 2a † a ln T2 2i 1 i!T2 i=0 j=0



†    gT1 a † a T3 κT1 a a exp : exp 2a † a ln T2 . = : exp 2 1 − gT1 1 − gT1 T2 (6.122)

= T3

Finally, using the operator identity in Eq. (1.21), we can prove ∞ 

Mi,† j Mi, j = 1,

(6.123)

i, j=0

which is just the normalization condition of the Kraus operator Mi, j . For a single-mode quantum state, its Wigner distribution function is defined as † W (α, α∗ ) = tr [ (α, α∗ ) ρ], where  (α, α∗ ) = π1 D (2α) (−1)a a is the singlemode Wigner operator in the coherent state representation, D (2α) being the displaced operator. In terms of the thermal field dynamics theory, the function W (α, α∗ ) can also be put into another form   1 W α, α∗ =  = 2α| ρ, π

(6.124)

its detailed derivation can be seen in Eq. (4.91), where | is another thermal state conjugate to the state |χ and shown in Eq. (1.107). Through direct calculation, the inner product of the states |χ and | reads χ | =

   1 1 ∗ exp χ − ∗ χ . 2 2

(6.125)

Substituting the completeness relation of the state |χ in Eq. (1.102) into Eq. (6.124) and using Eq. (6.125), we therefore have   W α, α∗ =



d2 χ  = 2α| χ χ| ρ π2 2   d χ = exp α∗ χ − αχ∗ χ| ρ. 2 2π

(6.126)

192

6 Evolution and Decoherence of Quantum States in Open Systems

So, once one know the inner product χ| ρ and perform the integration over χ in Eq. (6.126), the Wigner distribution function for the state ρ can be obtained directly. Inserting the inner product in Eq. (6.107) into Eq. (6.126), and using the integral formula in Eq. (1.30) to integrate χ, we can arrive at 

 −(κ−g)t  d2 χ A 2 ∗ ∗  ρ (0) χe exp − |χ| + α χ − αχ 2 2π 2 

2 2   2 2  d z = exp − α − ze−(κ−g)t  W z, z ∗ ; 0 , (6.127) A π A

  W α, α∗ ; t =



which is the relation between the evolved Wigner distribution function W (α, α∗ ; t) ∗ and the initial  function W (α, α ; 0) in the laser channel, A = (κ + g)  −2(κ−g)t / (κ − g). In particular, when g → κ¯n and κ → κ (¯n + 1), thus 1−e A → (2n¯ + 1) T , T = 1− e−2κt , Eq. (6.127) becomes   W α, α∗ ; t =

2 (2n¯ + 1) T



    d2 z  ∗  2 α − ze−κt 2 , W z, z ; 0 exp − π (2n¯ + 1) T

(6.128)

which is the analytical evolution of the Wigner distribution function W (z, z ∗ ; 0) in the thermal environment. However, for the case of g = 0, Eq. (6.127) reduces to the evolved Wigner distribution function in the amplitude damping channel, that is   2 W α, α∗ ; t = T





   2  d2 z −κt 2 W z, z ∗ ; 0 . exp − α − ze π T

(6.129)

So, to summarize, when a quantum state passes through the laser channel, the evolution of its density operator ρ with the time t under the lowest order approximation follows the quantum master equation (6.101) and the infinitive operator-sum representation of the evolved density operator ρ (t) in Eq. (6.111) gives the specific evolution law of this state in this channel, thus it is convenient for discussing the influence of decoherence effect on its nonclassical properties.

6.2.1 Squeezed Number States Theoretically, a squeezed number state can be obtained via operating the single-mode 2 †2 squeezed operator S1 (r ) = er (a −a )/2 on the number state ρn = |n n|, so its density operator ρsn reads [5] (6.130) ρsn = S1 (r )ρn S1−1 (r ). In experiment, the squeezed number state ρsn can be prepared via inputting the number state |n into a squeezed generating device (e.g., a parameter amplifier). To

6.2 Evolution of Quantum States in the Laser Process

193

derive the evolution law of the state ρsn in the laser channel, we now calculate the 2 †2 normal ordering product of the operator er (a −a )/2 a †n . For this purpose, substituting the completeness relation of coherent states |z into this operator, such that r (a 2 −a †2 )/2 †n

a

e



d2 z |z z| a †n π 2  d z ∗n 1/2 z : exp − |z|2 + za † sech r = sech r π ! 2 a †2 z ∗ † tanh r − a a : . + z a + tanh r − 2 2 r (a 2 −a †2 )/2

=e

(6.131)

Using the integral formula in Eq. (5.123) to carry out the integration (6.131), we therefore have   tanh r 2 r (a 2 −a †2 )/2 †n 1/2 †2 † (a − a ) + (sech r − 1)a a a = sech r : exp e 2

 [n/2]  tanh r k † n! × (a sech r + a tanh r )n−2k : . k!(n − 2k)! 2 k=0 (6.132) Further, operating Eq. (6.132) on the vacuum state |0 and using the generating functions for single-mode Hermite polynomials in Eq. (5.141), we thus obtain



tanh r †2 |0 . a exp − 2 − sinh 2r (6.133) Exactly, the state S1 (r ) |n can be viewed as a Hermite polynomial excited squeezed vacuum. Therefore, using Eq. (6.133) and the normal ordering of the vacuum projector in Eq. (1.7), we easily obtain the normal ordering product of the density operator ρsn , which is of the form S1 (r ) |n =

sech r (− tanh r )n Hn 2n n!

√

a†



  a† a sech r (− tanh r )n : H H √ √ n n 2n n! − sinh 2r − sinh 2r   tanh r †2 (a + a 2 ) − a † a : . × exp − (6.134) 2

ρsn =

Suppose that the state ρsn is input into the laser channel, thus using Eqs. (6.111) and (6.134) to leads to

194

6 Evolution and Decoherence of Quantum States in Open Systems

(1 − gT1 ) sech r (− tanh r )n  κi g j i+ j T 2n n! i! j! 1 i, j=0

  a† a † j a † a ln T2 i Hn √ ×a e a : Hn √ − sinh 2r − sinh 2r   tanh r †2 † (a + a 2 ) − a † a : a †i ea a ln T2 a j . × exp − 2

ρsn (t) =

(6.135)

Inserting the completeness relations of single-mode coherent states |α , |β into Eq. (6.135), we then give 2 2  κi g j i+ j d αd β i T1 a † j : α i! j! π2 i, j=0

   α∗ β Hn √ exp − |α|2 − |β|2 × Hn √ − sinh 2r − sinh 2r   tanh r  ∗2 α + β 2 + αT2 a † + β ∗ T2 a − a † a : β ∗i a j , − 2

ρsn (t) = A

(6.136)

where the parameter A is A=

(1 − gT1 )(− tanh r )n sech r . 2n n!

(6.137)

Using again the generating functions for single-mode Hermite polynomials in Eq. (5.141), thus Eq. (6.136) is given by

2 2 ∂ 2n d αd β 2α∗ τ − τ2 exp ρsn (t) = A n n : √ ∂τ ∂s π2 − sinh 2r    2βs tanh r  ∗2 2 α + β2 − s exp − |α|2 − |β|2 − +√ 2 − sinh 2r  † ∗ ∗ †  +αT2 a + β T2 a + κT1 αβ + (gT1 − 1)a a s=t=0 : . (6.138) Further, using the integral formula (1.30) to complete the integration (6.138), we thus obtain √   ∂ 2n exp Dτ − C 2 τ 2 ρsn (t) = A B : n ∂s n ∂τ     4BκT1 τ † 2 2  s−C s  × exp D − sinh 2r s=t=0     BT22 tanh r †2 (a + a 2 ) + BκT1 T22 tanh2 r + gT1 − 1 a † a : , × exp − 2 (6.139)

6.2 Evolution of Quantum States in the Laser Process

195

where −1  , B = 1 − κ2 T12 tanh2 r    C = B 1 − κ2 T12 , D=

2BT2 (a † − κT1 a tanh r ) . √ − sinh 2r

(6.140)

Using the generating function in Eq. (5.141), and the differential relation in Eq. (5.142), as well as the operator identity ea

†

a ln T2

|z = e−

|z|2 2

†

ezT2 a |0 ,

(6.141)

it then follows from Eq. (6.139) that

l 4BκT1 (n!)2 − 2 ρsn (t) = A BC l![(n − l)!]2 C sinh 2r l=0

    BT2 a − κT1 a † tanh r BT2 (a † − κT1 a tanh r ) × : Hn−l Hn−l e : (6.142) √ √ C − sinh 2r C − sinh 2r √

2n

n 

with     BT 2 tanh r †2 (a + a 2 ) + BκT1 T22 tanh2 r + gT1 − 1 a † a : , e = : exp − 2 2 (6.143) which is just the evolution formula of the density operator ρsn in the laser channel. Equation (6.142) shows that, after undergoing the laser channel, the initial pure squeezed number state ρsn evolves into a mixed state related to Hermite polynomials. Next, let us analyze several special cases. For g = 0, T1 = (1− e−2κt )/κ, T2 = e−κt , B = [1 − (1− e−2κt )2 tanh2 r ]−1 , thus Eq. (6.142) refers to the evolution of the state ρsn in the amplitude damping channel, which  is also a mixed state with the

decay rate κ. For the case of r = 0, B = 1, C = 1 − κ2 T12 , Eq. (6.142) becomes the evolution of the number state |n in the laser channel, that is

ρn (t) = κn T1n (1 − gT1 ) : Ln

−T22 a † a κT1



  exp (gT1 − 1)a † a : .

(6.144)

Further, using the operator identity in Eq. (1.21), thus Eq. (6.144) can be expressed as ρn (t) =

n  (1 − gT1 )n!κl T l T 2(n−l) l=0

l![(n

1 2 − l)!]2

  a †n−l exp a † a ln(gT1 ) a n−l .

(6.145)

196

6 Evolution and Decoherence of Quantum States in Open Systems

Comparing with the density operator of thermal state in Eq. (5.133) shows that, after passing through the laser channel, the number state |n evolves into a superposition of photon-added thermal states. Taking g = 0 in Eq. (6.145), we therefore obtain the evolution of the state |n for amplitude damping as ρn (t; g = 0) =

n   n−l −2lκt n  |l l| , e 1 − e−2κt l

(6.146)

l=0

which is exactly a binomial mixed state with the parameter e−2κt . However, for n = 0, the state ρsn decays to squeezed vacuum, thus Eq. (6.142) becomes √  BT 2 tanh r †2 (1 − gT1 ) B : exp − 2 (a + a 2 ) ρsv (t) = cosh r 2    + BκT1 T22 tanh2 r + gT1 − 1 a † a : .

(6.147)

On the other hand, at the initial time, i.e., t = 0, T1 = 0, T2 = 1,

A=

(− tanh r )n sech r , B = E = C = 1, 2n n!

(6.148)

thus Eq. (6.142) decays to the density operator ρsn as expected. For the limit case, t → ∞, noting that 1 (1 − g/κ) (− tanh r )n sech r , T2 → 0, A → , κ 2n n! B → cosh2 r, C → 0,

T1 →

(6.149)

thus the evolved state ρsn (t) is given by    g  g exp a † a ln , ρsn (∞) = 1 − κ κ

(6.150)

which just corresponds to a thermal field with the mean photon number g/(κ − g). A comparison of Eqs. (6.130) and (6.150) shows that, after a long interaction with the laser channel, the initial squeezed number state ρsn completely loses its nonclassicality and non-Gaussianity, and finally becomes a highly classical Gaussian thermal field. Next, we want to present the evolutions of the Wigner distribution functions for squeezed number states in the laser channel via the normal ordering product of the density operator ρsn (t) in Eq. (6.142). Using the coherent state representation of Wigner operator in Eq. (6.32), thus the Wigner distribution function for the state ρsn (t) reads Wsn (z, t) = tr[ρsn (t)(z, z ∗ )]. (6.151)

6.2 Evolution of Quantum States in the Laser Process

197

Substituting Eq. (6.142) into Eq. (6.151) leads to n 



  d2 z     −z : Hn−l h 3 a † + h 4 a 2 π l=0     × Hn−l h 3 a + h 4 a † exp[h 5 (a †2 + a 2 ) + h 6 a † a] : z  exp[2(zz ∗ − z ∗ z  )] 2  n      d z 2|z|2 h2 Hn−l −h 3 z ∗ + h 4 z  Hn−l h 3 z  − h 4 z ∗ = h1e 2 π l=0  2 × exp[h 5 (z ∗2 + z 2 ) − (h 6 + 2) z   + 2(zz ∗ − z ∗ z  )], (6.152)

Wsn (z, t) = h 1 e2|z|

2

h2

where √ h 1 = A BC 2n ,

(n!)2 22l B l κl T1l , l![(n − l)!]2 C 2l (− sinh 2r )l BT2 BκT1 T2 tanh r h3 = √ , h4 = − √ , C − sinh 2r C − sinh 2r BT 2 tanh r , h 6 = BκT1 T22 tanh2 r + gT1 − 1. h5 = − 2 2 h2 =

(6.153)

Further, using the generating functions for single-mode Hermite polynomials in Eq. (5.141) and the integral formula in Eq. (2.106), thus Wsn (z, t) becomes Wsn (z, t) = h 1 e2|z|

2

n  l=0

h2

∂ 2(n−l) ∂τ n−l ∂s n−l



d2 z  π2

  × exp 2(h 3 s + h 4 τ )z  − 2(h 4 s + h 3 τ )z ∗ − τ 2 − s 2 s=t=0  2 × exp[−(h 6 + 2) z   + h 5 (z ∗2 + z 2 ) + 2(zz ∗ − z ∗ z  )]  !   n  h2 4h 5 ∗2 4(h 6 + 2) 2 2 |z| exp (z + z ) + 2 − = h1 g g12 g12 l=0 1 ×

 ∂ 2(n−l) exp[−(1 − g2 )(τ 2 + s 2 ) + g3 sτ + g4 τ + g4∗ s]s=t=0 , n−l n−l ∂τ ∂s (6.154)

where g1 = [(h 6 + 2)2 − 4h 25 ]1/2 ,

g2 =

4 2 2 [h (h + h 24 ) − h 3 h 4 (h 6 + 2)], g12 5 3

4 [4h 3 h 4 h 5 − (h 6 + 2)(h 23 + h 24 )], g12  4  g4 = 2 [(h 6 + 2)h 4 − 2h 3 h 5 ]z + [(h 6 + 2)h 3 − 2h 4 h 5 ]z ∗ . g1 g3 =

(6.155)

198

6 Evolution and Decoherence of Quantum States in Open Systems

In a similar way to deriving Eq. (6.139), we finally obtain √ n n−l A BC 2n   (n!)2 22l B l κl T1l Wsn (z, t) = πg1 l!k![(n − l − k)!]2 l=0 k=0 

2  g4 (1 − g2 )n−l−k g3k   Hn−l−k × 2l √  l C (− sinh 2r ) 2 1 − g2   !   4h 5 ∗2 4(h 6 + 2) 2 2 |z| , × exp (z + z ) + 2 − g12 g12

(6.156)

which refers to the analytical evolution formula of the squeezed number state in the laser channel. Obviously, the Wigner distribution function Wsn (z, t) has a remarkable non-Gaussian feature because of the module square of Hermite polynomials, that is   Hn−l−k [g4 /(2√1 − g2 )]2 . Especially, taking g = 0 in Eq. (6.156), it becomes the evolved Wigner distribution function for the state ρsn in the amplitude damping channel. For r = 0, note that h3 = 

T2 (κ2 T12 − 1) sinh 2r

g1 = gT1 + 1,

g3 = −

,

h 4 = h 5 = g2 = 0, h 6 = gT1 − 1,

4h 23 , gT1 + 1

g4 =

4h 3 z ∗ , gT1 + 1

(6.157)

thus Eq. (6.156) can be put into n  n−l  n!(1 − gT1 )κl T1l (−1)k T22(n−l) 22(n−l−k) Wn (z, t) = πl!k![(n − l − k)!]2 (gT1 + 1)2(n−l)−k+1 l=0 k=0 !   4 |z|2 , × |z|2(n−l−k) exp 2 − gT1 + 1

(6.158)

which is just the analytical evolution of the Wigner distribution function for the number state in the laser channel. However, for n = 0, we arrive at the evolved Wigner distribution function for the squeezed vacuum in the laser channel, which is given by √  !   4h 5 ∗2 (1 − gT1 ) B 4(h 6 + 2) 2 2 |z| . exp (z + z ) + 2 − Wsv (z, t) = πg1 cosh r g12 g12 (6.159) Furthermore, in the case of t = 0, owing to

6.2 Evolution of Quantum States in the Laser Process

1 , h 4 = 0, h 5 = − tanh r, h 6 = −1, 2 − sinh 2r √ 1 2 2 (z ∗ + z tanh r ) cosh r, g1 = sech r, g2 = − = h 5 , g4 = √ g3 − tanh r

h3 = √

199

1

(6.160)

Equation (6.156) therefore becomes the Wigner distribution function for the state ρsn , that is n!(− tanh r )n  23k/2 (2 + tanh r )n−k/2 π22n k![(n − k)!]2 tanhk r k=0   ∗   2(z + z tanh r ) cosh r 2 × Hn−k √ −(2 + tanh r ) tanh r  × exp[−(z ∗2 + z 2 ) sinh 2r − 2 |z|2 cosh 2r ], n

Wsn (z, 0) =

(6.161)

which coincides quite well with the result (4.11) in Ref. [14]. In the limit case, t → ∞, h 6 = g/κ − 1, g1 = g/κ + 1, and other parameters is equal to zero, such that   2(κ − g) 2 κ−g |z| , (6.162) exp − Wsn (z, ∞) = π(κ + g) κ+g which is exactly a Gaussian distribution without the parameters n, r , corresponding to the Wigner distribution function for the thermal state ρsn (∞). To sum up, it can be seen that the laser noise makes the partial negativity of the non-Gaussian Wigner distribution function for the state ρsn disappear, and convert it into the Gaussian Wigner distribution function for the thermal state ρsn (∞), which means that the laser noise causes the loss of all the nonclassical properties of the state ρsn . In Fig. 6.5, it is shown that the evolved Wigner distribution function Wsn (z, t) for the state ρsn (t) in the laser channel is plotted for g = 1 and different values of the parameters n, r, κ and t. In the case of n = 0, Fig. 6.5a corresponds to the Wigner distribution function for the squeezed vacuum in the laser channel, which is just a Gaussian distribution stretched by the squeezing operation of along the Re z direction. For n = 0, the quantum interference structure between two main peaks can be clearly seen, and the frequency of quantum interference will increase as n increases. Moreover, the function Wsn (z, t) always shows some negative regions that gradually increase with the increase of n, which means that the squeezed number state with larger n possesses much longer decoherence time. For r = 0, Fig. 6.5b describes the evolved Wigner distribution function for the number state with n = 3 in the laser channel, it has obvious Gaussianity and partial negativity. A comparison of Fig. 6.5c and e clearly shows that the negative regions of the function Wsn (z, t) gradually disappear as the time t goes on, and the function Wsn (z, t) fully become positive and Gaussian in the whole phase space at long times, which refers to the complete loss of nonclassicality of the initial state ρsn . This is confirmed in terms of the analytical results outlined above. Besides, the negative regions of the function

200

6 Evolution and Decoherence of Quantum States in Open Systems

Fig. 6.5 In the laser channel, the evolutions of the Wigner distribution functions for the squeezed number states for g = 1 and a n = 0, r = 0.3, κ = 2, t = 0.01; b n = 3, r = 0, κ = 2, t = 0.01; c n = 3, r = 0.3, κ = 2, t = 0.01; d n = 6, r = 0.3, κ = 2, t = 0.01; e n = 3, r = 0.3, κ = 2, t = 0.08; f n = 3, r = 0.3, κ = 12, t = 0.01

Wsn (z, t) and the quantum interference phenomenon slowly disappear as κ increases, which shows that the increase of κ also leads to the loss of nonclassicality of the state ρsn [see Fig. 6.5c and f].

6.2.2 Squeezed Thermal States Similar to the definition of the density operator ρsn , the density operator of squeezed thermal state is defined as [5] ρst = S1 (r )ρth S1−1 (r ),

(6.163)

where ρth = (1− eλ )eλa a is the density operator of single-mode thermal state, λ = −ω/k B T . Substituting the density operator ρth into the Weyl ordering formula in Eq. (1.65), and using the operator identity in Eq. (1.37) and the integral formula in Eq. (1.30), we obtain †

6.2 Evolution of Quantum States in the Laser Process

  λ 2(1 − eλ ) : e −1 2 : 2 (Q + P ) . ρth = λ exp λ : e +1 : e +1

201

(6.164)

Note that the squeezed operator S1 (r ) yields the Bogoliubov transformations S1 (r )Q S1−1 (r ) = er Q, S1 (r )P S1−1 (r ) = e−r P,

(6.165)

 √  √    where Q = a + a † / 2, P = a − a † / i 2 are, respectively, the coordinate and momentum operators. Using the Weyl ordering invariance under similarity transformations, we can obtain the Weyl ordering product of the density operator ρst as   λ 2(1 − eλ ) : e − 1 2r 2 : −2r 2 (e Q + e P ) . ρst = λ exp λ : : e +1 e +1

(6.166)

Further, making the following variable substitutions Q → q and P → p in Eq. (6.166), thus the Weyl correspondence function for the density operator ρst reads  λ  e − 1 2r 2 2(1 − eλ ) −2r 2 exp λ (e q + e p ) . f st (q, p) = λ e +1 e +1

(6.167)

So, substituting the classical correspondence function fst (q, p) and the normal ordering product of the Wigner operator (q, p) in Eq. (1.52) into the Weyl quantization rule ∞ dqd p f st (q, p)(q, p), ρst = −∞

we obtain the normal ordering product of ρst as ρst = 2

"

 f −h 2 (a + a †2 ) − ( f + h)aa † : 2

(6.168)

1 1 , h= , (2n + 1)e2r + 1 (2n + 1)e−2r + 1

(6.169)

 f h : exp

with f =

where n = (eλ − 1)−1 is the mean photon number of single-mode thermal state ρth . In terms of Eq. (1.21), we find that the state ρst is a typical Gaussian nonclassical state, that is  

"   f − h †2 f −h 2 exp aa † ln[1 − ( f + h)] exp a a . ρst = 2 f h exp 2 2 (6.170) In terms of Eq. (6.111), we find that the anti-normal ordering product of the density operator ρst , in which all annihilation operators are to the left of creation operators, brings much convenience for deriving the evolution of the state ρst in the laser chan-

202

6 Evolution and Decoherence of Quantum States in Open Systems

nel. Therefore, substituting Eq. (6.168) into the formula in Eq. (1.36) and completing the integration to show that .  . ρst = A1 .. exp A2 (a 2 + a †2 ) + A3 a † a ..,

(6.171)

where the parameters A1 , A2 and A3 are, respectively, 1 A1 = " , 2 n − (2n + 1) sinh2 r −(2n + 1) sinh 2r , A2 = 4[n 2 − (2n + 1) sinh2 r ] 1 − (2n + 1) cosh 2r A3 = . 2[n 2 − (2n + 1) sinh2 r ]

(6.172)

Next, we investigate the evolution law of the state ρst as an initial state in the laser channel. Substituting Eq. (6.171) into Eq. (6.111) leads to ρst (t) = A1 (1 − gT1 )

 κi g j i+ j . † T1 a † j ea a ln T2 ..a i i! j! i, j=0

  . † × exp A2 (a 2 + a †2 ) + A3 a † a a †i ..ea a ln T2 a j .

