The Matrix Perturbation Method in Quantum Mechanics 3031485459, 9783031485459

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Francisco Soto-Eguibar Braulio Misael Villegas-Martínez Héctor Manuel Moya-Cessa

The Matrix Perturbation Method in Quantum Mechanics

The Matrix Perturbation Method in Quantum Mechanics

Francisco Soto-Eguibar • Braulio Misael Villegas-Martínez • Héctor Manuel Moya-Cessa

The Matrix Perturbation Method in Quantum Mechanics

Francisco Soto-Eguibar Instituto Nacional de Astrofísica, Óptica y Electrónica, (INAOE) San Andrés Cholula, Puebla, Mexico

Braulio Misael Villegas-Martínez Instituto Nacional de Astrofísica, Óptica y Electrónica, (INAOE) San Andrés Cholula, Puebla, Mexico

Héctor Manuel Moya-Cessa Instituto Nacional de Astrofísica, Óptica y Electrónica, (INAOE) San Andrés Cholula, Puebla, Mexico

ISBN 978-3-031-48545-9 ISBN 978-3-031-48546-6 https://doi.org/10.1007/978-3-031-48546-6

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 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 translation, 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 publisher, 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 publisher 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 publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

A las hermanas superiores Emma, Renée y Juju, y a Raulito. Francisco. Para mi madre Balbina, guia estrella de mi vida. Gracias por todo tu amor, apoyo y aliento. Braulio. Para Isabel y Leonardo. Héctor.

Preface

Introduction: Matrix Perturbation Method—A New Approach to Time-Independent Quantum Perturbation Theory Perturbation theory fell from the sky. In the seventeenth century, the world was confined to the six planets visible to the naked eye. The insatiable curiosity of the Western mind compelled the pursuit of comprehending the very universe it inhabited, thus propelling the quest for the elucidation of the celestial laws governing planetary motion. Johannes Kepler, based on Tycho Brahe’s meticulous observations, discovered the rules planets follow in their orbits around the Sun; yet, Kepler’s explanation was purely pragmatic [1,2]. The pivotal breakthrough came with Sir Isaac Newton’s discovery of the laws of motion and the law of universal gravitation, with which he could fundamentally explain why planets move the way they do [3]. The gravitational force keeps planets revolving around the Sun, and it is this gravitational force that holds satellites, the moons, revolving around planets. In fact, Newton’s explanation of “the World” marks one of humanity’s great milestones, an immense achievement for which he had to construct not only the necessary physics, but also a part of the mathematics. When only two bodies are considered to interact gravitationally, the solution to the equations of motion, the second law of Newton, can be found exactly and analytically. However, when a third body is present, it is very easy to determine the forces each of them experiences, as gravitational forces add linearly; still, the resulting equations of motion are very complicated and have not been solved exactly to date. The challenge becomes even greater when a fourth body is involved in the system; in such cases, describing the motion of any of the four bodies becomes an impressive task. Nevertheless, accounting for the influence of a third body in the motion of planets became necessary. Lagrange and Laplace were the first to realize that the constants describing the motion of planets around the Sun are perturbed by the influence of other planets, and these influences vary over time, hence the name perturbations. Uranus was discovered by William Herschel in 1781, and by the early nineteenth century, it was already evident that Uranus’s motion did vii

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not precisely match the predictions of Newtonian mechanics. Faced with such a dilemma, there were two alternatives: either the theory describing this phenomenon was incorrect, or the theory was correct, but not all necessary variables were being taken into account. The former hypothesis was absolutely unacceptable since, by then, Newton’s mechanics had been elevated to the status of dogma, as indicated by Laplace’s response to Napoleon when asked about God’s place in his book Treatise of Celestial Mechanics (Traité de mécanique céleste) [4]: Sir, I had no need of that hypothesis. The only viable option was the second: certain influences were not being considered in Uranus’s motion. The French astronomer Alexis Bouvard speculated that there should be an even more distant planet “disturbing” Uranus’s motion, and that this perturbation should explain the anomalies in Uranus’s movement. Urban Le Verrier and John Couch Adams seized upon this hypothesis, dedicating years to ascertain the location and mass of this hypothetical planet. Le Verrier sent his results to Johann Gottfried Galle, the director of the Berlin Observatory, who, on September 24, 1846, pointed his telescope to the indicated location and found Neptune within 8 arc minutes. As Francois Arago later remarked, Le Verrier discovered Neptune with the tip of his pen. Adams sent his conclusions to the Royal Greenwich Observatory, but they did not pay much attention, and all the “ glory” went to Le Verrier. It is worth noting that historians of science have studied this event, and currently, some credit is also given to Adams. How did Le Verrier and Adams calculate where this eighth planet should be? By the time they embarked on the adventure of searching for the eighth planet, Laplace and Lagrange had been joined by other physicist-mathematicians, such as Poisson and Gauss, in the study of the so-called theory of perturbations, and numerical calculations could be done with great precision. We have already mentioned that solving the motion of Uranus under the influence of the Sun is relatively straightforward; however, when the influence of a third body, another planet, is considered, the problem becomes formidable, unsolvable. In such problems, the need for perturbative methods arises. Evaluating the ratio of the force Neptune exerts on Uranus compared to the force the Sun exerts on Uranus, one finds it to be approximately .1.89 × 10−4 ; that is, the force Neptune exerts on Uranus is very small compared to that of the Sun. This leads us to assume that the solution to the problem of these bodies should be very similar to the solution of the problem with only two bodies. We can assume that the original solution will be slightly modified, slightly perturbed. This methodology was used by Le Verrier and Adams to make their determination. Perturbation theory continued to develop throughout the end of the nineteenth century and the beginning of the twentieth century. In particular, Lord Rayleigh, in his book The Theory of Sound, Volume I [5], studied the harmonic vibrations of a string perturbed by small inhomogeneities, where he developed perturbative methods. Of course, many mathematicians were also involved in the development of perturbation theory; among them was Henri Poincaré, who made valuable contributions in his book Les méthodes nouvelles de la mécanique céleste [6]. In general, perturbation theory is primarily used to find solutions for differential equations that undergo small modifications in some part of their structure. However,