(6.173)

Inserting the completeness relation of coherent states |z into Eq. (6.173), and using the operator identity in Eq. (6.20) and the integral formula in Eq. (2.106), we therefore have   A1 (1 − gT1 ) ρst (t) =  : exp (G − 1) a † a + H (a 2 + a †2 ) : F 2 − 4 A22

(6.174)

with F = 1 − A3 − κT1 , G =

F T22 A2 T22 , H = , (F 2 − 4 A22 ) + gT1 F 2 − 4 A22

(6.175)

which represents the analytical evolution of the squeezed thermal state ρst in the laser channel. In terms of Eqs. (1.21), (6.174) can therefore be rewritten as   A1 (1 − gT1 ) exp(H a †2 ) exp a † a ln G exp(H a 2 ). ρst (t) =  F 2 − 4 A22

(6.176)

Obviously, the output state ρst (t) is still a squeezed thermal state related to the cavity gain g and loss κ of the laser when the state ρst as an initial state passes

6.2 Evolution of Quantum States in the Laser Process

203

through the laser channel, not like the evolution of the squeezed number state ρsn in the laser channel. A comparison of Eqs. (6.142) and (6.176) shows that the normal ordering product of ρst (t) is more simple than that of ρsn (t), so it’s easier to study the decoherence behaviors of the squeezed thermal state in the laser channel. Particularly, for g = 0, F = e−2κt − A3 , H = A2 Le−2κt , G = L(e−2κt − A3 )e−2κt , L =

1 , (e−2κt − A3 )2 − 4 A22

(6.177)

it then follows from Eq. (6.176) that √   ρst (t; g = 0) = A1 L exp A2 Le−2κt a †2        × exp a † a ln L e−2κt − A3 e−2κt exp A2 Le−2κt a 2 ,

(6.178)

which is a squeezed thermal field with photon-number decay. For r = 0, 1 = −A3 , A2 = 0, H = 0, n nT22 n(1 − κT1 ) + 1 , G= , F= [n(1 − κT1 ) + 1] + gT1 n

A1 =

(6.179)

thus the evolution formula of the state ρth in the laser channel is easily obtained as !   nT22 1 − gT1 † exp a a ln + gT1 . ρth (t) = n(1 − κT1 ) + 1 n(1 − κT1 ) + 1

(6.180)

For n = 0, the squeezed thermal state becomes the squeezed vacuum, in such a case Eq. (6.174) reduces to Eq. (6.147). On the other hand, t = 0, ρst (t) becomes ρst as expected. In fact, the correctness of this result can be checked by taking t = 0 in Eq. (6.178). When t → ∞, note that F → −A3 , G → g/κ, H → 0,

(6.181)

thus ρst (t → ∞) becomes ρsn (∞), corresponding to the single-mode thermal state with the mean photon number g/(κ − g). Similarly, the decoherence behaviors of the squeezed thermal state in the laser channel can be also investigated via its evolved Wigner distribution function. Substituting the normal ordering product of ρst (t) into Eq. (6.151) and using the integral formula in Eq. (1.30), we thus get

204

6 Evolution and Decoherence of Quantum States in Open Systems

A1 (1 − gT1 )  π (F 2 − 4 A22 )[(G + 1)2 − 4H 2 ]  !  4(G + 1) 4H (z 2 + z ∗2 ) 2 |z| × exp − . (6.182) + 2 + (G + 1)2 − 4H 2 (G + 1)2 − 4H 2

Wst (z, t) =

Obviously, the Wigner distribution function Wst (z, t) for the squeezed thermal state still remains Gaussian. For n = 0, Eq. (6.182) becomes the Wigner distribution function for the squeezed vacuum in Eq. (6.159). When t = 0, Eq. (6.182) reduces to the Wigner distribution function for the squeezed thermal state, which is the same as Eq. (4.13) in Ref. [14]. However, for t → ∞, there must exists Wst (z, ∞) = Wsn (z, ∞). In Fig. 6.6, the evolution of the Wigner distribution function Wst (z, t) in the laser channel is plotted for given r , n¯ and g, and several different values of κ and t. Clearly, the function Wst (z, t) always remains positive and has a Gaussian distribution with maximum in the center of the phase space (i.e., z = z ∗ = 0). With the increase of t (or κ), the function Wst (z, t) slowly loses its squeezing and finally evolves into the shape of Wst (z, ∞). Comparing the evolution behaviors of the states ρst (t) and ρsn (t) show that, after undergoing the laser channel, the initial pure squeezed number state evolves into a mixed state with Hermite polynomials, however, the mixed squeezed thermal state always remains squeezed and thermal. On one hand, both of the functions Wsn (z, t) and Wst (z, t) always exhibit the squeezing in one of the quadratures. On the other hand, the laser noise can destroy the original non-Gaussianity of the function Wsn (z) and further contributes to more complicated negative regions and remarkable oscillating feature as shown in Fig. 6.6, while it never changes the Gaussianity of the function Wst (z, t). At long times, both the states ρsn (t) and ρst (t) lose their nonclassicality and become the same classical thermal state with the average photon number g/(κ − g) owing to the presence of the gain factor g of the laser. This is completely different from what happens in the amplitude damping channel because the decay of photons turns such states into single-mode vacuum in this channel.

Fig. 6.6 In the laser channel, the evolutions of Wigner distribution functions for the squeezed thermal states for r = 0.5, n¯ = 1, g = 1, and a κ = 2, t = 0.01; b κ = 2, t = 5; c κ = 200, t = 0.01

6.2 Evolution of Quantum States in the Laser Process

205

6.2.3 Multi-photon Subtracted Squeezed Vacuum States Making the following substitutions g → κ¯n and κ → κ (¯n + 1) in Eq. (6.111), we easily arrive at the analytical evolution formula of the density operator ρ in the thermal environment, that is ρ(t) = (1 − κ¯nT1 )

 [κ(¯n+1)]i (κ¯n) j i! j! i, j=0

i+ j † j a † a ln T2 i

× T1

a e

a ρ0 a †i ea

†

a ln T2

aj

(6.183)

or ρ(t) =



Mi, j ρ0 Mi,† j ,

i, j=0

Mi, j =

(1 − κ¯nT1 )

[κT1 (¯n+1)]i (κT1 n¯ ) j † j a † a ln T2 i a e a, i! j!

(6.184)

which is also the infinitive operator-sum representation of ρ(t) in the thermal environment, and the parameters T1 and T2 are, respectively, T1 =

T , κ (¯nT + 1)

T2 =

e−κt . n¯ T + 1

(6.185)

Equation (6.184) clearly shows that the dissipation behavior occurs in the thermal environment owing to the presence of the factor T = 1− e−2κt in the Kraus operator Mi, j . So, the entangled state representation provides us with a new way to explore the characteristics of dissipative processes. Next, we want to examine the decoherence behaviors of the multi-photon subtracted squeezed vacuum states with the density operator ρ p = a m ρs a †m in the thermal environment, where ρs = S1† (r ) |0 0| S1 (r ) is the density operator of squeezed vacuum in Eq. (5.31). Thus, the normal ordering product of ρs reads [6] 

  1 2 †2 † a + a tanh r − a a : . ρs = sech r : exp 2

(6.186)

Substituting Eq. (6.186) into the anti-normal ordering formula (1.36) leads to    2  . 1 .. 1 †2 † a + a + a a .., ρs = − . exp − i sinh r 2 tanh r it then follows from Eq. (6.187) that

(6.187)

206

6 Evolution and Decoherence of Quantum States in Open Systems

i(1 − κ¯nT1 )  [κ(¯n+1)]i (κ¯n) j i+ j † j a † a ln T2 T1 a e Nm sinh r i, j=0 i! j!   .. m+i . † 1 2 †2 † (a + a ) + a a a †m+i ..ea a ln T2 a j . × .a exp − 2 tanh r

ρ p (t) =

(6.188)

Using the completeness relation of coherent states |z and the operator identity in Eq. (4.44), we therefore have √

 m i B(1 − κ¯nT1 )  (m!)2 [−κ(¯n+1)T1 ]l B m coth r m−l ρ p (t) = Nm sinh r l![(m − l)!]2 2 l=0 

B tanh r T2 (κ(¯n+1)T1 a + a † coth r ) × : Hm−l − 2 

B tanh r † × Hm−l − T2 (κ(¯n+1)T1 a + a coth r ) 2 !  BT22 (a 2 + a †2 ) : , (6.189) × exp [κ¯nT1 − Bκ(¯n+1)T1 T22 − 1]a † a − 2 tanh r where B = [κ2 (¯n+1)2 T12 − coth2 r ]−1 . Equation (6.189) is just the analytical evolution formula of the state ρ p in the thermal environment, which is the same as the differential result (25) in Ref. [15]. Exactly, after undergoing a thermal environment, the pure initial state ρ p evolves into a mixed non-Gaussian state that is related to the product of two different Hermite polynomials within normal ordering. In particular, when κ¯n → 0, T1 = (1−e−2κt )/κ, T2 = e−κt , thus Eq. (6.189) becomes iC m+1/2 (m!)2 cothm r  [2(e−2κt − 1) tanh r ]l Nm 2m sinh r l![(m − l)!]2 l=0 

C tanh r −κt −2κt C −κt † e (e e a ) × : Hm−l − 1)a − 2 2 tanh r 

C tanh r −κt −2κt C † −κt e (e e a) ×Hm−l − 1)a − 2 2 tanh r  ! Ce−2κt 2 −2κt −2κt † †2 × exp [Ce (a + a ) : , (e − 1) − 1]a a − 2 tanh r (6.190) m

ρ p (t; κ¯n → 0) =

which is the analytical evolution of the state ρ p in the amplitude damping channel, C = [(1−e−2κt )2 − coth2 r ]−1 . For m = 0, ρ p (t) becomes

6.2 Evolution of Quantum States in the Laser Process

√  i B(1 − κ¯nT1 ) : exp [−Bκ(¯n+1)T1 T22 ρ p (t; m = 0) = sinh r !  BT22  2 a + a †2 : , +κ¯nT1 − 1]a † a − 2 tanh r

207

(6.191)

which indicates that, after an initial squeezed vacuum undergoes the thermal environment, the output state becomes a squeezed thermal state with the parameters n¯ and κ. At the initial time, κt = 0, T1 = 0, T2 = 1 and B = − tanh2 r , the state ρ p (t) returns to the initial density operator ρ p , as expected. For the case of κt → ∞, T1 → 1/κ(¯n + 1), T2 → 0 and B → − sinh2 r , thus Eq. (6.189) becomes the a highly classical thermal state with the mean photon number n¯ , that is

 1 n¯ † exp a a ln , ρ p (κt → ∞) = n¯ +1 n¯ +1

(6.192)

which is similar to the result in Eq. (6.150). Similarly, using the normal ordering product of ρ p (t) and the Wigner operator in the coherent state representation, as well as the way for calculating Eq. (6.189), we thus obtain the analytical evolution of the Wigner distribution function for the state ρ p in the thermal environment, which is of the form W p (α, t) = Cm (α, t)Ws (α, t),

(6.193)

where ws (α, t) is the Wigner distribution function for the squeezed vacuum in the thermal environment, i.e., √

B(1 − κ¯nT1 ) exp Ws (α, t) = √ π −h 1 sinh r



  4D 4E 2 2 ∗2 + 2 |α| + (α + α ) h1 h1

(6.194)

and the term Cm (α, t) results from the multi-photon subtraction operation, i.e., 1   (−1)m−n (m!)2 [κ(¯n+1)T1 ]l B m Cm (α, t) = Nm l=0 n=0 l!n![(m − l − n)!]2

2 m−l n 

 h 3  h4 h 2 coth r  Hm−l−n × √  2 h2 2 −h 2  m m−l

with D = Bκ(¯n+1)T1 T22 − κ¯nT1 − 1, BT22 , h 1 = D 2 − 4E 2 , 2 tanh r 2BT22 [Dκ(¯n+1)T1 + Eκ2 (¯n+1)2 T12 tanh r + E coth r ] − 1, h2 = h1 E =−

(6.195)

208

6 Evolution and Decoherence of Quantum States in Open Systems

Fig. 6.7 The evolution law of the Wigner distribution function for the multi-photon subtracted squeezed vacuum in the thermal environment for n¯ = 0.05 and a m = 2, r = 0.3, κt = 0.1; b m = 2, r = 0.3, κt = 0.5; c m = 5, r = 0.3, κt = 0.1; d m = 5, r = 0.3, κt = 0.5; e m = 2, r = 0.8, κt = 0.1; f m = 5, r = 0.8, κt = 0.1

2BT22 [Dκ2 (¯n+1)2 T12 tanh r + D coth r + 4Eκ(¯n+1)T1 ], h1 √ 2 2B tanh r T2 h4 = {[Dκ(¯n+1)T1 + 2E coth r ] α h1 + [D coth r + 2Eκ(¯n+1)T1 ] α∗ }. h3 =

(6.196)

Obviously, the function W p (α, t) is non-Gaussian owing to the presence of the term Cm (α, t). Particularly, taking κ¯n → 0, Eq. (6.193) refers to the evolved Wigner distribution function for the state ρ p in the amplitude damping channel. At the initial time, κt = 0, Eq. (6.193) reduces to the Wigner distribution function for the state ρ p , which is consistent with the result (24) in Ref. [6]. After a long enough time, that is κt → ∞, Eq. (6.193) reduces to W p (α, κt → ∞) =



2 1 |α|2 , exp − π(2n¯ +1) 2n¯ +1

(6.197)

which is just the Wigner distribution function for the thermal state ρ p (κt → ∞). A comparison of Eqs. (6.193) and (6.197) shows that, after undergoing the thermal environment, thermal noise can transform the Wigner distribution function from the non-Gaussian distribution of the state ρ p to the Gaussian distribution of the thermal state. Since the partial negativity of the Wigner distribution function for the state ρ p is converted to the positive definite of the thermal state, this means that the nonclassical properties of the state ρ p completely disappear in the thermal environment. In terms of Eq. (6.193), we plot the Wigner distribution function for the state ρ p in the thermal environment for given n¯ and several different values of m, r and κt in Fig. 6.7. With the increase of κt, the negativity of the function W p (α, t) decreases gradually and the quantum interference disappears slowly, which means the loss

6.2 Evolution of Quantum States in the Laser Process

209

of nonclassicality of the state ρ p because of the decoherence effect in the thermal environment. At long times, the function W p (α, t) becomes positive and Gaussian, corresponding to thermal, as the analytical result outlined in Eq. (6.197). Besides, the larger the parameter m (or r ) is, the slower the nonclassicality disappears, which indicates that the larger m (or r ) can lead to much larger decoherence time. As a comparison, now we briefly investigate the effect of phase damping on the state ρ p . When the initial quantum state is input into the phase damping channel, the corresponding evolved density operator with time t yields the following master equation, which is of the form dρ (t) = κ [2 Aρ (t) A† − A† Aρ (t) − ρ (t) A† A], dt

(6.198)

where A = a † a is the number operator, and κ refers to the decay rate. Similarly, using the IWOP method and the thermal entangled state representation, the evolution law of the initial density operator ρ0 in the phase damping channel is given by 

ρ(t) = e−κ t A

2

 (2κ t)n n=0

n!



An ρ0 An e−κ t A , 2

(6.199)

which is just the infinitive operator-sum representation of ρ(t) as the solutions of the master equation in Eq. (6.198). So, when the state ρ p as an initial state is input into the phase damping channel, in terms of the anti-normal ordering of the density operator ρ p in Eq. (6.186), we easily obtain ∞ (2k)!(2k  )! sech r  Nm k,k  =0 k!k  !(2k − m)!(2k  − m)!  

tanh r k+k −4κ t (k−k  )2 †2k−m 2k  −m e :a a exp(−a † a) : . × 2

ρp (t) =

(6.200)

Equation (6.200) shows that, the state ρp (t) is a mixed state with the exponential   2 decay factor e−4κ t (k−k ) (k, k  = 0, 1, 2 . . .) that only consists of even (or odd) number states when the photon-subtraction number m is even (or odd). For example, m = 0, ρp (t) is just the superposition of even number states, that is ρp (t; m = 0) = sech r 

 2

∞ 

1 ! k!k k,k  =0 

tanh r 2

k+k 

×e−4κ t (k−k ) : a †2k a 2k exp(−a † a) : ,

(6.201)

which is the evolution of squeezed vacuum in the phase damping channel. Owing to   2 the existence of the decay factor e−4κ t (k−k ) in Eq. (6.200), the off-diagonal terms in  the density operator ρ p (t) decay slowly. When κ t → ∞, we naturally obtain

210

6 Evolution and Decoherence of Quantum States in Open Systems ∞

[(2k)!]2 sech r  Nm k=0 (k!)2 [(2k − m)!]2 

tanh r 2k × : (a † a)2k−m exp(−a † a) : , 2

ρp (κ t → ∞) =

(6.202)

where only diagonal elements relying on the photon-subtraction number m survive. In fact, the state ρp (κ t → ∞) can also be viewed as a superposition of weighted number states. Next, in a similar way to deriving the function in Eq. (6.193), we can obtain the evolution of the Wigner distribution function for the state ρ p in the phase damping channel, i.e., W p (α, t)

 r k+k  ∞ sech r  (2k)!(2k  )! tanh 2 = π Nm k,k  =0 k!k  !(2k − m)!(2k  − m)! 

 2

×e−4κ t (k−k ) exp(−2 |α|2 )H2k  −m,2k−m (2α, 2α∗ ),

(6.203)

which exhibits the decay as the result of phase damping. For κ t → ∞, Eq. (6.203) becomes W p (α, κ t → ∞) = where

 ∞ tanh r 2k [(2k)!]2 sech r  W|2k−m2k−m| (α), Nm k=0 (k!)2 (2k − m)! 2 (6.204)

e−2|α| H2k−m,2k−m (2α, 2α∗ ) W|2k−m2k−m| (α) = π(2k − m)! 2

(6.205)

is the Wigner distribution function for the number state |2k − m. Therefore, W p (α, κ t → ∞) can be viewed as a sum of weighted Wigner distribution functions for all the number states. Using Eq. (6.203), we plot the evolution of the Wigner distribution function for the state ρ p in the phase damping channel for different values of m, r and κ t in Fig. 6.8. Clearly, for small squeezing r , the function W p (α, t) always has a certain negative region characterizing nonclassicality of the state ρ p in the center of phase space for small odd number m, however, has no chance to exhibit the negativity in the whole phase space for small even number m. This result is the same as the case of single-photon subtracted squeezed vacuum, but is completely different from the Wigner distribution function for ρ p in the thermal environment. For large m (or r ), the function W p (α, t) shows the negativity around the origin of the phase space at long times, and the larger m (or r ), the more closely the function W p (α, κ t → ∞) trends to the Wigner distribution function for a number state. This can be verified in terms of the analytical result in Eq. (6.204) because the weight of the function W|2k−m2k−m| (α) is greater in the superposition (6.204) as m (or r ) gets larger.

6.2 Evolution of Quantum States in the Laser Process

211

Fig. 6.8 The evolution law of the Wigner distribution function for the multi-photon subtracted squeezed vacuum in the phase damping channel for a m = 2, r = 0.3, κ t = 0.1; b m = 2, r = 0.3, κ t = 3; c m = 3, r = 0.3, κ t = 0.1; d m = 3, r = 0.3, κ t = 3; e m = 17, r = 0.3, κ t = 0.1; f m = 17, r = 0.3, κ t = 3; g m = 2, r = 0.8, κ t = 0.1; h m = 2, r = 0.8, κ t = 3

To sum up, comparing the decoherence effects in the two different channels means that, during undergoing the thermal environment, the initial pure state ρ p evolves into a mixed state related to the Hermite polynomials, however, it becomes another mixed state with the exponential decay factor in the phase damping channel. After a long enough time, the state ρ p loses completely its nonclassicality and becomes a classical thermal state in the thermal environment. This differs from what happens for phase damping because phase damping only changes such a state into a sum of weighted number states that still remains nonclassicality and non-Gaussianity to a certain extent.

6.2.4 Multi-photon Added Two-Mode Squeezed Thermal States Usually, a two-mode squeezed thermal state is defined via operating the two-mode squeezed operator S2 (r ) with the squeezing r on a two-mode thermal state ρth , so its density operator is expressed as [7] ρs = S2 (r )ρth S2−1 (r ),

(6.206)

212

6 Evolution and Decoherence of Quantum States in Open Systems

where ρth = (1− eλ )2 eλ(a a+b b) is the density operator of thermal state, λ = −ω/k B T . As a note, assume that the two-mode thermal field has the same frequency ω and temperature T in the definition of ρth . Generalizing the Weyl ordering formula to the case of two-mode, i.e., †

: ρ= 4 :



†

d2 αd2 β −α, −β| ρ |α, β π2

: × exp[2(α∗ a − αa † + a † a) + 2(β ∗ b − βb† + b† b)] , :

(6.207)

and substituting the following operator identity: eλ(a

†

a+b† b)

   = : exp eλ − 1 (a † a + b† b) :

(6.208)

into the integration in Eq. (6.207) and completing it, we thus have eλ(a

†

a+b† b)

=

  λ 4 e −1 2 : : 2 2 2 (Q + P + Q + P ) exp , a b b : (1 + eλ )2 : eλ + 1 a

(6.209)

where Q i , Pi (i = a, b) are, respectively, the coordinate operators and momentum operators of the ith mode. Noting the Weyl ordering invariance under similarity transformations, and using the following Bogoliubov transformations S2 (r )Q j S2−1 (r ) = Q j cosh r − Q k sinh r, S2 (r )P j S2−1 (r ) = P j cosh r + Pk sinh r

(6.210)

for j, k = a, b but j = k, we obtain the Weyl ordering product of ρs as   4 1  2 : Q a + Pa2 + Q 2b + Pb2 cosh 2r ρs = exp − 2 : (2n + 1) 2n + 1 : (6.211) +2(Pa Pb − Q a Q b ) sinh 2r ]} , : where n = (eλ − 1)−1 is the mean photon number of single-mode thermal state. Further, substituting the normal ordering product of two-mode Wigner operator (qa , pa ; qb , pb ) =

1 2 2 2 2 : e−(qa −Q a ) −( pa −Pa ) −(qb −Q b ) −( pb −Pb ) : , 2 π

(6.212)

and the Weyl classical function h(qa , pa ; qb , pb ) of the density operator ρs that can be obtained via making the following substitutions Q i → qi and Pi → pi in Eq. (6.212) into the Weyl quantization rule

6.2 Evolution of Quantum States in the Laser Process

ρs =

∞ −∞

dqa d pa dqb d pb h(qa , pa ; qb , pb )(qa , pa ; qb , pb ),

213

(6.213)

and using the IWOP method, we therefore give the normal ordering product of ρs as   ρs = A : exp g(Q a2 + Pa2 + Q 2b + Pb2 ) + h(Pa Pb − Q a Q b ) : ,

(6.214)

where 1 g = − A [(2n + 1) cosh 2r + 1] , 4 1 h = A(2n + 1) sinh 2r, 2 1 A= 2 . n + (2n + 1) cosh2 r

(6.215)

Or, we have   ρs = g1 : exp g2 (a † a + b† b) + g3 (ab + a † b† ) : ,

(6.216)

and g1 = [¯n2 + (2n¯ + 1) cosh2 r ]−1 , g1 g2 = − [(2n¯ + 1) cosh 2r + 1], 2 g1 g3 = (2n¯ + 1) sinh 2r, g1 = g22 − g32 . 2

(6.217)

Obviously, the density operator ρs in Eq. (6.214) is fully different from the direct product of two single-mode squeezed thermal states because of the presence of the entanglement part Pa Pb − Q a Q b . Basically, the entanglement part resulting from two-mode squeezing operation leads to the appearance of entanglement between a and b of two-mode thermal state. At this point, the squeezing operator S2 (r ) plays the role of entangling modes, thus the usual two-mode squeezed state is a good example as a typical entangled state. Repeatedly operating the photon creation operators a † and b† on a two-mode squeezed thermal state, thus the multi-photon added two-mode squeezed thermal state is given by −1 †m †n a b ρs a m bn , (6.218) ρ p = Nm,n where m and n (the arbitrary non-negative integers) are, respectively, the photon addition numbers of a and b mode, and Nm,n is the normalization constant for the state ρ p . For the convenience of calculating the factor Nm,n , using the completeness relation for two-mode coherent states |α, β and the normal ordering product of ρs

214

6 Evolution and Decoherence of Quantum States in Open Systems

in Eq. (6.216) to calculate the normalization factor of a general operator product a †m b†n ρa k bl , we arrive at Nm,n;k,l = tr(a †m b†n ρa k bl ) 2 2 d αd β α, β| : a †m b†n = g1 π2   × exp g2 (a † a + b† b) + g3 (ab + a † b† ) a k bl : |α, β 2 2 d αd β k ∗m l ∗n α α ββ = g1 π2 × exp[g2 (|α|2 + |β|2 ) + g3 (αβ + α∗ β ∗ )].

(6.219)

The integration may be evaluated by using the infinite integral of complex function in Eq. (5.136), we obtain the normalization factor Nm,n;k,l , i.e., Nm,n;k,l =

min(m,k) 

m!k!(l + m − i)!(−1)k+l g3m+k−2i

i=0

i!(m − i)!(k − i)!g2k−l g1m+l−i

δn+k−i,l+m−i .

(6.220)

Letting k = m and l = n, we have Nm,n;k,l =

m  (−1)m+n (m!)2 (n + m − i)!g 2(m−i) 3

i=0

i![(m − i)!]2 g m−n g1(m+n−i) 2

.

(6.221)

Comparing Eq. (6.221) with the newly found expression of the Jacobi polynomials (without loss of generality, assuming n ≥ m) (x) = P(0,n−m) m



m  m!(n + m − l)! x − 1 m−l , n!l![(m − l)!]2 2 l=0

(6.222)

Bn P(0,n−m) (C), (−2g)m m

(6.223)

we thus have Nm,n = m!n! where

1 [(2n + 1) cosh 2r + 1], 2 A C = {(2n + 1)[(2n + 1) cosh 4r + 2 cosh 2r ] + 1} . 4 B=

(6.224)

When m = n = 0, namely the case of no photon addition, so N0,0 = 1 as expected. For m = 0 and n = 0, P(0,n) (C) = 1, N0,n = n!B n . In the case of n → 0, g = −1/2, 0 2 (cosh 2r ), B = cosh r and C = cosh 2r , thus Nm,n;n=0 = m!n! cosh2n r P(0,n−m) m

6.2 Evolution of Quantum States in the Laser Process

215

which is just the normalization factor of the photon-added two-mode squeezed vacuum. However, for r = 0, the states ρ p become the photon-added two-mode thermal states, thus g = −1/[2(n + 1)], B = n + 1 and C = 1, Nm,n;γ=0 = m!n!(n + 1)m+n . Next, we want to investigate the influence of thermal noise on the nonclassical properties of the state ρ p via the evolution of the Wigner distribution functions in the thermal environment. For this purpose, we generalize the master equation (6.102) describing single-mode thermal environment to the case of two modes. Thus, when an initial two-mode quantum state is input into the thermal environment, the evolution of its density operator yields the following master equation dρ(t) = (L a + L b ) ρ(t), dt

(6.225)

where L i = κ(¯n + 1)[2iρ(t)i † − i † iρ(t) − ρ(t)i † i] + κ¯n[2i † ρ(t)i − ii † ρ(t) − ρ(t)ii † ], (i = a, b) .

(6.226)

For convenience, the dissipation coefficient κ and the mean thermal photon number n¯ of two modes are considered to be the same. Using the IWOP method and the thermal entangled representation, in a similar way to solve the master equation of the laser channel in Eq. (6.101), we thus arrive at the infinitive operator-sum representation of ρ(t) as the analytical solution of master equation (6.225), that is ρ(t) =

∞ 

† Mm,n,r,s ρ0 Mm,n,r,s ,

(6.227)

m,n,r,s=0

where Mm,n,r,s is the Kraus operator Mm,n,r,s =

1 n¯ T + 1

Tr1+s Tm+n † † 3 a †r b†s e(a a+b b) ln T2 a m bn , r !s!m!n!

(6.228)

where T2 can be found in Eq. (6.185), and T1 and T3 are, respectively, T1 =

n¯ T (¯n + 1) T , T3 = . n¯ T +1 n¯ T +1

(6.229)

Further, using two-mode Wigner operator ∗

∗



 d2 α d2 β         α −α , β , −β π4 ∗ ∗  ∗ exp[2(αα − α α + ββ − β ∗ β  )] (6.230)

(α, α ; β, β ) = exp(2 |α| + 2 |β| ) 2

2

216

6 Evolution and Decoherence of Quantum States in Open Systems

in the coherent state representation, and substituting Eq. (6.227) into the basic definition of Wigner distribution function, which is of the form W p (α, β, t) = tr[ρ(t)(α, α∗ ; β, β ∗ )],

(6.231)

we naturally obtain the evolution of the Wigner distribution function for the initial state ρ0 in the thermal environment. Instead of the definition (6.231), here using the integral formula between the evolved Wigner distribution function W (α, β, t) for a two-mode quantum state in the thermal environment and its initial Wigner distribution function W (α, β, 0), i.e., 2  2  d αd β 4 w(α , β  , 0) W (α, β, t) = (2n¯ + 1)2 T 2 π2        2 α − α e−κt 2 + β − β  e−κt 2 × exp − (2n¯ + 1)T

(6.232)

to investigate the influence of thermal environment on the state ρ p . For this, using the normal ordering product of ρs in Eq. (6.214) and the two-mode Wigner operator in Eq. (6.230), we thus obtain the Wigner distribution function for the state ρ p , i.e.,   A exp 2 |α|2 + 2 |β|2 W p (α, β) = Nm,n



d2 α d2 β  π4   ∗ ∗  ∗ ∗  × exp[2(αα − α α + ββ − β β )] −α , −β   : a †m b†n     × exp g(Q a2 + Pa2 + Q 2b + Pb2 ) + h(Pa Pb − Q a Q b ) a m bn : α , β   d2 α d2 β    2m   2n  (−1)m+n A α  β  exp 2 |α|2 + 2 |β|2 = Nm,n π4     2  2 × exp −2(g + 1) α  + β   − h(α β  + α∗ β ∗ )  +2(αα∗ − α∗ α + ββ ∗ − β ∗ β  ) . (6.233)

Using the integral formula in Eq. (5.136) and the generating functions for twovariable Hermite polynomials in Eq. (1.85), we therefore have     AG n−m exp −A1 |α|2 + |β|2 + A2 (αβ + α∗ β ∗ ) n+1 m,n K 2m   ∂ × m m Hn,n A4 α + A5 β ∗ + A6 t, A4 α∗ + A5 β + A6 t  ∂t ∂t    G2 × exp − tt  + (A7 β + A8 α∗ )t + (A7 β ∗ + A8 α)t   , (6.234) K t=t  =0

W p (α, β) =

where

π2 N

6.2 Evolution of Quantum States in the Laser Process

217

G = 2(g + 1) = A(2n + 1)(n + cosh2 r ), K = 4(g + 1)2 − h 2 = A(2n + 1)2

(6.235)

and 4G 4h 2 2h − 2, A2 = − , A3 = , A4 = √ , K K (2n¯ + 1)T KG √ √ √ 2 G −h 2 Gh 2G G A5 = √ , A6 = √ , A7 = , A8 = . (6.236) K K K K A1 =

Further, using Eq. (1.85), we thus have     AG n−m exp −A1 |α|2 + |β|2 + A2 (αβ + α∗ β ∗ ) 2 n+1 π Nm,n K  ∂ 2n ∂ 2m × m m n n exp −ss  + A6 (st + s  t  ) ∂t ∂t ∂s ∂s   G2 ∗ ∗  +(A4 α + A5 β )s + (A4 α + A5 β)s exp − tt  K  +(A7 β + A8 α∗ )t + (A7 β ∗ + A8 α)t  ]s=s  =t=t  =0 . (6.237)

W p (α, β) =

To facilitate the calculation of W p (α, β), letting f 1 = A4 α + A5 β ∗ ,

f 2 = A 7 β + A 8 α∗

(6.238)

and calculating the following differential    ∂ 2m ∂ 2n G2  exp A6 (st + s  t  ) − ss  − tt + f 1 s + f 1∗ s  + f 2 t + f 2∗ t   m m n n ∂t ∂t ∂s ∂s K s=s  =t=t  =0 2m 2n  1 ∂ ∂ p+q = s p s q t p t q A p!q! 6 ∂t m ∂t m ∂s n ∂s n p,q=0

  G2  × exp −ss  − tt + f 1 s + f 1∗ s  + f 2 t + f 2∗ t   K s=s  =t=t  =0 p+q p+q 2m 2n  1 ∂ ∂ ∂ ∂ p+q = A p!q! 6 ∂ f 1p ∂ f 1∗q ∂ f 2p ∂ f 2∗q ∂t m ∂t m ∂s n ∂s n p,q=0 

 G2  × exp −ss  − tt + f 1 s + f 1∗ s  + f 2 t + f 2∗ t   K s=s  =t=t  =0

218

6 Evolution and Decoherence of Quantum States in Open Systems

#√ $ √ 2m   1 ∂ p+q ∂ p+q K K ∗ p+q G A f2 , f = Hm,m Hn,n f 1 , f 1∗ p!q! 6 K m ∂ f p ∂ f ∗q ∂ f p ∂ f ∗q G G 2 1 1 2 2 p,q=0







 2m  m G m n n p!q! = m p q p q K p,q=0 #√ $ √ 

K K ∗ h p+q ∗ f2 , f Hn− p,n−q ( f 1 , f 1 )Hm− p,m−q (6.239) × − , G G G 2 

where we have used the differential relations of H p,q (x, y), i.e., ∂ H p,q (x, y) = pH p−1,q (x, y), ∂x

∂ H p,q (x, y) = qH p,q−1 (x, y). ∂y

(6.240)

Combining Eqs. (6.238) and (6.239), we finally obtain W p (α, β) = Cm,n (α, β)Ws (α, β),

(6.241)

where Ws (α, β) is the Wigner distribution function for two-mode squeezed thermal state Ws (α, β) =

A π2 K

    exp −A1 |α|2 + |β|2 + A2 (αβ + α∗ β ∗ ) ,

(6.242)

and the term Cm,n (α, β) comes from the multi-photon addition operation −1 Cm,n (α, β) = Nm,n

G K

× p!q! −

m+n min(m,n) 

h G 

× Hn− p,n−q

 p+q

p,q=0

m p



m q

   n n p q

  Hm− p,m−q A4 β + A5 α∗ , A4 β ∗ + A5 α  A 4 α + A 5 β ∗ , A 4 α∗ + A 5 β . (6.243)

Obviously, Eq. (6.241) is the analytical formula of the Wigner distribution function for the state ρ p , related to two-variable Hermite polynomials. In particular, for m = n = 0, C0,0 (α, β) = 1, thus Eq. (6.241) becomes the Wigner distribution function Ws (α, β); however for m = 0 and n = 0, Eq. (6.241) becomes     A exp −A1 |α|2 + |β|2 + A2 (αβ + α∗ β ∗ ) π 2 n!K 

  G n × Hm,m A4 β + A5 α∗ , A4 β ∗ + A5 α . (6.244) BK

W p,m=0 (α, β) =

6.2 Evolution of Quantum States in the Laser Process

219

For the case of n → 0, Eqs. (6.242) and (6.243), respectively, reduce to Ws;n=0 (α, β) =

    1 exp (2 − ζ 2 ) |α|2 + |β|2 − ξζ(αβ + α∗ β ∗ ) 2 π

(6.245)

and min(m,n)  m  m  n  n  ζ 2(m+n) Cm,n;n=0 (α, β) = 2(m+n) p q p q 2 Nm,n;n=0 p,q=0

   ξ p+q Hm− p,m−q ξβ + ζα∗ , ξβ ∗ + ζα × p!q! − ζ   × Hn− p,n−q ξα + ζβ ∗ , ξα∗ + ζβ , (6.246)

where ξ = 2 sinh r and ζ = 2 cosh r . Thus, the product Cm,n;n=0 (α, β)Ws;n=0 (α, β) is just the analytical formula of the Wigner distribution function for the multi-photon added two-mode squeezed vacuum. However, for r = 0, Ws;r =0 (α, β) =

   1 2  2 2 |α| |β| exp − + , π 2 (2n + 1)2 2n + 1

(6.247)

thus Eq. (6.241) becomes the Wigner distribution function for the multi-photon added two-mode thermal state, i.e., 1 Hm,m (χα∗ , χα) π 2 m!n!(2n + 1)m+n+2    2  2 |α| + |β|2 , × Hn,n (χβ ∗ , χβ) exp − 2n + 1

W p;r =0 (α, β) =

(6.248)

√ √ where χ = 2 n + 1/ 2n + 1. Using Eq. (6.241), the Wigner distribution function W p (α, β) is plotted for several different values of m, n, n and γ in Fig. 6.9. Clearly, squeezing in one of the quadratures is shown via comparing Fig. 6.9a and e. For any values of m, n, n and r , the non-Gaussian function W p (α, β) always shows the partial negativity in the phase space, which is an indicator of the nonclassicality of the state. Figure 6.9a–d show that, for given n, γ and different values of m, n, several interesting results are presented as follows: 1 With the increase of even (odd) m, there always exist two (three) main peaks and m − 2 (m − 3) secondary peaks, one (m − 1) main valley(s) and m (two) secondary valleys. The multi-peak feature represents the formation of quantum interference halfway between the two adjacent main peaks, and the interference frequency increases as m increases. This result comes from the two-mode squeezing operation in the state ρ p because it fully differs from the double-peak structure of the multi-photon added single-mode squeezed thermal state; 2 The negativity cannot monotonically change with the photon-subtraction number m because

220

6 Evolution and Decoherence of Quantum States in Open Systems

Fig. 6.9 The Wigner distribution functions for the multi-photon added two-mode squeezed thermal states for a m = 2, n = 1, r = 0.3, n = 0.1; b m = 3, n = 1, r = 0.3, n = 0.1; c m = 4, n = 1, r = 0.3, n = 0.1; d m = 5, n = 1, r = 0.3, n = 0.1; e m = 2, n = 1, r = 0.8, n = 0.1; f m = 2, n = 1, r = 0.3, n = 0.5

the function Ws (α, β) is always positive and the negativity of the function W p (α, β) fully depends on the term Cm,n (α, β); 3 The similar results are found for the case of n. Besides, as the parameter n (or T ) increases, the partial negativity of the function W p (α, β) decreases gradually and the interference structure is obviously destroyed, which refers to the loss of nonclassicality of the state ρ p [see Fig. 6.9a and f]. Substituting Eq. (6.237) into Eq. (6.232) leads to the form 2  2  ∂ 2m ∂ 2n d αd β A A23 G n−m π 2 Nm,n K n+1 ∂t m ∂t m ∂s n ∂s n π2      2   2 × exp −A1 α  + β   + A2 α β  + α∗ β ∗ exp[−ss      + A4 α + A5 β ∗ + A6 t s + A4 α∗ + A5 β  + A6 t  s  ]    2  2  G2 × exp −A3 α − α e−κt  + β − β  e−κt  exp − tt  K       ∗ ∗    + A7 β + A8 α t + A7 β + A8 α t t=t  =s=s  =0 . (6.249)

W p (α, β, t) =

6.2 Evolution of Quantum States in the Laser Process

221

Using the integral formula in Eq. (1.30), we thus have W p (α, β, t) = 

A23

2 A1 + A3 e−2κt − A22       A1 + A3 e−2κt A23 e−2κt × exp − A3 |α|2 + |β|2  2 A1 + A3 e−2κt − A22    A2 A23 e−2κt ∗ ∗ + αβ + α β 2 A1 + A3 e−2κt − A22 AG n−m ∂ 2m ∂ 2n π 2 Nm,n K n+1 ∂t m ∂t m ∂s n ∂s n  × exp −h 1 tt  − h 2 ss  + h 3 (st + s  t  ) + h 4 t  +h ∗ t  + h 5 s + h ∗ s  ]  ,  ×

4

5

s=s =t=t =0

(6.250)

where the parameters are, respectively, 2(2G − K )(2n¯ + 1)T + 2K e−2κt , K (2n¯ + 1)T √ 2 K (2 − G)(2n¯ + 1)T + Ge−2κt = √ , G (2G − K )(2n¯ + 1)T + K e−2κt √   2h G e−2κt − (2n¯ + 1)T = (2G − K )(2n¯ + 1)T + K e−2κt

A9 = A10 A11

(6.251)

and A2 G2 A9 A2 A2 A9 A2 − 8 − 2 11 2 , h 2 = 1 − 4 − 2 10 2 , K A9 A9 A9 − A2 A9 − A2 A4 A8 A9 A10 A11 h 3 = A6 + + 2 , A9 A9 − A22 

A3 A8 A2 A3 A11 −κt ∗ A3 A9 A11 −κt e α + 2 h4 = + 2 e β, A9 A9 − A22 A9 − A22 

A3 A4 A2 A3 A10 −κt A3 A9 A10 −κt ∗ e α+ 2 h5 = + 2 e β . A9 A9 − A22 A9 − A22 h1 =

(6.252)

Thus, using a similar way to calculating Eq. (6.241), we finally obtain the analytical evolution of the Wigner distribution function for the state ρ p in the thermal environment, that is W p (α, β, t) = Cm,n (α, β, t)Ws (α, β, t),

(6.253)

222

6 Evolution and Decoherence of Quantum States in Open Systems

where Ws (α, β, t) is the evolution of the Wigner distribution function for the twomode squeezed thermal state in the thermal environment, that is A A23

 2 π 2 K A1 + A3 e−2κt − A22       A1 + A3 e−2κt A23 e−2κt − A3 |α|2 + |β|2 × exp  2 A1 + A3 e−2κt − A22    A2 A23 e−2κt ∗ ∗ + αβ + α β (6.254) 2 A1 + A3 e−2κt − A22

Ws (α, β, t) =



and   min(m,n) G n−m m n  m m h h p q Nm,n K n 1 2 p,q=0  p+q

 

h3 n n × p!q! √ p q h1h2



 h ∗5 h4 h5 h ∗4 Hn− p,n−q √ , √ . × Hm− p,m−q √ , √ h1 h1 h2 h2

Cm,n (α, β, t) =

(6.255)

Specially, for the limit case, κ¯n → 0, W p (α, β, t) becomes the Wigner distribution function for the state ρ p in the amplitude damping channel, which can be obtained via making the following substitutions 2 2(2G − K )T + 2K e−2κt , A9 → , T √ KT √     2 K (2 − G)T + Ge−2κt 2h G e−2κt − T →√ , A → √ 11 (2G − K )T + K e−2κt G(2G − K )T + G K e−2κt (6.256)

A3 → A10

in Eq. (6.253). On the other hand, at the initial time, i.e., κt = 0, thus T = 0 and G2 , h 2 = 1, h 3 = A6 , K h 4 = A8 α∗ + A7 β, h 5 = A4 α + A5 β ∗ , h1 =

(6.257)

so Eq. (6.253) reduces to the Wigner distribution function in Eq. (6.241). For κt → ∞, note that T → 1, h 1 →

A2 G2 (A1 A7 + A2 A8 )2 − 8− , K A1 (A21 − A22 )A1

6.2 Evolution of Quantum States in the Laser Process

223

A24 (A1 A5 + A2 A4 )2  −  2 , A1 A1 − A22 A1 A4 A8 (A1 A7 + A2 A8 )(A1 A5 + A2 A4 )  2  h 3 → A6 + + , A1 A1 − A22 A1

h2 → 1 −

h 4 → 0,

h5 → 0

(6.258)

and use the new formula of Jocobi polynomials in Eq. (6.222) and the following identity (6.259) Hm,n (0, 0) = (−)m m!δm,n , thus Eq. (6.253) finally becomes    2  1 2 2 |α| + |β| , exp − W p (α, β, ∞) = 2 π (2n¯ + 1)2 (2n¯ + 1)

(6.260)

which is just the Wigner distribution function for two-mode thermal state with the same mean thermal photon number n¯ , and has nothing with the photon addition numbers m, n and the squeezing parameter r of the initial state ρ p . Similarly, after a long time interaction with the thermal environment, thermal noise can also change the state ρ p into a highly classical thermal state without entanglement and squeezing, similar to the result (6.197) in single-mode thermal environment. Using Eq. (6.253), we present the evolution of the Wigner distribution function W p (α, β, t) with given n, n and r for several different values of m, n¯ and κt in Fig. 6.10. Clearly, at short times κt or for small values of n¯ , the function W p (α, β, t)

Fig. 6.10 The evolved Wigner distribution functions for the multi-photon added two-mode squeezed thermal states in the thermal environment for n = 1, n = 0.1, r = 0.3 and a m = 2, n¯ = 0.1, κt = 0.01; b m = 2, n¯ = 0.1, κt = 0.6; c m = 2, n¯ = 3, κt = 0.01; d m = 5, n¯ = 0.1, κt = 0.01

224

6 Evolution and Decoherence of Quantum States in Open Systems

always exhibits the complicated quantum interference coming from the two-mode squeezing operation, as is in the absence of the multi-photon added single-mode squeezed thermal state, and has the partial negativity in the phase space. As κt or n¯ increases, the partial negativity of the function W p (α, β, t) slowly decreases and the quantum interference gradually disappears [see Fig. 6.10a–c]. When κt → ∞, the function W p (α, β, t) completely loses its partial negativity and interference feature becomes the Gaussian distribution of W p (α, β, ∞). Besides, with the increase of m or n, the decoherence time scales κt become longer.