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perturbation theory is a method employed to discover approximate solutions for various problems that can be mathematically represented by equations, whether they are differential or of any other type. Such approximations to the equations are derived from previously established exact solutions. Initially, any system is modeled by a set of equations (usually differential, but not necessarily), and an exact analytical solution is obtained. Subsequently, we identify variables in our system that have not been considered, either to simplify the system or due to lack of awareness. At a certain point, we realize that these variables might have significant and interesting effects. In a subsequent approximation, these variables must be taken into account. This process can also be somewhat reversed, implying that these small or negligible variables are identified from the outset. However, if we consider them, obtaining an exact analytical solution for the mathematical problem becomes impossible. Consequently, we choose to neglect them to attain an exact analytical solution. If we can quantify the potential effects of these initially disregarded variables and deduce that they are small, we can treat them as a perturbation. This revised system is referred to as the perturbed system. It is reasonable to assume that the solution to the new problem, the perturbed problem, should closely resemble the solution of the original problem, the unperturbed one. While the hypothesis that the solution to the perturbed problem closely resembles the solution of the unperturbed problem is not universally true, instances where this does not hold are particularly intriguing; these cases, however, often extend beyond the confines of the theories in which they originated and warrant a distinct form of analysis. In the context of quantum mechanics, perturbation theory was initially developed by Schrödinger for his stationary equation [7–11]. Schrödinger relied on Rayleigh’s studies of sound theory, and thus, it is commonly known as Rayleigh-Schrödinger perturbation theory. It is a theory of time-independent perturbations that essentially serves to find the eigenvalues and eigenfunctions of a Hamiltonian. In timeindependent perturbation theory, both the original, unperturbed Hamiltonian, and the perturbation are static, i.e., they do not depend on time. The success and utility of time-independent perturbation theory have been enormous, and it has been applied to countless fields, like solid-state physics, atomic physics, and quantum field theory. With the advent of powerful computers, its usefulness continues to grow. Timedependent perturbation theory is an extension of the standard perturbation theory in quantum mechanics, specifically tailored to address time-dependent Hamiltonians. This powerful approach has been instrumental in understanding the dynamic behavior of quantum systems subjected to external influences and has played a pivotal role in various areas of physics, including atomic and molecular physics, quantum optics, and condensed matter physics. Time-dependent perturbation theory was originally developed by Dirac in his studies of radiation theory [12]. In this book, an alternative perturbative approach, called the Matrix Perturbation Method, is proposed. This new scheme, based on the implementation of triangular matrices, allows for the approximate solution of the time-dependent Schrödinger equation in an elegant and straightforward manner. The method has demonstrated that the corrections to the wave function and energy can be contained in only one expression, unlike standard perturbation theory, where they need to be calculated

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separately. Moreover, the Matrix Perturbation Method can be employed even when it is not possible to find the eigenstates of the unperturbed Hamiltonian but a unitary evolution operator can be found. Furthermore, its approximated solutions not only present conventional stationary terms but also time-dependent factors, allowing us to understand the temporal evolution of the corrections. A remarkable feature is that the general expression to compute them does not distinguish if the Hamiltonian is degenerate or not. Additionally, the formalism offers an alternative way to express the Dyson series in a matrix form. Another essential aspect is that the Matrix Perturbation Method can be extended to the Lindblad master equation. Therefore, the Matrix Perturbation Method possesses many attractive characteristics that cannot be found in conventional treatments of perturbation theory. It is also worth noting that the Matrix Perturbation Method has been extended to provide normalized approximated solutions in each step, in contrast to standard perturbation theory, where the solutions are not normalized. The organization of this book is as follows: In Chap. 1, we present the traditional time-independent perturbation theory, the one developed by Schrödinger, based on Lord Rayleigh’s studies of sound theory. The idea is not to present this theory in detail, but to briefly show it for comparison with the Matrix Perturbation Method. We only present the case when the Hamiltonian spectrum is discrete and analyze both, non-degenerate and degenerate cases. In Chap. 2, we demonstrate the standard time-dependent perturbation theory; we study the method of variation of constants and the Dyson series; some specific cases are analyzed; and examples are presented. Again, the intention is not to present time-dependent perturbation theory in detail, as it is excellently done in many books; what we want is to have a reference for comparison with the Matrix Perturbation Method. In these first three chapters, we use the International System of Units (SI), in such a way, that .h¯ is present in many expressions. Due to the complexity of the expressions that appear in the Matrix Perturbation Method, in the following chapters, that is, from Chap. 3 onward, we use a system of units in which .h¯ = 1; this is in fact a common practice in quantum mechanics. In Chap. 3, we introduce the Matrix Perturbation Method; in this case, we have tried to be very detailed, showing all the steps of the method’s development. We present the solution to first order, then to second order, and then generalize it for order n; we also introduce the normalization constant, establish the connection with traditional time-independent perturbation theory, and finally, present the Dyson series in matrix form. Chapter 4 is devoted to studying examples of problems solved with the Matrix Perturbation Method; we present various systems and compare the results of the Matrix Perturbation Method with traditional methods and exact solutions, where they exist. In Chap. 5, we study concrete applications of the Matrix Perturbation Method; we analyze the case of an ion trapped in the high-intensity regime; as a corollary of the trapped ion, we analyze the Rabi model; and finally, we present a binary array of waveguides. In Chap. 6, we show the extension of the Matrix Perturbation Method to the case of the Lindblad master equation; as an example, we present a cavity with losses filled with a Kerr medium. Finally, in Chap. 7, we apply the Matrix Perturbation Method to time-dependent systems