References 1. Fan HY, Hu LY (2008) Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach. Mod Phys Lett B 22(25):2435–2468 2. Wu WF (2015) Analytical evolution of displaced thermal states for amplitude damping. Chin J Phys 53(4):17–24 3. Meng XG, Goan XS, Wang JS, Zhang R (2018) Nonclassical thermal-state superpositions: analytical evolution law and decoherence behavior. Opt Commun 411(7):15–20 4. Meng XG, Wang JS, Liang BL, Han CX (2018) Evolution of a two-mode squeezed vacuum for amplitude decay via continuous-variable entangled state approach. Front Phys 13(5):130322 5. Meng XG, Wang Z, Fan HY, Wang JS (2012) Squeezed number state and squeezed thermal state: decoherence analysis and nonclassical properties in the laser process. J Opt Soc Am B 29(7):1835–1843 6. Meng XG, Wang Z, Fan HY, Wang JS (2012) Nonclassicality and decoherence of photonsubtracted squeezed vacuum states. J Opt Soc Am B 29(11):3141–3149 7. Meng XG, Wang Z, Fan HY, Wang JS, Yang ZS (2012) Nonclassical properties of photonadded two-mode squeezed thermal states and their decoherence in the thermal channel. J Opt Soc Am B 29(7):1844–1853 8. Bonifacio R, Lugiato LA (1982) Dissipative systems in quantum optics: resonance fluorescence, optical bistability, superfluorescence. Springer, Berlin 9. Carmichael HJ (1999) Statistical methods in quantum optics 1: master equations and FokkerPlanck equations. Springer, New York 10. Jeong H, Ralph TC (2006) Transfer of nonclassical properties from a microscopic superposition to macroscopic thermal states in the high temperature limit. Phys Rev Lett 97(10):100401 11. Jeong H, Ralph TC (2007) Quantum superpositions and entanglement of thermal states at high temperatures and their applications to quantum-information processing. Phys Rev A 76(4):042103 12. Gerry CC, Hach EE III (1993) Generation of even and odd coherent states in a competitive two-photon process. Phys Lett A 174(3):185–189 13. Ourjoumtsev A, Jeong H, Tualle-Brouri R, Grangier P (2007) Generation of optical ‘Schröinger cats’ from photon number states. Nature 448(7155):784–787 14. Kim MS, de Oliveira FAM, Knight PL (1989) Properties of squeezed number states and squeezed thermal states. Phys Rev A 40(5):2494–2053 15. Hu LY, Fan HY (2008) Statistical properties of photon-subtracted squeezed vacuum in thermal environment. J Opt Soc Am B 25(12):1955–1964

Chapter 7

Generalized Binomial Theorems and Multi-variable Special Polynomials Involving Hermite Polynomials

Hermite polynomials as a kind of well-known special polynomials can be used widely in mathematics and physics because they possess some fundamental properties (e.g., orthogonality and completeness) and relevant identities (e.g., recurrence formula and generating function). Usually, the single-variable Hermite polynomials Hn (x) are defined by their generating functions [1, 2] ∞ n  t 2 Hn (x). e−t +2xt = (7.1) n! n=0 Physically, Hn (x) can solve the eigenvalue problems of the quantum (coupled) Harmonic oscillator, the fractional Fourier transformation [3], and generalized angular momentum systems [4–6]. For the two-variable case, Hermite polynomials Hn,m (x, y) are given as [7] 



e−tt +t x+t y =

∞  t n t m Hn,m (x, y), n!m! m,n=0

(7.2)

or their partial differential expressions and power-series expansions that respectively are ∂ n+m −tt  +t x+t  y e |t=t  =0 ∂t n ∂t m min(m,n)  n m  = l!(−1)l x n−l y m−l . l l

Hn,m (x, y) =

(7.3)

l=0

The physical explanations of Hn,m (x, y) are the transition amplitudes of Fock states driven by the forced Harmonic oscillator [8] or the eigenmodes of two-dimensional complex fractional Fourier transformation [3], and they are useful tools for analyzing the Talbot effect in a quadratic-index medium, quantum entanglement, and Bargmann © Science Press 2023 X.-G. Meng et al., Entangled State Representations in Quantum Optics, https://doi.org/10.1007/978-981-99-2333-5_7

225

226

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

transformation [9]. Moreover, the single- and two-variable Hermite polynomials are used to derive new asymptotic formulas [10] and Boson operator ordering products [11], and prepare new non-Gaussian quantum states as a class of critical quantum information entangled resources for realizing quantum metrology, communications and computation [12]. So far, several deformed Hermite polynomials (e.g., degenerate, holomorphic and q-deformed Hermite polynomials [2, 13, 14]) are successively constructed and widely used for many fields of mathematics and physics (e.g., probability theory and number theory). In this charter, by using the IWOP method and the entangled state representation with continuous variables, the ordinary binomial theorem is extended to the case involving two-variable Hermite polynomials that cannot obtain at all mathematically [7]; On the basis of ordinary Hermite polynomials, two multi-variable special polynomials and their generating functions are introduced, from which some new operator identities and integral formulas are obtained [15]; Besides, the applications of the generalized binomial theorems and multi-variable special polynomials in dealing with some quantum optical problems, such as the normalization of quantum states, photon-counting distribution and Wigner distribution function, are discussed in detail.

7.1 Generalized Binomial Theorems Involving Hermite Polynomials As a basic algebra theorem, binomial expansion is widely used in mathematics physics [16, 17]. Using the theorem, we expand the polynomial (x + y)n as a sum involving the terms x k y n−k , that is (x + y)n =

n    n k=0

k

x k y n−k ,

(7.4)

  n where are the binomial coefficients interpreted as combinatorial quantities that k can express the number of ways of selecting k objects out of n without replacement. Indeed, the multiple-angle identities, derivatives of some power function and probability mass function of the negative binomial distribution are introduced using the theorem in Eq. (7.4). Besides, the binomial theorem is true when x and y are complex. At present, one thinks it is necessary to develop the binomial theorem in order to deal with the power-series expansions of the coordinate operator n  a + a† , Q = √ 2n n

(7.5)

7.1 Generalized Binomial Theorems Involving Hermite Polynomials

227

† where  a, a are Boson annihilation and creation operators that obey the relation  † a, a = 1. In this case, we obtain the relation as follows

n  n   a + a† 1  n a k a †n−k . Q = √ = √ n n k 2 2 k=0 n

(7.6)

Using the completeness relation of coordinate eigenstates |q

∞

−∞

dq |q q| = 1, Q |q = q |q

(7.7)

and the IWOP method, we give the expansion of the operator Q n as Q =



∞

n

dqq |q q| =

∞

n

−∞

dq 2 √ q n : e−(q−Q) : π

−∞

= (−i/2)n : Hn (iQ) : .

(7.8)

Equation (7.8) can also be proved via another method as follows. Using the Glauber formula in Eq. (6.114), thus we have eλQ = eλ(a

†

√ +a )/ 2

=e

√λ a † 2

e

√λ a 2

†

a λ√ +λ √a2 + 14 λ2 2

e4λ =: e 1

2

:.

(7.9)

Further, using the generating functions for single-variable Hermite polynomials, we obtain eλQ = : e =

√λ (a † +a)−(−iλ/2)2 2

∞  (−iλ/2)n

n!

n=0

2

: = : e2(−iλ/2)(iQ)−(−iλ/2) :

: Hn (iQ) : =

∞  λn n=0

n!

Qn .

(7.10)

By comparing the coefficient of λn in Eq. (7.10), the proof of Eq. (7.8) is completed. Next, inserting a Hermite polynomial Hm−l,l (x, y) into the ordinary binomial expansion in Eq. (7.4), thus we obtain a new generalized binomial formula as  m   m l

l=0

τ m−l q l Hm−l,l (x, y)

(7.11)

or using the product of two binomial expansions  m   m l=0

l

τ

m−l l

q

n    n k=0

k

σ n−k p k

(7.12)

228

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

to introduce another generalized binomial expansion involving a Hermite polynomial Hm−l,n−k (x, y), that is  m   m l=0

l

τ m−l q l

n    n

k

k=0

σ n−k p k Hm−l,n−k (x, y),

(7.13)

where τ , q, σ, and p are arbitrary real parameters, x, y are the variables of Hm−l,n−k (x, y). Besides, by generalizing Eq. (7.13), we construct a new sum involving the product Hl,k (μ, ν)Hm−l,n−k (x, y), i.e.,   n  m   m n l=0 k=0

l

k

f l g k Hl,k (μ, ν) Hm−l,n−k (x, y),

(7.14)

where f, g are arbitrary and real, and (μ, ν) is a pair of function variables likes (x, y). To calculate the generalized binomial summations in Eqs. (7.11), (7.13), and (7.14), we first derive two lemmas that are respectively related to the normal ordering of Hm,n (a − b† , a † − b) and the anti-normal ordering of a †l a k . Lemma 1. We now prove the following operator identity Hm,n (a − b† , a † − b) = : (a − b† )m (a † − b)n : .

(7.15)

Indeed, using the completeness relation of the entangled states |η, that is

d2 η |η η| = π



d2 η −[η−(a−b† )][η∗ −(a † −b)] :e : =1 π

(7.16)

and the eigen-equations     a − b† |η = η |η , a † − b |η = η ∗ |η ,

(7.17)

we obtain Hm,n (a − b† , a † − b) =

d2 η † ∗ † Hm,n (η, η ∗ ) : e−[η−(a−b )][η −(a −b)] : . π

(7.18)

Further, using the integral expression of two-variable Hermite polynomials, that is Hm,n (η, η ∗ ) = im+n e|η|

2



d2 z n ∗m −|z|2 −iηz−iη∗ z ∗ z z e π

and the IWOP method, then Eq. (7.18) becomes

(7.19)

7.1 Generalized Binomial Theorems Involving Hermite Polynomials

Hm,n (a − b , a − b) = i †

†

m+n

229

d2 z n ∗m z z : δ(a † − b − iz) π

× δ(a − b† − iz ∗ )e−|z|

2

−(a−b† )(a † −b)

:

= : (a − b ) (a − b) : , † m

†

n

(7.20)

thus the proof of the Lemma 1 is completed. Lemma 2. The anti-normal ordering of a †l a k reads  . . a †l a k = ..Hl,k a † , a ...

(7.21)

.. Noticing that the operators a and a † are commutative within the symbol .. .., and introducing the real parameter λ, σ and using Eq. (7.2), we arrive at ∞  λl σ k †l k † † † a a = eλa eσa = eσa eλa e[λa ,σa ] l!k! l,k=0 ∞  . .  .. λl σ k . † Hl,k a † , a ... . = ..eλa +σa−λσ .. = l!k! l,k=0

(7.22)

By comparing the coefficient of λl σ k in Eq. (7.22), we obtain the Lemma 2. 1

Generalized binomial theorem involving Hm−l,l (x, y)

To calculate the binomial sum in Eq. (7.11), using the operator Hermite polynomial Hm−l,l (a − b† , a † − b) and the Lemma 1, we have the operator identity  m   m

τ m−l q l Hm−l,l (a − b† , a † − b) l l=0  m   m τ m−l q l : (a − b† )m−l (a † − b)l : = l l=0 m  = : τ (a − b† ) + q(a † − b) : .

(7.23)

 Further, in order to deprive of the symbol : : from the both sides of : τ (a − b† )+  m q(a † − b) : , we use the generating function for Hm (x) to calculate the sum

230

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …



  ∞  τ (a − b† ) + q(a † − b) sm τ (a − b† ) + q(a † − b) 2 Hm = exp −s + s √ √ m! 2 qτ qτ m=0 

† † τ (a − b ) + q(a − b) : = : exp s √ qτ

m ∞  τ (a − b† ) + q(a † − b) sm = : :. √ m! qτ m=0 (7.24) By comparing the first term and the last term of Eq. (7.24), we obtain

:

τ (a − b† ) + q(a † − b) √ qτ



m : = Hm

 τ (a − b† ) + q(a † − b) . √ 2 qτ

(7.25)

So combining Eqs. (7.23) and (7.25) leads to  m   m l=0

=

l

τ m−l q l Hm−l,l (a − b† , a † − b)

√ m qτ Hm



 τ (a − b† ) + q(a † − b) . √ 2 qτ

(7.26)

Noting that [a − b† , a † − b] = 0 and making the substitutions (a − b† , a † − b) → (x, y) and (q, τ ) → ( f, g) in Eq. (7.26), we obtain  m   m l=0

l

f l g m−l Hm−l,l (x, y) =



fg



m Hm

gx + f y √ 2 fg

 ,

(7.27)

which is just the result of summation in Eq. (7.11) only related to the polynomials Hm (·). Specially, for the case of f = 1 and g = 1, Eq. (7.27) becomes  m   m l=0

2

l

 Hm−l,l (x, y) = Hm

x+y 2

 .

(7.28)

Generalized binomial theorem involving Hm−l,n−k (x, y)

Now we calculate the result of summation in Eq. (7.13). Replacing the polynomial . . H (x, y) by the anti-normal ordering product ..H (a † , a).. in Eq. (7.13) m−l,n−k

and using the Lemma 2, we arrive at

m−l,n−k

7.1 Generalized Binomial Theorems Involving Hermite Polynomials

 m   m

231

n    n

. . τ q σ n−k p k ..Hm−l,n−k (a † , a).. l k l=0 k=0  m  n     m n τ m−l q l σ n−k p k : a †m−l a n−k : = l k m−l l

l=0

k=0

= : (q + τ a ) ( p + σa)n : . † m

(7.29)

Thus, making the summation for the right-hand side of Eq. (7.29) leads to ∞  t m sn † : (q + τ a † )m ( p + σa)n : = et (q+τ a ) es( p+σa) . m!n! m,n=0

(7.30)

Using the generating function in Eq. (7.2) and the Glauber formula in Eq. (6.114), we obtain the summation for the left-hand side of Eq. (7.29) as  ∞ m  n     . . t m sn  m n τ m−l q l σ n−k p k ..Hm−l,n−k (a † , a).. k m!n! l=0 l m,n=0 k=0 . . † = ..es( p+σa) et (q+τ a )−tτ sσ .. . . p q † = ..esσ( σ +a) etτ ( τ +a )−(tτ )(sσ) .. ∞ . q  (tτ )m (sσ)n .. p + a † , + a ... = .Hm,n m!n! τ σ m,n=0

(7.31)

Comparing the term t m s n on the both sides of Eq. (7.31), we find  m   m l=0

l

τ

m−l l

q

n    n k=0

k

. . σ n−k p k ..Hm−l,n−k (a † , a)..

. q . p + a † , + a ... = τ m σ n ..Hm,n τ σ

(7.32)

Noticing that the both sides of Eq. (7.32) are in anti-normal ordering, so we can make the substitutions (a † , a) → (x, y) and (q/τ , p/σ) → ( f, g) in Eq. (7.32) that leads to the new binomial theorem, i.e.,   m  n   m n l=0 k=0

l

k

f l g k Hm−l,n−k (x, y) = Hm,n ( f + x, g + y) .

Specially, when f = g = 1, we obtain

(7.33)

232

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

  m  n   m n l=0 k=0

3

l

k

Hm−l,n−k (x, y) = Hm,n (1 + x, 1 + y) .

(7.34)

Generalized binomial theorem involving Hl,k (μ, ν)Hm−l,n−k (x, y)

Next we turn to the summation in Eq. (7.14). For this, replacing Hl,k (μ, ν) by the anti . . normal ordering product ..H a † , a .. in (7.14), and using Eqs. (7.21) and (7.33), l,k

we have  n   m    . m  n .. .Hl,k a † , a .. f l g k Hm−l,n−k (x, y) l k l=0 k=0   m n    m  n = : a †l a k : f l g k Hm−l,n−k (x, y) l k l=0 k=0   = : Hm,n f a † + x, ga + y : .

(7.35)

Further, using Eq. (7.2) we obtain ∞    t m t n  †  : Hm,n f a † + x, ga + y : = : e−tt +t ( f a +x)+t (ga+y) : m!n! m,n=0 





= e−tt et f a et ga et x+t y . .  †  = ..e−tt (1+ f g)+t ( f a +x)+t (ga+y) ... †

(7.36)

√ √ Taking t 1 + f g = s and t  1 + f g = s  , thus Eq. (7.36) becomes ∞    t m t n : Hm,n f a † + x, ga + y : m!n! m,n=0

. −ss  +s √f a† +x +s  √ga+y .. 1+ f g 1+ f g . = ..e   ∞  f a † + x ga + y .. s m s n .. ,√ = .Hm,n √ . m!n! 1+ fg 1+ fg m,n=0   ∞  f a † + x ga + y .. t m t n m+n . . 2 ,√ .Hm,n √ .. = (1 + f g) m!n! 1+ fg 1+ fg m,n=0

(7.37)

Comparing the term t m t n on the both sides of (7.37), we reach to the operator identity 



: Hm,n f a + x, ga + y : = (1 + f g) †

m+n 2

.. .Hm,n



 f a † + x ga + y .. ,√ .. √ 1+ fg 1+ fg (7.38)

7.1 Generalized Binomial Theorems Involving Hermite Polynomials

233

Combining Eqs. (7.35) and (7.38), we see  n   m    . m  n .. .Hl,k a † , a .. f l g k Hm−l,n−k (x, y) l k l=0 k=0   f a † + x ga + y .. m+n . = (1 + f g) 2 ..Hm,n √ ,√ .. 1+ fg 1+ fg

(7.39)

Considering that both sides of (7.39) are in anti-normal ordering, so making the substitutions a † → μ and a → ν, we give   m  n   m n l

l=0 k=0

k

f l g k Hl,k (μ, ν) Hm−l,n−k (x, y) 

= (1 + f g)

m+n 2

Hm,n

 fμ+x gν + y ,√ , √ 1+ fg 1+ fg

(7.40)

which is just another generalized binomial theorem involving the product Hl,k (μ, ν) Hm−l,n−k (x, y). Particularly, for the case of f = g = 1, Eq. (7.40) reduces to   n  m   m n

Hl,k (μ, ν) Hm−l,n−k (x, y) k   √ μ+x ν+y m+n . = 2 Hm,n √ , √ 2 2 l

l=0 k=0

(7.41)

According to the latest generalized binomial theorem in (7.33), we can change the † † † † multiple-photon-subtracted squeezed state a m bn esa b +ra +tb |00 denoted as |sub into the two-variable Hermite polynomial excitation on squeezed state that can realize quantum information tasks such as quantum teleportation and quantum key distribution and so on, where s, r , and t are arbitrary and real. Indeed, the state |sub can be implemented physically via conditional measurements on a beam splitter [18]. Using the Baker-Hausdorff formula (1.27) to express the operator † † † † a m bn esa b +ra +tb as a m bn esa = esa

b +ra † +tb†

† †

b +ra † +tb†

† †

m  n  a + sb† + r b + sa † + t .

(7.42)

Noticing that both the operators a + sb† and b + sa † are commutative, thus we can † † expand the exponential operator eτ (a+sb )+υ(b+sa ) as † † eτ (a+sb )+υ(b+sa ) =

∞  m  n τ m υn  a + sb† b + sa † , m!n! m,n=0

(7.43)

234

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

where τ , υ are also arbitrary and real. On the other hand, using the generating function † † for Hm,n (x, y), the operator eτ (a+sb )+υ(b+sa ) can be rewritten as † † † † eτ (a+sb )+υ(b+sa ) = : esτ υ+τ (a+sb )+υ(b+sa ) :

  ∞  a + sb† b + sa † τ m υn m+n 2 :. = : Hm,n , √ √ (is) m!n! is is m,n=0

(7.44)

 m Comparing Eqs. (7.43) and (7.44), we find that the normal ordering of a + sb†   n b + sa † reads m  n  m+n b + sa † = (is) 2 : Hm,n a + sb†



a + sb† b + sa † , √ √ is is

 :.

(7.45)

Thus, using the generalized binomial expansion in Eq. (7.33), we have  m  n a + sb† + r b + sa † + t   n  m    l  k m n r m−l t n−k a + sb† b + sa † = l k l=0 k=0     n m   l+k m n = (is) 2 r m−l l k l=0 k=0   a + sb† b + sa † n−k : ×t : Hl,k , √ √ is is   r + a + sb† t + b + sa † m+n :. = (is) 2 : Hm,n , √ √ is is

(7.46)

Inserting Eq. (7.46) into Eq. (7.42) shows that the state |sub can be rewritten as a Hermite-polynomial-weighted quantum state, i.e., |sub = esa

b +ra † +tb†

† †



× : Hm,n = (is)

m+n 2

(is)

m+n 2

 r + a + sb† t + b + sa † : |00 , √ √ is is   r + sb† t + sa † sa † b† +ra † +tb† |00 . e Hm,n , √ √ is is

(7.47)

From above, we can see that the state |sub is a new non-Gaussian  entangled state † † √ , t+sa √ on the Gaussian after acting the operator Hermite polynomial Hm,n r +sb is is  † † † † † † √ , t+sa √ squeezed state esa b +ra +tb |00. So, the operator Hm,n r +sb is a kind of is is two-mode non-Gaussian operation. Further, in terms of Eq. (7.47), the normal ordering product of the density operator of the state |sub is obtained as

7.1 Generalized Binomial Theorems Involving Hermite Polynomials

235

 r + sb† t + sa † , √ √ is is   r + sb t + sa sa † b† +ra † +tb† ×e Hm,n √ , √ −is −is 

|sub sub| = s m+n : Hm,n

× e−a

†

a−b† b sab+ra+tb

e

:,

(7.48)

which provides convenience for calculating the normalization coefficient, EPR correlation and quantum teleportation of the state |sub. Further, we use the generalized binomial theorems (7.27) and (7.40) to obtain the Wigner distribution function for the atomic coherent state |τ  and its marginal distributions. Using the Schwinger Boson realization of angular momentum operators, the normalized atomic coherent state |τ  reads |τ  = D

2 j  1/2  2j n=0

n

τ n |2 j − n, n ,

(7.49)

where D = (1 + |τ |2 )− j is the normalization coefficient. Using the expansion of the entangled state |η in Eq. (1.86), the entangled state |η representation of two-mode Wigner operator, that is (σ, γ) =

d2 η ∗ ∗ |σ − η σ + η| eηγ −η γ , 3 π

(7.50)

and the complex conjugate relation H∗m,n ( , ∗ ) = Hn,m ( , ∗ ), we give the inner product η| τ  as η| τ  = De−|η|

2

/2

√   (−τ )n (2 j)! . Hn,2 j−n η, η ∗ n!(2 j − n)! n=0

2j 

(7.51)

Thus, using Eqs. (1.86), (7.3), (7.49), and (7.51), we obtain the Wigner distribution function for the state |τ  as     W σ, γ = tr[ σ, γ |τ  τ |]   2 2j  D2 e−|σ|  2 j 2j (−1)m+n τ ∗m τ n = n (2 j)! m,n=0 m 2  ∂4 j d η × 2 j−m m n 2 j−n exp − |η|2 + ηγ ∗ ∂t ∂t ∂r ∂r π3   − η ∗ γ − tt  + t (σ − η) + t  σ ∗ − η ∗   −rr  + r (σ + η) + r  σ ∗ + η ∗ |t=t  =r =r  =0 .

(7.52)

236

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

Using the integral formula in Eq. (1.30) to integrate (7.52) over all complex values of η and completing the high-order differentiation, we have   2 2 2j    D2 e−|σ| −|γ|  2 j 2j W σ, γ = (−1)m+n τ ∗m τ n n π 2 (2 j)! m,n=0 m     × H2 j−m,2 j−n σ + γ, σ ∗ + γ ∗ Hm,n σ ∗ − γ ∗ , σ − γ .