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through the factorization of the temporal dependence; we present the cases of a particle with pulsating mass and a mass changing exponentially in a linear potential; some other examples are also shown. Puebla, Mexico

Francisco Soto-Eguibar Braulio Misael Villegas-Martínez Héctor Manuel Moya-Cessa

References 1. J. Kepler, Johannes Kepler New Astronomy (Cambridge University Press, Cambridge, 1993) 2. J. Kepler, Harmonice Mundi (Johann Planck, Linz, 1619) 3. I. Newton, Philosophiœ Naturalis Principia Mathematica (Edmund Halley, London, 1687) 4. P.S. Laplace, Traité de mécanique céleste (Charles Crapelet, Paris, 1798–1825) 5. J.W. Strutt, Theory of Sound (Cambridge University Press, Cambridge, 2011) 6. H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Gauthier-Villars et Fils, 1892) 7. E. Schrödinger, Quantisation as a problem of proper values I. Ann. Phys. 79(4), 361–367 (1926) 8. E. Schrödinger, Quantisation as a problem of proper values II. Ann. Phys. 79(4), 489–527 (1926) 9. E. Schrödinger, Quantisation as a problem of proper values III. Ann. Phys. 80(4), 437–490 (1926) 10. E. Schrödinger, Quantisation as a problem of proper values IV. Ann. Phys. 81(4), 109–139 (1926) 11. E. Schrödinger, Collected Papers on Wave Mechanics (Blackie and Son Limited, Glasgow, 1928) 12. P.A.M. Dirac, The quantum theory of the emission and absorption of radiation. Proc. R. Soc. London Ser. A 114(767), 243–265 (1927)

Contents

1

2

3

Standard Time-Independent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Discrete Non-degenerated Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Example: The One Dimensional Harmonic Oscillator with a Cubic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Discrete Degenerated Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Example: The Three-Dimensional Isotropic Harmonic Oscillator with an xy Perturbation . . . . . . . . . . . . . . . . . 1.3.2 Example: The Stark Effect in the Hydrogen Atom. . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 8 14 16 18 20

Standard Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Method of Variation of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Time-Dependent Perturbation Theory When the Spectrum Is Discrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Time-Dependent Perturbation Theory When the Spectrum Is Discrete and Continuous . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Method of Dyson Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Constant Perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Constant Perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 23

The Matrix Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Matrix Approach to the Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 First-Order Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Second-Order Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Higher Order Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Normalization Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 48 49 55 60 61

23 33 40 43 44 45 46

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3.4 Connection with the Standard Time-Independent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Dyson Series in the Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 66 67

4

Examples of the Matrix Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term . . 70 4.2.1 The Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.2 The Approximated Perturbative Solution. . . . . . . . . . . . . . . . . . . . . . 75 4.2.3 Comparison Between the Exact and the Approximated Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 The Harmonic Oscillator Plus a Cubic Potential . . . . . . . . . . . . . . . . . . . . . . 83 4.3.1 First Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.2 Second Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.3 Initial Condition Equal to a Number State. . . . . . . . . . . . . . . . . . . . . 89 4.4 The Repulsive Quadratic Potential Plus a Linear Term . . . . . . . . . . . . . . . 90 4.4.1 Exact Solution of the Repulsive Quadratic Potential. . . . . . . . . . 90 4.4.2 Exact Solution to the Quadratic Repulsive Potential Plus a Linear Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.3 Perturbative Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.4 Comparison of the Exact and the Perturbative Solutions . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5

Applications of the Matrix Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Trapped Ion Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 High Intensity Regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Perturbative Solution for the Rabi Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Binary Waveguide Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Small Rotation Approximated Solution . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Matrix Perturbative Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Comparison of the Perturbative Solution with the Exact Solution and with the Small Rotation Solution. . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

The Matrix Perturbation Method for the Lindblad Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Lindblad Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 First-Order Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Second-Order Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Lossy Cavity Filled with a Kerr Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 109 110 121 122 124 124 125 131 133 137 137 138 139 141 142 143 143

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6.3.2 Perturbative Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.4 Comparison Between the Exact and the Approximated Solution. . . . . 148 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7

Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 First Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 A Particle with Strongly Pulsating Mass Moving in a Linear Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 A Particle with Exponentially Increasing Mass in the Presence of a Linear Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Second Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 158 161 166 173 175 179 187

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Chapter 1

Standard Time-Independent Perturbation Theory

Abstract We present standard time-independent perturbation theory in this chapter. Since the degeneracy of the spectrum fundamentally influences the solution procedure, we have divided the non-degenerate case from the degenerate one. For nondegenerate systems, we provide second-order corrections for both the eigenvalues and the eigenfunctions. Additionally, we briefly analyze a simple example. In the case of degenerate systems, given their complexity, we present a method to calculate only the first correction to the energy along with the corresponding wave functions at zero order. We also provide two example cases for illustration.