(7.53)

According to the new binomial theorem in Eqs. (7.40), (7.53) is changed to the compact form 2 2   e−|γ| −|σ| H2 j,2 j (ϑ, ϑ∗ ), W σ, γ = 2 (7.54) π (2 j)! where ϑ=

(σ + γ) − τ ∗ (σ ∗ − γ ∗ )

. 1 + |τ |2

(7.55)

Further, using the relation between the Hermite polynomials Hm,n (λ, λ∗ ) and Laguerre polynomials Llp (r 2 ) which yields Hm,n (λ, λ∗ ) = ei(m−n)θ Hm,n (r, r ) = ei(m−n)θ (−1) p p!r l Llp (r 2 ),

(7.56)

where λ = r eiθ , p = min(m, n) and l = |m − n|, we obtain   (−1)2 j e−|γ| W σ, γ = π2

2

−|σ|2

  L2 j |ϑ|2 ,

(7.57)

  which shows that the Wigner distribution function W σ, γ for the state |τ  is only  2 related to the Laguerre polynomials L2 j |ϑ| . In terms of the original definition of   L2 j |ϑ|2 , we conclude that the state |τ  may exhibit obvious non-Gaussian features.  Eqs. (1.86) and (7.27), the concise form of the marginal distribution of Using W σ, γ in the γ-direction is given by

  D2 |τ |2 j e−|γ| d σW σ, γ = π(2 j)! 2

2

 2  ∗   H2 j γ √− τ γ  .  2 −τ 

(7.58)

Similarly, using the expansion of the entangled state |ζ in Eq. (1.91) that is |ζ = e−|ζ|

2

/2

∞  Hm,n (ζ, ζ ∗ ) |m, n , √ m!n! m,n=0

(7.59)

  we arrive at another marginal distribution of W σ, γ in the σ-direction, that is

7.2 Multi-variable Special Polynomials and Their Generating Functions



  D2 |τ |2 j e−|σ| d γW σ, γ = π(2 j)! 2

2

   ∗  τ σ 2 H2 j σ + √  .  2 τ

237

(7.60)

It can see from Eqs. (7.58) and (7.60) that the marginal distributions of   clearly W σ, γ are just related to the module square of the Hermite polynomials H2 j (·).

7.2 Multi-variable Special Polynomials and Their Generating Functions By virtue of the power-series expansions of the ordinary Hermite polynomials in Eq. (7.3), we now introduce two new multi-variable special polynomials and their generating functions. For this purpose, replacing the terms (−1)l and x n−l y m−l by a more general power function ϑl (ϑ being a arbitrary parameter) and the product of two different Hermite polynomials that is Hn−l (x/2)Hm−l (y/2) in Eq. (7.3), we have min(n,m) x  y  n m  Hm−l . (7.61) l!ϑl Hn−l l l 2 2 l=0 Given Eq. (7.61), here we show a new useful special polynomial and find its specific applications in quantum physics. Besides, we present another new special polynomial when x n−l y m−l is replacedby theproduct of two-variable Hermite polynomials, that is Hn−i,m− j (x, y)Hl−i,k− j x  , y  .

7.2.1 Three-variable Case Using the original definition of Hn,m (x, y) in Eq. (7.3), we expand the polynomials Hn,m (x, y; ϑ) as ∂ n+m exp (ϑsτ + sx + τ y)|s=τ =0 ∂s n ∂τ m min(n,m)  n m  = l!ϑl x n−l y m−l . l l

Hn,m (x, y; ϑ) =

(7.62)

l=0

Obviously, √ Hn,m (x, y; ϑ) become Hn,m (x, y) via making the substi√ the polynomials  and i → s tutions i ϑs √   ϑτ → τ√in Eq. (7.62).  Introducing the superposed operators X = 2 a + a † and Y = 2 b + b† that yield the relation [X, Y ] = 0, where a † , b† are respectively the Boson creation operators of two modes, and replacing (x, y) by (X, Y ) in Eq. (7.62), we give the operator identity as

238

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

∂ n+m exp (ϑsτ + s X + τ Y )|s=τ =0 ∂s n ∂τ m min(n,m)  n m  = l!ϑl X n−l Y m−l . l l

Hn,m (X, Y ; ϑ) =

(7.63)

l=0

Using the anti-normal ordering products of X n and Y m , i.e., . X = ..Hn



n

X 2



. .. ., Y m = ..Hm

  Y .. ., 2

(7.64)

we rewrite Eq. (7.63) as Hn,m (X, Y ; ϑ) =

min(n,m)   l=0

n l



m l



. l!ϑl ..Hn−l



X 2

 Hm−l

  Y .. .. 2

(7.65)

On the other hand, using the Glauber formula in Eq. (6.114), we obtain .  . exp (ϑsτ + s X + τ Y ) = .. exp s X + τ Y + ϑsτ − s 2 − τ 2 ...

(7.66)

Further, combining Eqs. (7.63), (7.65), and (7.66), we obtain the operator identity as   .  ∂ n+m .. 2 2 . . − τ . exp s X + τ Y + ϑsτ − s  ∂s n ∂τ m s=τ =0     min(n,m)  n m  . X Y .. l!ϑl ..Hn−l Hm−l = .. l l 2 2

(7.67)

l=0

Noticing that both sides of Eq. (7.67) are in anti-normal ordering, thus replacing (X, Y ) by (x, y) in Eq. (7.67) and comparing with Eq. (7.3), we easily obtain ∞    sn τ m Hn,m (x, y; ϑ) , exp −s 2 − τ 2 + ϑsτ + sx + τ y = n!m! n,m=0

(7.68)

where   ∂ n+m exp sx + τ y + ϑsτ − s 2 − τ 2 s=τ =0 n m ∂s ∂τ min(n,m) x  y  n m  l!ϑl Hn−l Hm−l = l l 2 2 l=0

Hn,m (x, y; ϑ) =

(7.69)

is a new two-index, three-variable special polynomial whose generating function is exp(−s 2 − τ 2 + ϑsτ + sx + τ y).

7.2 Multi-variable Special Polynomials and Their Generating Functions

239

Particularly, for ϑ = 0, the polynomial Hn,m (x, y; ϑ) reduces to the product Hn (x/2)Hm (y/2). Further, using the differential relation Hm (x) = 2mHm−1 (x), we obtain the partial differential of Hn,m (x, y; ϑ) with respect to the variables x and y, i.e., ∂ Hn,m (x, y; ϑ) = nHn−1,m (x, y; ϑ) , ∂x ∂ Hn,m (x, y; ϑ) = mHn,m−1 (x, y; ϑ) . ∂y

(7.70)

So, its high-order partial differential equation reads ∂ k+l n!m! Hn−k,m−l (x, y; ϑ) , Hn,m (x, y; ϑ) = ∂x k ∂ y l (n − k)! (m − l)!

(7.71)

which has the same form as the well-known differential equation of Hn,m (x, y).

7.2.2 Six-variable Case To derive six-variable special polynomials, we first introduce the product Hn,l (x, x  ; ν)Hm,k (y, y  ; υ) denoted as Fn,m,l,k (x, y, x  , y  ; ν, υ), so its partial differential form reads  ∂ l+k ∂ m+n exp νss  + υτ τ  ∂s n ∂τ m ∂s l ∂τ k  +sx + s  x  + τ y + τ  y  s=s  =τ =τ  =0 ,

Fn,m,l,k (x, y, x  , y  ; ν, υ) =

(7.72)

where ν, υ are arbitrary parameters. Introducing the following four superposed operators (7.73) W = a + b† , Z = a † + b, W  = c + d † , Z  = c† + d that commute with each other, where a † , b† , c† , d † are respectively the Boson creation operators of four modes, and making the substitutions x → W , y → Z , x  → W  and y  → Z  in Eq. (7.72), thus we obtain  ∂ l+k ∂ m+n exp νss  + υτ τ  n m l k ∂s ∂τ ∂s ∂τ  +sW + s  W  + τ Z + τ  Z  s=s  =τ =τ  =0 min(n,m,l,k)  n m l k  = i j i j

Fn,m,l,k (W, Z , W  , Z  ; ν, υ) =

i. j=0

× i! j!ν i υ j W n−i W l−i Z m− j Z k− j .

(7.74)

240

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

Further, using the s-ordered expansions of operators and the completeness relation of the entangled states |ζ in Eq. (1.91) as the common eigenstates of the operator pair (W, Z ) or (W  , Z  ) to arrive at the following operator identities  . . . . W n Z m = ..Hn,m (W, Z ) .., W l Z k = ..Hl,k W  , Z  ..,

(7.75)

thus Eq. (7.74) is rewritten as follows Fn,m,l,k (W, Z , W  , Z  ; ν, υ) =

min(n,m,l,k)   i. j=0

n i



m j

   l k i! j! i j

 . . × ν i υ j ..Hn−i,m− j (W, Z ) Hl−i,k− j W  , Z  .., (7.76) which refers to the anti-normal ordering of the operator function Fn,m,l,k (W, Z , W  , Z  ; ν, υ). On the other hand, using the Glauber formula in Eq. (6.114), we obtain   exp νss  + υτ τ  + sW + s  W  + τ Z + τ  Z   . . = .. exp −sτ − s  τ  + νss  + υτ τ  + sW + s  W  + τ Z + τ  Z  ...

(7.77)

By comparison of Eqs. (7.74), (7.76), and (7.77) and making the substitutions W → x, Z → y, W  → x  and Z  → y  in the result after comparison, we define a new four-index, six-variable special polynomial as  ∂ l+k ∂ m+n exp −sτ − s  τ  + νss  ∂s n ∂τ m ∂s l ∂τ k  +υτ τ  + xs + x  s  + yτ + y  τ  s=s  =τ =τ  =0 min(n,m,l,k)  n m l k  = i! j! i j i j i. j=0   × ν i υ j Hn−i,m− j (x, y) Hl−i,k− j x  , y  ≡ Fn,m,l,k (x, y, x  , y  ; ν, υ)

(7.78)

with the following generating function   exp −sτ − s  τ  + νss  + υτ τ  + xs + x  s  + yτ + y  τ  =

∞ 

s n τ m s l τ k Fn,m,l,k (x, y, x  , y  ; ν, υ). n!m!l!k! n,m,l,k=0

(7.79)

Similarly, using the differential relations of Hn,m (x, y), we also obtain the partial of Fn,m,l,k (x, y, x  , y  ; ν, υ) with respect to the variables x, y, x  and y  that is

7.2 Multi-variable Special Polynomials and Their Generating Functions 

241



∂ i+ j ∂i + j Fn,m,l,k (x, y, x  , y  ; ν, υ) ∂x i ∂ y j ∂x i  ∂ y  j  n!m!l!k! = (n − i)! (m − j)!(l − i  )!(k − j  )! × Fn−i,m− j,l−i  ,k− j  (x, y, x  , y  ; ν, υ).

(7.80)

7.2.3 New Operator Identity and Integral Formula Using Eqs. (7.65) and (7.69), we obtain a new operator identity as . . Hn,m (X, Y ; ϑ) = ..Hn,m (X, Y ; ϑ) ..,

(7.81)

which can help to obtain some meaningful identities about the polynomials Hn,m (x, y; ϑ). For example, according to the generating function for Hn,m (X, Y ; ϑ), we get the recurrence formula of Hn,m (X, Y ; ϑ) that is nmHn−1,m−1 (X, Y ; ϑ) − n X Hn−1,m (X, Y ; ϑ) + nHn,m (X, Y ; ϑ) = 0.

(7.82)

Inserting Eq. (7.81) into Eq. (7.82) leads to . . . . . . nm ..Hn−1,m−1 (X, Y ; ϑ) .. − n X ..Hn−1,m (X, Y ; ϑ) .. + n ..Hn,m (X, Y ; ϑ) .. = 0, (7.83) thus the recurrence formula of Hn,m (x, y; ϑ) reads nmHn−1,m−1 (x, y; ϑ) − nxHn−1,m (x, y; ϑ) + nHn,m (x, y; ϑ) = 0,

(7.84)

which has the same form as the ordinary Hermite polynomials Hn,m (x, y). From Eqs. (7.63) and (7.68), we then find ∞  sn τ m 2 2 Hn,m (X, Y ; ϑ) = e−s −τ +ϑsτ +s X +τ Y n!m! n,m=0

= : eϑsτ +s X +τ Y : =

∞  sn τ m : Hn,m (X, Y ; ϑ) : , n!m! n,m=0

(7.85)

which leads to another operator identity Hn,m (X, Y ; ϑ) = : Hn,m (X, Y ; ϑ) : .

(7.86)

242

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

When acting both sides of Eq. (7.86) on a two-mode vacuum, we get a new quantum √ † √ † 2a , 2b ; ϑ |00. Using Eq. (7.86) and the normal ordering state, that is Hn,m of the completeness of coordinate eigenstates, we have ∞ 1 2 2 dq1 dq2 : e−(q1 −Q 1 ) −(q2 −Q 2 ) : Hn,m (2q1 , 2q2 ; ϑ) Hn,m (X, Y ; ϑ) = π −∞ = : Hn,m (X, Y ; ϑ) : , (7.87) which leads to a new integral formula 1 π



∞ −∞

dq1 dq2 e−(q1 −x)

2

−(q2 −y)2

Hn,m (2q1 , 2q2 ; ϑ) = Hn,m (2x, 2y; ϑ) .

On the other hand, making the substitutions x → Eq. (7.68), we thus give e−s

2

√ √ −τ 2 +|ϑ|sτ + ϑsW + ϑ∗ τ Z

=

(7.88)

√ √ ϑW and y → ϑ∗ Z in

∞ √   √ sn τ m Hn,m ϑW, ϑ∗ Z ; ϑ . n!m! n,m=0

(7.89)

Further, using the generating function for Hn (x) in Eq. (7.1) and the Glauber formula (6.114), Eq. (7.89) therefore becomes ∞  .. −s 2 −τ 2 +√ϑsW +√ϑ∗ τ Z .. s n τ m .. .e .= .Hn n!m! n,m=0

√

ϑW 2



√  ϑ∗ Z .. Hm .. 2

(7.90)

By comparing Eqs. (7.89) and (7.90), we arrive at Hn,m

√

ϑW,

√

 . ϑ∗ Z ; |ϑ| = ..Hn

√ √   ϑW ϑ∗ Z .. Hm .. 2 2

(7.91)

Besides, based on Eqs. (7.1) and (7.68), we have √ √ −s −τ + ϑsW + ϑ∗ τ Z

e

2

2

∞  sn τ m Hn = n!m! n,m=0

= : e−s

2

√

ϑW 2



√  ϑ∗ Z Hm 2

√ √ −τ 2 +|ϑ|sτ + ϑsW + ϑ∗ τ Z

:  √ √ s τ : Hn,m ϑW, ϑ∗ Z ; |ϑ| : , = n!m! n,m=0 ∞ 

n m

which leads to the new operator identity

(7.92)

7.2 Multi-variable Special Polynomials and Their Generating Functions

√ √   √  √ ϑW ϑ∗ Z Hn ϑW, ϑ∗ Z ; |ϑ| : Hm = : Hn,m 2 2

243

(7.93)

√  √ or another new quantum state Hn,m ϑb† , ϑ∗ a † ; |ϑ| |00. Further, using the completeness relation of |ζ, where |ζ obey the following eigenvalue equations, i.e., W |ζ = ζ |ζ and Z |ζ = ζ ∗ |ζ, we obtain √ Hn

ϑW 2

√

 Hm

ϑ∗ Z 2



√  √  ϑζ ϑ∗ ζ ∗ d2 ζ −(ζ−W )(ζ ∗ −Z ) :e : Hn = Hm π 2 2 √  √ = : Hn,m ϑW, ϑ∗ Z ; |ϑ| : ,

(7.94)

thus we obtain another new integral formula as √  √  √  √ ϑζ ϑ∗ ζ ∗ ϑσ, ϑ∗ σ ∗ ; |ϑ| , Hm = Hn,m 2 2 (7.95) which is useful in calculating the normalization coefficient of the photon-modulated m vacuum ta + ra † |0 as seen below. In a similar way, we can also obtain some new related identities about the polynomials Fn,m,l,k (x, y, x  , y  ; ν, υ). For example, comparison of Eqs. (7.76)–(7.78) leads to

d2 ζ −(ζ−σ)(ζ ∗ −σ∗ ) Hn e π

. . Fn,m,l,k (W, Z , W  , Z  ; ν, υ) = ..Fn,m,l,k (W, Z , W  , Z  ; ν, υ)...

(7.96)

7.2.4 Applications 1

Normalization

Normalization is important for describing the probability of successful preparation of a quantum state and further exploring the basic properties and physical applications of this state. The multi-photon-modulated states a m S (r ) |0 can be obtained theoretically by operating the annihilation operator a (i.e., single-photon subtraction) on a squeezed vacuum S (r ) |0 for m times. And, it should be feasible to prepare a m S (r ) |0 experimentally when m is small, especially the single-photon subtraction has been produced by virtue of the high transmittance beam splitter [19]. Therefore, single-photonmodulated state aS (r ) |0 has been successfully realized in a periodically-poled KTiOPTO4 crystal [20]. According to the Lemma 2 in Eq. (7.21), we have

244

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

. . a †m a m = ..Hm,m (a † , a)..,

(7.97)

thus the normalization coefficient of a m S (r ) |0 reads     .. . 1 2 1 †2 † Nm = sech r 0| . exp a tanh r Hm,m (a , a) exp a tanh r .. |0 . (7.98) 2 2 Substituting the completeness of coherent states into Eq. (7.98), and using the generating function for Hn,m (x, y) and the integral formula in Eq. (1.30), we have Nm =

 ∂ 2m 1 2  2 2 4 (τ +s ) sinh 2r +τ s sinh r  e . s=τ =0 ∂s m ∂τ m

(7.99)

Comparing with the generating function for Hn,m (x, y; ϑ) leads to m  1 Hm,m (0, 0; −2 tanh r ) . Nm = − sinh 2r 4

(7.100)

Indeed, the normalization coefficient Nm is also expressed as the form of Legendre polynomials Pm (x) in terms of the new expansion of the polynomials Pm (x) [21]. So, by comparing these two expressions, we find that the polynomials Hn,m (x, y; ϑ) and Pm (x) obeys the following relation Hm,m (0, 0; −2 tanh r ) = m!(i2 sech r )m Pm (i sinh r ).

(7.101)

In particular, when m = 0, H0,0 (0, 0; −2 tanh r ) = 1, thus N0 = 1. For m = 1, H1,1 (0, 0; −2 tanh r ) = −2 tanh r , so N1 = sinh2 r , which is just the normalization coefficient of aS (r ) |0.  m Another example is the photon-modulated coherent state ta + ra † |γ, which can be obtained via operating the elementary coherent superposition of photon substraction and addition (i.e., ta + ra † ) to a coherent state |γ for m times, t, r † are the ratios of the photon subtraction and  addition in the operation ta + ra ,  2 2 † t + r = 1. In theory, the state ta + ra |γ can be prepared physically via a parametric down amplifier and two beam splitters [22]. Using the normal ordering m product of ta + ra † that is   m     ta + ra † rt † m :, ta + ra = −i : Hm i √ 2 2r t

(7.102)

 m we rewrite the state ta + ra † |γ as   m     tγ + ra † rt † m |γ . |γ = −i ta + ra Hm i √ 2 2r t

(7.103)

7.2 Multi-variable Special Polynomials and Their Generating Functions

245

Using the completeness of coherent states |β and function for m  the generating Hn,m (x, y; ϑ), thus the normalization coefficient of ta + ra † |γ reads      2 tγ ∗ + r β tγ + r β ∗ rt m d β −|β|2 −|γ|2 +β ∗ γ+βγ ∗ Hm i √ Hm −i √ e 2 π 2r t 2r t   m 2m  rt ∂ 2 2 ∗ = e−s −τ +2r sτ /t+ρs+ τ  2 ∂s m ∂τ m s=τ =0    m rt 2r , (7.104) = Hm,m , ∗ ; 2 t 

Nm =

√ √ where  =i 2 (tγ + r γ ∗ ) / r t. Specially, in the case of m = 0, H0,0 (, ∗ ; 2r/t) = 1, thus N0 = 1. For m = 1, H1,1 (, ∗ ; 2r/t) = 2r/t + ||2 , so we have  2 N1 = r 2 + tγ + r γ ∗  .

(7.105)

m When γ = 0, the coefficient Nm becomes (r t/2)  Hm,m (0, 0; 2r/t), which is the  † m |0. Indeed, the coefficient Nm normalization coefficient of the state ta + ra can also be obtained using the new integral formula in Eq. (7.95) because

 √   √ † ra ra Hm i √ Hm −i √ 2t 2t  √   √ ∗ 2 d α −(α−a)(α∗ −a † ) rα rα : = Hm i √ Hm −i √ :e π 2t 2t  √  √ 2r 2r 2r = : Hm,m −i √ a, i √ a † ; :. t t t

(7.106)

  for a two-mode entangled state Hn,m f a † , gb† S2 (r ) |00, where Hn,m  Finally,  f a † , gb† is a two-variable operator Hermite polynomial, f, g are arbitrary and  † †  a b − ab ] is two-mode squeezing operator. Experimentally, real, S2 (r ) = exp[r   the state Hn,m f a † , gb† S2 (r ) |00 can be produced by inputting the squeezed vacuum S2 (r ) |00 into two tunable paralleled beam splitters and performing the conditional multi-photonmeasurement  on each beam splitter [23]. Here, the normalization coefficient of Hn,m f a † , gb† S2 (r ) |00 is   † † Nn,m = sech2 r 00| eab tanh r Hn,m ( f a, gb) Hn,m f a † , gb† ea b tanh r |00 . (7.107) Substituting the completeness of two-mode coherent states |αβ into Eq. (7.107) and using the integral formula in Eq. (1.30), we obtain

246

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

 ∂ n+m ∂ n+m sτ + s  τ  exp − n m n m ∂s ∂τ ∂s ∂τ f  2  2 2  2  + f ss cosh r + g τ τ cosh r 

Nn,m =

s=τ =s =τ  =0

,

(7.108)

where f = 2/(2 − f g sinh 2r ). Further, using the original definition of the polyno  mials Fn,m,l,k (x, y, x , y ; ν, υ), thus the coefficient Nn,m reads Nn,m

  Fn,m,n,m 0, 0, 0, 0; f f 2 cosh2 r, fg 2 cosh2 r = . fn+m

(7.109)

  0, 0, 0, 0; f 2 , g 2 , which is For r = 0, f = 1, the coefficient Nn,m becomes  †Fn,m,n,m  just the normalization coefficient of Hn,m f a , gb† |00, however for f = g = 0, Nn,m = (−1)n+m n!m!. 2

Photon-counting distribution

The quantum photon-counting distribution in the sense of complete quantum mechanics was first introduced by Kelley and Kleiner [24], which is a probability statistical distribution of registering n photons within an interval. For a single-mode quantum state ρ, its photon-counting distribution is calculated as 2    d α −ξ|α|2  2   ξn   e Ln |α| 1 − ξα ρ  1 − ξα , (7.110) P(n) = n π (ξ − 1) √  which is indeed relevant to the average value of ρ in the coherent state  1 − ξα (i.e., Q distribution function in phase space), where ξ is the probability of detecting single photon in a certain interval. In the ideal situation, that is ξ = 1, P(n) is the photon number distribution of ρ. Using the normal ordering√ product of √ density  the operator of a m S (r ) |0, we easily calculate the average value 1 − ξα ρ  1 − ξα as 

   sech r tanhm r       1 − ξα ρ  1 − ξα = Hm α∗ Hm ∗ α N m 2m 2 1−ξ ∗2 2 × e−(1−ξ)|α| + 2 (α +α ) tanh r ,

(7.111)

where  = 2i [(1 − ξ) tanh r ]1/2 . Substituting Eq. (7.111) into Eq. (7.110), and using the relation (7.3) we obtain ξ n sech r tanhm r ∂ 2m ∂ 2n 2 2   e−s −τ −s τ n m m m Nm 2 n! (1 − ξ) ∂s ∂τ ∂s n ∂τ n 2 d α −|α|2 + 1−ξ (α∗2 +α2 ) tanh r 2sα∗ +2τ ∗ α+s  α+τ  α∗  2 e × e . (7.112)  s=τ =s  =τ  =0 π

P(n) =

Further, using the integral formula (2.106), we have

7.2 Multi-variable Special Polynomials and Their Generating Functions

247

∂ 2n ξ n sech r tanhm r ω 1/2 ∂ 2m n Nm 2m n! (1 − ξ) ∂s m ∂τ m ∂s n ∂τ n  exp (ω − 1) s  τ  + 4ω ||2 sτ + 2ω∗ τ τ 

P(n) =

+ 2ωss  − 4εωsτ  − 4εω∗ s  τ − εωs 2 − εωτ 2      − 1 + 4εω2 s 2 − 1 + 4εω∗2 τ 2 s=τ =s  =τ  =0 , where ω=

1 , 1 − 4ε2

ε=

(7.113)

(ξ − 1) tanh r . 2

(7.114)

Using the original definition of Hn,m (x, y; ϑ), we can change Eq. (7.113) into ξ n ω 1/2 sech r tanhm r P(n) = Nm 2m n! (1 − ξ)n

∞ 







k+l ∗l+k (2ω)l+k (−4ωε)l +k l!k!l  !k  ! l,k,l  k  =0

∂ 2m ∂ 2n     s k+l τ l+k s k+k τ l+l m m n n ∂s ∂τ ∂s ∂τ × exp (ω − 1) s  τ  + 4ω ||2 sτ − εωs 2 − εωτ 2 − λ−2 s 2  −λ∗−2 τ 2 + sx + τ y + s  x  + y  τ  ] ,    



×

s=τ =s =τ =x=y=x =y =0

(7.115)

√ where λ = 1/ 1 + 42 ωε and we have added the term exp(sx + τ y + s  x  + y  τ  )x=y=x  =y  =0 in order to meet the original definition of Hn,m (x, y; ϑ). Using the differential identity of Hn,m (x, y; ϑ) in Eq. (7.70), we obtain the analytical formula of photon-counting distribution of the state a m S (r ) |0, that is P(n) = ×

ξ n ω 1/2 sech r tanhm r n! (m!)2 (εω)n Nm 2m (1 − ξ)n |λ|2m     ∞  k+l ∗l+k (2ω)l+k (−4ωε)l +k

l!k!l  !k  ! (m − l − k  )!   Hm−k−l  ,m−l−k  0, 0; 4 |λ|2 ω × (m − k − l  )! (n − l − l  )! (n − k − k  )!   ω−1 , × Hn−k−k  ,n−l−l  0, 0; εω l,k,l  ,k  =0

(7.116)

which √is just related to the product of two different three-variable polynomials, = λ/ εω. However, the previous work only gives its high-order partial differential form that cannot be calculated [25]. For m = 0, the photon-counting distribution of the state S (r ) |0 reads   ω−1 ξ n ω 1/2 (εω)n sech r Pm=0 (n) = . Hn,n 0, 0; n! (1 − ξ)n εω

(7.117)

248

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

For ξ = 1, ε =  = 0, λ = ω = 1, thus the distribution P(n) becomes the photon number distribution of a m S (r ) |0. Next, we want to calculate the photon-counting distribution evolution of the state |m in the thermal channel. For this purpose, we introduce another new formula about the Wigner distribution function w (α) for single-mode quantum state ρ, that is 4 (−ξ)n P(n) = (2 − ξ)n+1

d αe 2

−2ξ|α|2 /(2−ξ)

 Ln

4 |α|2 2−ξ

 W (α) .

(7.118)

Noticing that the Wigner distribution function evolution of the number state |m in the thermal channel reads     2 |α|2 [2 (n¯ + 1) T − 1]m 2 |gα| Wm (α, t) = , (7.119) exp − L m 2nT ¯ +1 π (2nT ¯ + 1)m+1 where

2e−κt g= √ , T = 1 − e−2κt , ¯ + 1) [1 − 2 (n¯ + 1) T ] (2nT

(7.120)

κ is decay rate, n¯ is the average photon number of the thermal field, and substituting Eq. (7.119) into Eq. (7.118), and using a method similar to that used in deriving Eq. (7.112), we obtain ∂ 2n ∂ 2m 4ξ n [1 − 2 (n¯ + 1) T ]m   e−sτ −s τ n+1 m+1 ∂s n ∂τ n ∂s m ∂τ m n!m! (2 − ξ) (2nT ¯ + 1)  2 ∗τ) d α −g |α|2 + 2(αs+α √ +gατ  +gα∗ s   2−ξ e × , (7.121)  π s=τ =s  =τ  =0

P(n, t) =

where g =

2 2ξ + . 2−ξ 2nT ¯ +1

(7.122) 



Further, using the original definition of the polynomials Fn,m,l,k (x, y, x , y ; ν, υ), we rewrite the distribution P(n, t) as P(n, t) =

 n 4ξ n [1 − 2 (n¯ + 1) T ]m g (2 − ξ) − 4

n!m! (2 − ξ)2n+1 (2nT ¯ + 1)m+1 gn+m+1    m × g − g2 Fn,n,m,m (0, 0, 0, 0; h, h) ,

where

2g . h =   g − g2 [g (2 − ξ) − 4]

Obviously, when t → ∞,

(7.123)

(7.124)

7.2 Multi-variable Special Polynomials and Their Generating Functions

T → 1, g → 0, h →0,

g →

2 2ξ + , 2−ξ 2n¯ + 1

249

(7.125)

thus P(n, t) becomes the photon-counting distribution of the thermal field with the average photon number n, ¯ i.e., P(n, ∞) →

¯ n (ξ n) . (ξ n¯ + 1)n+1

(7.126)

For ξ = 1, P(n, t) becomes the photon number distribution evolution of the state |m in the thermal channel, i.e., Pξ=1 (n, t) =

 m n  4 [1 − 2 (n¯ + 1) T ]m g − 4 g − g2

n!m! (2nT ¯ + 1)m+1 gn+m+1   × Fn,n,m,m 0, 0, 0, 0; h , h ,

where g = 2 +

2 , 2nT ¯ +1

2g  . h =   2 g − g (g − 4)

(7.127)

(7.128)

For the limit case, t → ∞, g → 2 + 2/(2n¯ + 1), h →0, thus we have Pξ=1 (n, ∞) →

n¯ n , (n¯ + 1)n+1

(7.129)

which is a Bose-Einstein statistical distribution. 3

Wigner distribution function

Using the definition of the Wigner distribution function for single-mode quantum state ρ, that is W (α) = tr[ρ(α)], where (α) is the coherent state representation of single-mode Wigner operator. So, using we can express the Wigner m  Eq. (7.103), distribution function W (α) for the state ta + ra † |γ as   2 2 tγ ∗ + r α d2 α  (r t)m e2|α| −|γ| W (α) = Hm −i √ 2m Nm π2 2r t   tγ − r α∗ −|α |2 +(2α−γ)α∗ −(2α∗ −γ ∗ )α × Hm i √ e . 2r t

(7.130)

Using the generating function for Hn (x) and the integral formula in Eq. (1.30), we have W (α) =

(r t)m −2|α−γ|2 e π2m Nm

250

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

   ∂ 2m 2r 2 2 ∗ × m m exp −s − τ − sτ + κs + κ τ  ∂s ∂τ t s=τ =0   m −2|α−γ|2 2r (r t) e , = Hm,m κ, κ ∗ ; − π2m Nm t

(7.131)

√ √ where κ = i 2 (tγ − r γ ∗ + 2r α∗ ) / r t. For m = 0, H0,0 (κ, κ ∗ ; −2r/t) = 1, so 2 W0 (α) = π −1 e−2|α−γ| , which refers to the Wigner distribution function for the coherent state |γ. However, for m = 1, H11 (κ, κ ∗ ; −2r/t) = |κ|2 − 2r/t, thus we have W1 (α) =

|tγ − r γ ∗ + 2r α∗ |2 − r 2 W0 (α). r 2 + |tγ + r γ ∗ |2

(7.132)

The Wigner distribution functions W (α) for three different parameters m, r and γ are plotted in Fig. 7.1. From Fig. 7.1, we can clearly see that the negative volume of W (α) increases with the ratio r and decreases with the amplitude γ, but has a irregular change with the order m. Therefore, we need a coherent superposition with

W( )

W( )

(a)

Re

(b)

Im

Re

Im

W( )

W( )

(d )

(c )

Re

Im

Re

W( )

Im

W( )

( e)

Re

Im

(f)

Re

Im

 m Fig. 7.1 Wigner distribution functions for the photon-modulated states ta + ra † |γ (r 2 + t 2 = 1) for different parameters m, r and γ, where the values of (m, r, γ) are respectively a (1, 0.5, 0.2), b (3, 0.5, 0.2), c (4, 0.5, 0.2), d (3, 0.5, 0.1), e (3, 0.5, 0.5), f (3, 0.2, 0.2)

7.2 Multi-variable Special Polynomials and Their Generating Functions

251

an exact order m and a higher ratio r, and an initial coherent state with a smaller γ  † m |γ. for improving the nonclassicality of ta + ra   For the state Hn,m f a † , gb† S2 (r ) |00, using its density operator and two-mode Wigner operator (α, β), we have 2  2      sech2 r 2(|α|2 +|β|2 ) d αd β e Hn,m − f α∗ , −gβ ∗ Hn,m f α , gβ  W (α, β) = 2 2 π Nn,m π   2     ∗ ∗ × exp α β + α β tanh r + 2 αα∗ + ββ ∗ − α     2 −2 β   + β ∗ β  + α∗ α . (7.133) Further, using the generating function for Hn,m (x, y) and the integral formula in Eq (2.106) to integrate over the variables α , β  in Eq. (7.133), we obtain the analytical function W (α, β) as n+m ∂ n+m 1 2(|β|2 −|α|2 )−|G|2 cosh2 r ∂ e π 2 Nn,m ∂s n ∂τ m ∂s n ∂τ m  −1     2  exp −f sτ + s τ − g τ τ cosh2 r

W (α, β) =

− f 2 ss  cosh2 r − gG∗ τ cosh2 r − gGτ  cosh2 r   + 2 f cosh r α cosh r − β ∗ sinh r s    +2 f cosh r α∗ cosh r − β sinh r s  s=τ =s  =τ  =0       1 = 2 exp −2 |α|2 + |β|2 cosh 2r + 2 αβ + α∗ β ∗ sinh 2r π Nn,m fn+m   (7.134) × Fn,m,n,m  , κ  , ∗ , κ ∗ ; − f 2 f cosh2 r, −g 2 f cosh2 r , where

  f cosh r α cosh r − β ∗ sinh r ,

κ  = −g fG∗ cosh2 r, G = 2α tanh r − 2β ∗ .  = 2 f

(7.135)

For the case of r = 0, f = 1, G = −2β ∗ ,  = 2 f α and κ  = 2gβ, thus  the function  W (α, β) becomes the Wigner distribution function for the state Hn,m f a † , gb† |00, that is 2 2 e−2(|α| +|β| )   W0 (α, β) = 2 π Fn,m,n,m 0, 0, 0, 0; f 2 , g 2   × Fn,m,n,m 2 f α, 2gβ, 2 f α∗ , 2gβ ∗ ; − f 2 , −g 2 .

(7.136)

When f = g = 0,  = κ  = 0 and f = 1, so the function W (α, β) becomes the Wigner distribution function for two-mode squeezed vacuum, i.e.,

252

7 Generalized Binomial Theorems and Multi-variable Special Polynomials …

W( , )

W( , ) (a)

Re W( , )

Im

(b)

Re

Im

W( , ) (c ) (d )

Re W( , )

Im

Re W( , )

Im

( e) (f)

Re

Im

Re

Im

  Fig. 7.2 Wigner distribution functions for the states Hn,m f a † , gb† S2 (r ) |00 for different parameters f, g, n, m and r , where the values of ( f, g, n, m, r ) are respectively a (0.5, 0.5, 2, 2, 0.2), b (0.5, 0.5, 5, 5, 0.2), c (0.5, 0.5, 5, 2, 0.2), d (0.5, 0.5, 5, 4, 0.2), e (1.5, 1.5, 5, 4, 0.2), f (0.5, 0.5, 5, 4, 0.9)

      1 exp −2 |α|2 + |β|2 cosh 2r + 2 αβ + α∗ β ∗ sinh 2r . 2 π (7.137) Similarly, we plot how the parameters f, g, n, m and r influence the nonclas sicality of the states Hn,m f a † , gb† S2 (r ) |00 in Fig. 7.2. Figure 7.2 shows that the Wigner distribution functions W (α, β) have two negative dips on both sides of the upward main peak when m = n = 0 and the negative dips slowly increase with increasing m = n. Also, the squeezing and the negative region of the functions W (α, β) increases with the squeezing r . However, the negativity has the uncertain change with the increase of m = n or the parameters f, g. In brief, the increase of m = n and r can monotonically improve the nonclassicality, but the increase of f, g and m = n has irregular influences on it. W0,0 (α, β) =

References

253

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24. Kelley PL, Kleiner WH (1964) Theory of electromagnetic field measurement and photoelectron counting. Phys Rev 136(2A):316–334 25. Hu LY, Fan HY (2008) Statistical properties of photon-subtracted squeezed vacuum in thermal environment. J Opt Soc Am B 25(12):1955–1964

Chapter 8

Quantum Theory of Mesoscopic Circuit Systems

In recent year, with the rapid development of nanotechnology and microelectronics, mesoscopic circuits have attracted extensive attention of physicists [1–4]. For the most basic non-dissipative mesoscopic LC circuit including a source, Louisell quantized the amount of charge q on a certain electrode plate of the capacitor as the coordinate operator Q, and quantized the magnetic flux  (= L I ) passing through the coil as the momentum operator P [5]. Thus, the mesoscopic LC circuit can be regarded as a one-dimensional simple quantum harmonic oscillator. At present, with the progress of quantum information technology, one can find that mesoscopic circuits including Josephson junctions are increasingly able to meet the demand for solid-state qubits in quantum information. This is mainly because Josephson junctions have three advantages: 1 Josephson junctions exhibit good nonlinearity, from which quantum entanglement can be realized. What’s more, quantum entanglement is an important physical resource in quantum information and many protocols are implemented based on quantum entangled states; 2 Qubits can be realized by Josephson junctions that have the advantages of easy integration to form large-scale circuits, so they are considered to be a class of the ideal carriers most likely to realize quantum computers; 3 The short-time decoherence of a single Josephson junction charge qubit provides a good choice for manufacturing quantum computer hardware [6]. In principle, the Josephson junction is a structure formed by weak connection of two superconductors. For this, Feynman pointed out that, for such a junction, its electron pairs can be regarded as Bosons, ......, and almost all pairs are precisely locked in the same lowest energy state, here the electron pairs were named Cooper pairs [7]. Also, Feynman believed that the superconducting state of each superconductor was √ described by the wave function ψi = ρi eiθi (i = 1, 2), in which θi was the phase of the corresponding superconducting sequence and ρi was the electron density on the ith plate, and stressed that the junction current is related to the phase difference θ = θ2 − θ1 across the two plates. Subsequently, Vourdas believed that the above phase difference θ should be a quantum variable that can well describe the quantum mechanical properties of the system [8].

© Science Press 2023 X.-G. Meng et al., Entangled State Representations in Quantum Optics, https://doi.org/10.1007/978-981-99-2333-5_8

255

256

8 Quantum Theory of Mesoscopic Circuit Systems

Based on Feynman’s hypothesis of Cooper pairs and Vourdas’s quantum explanation of phase difference, Fan introduced the Boson form of the phase operator and obtained the Boson-Hamiltonian model of the Josephson junction [9–11]. In this chapter, with the help of the continuous-variable entangled state representation and the Boson representation of the phase operator, we present the Cooper-pair number-phase quantization schemes for several mesoscopic circuits with or without Josephson junctions, and develop further discussion of the characteristic oscillation frequency, quantum noise, modified Josephson operator equations [12–16], etc.

8.1 Quantum Theory of Mesoscopic LC Circuits 8.1.1 Single Mesoscopic LC Circuit In a mesoscopic LC circuit, if the amount of charge q on a certain electrode plate of the capacitor is regarded as a generalized coordinate, and q = en (n is the number of electrons) is taken into account, the potential energy of the system (considered as the electrostatic energy stored in the capacitor) is V =

1 2 2 q2 = e n , 2C 2C

(8.1)

and the kinetic energy of the system (i.e., the magnetic energy stored in the inductor) is  2 1 dq 1 1 T = LI2 = L = Le2 n˙ 2 . (8.2) 2 2 dt 2 So, the Lagrangian function L of the system is given by L=T −V =

1 2 2 e2 n 2 Le n˙ − . 2 2C

(8.3)

Accordingly, the classical canonical momentum p conjugated to the number n of electrons is ∂L p= = Le2 n. ˙ (8.4) ∂ n˙ In addition, there exists the following relationship: en˙ = I =

φ , L

(8.5)

where φ is the magnetic flux inside the inductor. Substituting Eq. (8.5) into Eq. (8.4) leads to the relationship between the classical canonical momentum p and the magnetic flux φ, that is

8.1 Quantum Theory of Mesoscopic LC Circuits

p = eφ.

257

(8.6)

According to Faraday’s law of electromagnetic induction, the relationship between the voltage u and the time evolution of the magnetic flux φ reads dφ = u, dt

(8.7)

which is also equal to the voltage across the capacitor in the mesoscopic LC circuit without power supply. From the view of wave function of quantum mechanics, this voltage is related to the phase difference between the two plates of the capacitor in a certain time interval dt. Suppose that the wave function of each plate is ψi = φi eiEi t/ = φi eiωi t = φi eiθi , (i = 1, 2),

(8.8)

where θi is the phase of the wave function of the ith plate. Note that dθi =

Ei dt, 

(8.9)

it then follows that dθ = d (θ2 − θ1 ) =

eu E2 − E1 dt = − dt,  

(8.10)

where θ = θ2 − θ1 is the phase difference of the wave functions of the two plates. Substituting Eq. (8.7) into Eq. (8.10) and using Eq. (8.6), we thus have p = θ.

(8.11)

So, we can say surely that θ is the canonical momentum and the particle n is its canonical conjugate coordinate. Therefore, for a single mesoscopic LC circuit, the standard quantization condition is

or

  n, ˆ θˆ = i

(8.12)

ˆ = i. [n, ˆ θ]

(8.13)

And, the number operator nˆ of particles and the phase operator θˆ obey the uncertainty relation 1 (8.14) n ˆ θˆ  , 2 which means the presence of quantum fluctuations. Notedly, the quantum mechanical operators corresponding to the observable physical variables n, θ discussed above are

258

8 Quantum Theory of Mesoscopic Circuit Systems

just the number nˆ of electrons on one plate of the capacitor and the phase difference θˆ of the wave function between the two plates, respectively. Therefore, the quantization of a single mesoscopic LC circuit actually belongs to the context of number-phase quantization. In addition, according to Eqs. (8.6) and (8.11), the magnetic flux φ is quantized ˆ as the operator φˆ = θ/e, which is consistent with Vourdas’ analysis of the coupling system between the external microwave and the superconducting ring (that is, the total magnetic flux is related to the phase difference across the two plates of ˆ thus the Hamiltonian the superconducting junction). Using the operators nˆ and θ, operator of the mesoscopic LC circuit can be obtained as e2 nˆ 2 1 2 θˆ2 + . H = T + V → Hˆ = 2 e2 L 2C

(8.15)

In the Heisenberg picture, by use of the Heisenberg equation, we then get dnˆ θˆ = 2 , dt e L



dθˆ e2 nˆ =− , dt C

(8.16)

which correspond to the current equation and Faraday’s law of electromagnetic induction, respectively. From Eq. (8.16), we have d2 nˆ nˆ =− , dt 2 LC whose standard solution is

√

nˆ = nˆ (t = 0) eit/

(8.17)

LC

.