1.1 Introduction The main equation in non-relativistic quantum mechanics is the Schrödinger equation because its solution, the wave function, contains all the relevant information about the behavior of a physical quantum system [1–6]. When the Hamiltonian is time independent, the Schrödinger equation reduces to its stationary version, the solution of which essentially involves an eigenvalue problem. The eigenvalues are the energy values of the different levels, which, in turn, are represented by the corresponding stationary wave functions. Since its introduction, the stationary Schrödinger equation has been widely studied. However, there are only a few cases in which it can be solved exactly, such as the infinite well, the harmonic oscillator, the hydrogen atom [1–6], and the Morse potential [7]. These are typical examples of potentials where an exact analytical solution is known. The great majority of problems involving the stationary Schrödinger equation are very complex and cannot be solved exactly. We are then forced to apply approximation methods [1– 6], which, when correctly used, give us a very good understanding of the behavior of quantum systems. Time-independent perturbation theory, also known as Rayleigh–Schrödinger perturbation theory, has its roots in the works of Rayleigh and Schrödinger [8, 9], but the mathematical foundations were only established by Rellich in the late thirties of the past century (see [10] and the references therein). This method has been applied with great success to solve a vast variety of problems. Through its continuous © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6_1

1

2

1 Standard Time-Independent Perturbation Theory

implementation, a lot of techniques have been developed, ranging from numerical methods [11] to more mathematical and fundamental aspects, such as convergence problems [12, 13]. The Rayleigh–Schrödinger theory is appropriate when we have a time-independent Hamiltonian, which can be separated into two parts, as follows: Hˆ = Hˆ 0 + λHˆ p ,

.

where .Hˆ 0 is called non-perturbed Hamiltonian, and it is usually assumed to have known solutions, i.e., its eigenvalues and eigenvectors are known. The second part of the Hamiltonian, .Hˆ p , is supposed to be small compared to .H0 ; thus .Hˆ p is called the “perturbation” because its effect in the energy spectrum and in the eigenfunctions will be small. To keep track of the “size” of the terms that appear in the development of this method, it is usual to write .Hˆ p in terms of a dimensionless positive real parameter .λ, which is considered very small compared to one. As we already said, we want to solve the perturbed eigenvalue problem   Hˆ |n〉 = Hˆ 0 + λHˆ P |n〉 = En |n〉

.

(1.1)

knowing the complete analytic solution of the unperturbed equation       Hˆ 0 n(0) = En(0) n(0) ,

.

(1.2)

where n indexes the eigenvalues, and can represent a set of variables, which in turn can be discrete variables, continuous or variables that are a combination of discrete ˆ ˆ and continuous. Both operators,  (0)  .H0 and .HP , are supposed to be Hermitian [14].  The wave functions . n are assumed to be orthonormal, .

   m(0) n(0) = δm,n ,

(1.3)

 

  n(0) n(0) .

(1.4)

and to form a complete set, i.e., .

Iˆ =

n

As the perturbation parameter .λ is supposed to be small and the Schrödinger equation is linear, it is logical to assume that the solution of the perturbed problem should not be very different from the solution of the unperturbed one; thus, we propose that

.

|n〉 =

∞

j =0

   λj n(j ) ,

(1.5)

1.2 Discrete Non-degenerated Spectrum

3

and that En = En(0) +

∞

.

λk ΔEn(k) ,

(1.6)

k=1 (j )

and our goal is to determine the corrections   to the energy .ΔEn , j = 1, 2, 3, . . . and the corrections to the eigenvectors .n(j ) , j = 1, 2, 3, . . . . The spectrum of both problems, the non-perturbed and the perturbed, can be discrete, continuous or a mixed spectrum, part discrete and part continuous; also both spectra can be non-degenerated or degenerated. As the procedure to determine the corrections to the energy and to the eigenstates depends on degeneracy, we treat first the case of a discrete non-degenerated spectrum, and later we will address the case of a discrete degenerated spectrum. The organization of this chapter is the following: In Sect. 1.2, we present the standard time-independent perturbation theory for non-degenerated systems; an example is succinctly analyzed; in Sect. 1.3, the standard time-independent perturbation theory for degenerated systems is studied; two examples are briefly shown.

1.2 Discrete Non-degenerated Spectrum In quantum mechanics, a non-degenerated system is such that with every eigenvalue En(0) there is associated only one eigenvector, .|n(0) 〉. In the case of bound states in one dimension, the spectrum is always non-degenerated [5]; however, in more than one dimension it is usual for systems to be degenerate. Therefore, in the understanding that the system is non-degenerated, substituting the proposals (1.5) and (1.6) into the perturbed eigenvalue problem (1.1), we obtain

.