(8.18)

Clearly, there exist the electromagnetic resonance characteristics in a single mesoscopic LC circuit. Next, we compare the number-phase quantization of a single mesoscopic LC circuit with that of the Josephson junction. Feynman believed that a Cooper pair can be acted as a bound pair that has the behaviors of Bosons. Later, Tinkham gave the Hamiltonian describing the Josephson junction as [17] 1 H = − E c ∂ϕ2 + E j (1 − cos ϕ) , 2

(8.19)

where E j is the Josephson coupling energy, E c = q 2 /C is the Coulomb coupling energy with the junction equivalent capacitance C and the charge q = 2e of a Cooper pair, and ϕ is the phase difference across the two plates of the Josephson junction. For convenience, here we have taken the reduced Planck constant  = 1. In addition, Vourdas et al. pointed out that, in order to describe the quantum properties of the system, the phase difference between the two plates of the Josephson junction shall

8.1 Quantum Theory of Mesoscopic LC Circuits

259

be a quantum parameter. Inspired by the above results, we introduce the Cooper-pair number operator [9–11] (8.20) Nˆ d ≡ a † a − b† b ˆ

and the phase operator ei



a − b† ˆ ≡ ei . a† − b

(8.21)

ˆ the phase ϕ, and since  operator  corresponds to the phase difference Notedly, √ at the same a − b† , a † − b = 0, the two operators can appear in a root sign time. Thus, the Hamiltonian operator describing the Josephson junction reads



1 iˆ Ec ˆ 2 ˆ ˆ , cos  ˆ = e + e−i , Hˆ = Nd + E j 1 − cos  2 2

(8.22)

where E j = Icr /2e, Icr is the critical current of the Josephson junction, E c Nˆ d2 /2 ˆ is the tunneling is the energy stored in the junction equivalent capacitor, E j cos  ˆ coupling energy of Cooper pairs, and cos  is the phase operator function to describe the phase difference ϕ between two plates of the Josephson junction. By use of the Hamiltonian operator (8.22) and the Heisenberg equation, the Josephson operator  equation can be obtained. Using the Boson operator commutation relations a, a † =   b, b† = 1, we therefore arrive at 

 ˆ = i, Nˆ d , 

(8.23)

which is similar to the relation (8.13). Actually, Eq. (8.23) can be verified by the entangled state |η representation in Eq. (1.72). Noting that the states |η yield the completeness relation [18, 19] 1 π

 d 2 η |η η| = 1

(8.24)

and the eigenequations



a − b† |η = η |η , b − a † |η = −η ∗ |η . So, we obtain

ˆ

η| e−i = η| e−iϕ ,

ˆ = ϕ η| , η| 

(8.25)

(8.26)

ˆ or e−iˆ is a phase operator. Also, in the entangled state η| which shows that  representation, the operator Nˆ d exactly corresponds to a differential relation with the variable ϕ, i.e.,

260

8 Quantum Theory of Mesoscopic Circuit Systems

  −iϕ

|η|2 ∂ iϕ ∗ ˆ η| Nd = |η| 00| e a + e b exp − η| . + η a − ηb + ab = i 2 ∂ϕ (8.27) Therefore, in the entangled state η| representation, there exist the commutation relations     ∂ ˆ ˆ (8.28) Nd ,  = i , ϕ = i. ∂ϕ Projecting the Hamiltonian operator Hˆ into the state η|, we thus have   1 η| Hˆ = − E c ∂ϕ2 + E j (1 − cos ϕ) η| = H η| . 2

(8.29)

8.1.2 Mutual Inductance Coupling Mesoscopic LC Circuit 1

Charge-magnetic flux quantization scheme

The mesoscopic LC circuit with mutual inductance coupling is shown in Fig. 8.1. If q1 and q2 are selected as generalized coordinates, the Lagrangian function of the system can be represented as

1 1 2 l1 I1 + l2 I22 + m I1 I2 − L= 2 2



q12 q2 + 2 c1 c2

 ,

(8.30)

where m is the mutual inductance coupling coefficient between the inductances l1 √ and l2 , 0 < m < l1l2 . From Eq. (8.30), we can obtain the generalized momenta of the system as p1 =

∂L = l 1 I1 + m I2 , ∂ q˙1

Fig. 8.1 Mutual inductance coupling mesoscopic LC circuit

p2 =

∂L = l 2 I2 + m I1 . ∂ q˙2

(8.31)

8.1 Quantum Theory of Mesoscopic LC Circuits

261

Obviously, the above two generalized momenta p1 and p2 are essentially magnetic fluxes. Thus, the Hamiltonian H of the system can be obtained as H = p1 q˙1 + p2 q˙2 − L  

1 q12 q22 1 2 2 l 1 I1 + l 2 I2 + m I1 I2 + + = 2 2 c1 c2    2  p1 1 p22 1 q12 q2 m = + p1 p2 + + 2 , − 2 A l1 l2 Al1l2 2 c1 c2

(8.32)

where

m2 , m 2 < l1 l2 . (8.33) l1 l2   Endowing with the quantization condition qˆi , pˆ j = iδi, j in Eq. (8.32), the classical Hamiltonian H is quantized as the Hamiltonian operator Hˆ . Also, the existence of the coupling term Alm1 l2 pˆ 1 pˆ 2 of the momentum operator in Eq. (8.32) implies the existence of quantum entanglement in the system. In order to disentangle the Hamiltonian operator Hˆ , the unitary operator Uˆ is constructed as       ∞  q1 q1  , det u = 1 (8.34) dq1 dq2 u Uˆ = q2 q2  −∞ A =1−

 in the coordinate state |qi  representation, where

 q1  ≡ q1 , q2 | is the two-mode q2 

coordinate eigenstate, we therefore have        q1  q u 11 u 12  = qi  1 , u = . qˆi  q2 q2 u 21 u 22

(8.35)

Equation (8.34) shows that the unitary operator Uˆ of quantum mechanics in the Hilbert space is mapped by the classical matrix u. According to Eq. (8.34), the transformation rules of the operator qˆi can be given by Uˆ



qˆ1 qˆ2



Uˆ † = u −1



qˆ1 qˆ2

 .

(8.36)

Using the completeness relation of two-mode momentum eigenstates 

     p p1  = 1, d p1 d p2  1 p2 p2  −∞ ∞

(8.37)

the form of the unitary operator Uˆ in the momentum representation can also be defined as

262

8 Quantum Theory of Mesoscopic Circuit Systems

       p

q1  exp −i u T p j q j dq1 dq2  1  p2 q2 −∞ −∞       ∞  p p1  , (8.38) uT d p1 d p2  1 p2 p2  −∞

1 Uˆ = 2π  =





∞

∞

d p1 d p2

it then follows from Eq. (8.38) that the transformation relation is Uˆ



pˆ 1 pˆ 2



Uˆ † = u T



pˆ 1 pˆ 2

 .

(8.39)

By selecting the following ansatz solution,  u=

1 E G H

 ,

det u = H − E G = 1,

(8.40)

where H , E, G are to be determined, the following matrix transformation relationship can be given by   T −1 H −G = u . (8.41) −E 1 Under the action of unitary operator Uˆ , the unitary transformations of the operators qˆ1 , qˆ2 , pˆ 1 , and pˆ 2 read qˆ1 → Uˆ † qˆ1 Uˆ = qˆ1 + E qˆ2 , qˆ2 → Uˆ † qˆ2 Uˆ = G qˆ1 + H qˆ2 , pˆ 1 → Uˆ † pˆ 1 Uˆ = H pˆ 1 − G pˆ 2 , pˆ 2 → Uˆ † pˆ 2 Uˆ = −E pˆ 1 + pˆ 2 .

(8.42)

With the help of Eq. (8.42), the unitary transformation of the Hamiltonian operator Hˆ can be written as

2

2



−E pˆ 1 + pˆ 2 H pˆ 1 − G pˆ 2 m H pˆ 1 − G pˆ 2 −E pˆ 1 + pˆ 2 + − U Hˆ Uˆ = 2 Al1 2 Al2 Al1l2

2

2 G qˆ1 + H qˆ2 qˆ1 + E qˆ2 + . (8.43) + 2c1 2c2

ˆ†

To remove the entanglement terms pˆ 1 pˆ 2 and qˆ1 qˆ2 in Eq. (8.43), we need to let l2 H G + l1 E + m (G E + H ) = 0

(8.44)

c2 E + c1 G H = 0.

(8.45)

and Further, combining the condition H − E G = 1, we get

8.1 Quantum Theory of Mesoscopic LC Circuits

263

H=

c2 , c2 + c1 G 2

(8.46)

E=

−Gc1 , c2 + c1 G 2

(8.47)

and

as well as HG =

c2 G c2 = − E. 2 c2 + c1 G c1

(8.48)

Substituting Eq. (8.48) into Eq. (8.44), we then obtain −

l2 c2 E + l1 E + m (2E G + 1) = 0. c1

(8.49)

Similarly, inserting Eq. (8.47) into Eq. (8.49), we thus have mc1 G 2 + G (l1 c1 − l2 c2 ) − mc2 = 0

(8.50)

with the solutions G=

(l2 c2 − l1 c1 ) ±



(l2 c2 − l1 c1 )2 + 4m 2 c1 c2 . 2mc1

(8.51)

Without loss of generality, we only discuss the case of taking the minus sign “−”, and let (8.52) (c2 l2 − c1 l1 )2 + 4m 2 c2 c1 ≡ , thus three undetermined parameters H , E, G read c2 l2 − c1 l1 − G= 2mc1 c1 m E=√ , 

√ 

−2mc2 =√ ,  + c2 l2 − c1l1 √  − (c1 l1 − c2 l2 ) H= . √ 2 

(8.53)

So, the specific form of the unitary operator Uˆ is Uˆ =



       1 E q1 q1  . dq1 dq2  G H q2 q2  −∞ ∞

(8.54)

Next, we calculate the characteristic oscillation frequency of the inductance coupling mesoscopic LC circuit. Using the unitary operator Uˆ , the Hamiltonian operator Hˆ becomes

264

8 Quantum Theory of Mesoscopic Circuit Systems

pˆ 22 2 pˆ 12 l2 H 2 + l1 E 2 + 2m H E + l2 G + l1 + 2mG 2 Al1l2 2 Al1l2     qˆ 2 1 G2 H2 qˆ 2 E 2 + 1 + + + 1 . (8.55) 2 c1 c2 2 c1 c2

Uˆ † Hˆ Uˆ =

From Eq. (8.53), we thus have m 2 c1 G − mc2 l2 , √ G  √ (l2 G + m)  =− , mc1 c2 + c1 G 2 G = =− , c1 c2 Ec2 m =− √ . G 

l2 H 2 + l1 E 2 + 2m H E = l2 G 2 + l1 + 2mG 1 + c1 E2 + c1

G2 c2 H2 c2

(8.56)

Substituting Eq. (8.56) into Eq. (8.55), thus Eq. (8.55) becomes

2 m c1 G − mc2 l2 2 G 2 qˆ pˆ 1 − √ 2Ec2 1 2 Al1l2 G  √ m (l2 G + m)  2 − pˆ 2 − √ qˆ22 . 2 Al1l2 mc1 2G 

Uˆ † Hˆ Uˆ =

(8.57)

By substituting the parameters in Eq. (8.53) into Eq. (8.57), and comparing Eq. (8.57) with the Hamiltonian operator pˆ 2 /2μ + μω 2 qˆ 2 /2 of the standard harmonic oscillator, thus two characteristic frequencies read − and

√ c2 l2 + c1l1 +  G m 2 c1 G − mc2 l2 2 ≡ ω+ = √ Ec2 Al1l2 G  2 Al1l2 c1 c2 m √ G  mc1 √ Al1l2 (l G+m)  2

√ c2 l2 + c1l1 −  2 = ≡ ω− . 2 Al1l2 c2 c1

(8.58)

(8.59)

Owing to A = 1 − m 2 /l1l2 , the above two Eqs. (8.58) and (8.59) can be written together, i.e., √ c2 l2 + c1l1 ±  2 . (8.60) ω± = 2c2 c1 (l1l2 − m) Finally, using the IWOP method to investigate the quantum noises of the circuit system in a squeezed vacuum state. For this purpose, it is necessary to integrate Eq. (8.54) to obtain the explicit expression of the unitary operator Uˆ . Introducing the

8.1 Quantum Theory of Mesoscopic LC Circuits

265

following creation and annihilation operators ai =

qˆi + i pˆ i , √ 2

ai† =

qˆi − i pˆ i , √ 2

(8.61)

where the natural units have been used, i.e., letting m i = ωi =  = 1, and reviewing the expression of the coordinate eigenstate qi | in the Fock space qi | =

1 π 1/4

  2 √ q a2 0| exp − i + 2qi ai − i , 2 2

(8.62)

which yields the completeness relation 

∞ −∞

dqi |qi  qi | = 1,

(8.63)

thus using the two-mode coordinate eigenstates in the Fock space  

√ 1 2 1 1 2 1 2 2 q1 , q2 | = √ 00| exp − q1 + q2 + 2 (q1 a1 + q2 a2 ) − a1 − a2 2 2 2 π (8.64) and the normal ordering product of the projection operator of two-mode vacuum state, that is,

|00 00| = : exp −a1† a1 − a2† a2 : , (8.65) we thus obtain the normal ordering product of Uˆ as        1 E q1 q1  Uˆ = dq1 dq2  G H q2 q2  −∞   ∞  1 1 = dq1 dq2 : exp − (q1 + Eq2 )2 + (Gq1 + H q2 )2 π −∞ 2 √ √

1 2 q1 + q22 − 2 (q1 + Eq2 ) a1† + 2 (Gq1 + H q2 ) a2† − 2

2 1

2  √ 1 † : a1 + a1 − a2 + a2† + 2 (q1 a1 + q2 a2 ) − 2 2 



1  2 1 + E 2 − G 2 − H 2 a1†2 − a2†2 + 4 (G + E G) a1† a2† = √ exp 2L L   

a : × : exp a1† a2† (g − 1) 1 a2  



 1  2 E + H 2 − 1 − G 2 a12 − a22 + 4 (G + E H ) a1 a2 , (8.66) × exp 2L 

∞

266

8 Quantum Theory of Mesoscopic Circuit Systems

where L = E 2 + G 2 + H 2 + 3,     2 1+ H E −G 10 g= , 1= . 01 L G− E 1+ H

(8.67) (8.68)

Acting the operator Uˆ on the two-mode vacuum |00, we naturally obtain 

1  2 1 + E 2 − G2 − H 2 Uˆ |00 = √ exp 2L L

 †2 †2 × a1 − a2 + 4 (G + E H ) a1† a2† |00 ,

(8.69)

which is obviously a generalized two-mode squeezed state. Also, the state Uˆ |00 can be considered as anentangled state because of the presence of the coupling term exp 4 (G + E H ) a1† a2† . In order to investigate the fluctuation characteristics of the quantum state Uˆ |00, the following two components are defined as X1 =

1 1 qˆ1 + qˆ2 , X 2 = pˆ 1 + pˆ 2 . 2 2

(8.70)

Noting that 00| qˆi |00 = 0 and 00| pˆ i |00 = 0, we therefore have 00| qˆi2 |00 =

1 1 , 00| pˆ i2 |00 = . 2 2

(8.71)

Further, using the relations 1 1 Uˆ † X 1 Uˆ = (1 + G) qˆ1 + (H + E) qˆ2 , 2 2 1 1 Uˆ † X 2 Uˆ = (H − E) pˆ 1 + (1 − G) pˆ 2 , 2 2

(8.72)

so the mean square fluctuations of the components X 1 and X 2 can be obtained as  2 X 12 = 00| Uˆ † X 12 Uˆ |00 − 00| Uˆ † X 1 Uˆ |00  1 (H + E)2 + (1 + G)2 , 8  2 X 22 = 00| Uˆ † X 22 Uˆ |00 − 00| Uˆ † X 2 Uˆ |00 =

=

 1 (H − E)2 + (1 − G)2 . 8

(8.73)

(8.74)

8.1 Quantum Theory of Mesoscopic LC Circuits

267

Fig. 8.2 The changes of the variances X 12 , X 22 and the quantum noise X 1 X 2 of the components X 1 , X 2 for the mutual inductance coupling mesoscopic LC circuit with the mutual inductance coefficient m

It then follows that the uncertainty relation (quantum noise) for X 1 and X 2 can be obtained as 

2  1 + E 2 − G 2 − H 2 + 4, (8.75) X 1 X 2 = 8 where the constant  has been recovered, and the parameters E, G, H can be found in Eq. (8.53). Roughly, the large the mutual inductance coefficient m, the more quantum noise. In terms of Eqs. (8.73)–(8.75), we plot three figures in Fig. 8.2 to show how the mutual inductance coefficient m affects the quantum noise. Here, we choose the circuit components that are easy to realize in current technology, such as c1 = c2 = 6 × 10−17 F, l1 = 10−8 H and l2 = 10−7 H. In terms of Fig. 8.2a and b, we find that, when X 12 decreases, its partner X 22 increases correspondingly, which shows the presence of the squeezing of the two components X 1 and X 2 . Figure 8.2c shows that quantum noise of squeezed vacuum increases with the increase of the coefficient m, which is the same as the analytical result in Eq. (8.75). Considering that the quantum entanglement of the circuit comes from the mutual inductance, so we can see from Fig. 8.2c that the large quantum entanglement of the circuit, the more quantum noise. 2

Particle number-phase quantization scheme

In this section, the number-phase quantization scheme will be used to quantize the mesoscopic LC circuit with mutual-inductance coupling. Similar to the quantization of a single mesoscopic LC circuit, suppose that the charge number n i (qi = en i , i = 1, 2) on a certain electrode plate of the ith capacitor is regarded as a generalized coordinate, the potential energy of the system is V =

q2 1 2 2 1 2 2 q12 + 2 = e n1 + e n2, 2C1 2C2 2C1 2C2

(8.76)

268

8 Quantum Theory of Mesoscopic Circuit Systems

and the total kinetic energy of the system is T =

1 1 1 1 L 1 I12 + L 2 I22 + m I1 I2 = L 1 e2 n˙ 21 + L 2 e2 n˙ 22 + me2 n˙ 1 n˙ 2 . 2 2 2 2

(8.77)

So, the Lagrangian function reads 1 1 L 1 e2 n˙ 21 + L 2 e2 n˙ 22 2 2 1 2 2 1 2 2 2 + me n˙ 1 n˙ 2 − e n1 − e n2. 2C1 2C2

L=T −V =

(8.78)

Accordingly, the generalized momenta of canonical conjugation, respectively, with the charge numbers n 1 and n 2 are ∂L = L 1 e2 n˙ 1 + me2 n˙ 2 , ∂ n˙ 1 ∂L p2 = = L 2 e2 n˙ 2 + me2 n˙ 1 . ∂ n˙ 2 p1 =

(8.79)

Note that en˙ 1 L 1 = φ1 , en˙ 2 m = φ21 , en˙ 2 L 2 = φ2 , en˙ 1 m = φ12 ,

(8.80)

where φi and φi, j (i, j = 1, 2, but i = j) are, respectively, the magnetic flux caused by self-inductance of the ith inductor and its mutual-inductance with the jth inductor. Substituting Eq. (8.80) into Eq. (8.79) leads to

pi = e φi + φi, j = eφi,T ,

(8.81)

which is the relation between the canonical momentum pi and the total magnetic flux φi,T . Similarly, by virtue of Faraday’s law of electromagnetic induction, the terminal voltage u i of the ith inductor can be obtained as dφi,T = ui . dt

(8.82)

Similar to the calculation of Eqs. (8.11)–(8.14), we can obtain the generalized momentum pi pi = θi (8.83) and the canonical quantization condition   nˆ i , θˆ j = iδi j ,

(8.84)

8.1 Quantum Theory of Mesoscopic LC Circuits

269

as well as the uncertain relation nˆ i θˆi  1/2. So, the total magnetic flux can also be quantized as an operator  (8.85) φˆ i,T = θˆi , e and the commutation relation between the charge number operator nˆ i and the magnetic flux operator φˆ i,T can be written as     nˆ i , eφˆ j,T = qˆi , φˆ j,T = iδi j ,

(8.86)

where qˆi is the charge operator. According to the quantization condition given in Eq. (8.84), the Hamiltonian operator Hˆ of the system can be given by H = T + V → Hˆ = −L + 2 = 2D



θˆ2 θˆ12 + 2 L1 L2



pi q˙i

i

 −

e2 2 e2 2 m2 ˆ ˆ nˆ 1 + nˆ , θ1 θ2 + DL 1 L 2 2C1 2C2 2

(8.87)

where D = 1 − m 2 /(L 1 L 2 ). Obviously, the form in Eq. (8.87) is similar to the Hamiltonian operator of a harmonic oscillator system with motion coupling. Similar to treating with the Hamiltonian operator corresponding to H in Eq. (8.32), we can m2 ˆ ˆ θ1 θ2 in the introduce a new unitary operator to get rid of the coupling term − DL 1 L2 operator Hˆ in Eq. (8.87) and realize the diagonalization of the operator Hˆ . Next, we want to discuss the time evolution of the charge number operator nˆ i and phase difference operator θˆi . In the Heisenberg picture, the equations of motion of the charge number operators nˆ 1 and nˆ 2 are 1  ˆ m ˆ dnˆ 1 = [nˆ 1 , Hˆ ] = θ1 − θ2 , dt i DL 1 DL 1 L 2 1 dnˆ 2  ˆ m ˆ = [nˆ 2 , Hˆ ] = θ2 − θ1 , dt i DL 2 DL 1 L 2

(8.88)

which show that the change of current in each LC circuit is affected by the two phase operators θˆ1 and θˆ2 owing to the mutual-inductance coupling, and the equations of motion of the phase operator θˆ1 and θˆ2 read dθˆ1 1 e2 = [θˆ1 , Hˆ ] = − nˆ 1 , dt i C1

1 dθˆ2 e2 = [θˆ2 , Hˆ ] = − nˆ 2 . dt i C2

Using Eqs. (8.85) and (8.89), we therefore have

(8.89)

270

8 Quantum Theory of Mesoscopic Circuit Systems

dφˆ 1,T e = − nˆ 1 = uˆ 1 , dt C1

dφˆ 2,T e = − nˆ 2 = uˆ 2 , dt C2

(8.90)

which are just the operator of Faraday’s law of electromagnetic induction of the mesoscopic LC circuit with mutual-inductance coupling, uˆ i is the voltage operator. Using Eqs. (8.88) and (8.89), we therefore have d2 nˆ 1 = G nˆ 2 + H nˆ 1 , dt 2

d2 nˆ 2 = J nˆ 1 + K nˆ 2 , dt 2

(8.91)

where G=

me2 e2 , H =− , DL 1 L 2 C2 DL 1 C1

J=

me2 e2 , K =− . (8.92) DL 1 L 2 C1 DL 2 C2

Rewritten Eq. (8.91) as a matrix form and acting the unitary matrix U −1 on it, we thus get       d2 −1 nˆ 1 H G ˆ1 −1 −1 n = U . (8.93) U UU nˆ 2 nˆ 2 J K dt 2 In Eq. (8.93), the following condition U −1



H G J K



 U=

λ1 0 0 λ2

 (8.94)

is required, where the unitary matrix U reads  U=

Gλ 1 Gλ 2 λ1 −H λ2 −H λ 1 λ 2

 λi =

,



G λi − H

2 +1

(8.95)

and its two discrete eigenvalues are, respectively, λ1,2 =

(H + K ) ±

 (H − K )2 + 4G J . 2

(8.96)

Using the following equations, det |U | = and

d2 −1 U dt 2



nˆ 1 nˆ 2

G (λ2 − λ1 ) λ 1 λ 2 (λ1 − H ) (λ2 − H )



 =

λ1 0 0 λ2



U −1



nˆ 1 nˆ 2

(8.97)  ,

(8.98)

8.1 Quantum Theory of Mesoscopic LC Circuits

271

we can obtain the time evolution of particle number operators Nˆ and Nˆ as √

Nˆ = Nˆ 0 e

λ1 t

,

√

Nˆ = Nˆ 0 e

λ2 t

,

(8.99)

where Nˆ 0 , Nˆ 0 are the initial particle number operators, and the operators Nˆ , Nˆ yield the following relations: nˆ 1 +

G Nˆ nˆ 2 = , λ2 − H λ2

nˆ 1 +

G Nˆ

nˆ 2 = . λ1 − H λ1

(8.100)

Therefore, the time evolution of the charge number nˆ i on a certain plate of the ith capacitor is G nˆ 1 = det |U |



λ 1 Nˆ λ Nˆ

− 2 λ1 − H λ2 − H

 ,

nˆ 2 =

1 ˆ

λ2 N − λ 1 Nˆ . (8.101) det |U |

On the other hand, using Eqs. (8.88) and (8.89) and a similar way to obtaining (8.101), we can get the time evolution of the phase difference θˆi across the two plates of the ith capacitor, i.e.,   λ 2 Nˆ

e2 G λ 1 Nˆ −√ , √ C1 det |U | λ1 (λ1 − H ) λ2 (λ2 − H )  

ˆ

ˆ

2 λ N N e λ 1 2 θˆ2 = , √ − √ C2 det |U | λ1 λ2 θˆ1 = −

(8.102)

where 1/2  1/2 4m 4 C14 4m 4 C14

= +1 , λ2 = +1 , (C + D)2 (C − D)2 C = (L 2 C2 − L 1 C1 )2 + 2m 2 C1 C2 ,  D = (L 2 C2 − L 1 C1 ) C + 2m 2 C1 C2 .