∞

∞   

   λj Hˆ 0 n(j ) + λj +1 Hˆ P n(j )

j =0

j =0

.

= En(0)

∞

∞ ∞

  

   λj n(j ) + λj +k ΔEn(k) n(j ) ;

j =0

(1.7)

k=1 j =0

using the fact that [15] ∞ ∞

.

k=1 j =0

⎞ ⎛ j ∞   



  λj +k ΔEn(k) n(j ) = λj ⎝ ΔEn(k) n(j −k) ⎠ , j =1

k=1

(1.8)

4

1 Standard Time-Independent Perturbation Theory

and changing the index in the second sum in the left side, we cast Eq. (1.7) as ∞

.

∞ ∞    

 

   λj Hˆ 0 n(j ) + λj Hˆ P n(j −1) =En(0) λj n(j )

j =0

j =1

j =0

+

∞

⎞ ⎛ j  

 λj ⎝ ΔEn(k) n(j −k) ⎠ .

j =1

k=1

(1.9) Taking apart the term .j = 0, we get the unperturbed problem .Hˆ 0 |n0 〉 = En0 |n0 〉; then, the series reduces to ⎞ ⎛ j ∞       



    . λj ⎝Hˆ 0 n(j ) + Hˆ P n(j −1) − En(0) n(j ) − ΔEn(k) n(j −k) ⎠ = 0. j =1

k=1

(1.10) As the .λ powers are linearly independent, we arrive to j       

  (j )  (j −1)  (0)  (j ) ˆ ˆ + HP n − En n − .H0 n ΔEn(k) n(j −k) = 0.

(1.11)

k=1

Now, we multiply this last equation scalarly by .〈n(0) | to obtain          〈n(0) |Hˆ 0 n(j ) +〈n(0) |Hˆ P n(j −1) − En(0) n(0) n(j )

.

−

j

   ΔEn(k) n(0) n(j −k) = 0;

(1.12)

k=1

   (0)  using the hermiticity of the operator .Hˆ 0 and the fact that .Hˆ 0 n(0) = En n(0) , we can cancel the first and third term and get j   

   〈n(0) |Hˆ P n(j −1) − ΔEn(k) n(0) n(j −k) = 0;

.

(1.13)

k=1

taking .j = 1,    〈n(0) |Hˆ P |n(0) 〉 − ΔEn(1) n(0) n(0) = 0;

.

(1.14)

1.2 Discrete Non-degenerated Spectrum

5

and from the orthogonality of the eigenvectors, Eq. (3.4), ΔEn(1) = 〈n(0) |Hˆ P |n(0) 〉,

.

(1.15)

which is the first-order correction to the energy. Now, we are going to calculate the first-order correction to the eigenvectors; for that, we use the closure relation (3.5), so we write          

  (1)   (0) (0) = 〈m | n(1) = m(0) n(1) m(0) ; . n m m

(1.16)

m

using formula (1.11) for .j = 1,             Hˆ 0 n(1) + Hˆ P n(0) − En(0) n(1) − ΔEn(1) n(0) = 0,

.

(1.17)

and multiplying it scalarly by .〈m(0) | with .m /= n,       〈m(0) |Hˆ 0 n(1) +〈m(0) |Hˆ P |n(0) 〉 − En(0) m(0) n(1)    − ΔEn(1) m(0) n(0) = 0;

.

(1.18)

from the hermiticity of the Hamiltonian .Hˆ 0 , and from the orthonormality condition Eq. (3.4), as .m /= n, the last term is zero, so .

     (0) Em − En(0) m(0) n(1) + 〈m(0) |Hˆ P |n(0) 〉 = 0.

(1.19)

There are now two possibilities; first, that .〈m(0) |Hˆ P |n(0) 〉 be null, and second, that it be different from zero. We will discuss briefly the first possibility at the end of the section dedicated to degenerated systems; in the second situation, as we are treating the non-degenerated case, and we also have assumed that .m /= n, we get .

  〈m(0) |Hˆ |n(0) 〉 P  m(0) n(1) = , (0) (0) En − Em

(1.20)

and substituting it into Eq. (1.16),

.

  

〈m(0) |Hˆ P |n(0) 〉   (1) (0) m = , n  (0) (0) m, m/=n En − Em

which is the first-order correction to the eigenvectors.

(1.21)

6

1 Standard Time-Independent Perturbation Theory

Summarizing, the first-order correction to the energy results to be En ≈ En(0) + λ〈n(0) |Hˆ P |n(0) 〉,

.

(1.22)

and the first-order correction to the wave function is .

  

〈m(0) |Hˆ P |n(0) 〉   (0) |n〉 ≈ n(0) + λ m .  (0) (0) m, m/=n En − Em

(1.23)

Normally, first-order fixes are sufficient; furthermore, higher order corrections are obtained by particular procedures and turn out to be too complicated. However, to illustrate the method, and to show its complications, we will now calculate the second-order corrections; for that, we take again expression (1.11), but now for .j = 2, to arrive to                Hˆ 0 n(2) + Hˆ P n(1) − En(0) n(2) − ΔEn(1) n(1) − ΔEn(2) n(0) = 0;

.

(1.24)

we make the dot product by the left with .〈n(0) |, apply the Hamiltonian .Hˆ 0 , use the orthonormality relations (3.4), and cancel terms to obtain       ΔEn(2) = 〈n(0) |Hˆ P n(1) − ΔEn(1) n(0) n(1) .

.

(1.25)

  We can show that . n(0) n(1) = 0; indeed, using (1.21) .

   

〈m(0) |Hˆ P |n(0) 〉  (0)  (0) m n(0) n(1) = n  (0) (0) m, m/=n En − Em =

〈m(0) |Hˆ P |n(0) 〉 m, m/=n

(0)

(0)

En − Em

ΔEm,n = 0;

(1.26)

thus,    ΔEn(2) = 〈n(0) |Hˆ P n(1) ,

.

(1.27)

and using again Eq. (1.21), we find that the second-order correction to the energy is

ΔEn(2)

.

   (0) ˆ  (0) 2   |HP m  〈n

 . = 〈n(0) |Hˆ P n(1) = (0) (0) En − Em m, m/=n

(1.28)

1.2 Discrete Non-degenerated Spectrum

7

We proceed now to calculate the second-order correction to the eigenvectors; we use again Eq. (1.11), with .j = 2, to write                Hˆ 0 n(2) + Hˆ P n(1) − En(0) n(2) − ΔEn(1) n(1) − ΔEn(2) n(0) = 0;

.

(1.29)

multiplying from the left by the bra .〈m(0) |, with .m /= n, using the hermiticity of the Hamiltonian .Hˆ 0 , and the orthonormality condition (3.4), we get .

       (0) Em − En(0) m(0) n(2) + 〈m(0) |Hˆ P n(1) − ΔEn(1) m(0) n(1) = 0;

(1.30)

  (1) substituting .ΔEn from (1.15), and . m(0) n(1) from (1.20), m

.

(0)



(2)

n

=

 (0)   (1)  m Hˆ P n (0)

(0)

En − Em

−

〈n(0) |Hˆ P |n(0) 〉〈m(0) |Hˆ P |n(0) 〉 .   (0) (0) 2 En − Em

(1.31)

From the closure relation (3.5), we have          

  (2)   = m(0) n(2) m(0) ; . n m(0) 〈m(0) | n(2) = m

(1.32)

m

replacing in this last expression (1.31), we obtain ⎤ ⎡  (0)   (1)    (0) (0) (0) (0)   ˆ

ˆ ˆ 〈n |HP |n 〉〈m |HP |n 〉 ⎥  (0)   (2) ⎢ m HP n . = + . n ⎦ m ⎣ (0)   (0) (0) (0) 2 E − E n m m, m/=n E −E n

m

(1.33) Finally, if we use expression (1.21) above, we have the second-order correction to the energy  

 (2) = . n m, m/=n

       

m(0) H k (0) k (0) H |n(0) 〉  P P    m(0) (0) (0) (0) (0) En − Em En − Ek k, k/=n

− 〈n(0) |Hˆ P |n(0) 〉

〈m(0) |Hˆ P |n(0) 〉  (0)  .   m (0) (0) 2 m, m/=n En − Em

(1.34)

Summarizing, up to second-order correction, the energy is En ≈

.

En(0)

+ λ〈n |Hˆ P |n 〉 + λ (0)

(0)

2

 

|〈n(0) |Hˆ P m(0) |2 m, m/=n

(0)

(0)

En − Em

,

(1.35)

8

1 Standard Time-Independent Perturbation Theory

and also, up to second-order correction, the wave function is

.



〈m(0) |Hˆ P |n(0) 〉  (0) m (0) (0) m, m/=n En − Em          

m(0) H P n(1) − n(0) H P n(1)

|n〉 ≈|n(0) 〉 + λ

+ λ2

(0)

(0)

En − Em

m, m/=n

⎤

〈n(0) |Hˆ P |n(0) 〉〈m(0) |Hˆ P |n(0) 〉 ⎥  (0)  . + ⎦ m   (0) (0) 2 En − Em

(1.36)

1.2.1 Example: The One Dimensional Harmonic Oscillator with a Cubic Perturbation Let us consider as non-perturbed system a particle of mass .μ subject to a one dimensional harmonic potential; so, the unperturbed Hamiltonian is μω2 2 pˆ 2 xˆ , + Hˆ 0 = 2 2μ

.

(1.37)

where .xˆ = x is the position operator, .pˆ = −i h¯ d/dx is the momentum operator, and .ω is the oscillator angular frequency. It is well known [1–6] that the eigenvalues of the above Hamiltonian are   1 (0) , n = 0, 1, 2, . . . , .En = h¯ ω n + (1.38) 2 and the eigenfunctions, in the coordinate representation, are         μω 1/4 μω 1 μωx 2  Hn exp − ϕn(0) (x) = x n(0) = √ x , 2h¯ h¯ 2n n! π h¯

.

n = 0, 1, 2, . . . ,

(1.39)

where .Hn (z) denotes the Hermite polynomials [15–18]. To compute all the matrix elements of the perturbed Hamiltonian that we will need, it is much easier to introduce the usual harmonic oscillator’s raising and lowering operators; these operators are defined as [2, 4, 6]  aˆ =

.

  i μω pˆ , xˆ + μω 2h¯

 aˆ = †

  i μω xˆ − pˆ . 2h¯ μω

(1.40)

1.2 Discrete Non-degenerated Spectrum

9

Fig. 1.1 The harmonic oscillator potential (blue dashed) and the resulting perturbed potential (continuous black)

We consider now as perturbation a cubic potential Hˆ P (x) = ηxˆ 3 ,

.