λ 1



(8.103)

Obviously, from Eqs. (8.101) and (8.102), we can find that the charge number operator nˆ i and phase difference operator θˆi in the ith mesoscopic LC circuit are related to all self-inductance coefficients L 1 , L 2 , capacitance parameters C1 , C2 and mutualinductance coupling coefficient m in the whole circuit (Fig. 8.3).

272

8 Quantum Theory of Mesoscopic Circuit Systems

8.2 Quantum Theory of Mesoscopic Circuits Containing Josephson Junction 8.2.1 Mutual-Inductance Coupling Mesoscopic Circuit Including Josephson Junction In a mesoscopic circuit, for a single Josephson junction, there are two equations, i.e., the current equation and the junction voltage equation Ii = Ici sin ϕi ,

ϕ˙ i =

2e u i , (i = 1, 2), 

(8.104)

where, for the ith Josephson junction, Ici is its Josephson critical current and u i , ϕi are, respectively, the voltage and the phase difference across its two superconducting plates, 2e is the charge quantity of a single Cooper pair. Therefore, when the Cooper pair is tunneling, the work done by the electric field force between the two junction plates on the Cooper pairs on the ith electrode plate is  −

t

u i Ici sin ϕi dt = E ji (cos ϕi − 1),

(8.105)

0

where E ji = Ici /(2e) is the Josephson coupling constant. For the non-dissipative mesoscopic circuit, using Faraday’s law of electromagnetic induction, the inductance voltage u L i is (8.106) u L i = φ˙ L i , where φ L i is the total magnetic flux caused by self-inductance and mutual-inductance through the ith inductor. From Eqs. (8.104) and (8.106), we therefore obtain ϕ˙ i =

2e ˙ φL .  i

(8.107)

Integrating Eq. (8.107) leads to ϕi − ϕi (0) =

 2e  φ L i − φ L i (0) . 

(8.108)

Suppose that the circuit is excited by a pulse power supply and the excitation time t → 0, then φ L i (0) = 0. If taking the initial phase difference ϕi (0) = 0, we thus have 2e (8.109) ϕi = φ L i .  Now, by virtue of the canonical quantization method, we quantize the mesoscopic circuit coupled by mutual-inductance including the Josephson junction in Fig. 8.3,

8.2 Quantum Theory of Mesoscopic Circuits Containing Josephson Junction

EJ

273

EJ

1

L1

2

L2

CJ

CJ

1

2

Fig. 8.3 Mutual-inductance coupling mesoscopic circuit including Josephson junction

that is, obtaining the Hamiltonian operator of the circuit system. If ϕ1 and ϕ2 are regarded as generalized coordinates, the potential energy of the system is  2  2  2    1 m 1 ϕ21 + ϕ22 + ϕ1 ϕ2 2L 1 2e 2L 2 2e L 1 L 2 2e + E J1 (1 − cos ϕ1 ) + E J2 (1 − cos ϕ2 ),

V =

(8.110)

where the first three terms on the right of Eq. (8.110) represent the magnetic energy stored in the inductors and the last two terms are the Cooper-pair tunneling coupling energy. And, the kinetic energy of the system is T =

C1 2



 2e

2 ϕ˙ 21 +

C2 2



 2e

2 ϕ˙ 22 ,

(8.111)

which is actually the energy stored on the equivalent capacitance of two junctions. So, the Lagrangian function of the whole circuit system reads L=T −V  2  2       1 1 C1  2 2 C2  2 2 2 ϕ˙ 1 + ϕ˙ 2 − ϕ1 − ϕ22 = 2 2e 2 2e 2L 1 2e 2L 2 2e  2  m − ϕ1 ϕ2 − E J1 (1 − cos ϕ1 ) − E J2 (1 − cos ϕ2 ). (8.112) L 1 L 2 2e Correspondingly, the generalized momenta of canonical conjugation with the generalized coordinates ϕ1 and ϕ2 are, respectively,

274

8 Quantum Theory of Mesoscopic Circuit Systems

∂L p1 = = ∂ ϕ˙ 1 p2 =

∂L = ∂ ϕ˙ 2

 

 2e  2e

2 C1 ϕ˙ 1 =

 Q1, 2e

C2 ϕ˙ 2 =

 Q2. 2e

2

(8.113)

From Eq. (8.113), it can be seen clearly that the generalized momentum pi is proportional to the charge amount Q i on a certain plate of the ith Josephson junction. Noting that Q i = 2en i , thus Eq. (8.113) reveals the possibility of implementing number-phase quantization for this circuit system (Fig. 8.4). From Eqs. (8.112) and (8.113), we can arrive at the Hamiltonian H of the circuit system, which is of the form H = −L +



pi ϕ˙ i

i

=

2   E Ci n 2 i

i=1

2

 2 ϕi2 k2 ϕ1 ϕ2 + 2 + E (1 − cos ϕ ) − , √ J i i 8e (1 − k 2 )L i 4e2 (1 − k 2 ) L 1 L 2 (8.114)

where E Ci = (2e)2 /Ci is the Coulomb coupling energy and Ci is the equivalent capacitance of the ith Josephson junction. Next, we will employ the number-phase quantization scheme to quantize the circuit system so as to obtain its Hamiltonian operator Hˆ . To this end, based on Feynman’s assumption that is a Cooper pair acted as a Boson, we quantize the classical Hamiltonian H given by Eq. (8.114) by using the Boson operator model. As mentioned above, the charge 2en i on a superconducting plate of the ith Josephson junction can be quantized. So, after introducing the Boson creation operators aˆ i† (bˆi† ) and annihilation operators aˆ i (bˆi ), we can quantize the Cooper-pair number n i into the number difference operator Nˆ i between the two plates of the ith Josephson junction, i.e., (8.115) n i → Nˆ i ≡ aˆ i† aˆ i − bˆi† bˆi . In order to implement the number-phase quantization, we introduce the following entangled state |ηi   1 |ηi = exp − |ηi |2 + ηi ai† − ηi∗ bi† + ai† bi† |00i , 2

(8.116)

where ηi = |ηi |eiϕi , |00i is the two-mode vacuum state. Using the commutation relations [aˆ i , aˆ i† ] = [bˆi , bˆi† ] = 1, it is easy to prove that the state |ηi satisfies the following eigen-equations



aˆ i − bˆi† |ηi = ηi |ηi , bˆi − aˆ i† |ηi = −η ∗ |ηi ,

(8.117)

8.2 Quantum Theory of Mesoscopic Circuits Containing Josephson Junction

275

and the set of the states |ηi forms a complete representation because of 

d2 ηi |ηii η| = 1. (8.118) π   So, using Eq. (8.117) and the commutation relation aˆ i† − bˆi , aˆ i − bˆi† = 0, we can introduce the following Boson phase operator of Josephson junction ˆ

eii

   aˆ i − bˆ † i = † , aˆ i − bˆi

ˆ

e−ii

  †  aˆ − bˆi = i , aˆ i − bˆi†

ˆi = cos 

1 iˆ i ˆ (e + e−ii ). 2

(8.119)

ˆ This is also because, in the entangled state |ηi representation, eii behaves like a phase ˆ ˆ (8.120) eii |ηi = eiϕi |ηi , e−ii |ηi = e−iϕi |ηi .

Therefore, we have ˆi = 

aˆ i − bˆi† 1 ln † , i2 aˆ i − bˆi

ˆ i |ηi = ϕi |ηi . 

(8.121)

Further, we obtain

Nˆ i |ηi ≡ aˆ i† aˆ i − bˆi† bˆi |ηi



  = aˆ i† ηi + bˆi† − bˆi† ai† − ηi∗ |ηi

= |ηi | aˆ i† eiϕi + bˆi† e−iϕi |ηi = −i

∂ |ηi , ∂ϕi

(8.122)

which means that the number difference operator Nˆ i can be equivalent to an operator that differentiates the phase angle ϕi . Thus, we give i

ˆ i , Nˆ i ] = η| [

i

  ∂ ˆ i ] = i, η| ϕi , i = −i i η| → [Nˆ i ,  ∂ϕi

(8.123)

ˆ i are a pair of canonical conjugate operawhich implies that the operators Nˆ i and  ˆ i ] = i. It is consistent with the tors that can satisfy the commutation relation [Nˆ i ,  ˆ = i2e that is satisfied by the phase operator θˆ and the quantization condition [q, ˆ θ] net Cooper-pair charge operator qˆ proposed by Vourdas. Further, using Eqs. (8.120) and (8.122), we can give the following two commutation relations as

276

8 Quantum Theory of Mesoscopic Circuit Systems

ˆ i , Nˆ i ] = i sin  ˆ i , [sin  ˆ i , Nˆ i ] = −i cos  ˆ i. [cos 

(8.124)

By analogy to the classical case given in Eq. (8.109), and using Eq. (8.121), the following Boson magnetic flux operator can be defined as aˆ i − bˆi†   ˆi = ln † . φˆ L i =  2e i4e aˆ i − bˆi

(8.125)

From Eqs. (8.123) and (8.125), we can arrive at the commutation relation between the Cooper-pair charge operator Qˆ i = 2eNˆ i and the magnetic flux operator φˆ L i , which is of the form (8.126) [ Qˆ i , φˆ L i ] = i, which is the same as the expected form. From Eqs. (8.115) and (8.120), the Hamiltonian H given by Eq. (8.114) can be quantized as the following Hamiltonian operator   2 2 ˆ2  ˆ 1 ˆ2   E k2  C i i 2 ˆ i) − Nˆ i + 2 + E Ji (1 − cos  . Hˆ = √ 2 2 8e (1 − k )L i 4e2 (1 − k 2 ) L 1 L 2 i=1 (8.127) Using Eqs. (8.120) and (8.122), and projecting the Hamiltonian operator Hˆ into the basis vector 1 η|2 η|, we therefore obtain  2   2 ϕi2 E Ci ∂ 2 − + 2 + E Ji (1 − cos ϕi ) 2 ∂ϕi2 8e (1 − k 2 )L i i=1  k2 ϕ1 ϕ2 (8.128) − 2 √ 1 η|2 η| . 4e (1 − k 2 ) L 1 L 2

ˆ 1 η|2 η| H =

8.2.2 Mesoscopic LC Circuit with Josephson Junction Coupled by Mutual Inductance For non-dissipative mesoscopic circuit, the voltage u L l on the lth inductor can be obtained by Faraday’s law of electromagnetic induction, that is u L l = φ˙ L l = L l I˙l + m I˙k ,

(8.129)

where φ L l is the total magnetic flux caused by self-inductance of the lth inductor and its mutual-inductance with the kth inductor (l, k = 1, 2, but l = k), Il and Ik are, respectively, the current flowing through the lth and kth inductor. For the lth Josephson junction, the current equation and voltage equation are, respectively,

8.2 Quantum Theory of Mesoscopic Circuits Containing Josephson Junction Fig. 8.4 Mesoscopic LC circuit with Josephson junction coupled by mutual inductance

L1 EJ CJ

277

L2 EJ

1

2

CJ

1

Cc

Cc

1

I Jl = Icl sin ϕ Jl , ϕ˙ Jl =

2

2

2e uJ ,  l

(8.130)

where 2e is the charge amount of a single Cooper pair, E Jl and Icl = 2eE Jl / are, respectively, the coupling energy and the critical current of the lth Josephson junction, and u Jl and ϕ Jl are, respectively, the voltage and the phase difference across the two plates of the lth Josephson junction. Therefore, when the Cooper-pair tunneling occurs, the work done by the electric field force between the two plates of the lth junction on the Cooper pairs is 

t

−

u Jl Icl sin ϕ Jl dt = E Jl (cos ϕ Jl − 1).

(8.131)

0

Next, by using the canonical quantization method, we will quantize the mesoscopic LC circuit with Josephson junction coupled by the mutual inductance in Fig. 8.4. If ϕ Jl and Il are regarded as generalized coordinates, the potential energy of the system is 1

1 L l Il2 + E Jl 1 − cos ϕ Jl + m Il Ik . (8.132) V = 2 2 l On the other hand, the electrostatic energy stored in all the capacitors reads T =

1 l

2

1 Ccl u 2cl + C Jl u 2Jl , 2

(8.133)

where Ccl and C Jl are, respectively, the coupling capacitance and junction capacitance. According to the node voltage equation of the circuit, we thus give u cl = u Jl + u L l . Substituting Eqs. (8.129), (8.130) and (8.134) into Eq. (8.133) leads to

(8.134)

278

8 Quantum Theory of Mesoscopic Circuit Systems

T =

1 2

l

 Ccl

 ϕ˙ J + L l I˙l + m I˙k 2e l

2

1 + C Jl 2



 2e

2 ϕ˙ 2Jl ,

(8.135)

which is the kinetic energy because the voltage u Jl and u L l are, respectively, related to the generalized velocity ϕ˙ Jl and I˙l . Thus, the Lagrangian function of the system can be written as L=T −V  2  2 1   1 ˙ ˙ Ccl ϕ˙ Jl + L l Il + m Ik + C Jl ϕ˙ 2Jl = 2 2e 2 2e l −



1 1 L l I 2 − E Jl 1 − cos ϕ Jl − m Il Ik . 2 l 2

(8.136)

From the principle of current continuity on the circuit nodes, that is u Jl C Jl − 2n l e = −u cl Ccl ,

(8.137)

and using Eqs. (8.129), (8.130) and (8.134), we therefore obtain



ϕ˙ Jl + Ccl L l I˙l + m I˙k , 2n l e = u Jl C Jl + u cl Ccl = C Jl + Ccl 2e

(8.138)

where n l is the net Cooper-pair number in the charge island. From Eqs. (8.129), (8.130), (8.136) and (8.138), we can obtain the generalized momenta of canonical conjugation with ϕ Jl and Il as  



∂L  ˙ ˙ C Jl + Ccl ϕ˙ J + Ccl L l Il + m Ik = n l , = p Jl = ∂ ϕ˙ Jl 2e 2e l  ∂L  pLl = = Ccl L l ϕ˙ Jl + Cck m ϕ˙ Jk 2e 2e ∂ I˙l

(8.139) + Ccl L l L l I˙l + m I˙k + Cck m L k I˙k + m I˙l . It is worth noting that, the generalized momentum p Jl is proportional to the number n l of net Cooper pairs in the island, which implies the possibility of number-phase quantization of the Josephson junction. From Eq. (8.136), we can give the Hamiltonian of the system, i.e., H=



p Jl ϕ˙ Jl + p L l I˙l − L = H J + HL + Hint ,

(8.140)

l

where HJ =

 l



E c(Jl ) n l2 + E Jl 1 − cos ϕ Jl

(8.141)

8.2 Quantum Theory of Mesoscopic Circuits Containing Josephson Junction

279

is the Hamiltonian of the two Josephson junction, and HL is similar to the Hamiltonian of two coupled harmonic oscillators, i.e., HL =

 p 2L l l

2Ml

+

1 1 1 L l Il2 + M3 p L l p L k + m Il Ik , 2 2 2

(8.142)

and Hint is the interaction Hamiltonian, that is Hint =

m Lk nl p L k − nl p L l , T Tl l l

(8.143)

in which 2e2 gl gk 2m(g1 L 1 + g2 L 2 ) , M3 = − , Ml = , C Jl g1 g2 2(m 2 gl + L 2k gk )



2 C 2Jl L 1 L 2 − m 2 2C Js Ccs L 1 L 2 − m 2

Tl = , s = l, k. (8.144) , gs = C Js + Ccs 2e C Jl + Ccl

E c(Jl ) =

Noting that a single mesoscopic LC circuit can be viewed as a simple quantum harmonic oscillator, so we can obtain the Boson operator form belonging to the classical Hamiltonian H . First of all, replacing p L l , Il by the corresponding operators pˆ L l and Iˆl , and using the quantum quantization condition [ Iˆl , pˆ L l ] = i, thus the Boson operator form corresponding to the Hamiltonian HL of the system reads



Hˆ L = ω1 cˆ1† cˆ1 + 1/2 + ω2 cˆ2† cˆ2 + 1/2

 (1 − M1 M2 ω1 ω2 ) † † cˆ1 cˆ2 + cˆ1 cˆ2 + √ 2 M 1 M 2 ω1 ω2

 (1 + M1 M2 ω1 ω2 ) † cˆ1 cˆ2 + cˆ1 cˆ2† , + √ 2 M 1 M 2 ω1 ω2

(8.145)

 where ωl = 2L l (m 2 gl + L 2k gk )/gl gk is the characteristic frequency of the lth mesoscopic LC circuit, and 1 (Ml ωl Iˆl − i pˆ L l ), cˆl† = √ 2Ml ωl

cˆl = √

1 (Ml ωl Iˆl + i pˆ L l ) 2Ml ωl

(8.146)

are the Boson creation and annihilation operators that satisfy the commutation relation [cˆl , cˆl† ] = 1. Further, using Eqs. (8.120)–(8.123), the classical Hamiltonians H J and Hint can be quantized as the following operators Hˆ J =

 l



ˆ Jl E c(Jl ) Nˆ l2 + E Jl 1 − cos 

(8.147)

280

8 Quantum Theory of Mesoscopic Circuit Systems

and  !

M1 ω1 † m ˆ L2 ˆ cˆ1 − cˆ1 N2 − N1 i T2 T1 2  ! 

M2 ω2 † m ˆ L1 ˆ cˆ2 − cˆ2 , N1 − N2 i + T1 T2 2

Hˆ int =



(8.148)

ˆ jl can be found in Eqs. (8.115) and (8.121). where Nˆ l and  Next, we want to obtain the modified Josephson operator equations in the Heisenberg picture. By virtue of the commutation relations given in Eqs. (8.123) and (8.124), the time evolution of the difference operator Nˆ l of the lth junction can be obtained as 1 EJ d ˆ ˆ Jl , Nl = [Nˆ l , Hˆ J ] = − l sin  (8.149) dt i  which is equivalent to the following current operator equation of a single Josephson junction # d " ˆ # 2eE Jl " ˆ Jl = Il , Qˆ l = 2eNˆ l . − Ql = sin  (8.150) dt  ˆ jl conjugated with Similarly, the time evolution of the phase difference operator  ˆ the Cooper-pair difference operator Nl reads   ˆ Jl d 1 1 m Lk ˆ Jl , Hˆ J + Hˆ int ] = = [ 2E c(Jl ) Nˆ l + pˆ L k − pˆ L l . dt i  Tl Tl

(8.151)

Obviously, owing to the presence of the coefficient m, the voltage operator equation of Josephson junction is affected by the inductance coupling. Substituting Eqs. (8.139) and (8.144) into Eq. (8.151) leads to ˆ Jl 2eTl d = dt Ccl (L l L k − m 2 ) + 2eTl

 2E c(Jl ) Nˆ l

Cc dφˆ L l − l (L l L k − m 2 ) Tl dt

 ,

(8.152) which is the modified Josephson voltage operator equation caused by the inductance coupling. Actually, we can also use the Faraday operator equation obtained by using Heisenberg equation to verify the correctness of Eq. (7.126). The detailed calculation is as follows. Noting that 1 dφˆ L l = [L l Iˆl + m Iˆk , Hˆ L + Hˆ int ] dt i     Ll m Ll L k − m2 ˆ Nl , = + m M3 pˆ L l + + M3 L l pˆ L k − Ml Mk Tl

uˆ L l =

(8.153)

8.2 Quantum Theory of Mesoscopic Circuits Containing Josephson Junction

281

and substituting Eqs. (8.139) and (8.144) into Eq. (8.153), we thus have   ˆ Jl d 4e2 Tl dφˆ L l  2eT l

Nˆ l −

= , uˆ L l = 1 + dt 2e dt Ccl C Jl L l L k − m 2 Ccl L l L k − m 2 (8.154) which is just the same as Eq. (8.152). Also, Eq. (8.152) also shows that dφˆ L l /dt and ˆ Jl /dt are closely related. In other words, with the Faraday voltage operator equad tion being modified, the Josephson operator equation is also modified. Differentiating the time t on both sides of Eq. (8.152), we thus give ˆ Jl d2  2eTl =− dt 2 Ccl (L l L k − m 2 ) + 2eTl   2E c(Jl ) E Jl Ccl d2 φˆ L l 2 ˆ Jl + sin  × (L l L k − m ) 2 .  Tl dt

(8.155)

From Eq. (8.155), it can be seen clearly that, since the contribution of the lth inductor is equivalent to adding a controllable bias voltage in the circuit, the voltage across the lth junction can be changed by adjusting the inductance coupling coefficient m, and the bias voltage across the inductance L l and the capacitance Ccl . ˆ Jl Finally, we investigate the time evolution of the phase difference operator  of the lth Josephson junction when additional energy (e.g., light radiation) acts on the lth junction. Comparing Eqs. (8.130) and (8.151), we can see that the effective voltage between the two plates of the first junction reads uˆ J1

  1 m L2 (J ) ˆ 2E c1 N1 + = pˆ L − pˆ L . 2e T1 2 T1 1

(8.156)

For convenience, we choose the interaction picture. In this picture, the corresponding Hamiltonian operator is taken as EJ Hˆ 1 = 1 

  m L2 (J ) ˆ ˆ J1 , 2E c1 N1 + pˆ L − pˆ L sin  T1 2 T1 1

(8.157)

which is actually the work done by the first Josephson junction current in unit time interval. According to the Heisenberg equation of motion, we thus have

E (J ) E d 1 ˆ J1 = [sin  ˆ J1 , Hˆ 1 ] = c1 J1 sin 2 ˆ J1 sin  dt i 2 and

2E (J ) E d 1 ˆ J1 . ˆ J1 = [cos  ˆ J1 , Hˆ 1 ] = − c1 J1 sin2  cos  dt i 2

(8.158)

(8.159)

282

8 Quantum Theory of Mesoscopic Circuit Systems

So, we give ˆJ  d d tan 1 = dt 2 dt



ˆ J1 1 − cos  ˆ J1 sin 

 =

ˆJ 2E c(J1 ) E J1  tan 1 , 2  2

(8.160)

whose solution reads   ˆ J1 ˆ J (0) 2E c(J1 ) E J1   tan = exp t tan 1 . 2 2  2

(8.161)

ˆ J1 of the Equation (8.161) shows that the variation of the phase difference operator  ˆ first junction with time t is independent of the phase difference operator  J2 . Further, ˆ J2 of the second junction also the time evolution of the phase difference operator  reads  d 1  ˆ J2 = ˆ J2 , Hˆ 1 = 0, sin  sin  (8.162) dt i which means that, when the external energy acts on the first junction, the phase ˆ J2 of the second junction does not change with time t. Similarly, difference operator  when external energy only acts on the second junction, using Eqs. (8.130) and (8.151), we also obtain the corresponding Hamiltonian operator as EJ Hˆ 2 = 22 

  m L1 (J ) ˆ ˆ J2 . 2E c2 N2 + pˆ L − pˆ L sin  T2 1 T2 2

(8.163)

ˆ J2 reads Therefore, the time evolution of the phase difference operator    ˆ J (0) ˆ J2 2E c(J2 ) E J2   = exp t tan 2 . tan 2 2  2

(8.164)

ˆ J2 is independent of the Obviously, the evolution of the phase difference operator  first junction. Moreover, its time evolution law is the same as that of the phase ˆ J1 in form. difference operator 

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