(1.41)

where .η is a constant that has units of energy divided by units of length cubed (Fig. 1.1). In terms of the ladder operators, (1.40), the perturbation can be written as  Hˆ P (x) = α

.

3 h¯  † aˆ + aˆ , 2μω

(1.42)

and since aˆ |n〉 =

.

√ n |n − 1〉 ,

aˆ † |n〉 =

√ n + 1 |n + 1〉 ,

(1.43)

it is easy to find that the matrix elements of the perturbation potential are [2–4, 6] 3/2     h¯  m (m − 1) (m − 2) δn+3,m + 3m3/2 δn+1,m 〈n(0) |Hˆ P m(0) = α 2μω   +3 (m + 1)3/2 δn−1,m + (m + 3) (m + 2) (m + 1) δn−3,m ,

.

(1.44) where .δk,j is the Kronecker delta [15].

10

1 Standard Time-Independent Perturbation Theory

Thus, for the first-order correction to the energy, we obtain ΔEn(1) = 〈n(0) |Hˆ P |n(0) 〉 = 0,

.

n = 0, 1, 2, . . . .

(1.45)

Since the energy first order correction is zero, in this case second order correction is essential; for the second order correction to the energy, we have (2) .ΔEn

=

 

|〈n(0) |Hˆ P m(0) |2 m, m/=n

=−

(0)

(0)

En − Em

 h¯ 2 α 2  30n2 + 30n + 11 , n = 0, 1, 2, . . . . 3 4 8μ ω

(1.46)

Summarizing, the energy up to second order is    h¯ 2 α 2  1 − λ2 3 4 30n2 + 30n + 11 , En(2) = h¯ ω n + 2 8μ ω

.

n = 0, 1, 2, . . . . (1.47)

In Fig. 1.2, we depict the first five energy levels as function of the perturbation parameter .λ, given by expression (1.47); a full analysis of this example can be found in [19, Problem 35, page 80]. In order to have something with which to compare the perturbative results found for the eigenvalues of the energy, we have also calculated them numerically, using the function NDEigensystem of the Mathematica software [20]. In Table 1.1, we show what we have obtained for .λ = 0.01; for such a value of .λ, we see that

Fig. 1.2 The first five energy levels as function of the perturbation parameter .λ. The dotted gray lines are the energy levels of the unperturbed harmonic oscillator

1.2 Discrete Non-degenerated Spectrum

11

Table 1.1 Table of the eigenvalues of the perturbed Hamiltonian, using the perturbative secondorder energy, Eq. (1.47), and the numerical values obtained using the Mathematica software, for .λ = 0.01 = 0.01

.λ

Perturbative value Numerical value

.n

=0

0.500 0.500

.n

=1

1.499 1.499

.n

=2

2.498 2.498

.n

=3

3.495 3.495

Table 1.2 Table of the eigenvalues of the perturbed Hamiltonian, using the perturbative secondorder energy, Eq. (1.47), and the numerical values obtained using the Mathematica software, for .λ = 0.05 = 0.05 Perturbative value Numerical value

.λ

=0 0.497 0.496

.n

=1 1.478 1.477

.n

=2 2.440 2.435

.n

=3 3.384 3.368

.n

no difference is noticeable between the second-order value and the value obtained numerically. In Table 1.2, we present the same results, but for .λ = 0.05; in this case we can see how differences are beginning to be noticed between the second-order energy values and those values obtained numerically, which is to be expected, because the method, being perturbative, rests on the smallness of .λ. We can also calculate, without much difficulty, the first-order correction to the state vector of the system; we have from (1.23) and (1.44),

.

  

〈m(0) |Hˆ P |n(0) 〉   (1) (0) =λ n m (0) (0) m, m/=n En − Em      α h¯   3/2  (0) (0) = n − 1) − 2) + 9n − 1) − 3) (n (n (n (n 6 2μ3 ω5        −9 (n + 1)3/2 (n + 1)(0) − (n + 1) (n + 2) (n + 3) (n + 3)(0) , n = 0, 1, 2, . . . ,

(1.48)

where it is understood that the kets .|n(0) 〉 with negative value inside are zero. The complete wave function to first order is .

       (1)   = n(0) + λ n(1) , ϕ

(1.49)

and to get an idea of how good this approximation is, we plot the normalized solution, together with the numerical solution. To achieve this, we must write the wave function (1.49) in configuration space, which is easily accomplished by substituting in (1.48) each ket .|k〉 for the corresponding function .ϕk (x) from Eq. (1.39). The numerical solution is obtained using the function NDEigensystem

12

1 Standard Time-Independent Perturbation Theory

( ) ( )

( ) ( )

( )

   (1)(x) 2 Fig. 1.3 Probability density, .ϕn  , of the normalized wave function to first order (black  2 continuous line), and the probability density, .ϕn,num (x) , of the numeric solution (red dashed line), for .λ = 0.01. The values of the parameters are .h¯ = 1, μ = 1, ω = 1, α = 1. (a) .n = 0. (b) .n = 1. (c) .n = 2. (d) .n = 3

of Mathematica [20]; as Mathematica delivers normalized numerical solutions, we have normalized each one of the first-order approximation wave functions. In  2   Figs. 1.3 and 1.4, we plot the probability density, .ϕn(1)(x)  , of the normalized wave function to first order, and as a reference, we also graph the probability density,  2 .ϕn,num (x) , of the numerical wave function; in Fig. 1.3, we use .λ = 0.01, and in Fig. 1.4, .λ = 0.05. It is clear that for very small values of .λ the first-order perturbative approximation is very good; note also that for the same value of .λ, when the value of n grows, the approximation is a little worse, because as the amplitude of the oscillation is bigger, the particle travels to the part where the potential changes are more notorious. When .λ becomes larger, we begin to see differences between the perturbative solution and the numerical solution; again this result is within the logic of the method, which has as a fundamental assumption that .λ is small. The second-order correction to the state vector is of little use and the procedure to obtain it is long and tedious; however, at present, with symbolic calculation software (such as Mathematica [20] or Maple [21], among many others), they can be obtained easily, quickly, and precisely. We substitute the wave function, given by expression (1.48), in Eq. (1.33), taking into account the matrix elements given

1.2 Discrete Non-degenerated Spectrum

13

   (1)(x) 2 Fig. 1.4 Probability density, .ϕn  , of the normalized wave function to first order (black  2 continuous line), and the probability density, .ϕn,num (x) , of the numeric solution (red dashed line), for .λ = 0.05. The values of the parameters are .h¯ = 1, μ = 1, ω = 1, α = 1. (a) .n = 0. (b) .n = 1. (c) .n = 2. (d) .n = 3

in (1.44), to obtain    (2) =−α . n + + + + +



h¯ 3 μ ω5

√

(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6) |n − 6〉 √ 12 2

√ 3 (n + 1)(n + 2)(n + 3)(n + 4)(4n + 7) |n − 4〉 √ 8 2 ! " √ 3 (n + 1)(n + 2) 7n2 + 33n + 27 |n − 2〉 √ 4 2 ! " √ 3 n(n − 1) 7n2 − 19n + 1 |n + 2〉 √ 4 2 √ 3 n(n − 1)(n − 2)(n − 3)(4n − 3) |n + 4〉 √ 8 2 √ n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) |n + 6〉 √ 12 2

(1.50)

14

1 Standard Time-Independent Perturbation Theory

   (2)(x) 2 Fig. 1.5 Probability density, .ϕn  , of the wave function to second order (black continuous 2  line), and the probability density, .ϕn,num (x) , of the numeric solution (red dashed line), for .λ = 0.01. The values of the parameters are .h¯ = 1, μ = 1, ω = 1, α = 1. (a) .n = 0. (b) .n = 1. (c) .n = 2. (d) .n = 3

for .n = 0, 1, 2, . . . . The complete wave function up to second order is .

           (2)  = n(0) + λ n(1) + λ2 n(2) , ϕ

(1.51)

and in Figs. 1.5 and 1.6, we present the corresponding probability densities,   (2)  .ϕn (x), for the first four eigenvalues. In Fig. 1.5, we use .λ = 0.01 for the perturbation parameter, and in Fig. 1.6, we use .λ = 0.05,

1.3 Discrete Degenerated Spectrum As we already mentioned, the most common thing is that the spectrum of a quantum systems is degenerated. If the spectrum of the non-perturbed system is degenerated, (0) with every eigenvalue .En there are associated d eigenvectors .|n(0) , α〉 where .α = 1, 2, 3, . . . , d; thus, we need to write     ˆ 0 n(0) , α = En(0) n(0) , α , .H (1.52)

1.3 Discrete Degenerated Spectrum

15

   (2)(x) 2 Fig. 1.6 Probability density, .ϕn  , of the wave function to second order (black continuous 2  line), and the probability density, .ϕn,num (x) , of the numeric solution (red dashed line), for .λ = 0.05. The values of the parameters are .h¯ = 1, μ = 1, ω = 1, α = 1. (a) .n = 0. (b) .n = 1. (c) .n = 2. (d) .n = 3

  where n runs over the integers. The eigenvectors .n(0) , α satisfy the additional orthogonality condition    n(0) , α n(0) , β = ΔEα,β .

.

(1.53)

The complication in this case arises from the fact that we do not know which of all the functions  is the  right one to be the zero-order approximation. In fact, the set of vectors .n(0) , α , α = 1, 2, 3, . . . , d spans a vector subspace and any of the functions in that space may be the most suitable to be said zero-order approximation; we have to determine which one; for that, we suggest |n(0) , α〉 =

d

.

cα |n(0) , α〉,

(1.54)

α=1

and En = En(0) + λEn(1) ,

.

and we have to calculate the coefficients c.

(1.55)

16

1 Standard Time-Independent Perturbation Theory

The complexity mentioned above makes it very difficult to find general expressions to first and second order, both for the eigenvalues of the energy and for the eigenvectors. Therefore, in the case of degenerate systems we will restrict ourselves to the calculation of the first-order energy and to the determination of which are the appropriate zero-order functions. Once the appropriate zero-order functions have been determined, what should follow is an analysis similar to the non-degenerate case; however, as we will explain later, in most cases this is not necessary, because there are other options. Thus, we replace the suggestions (1.54) and (1.55) in Eq. (3.2), to get d

.

α=1

cα Hˆ P |n(0) , α〉 = En(1)

d

cα |n(0) , α〉,

(1.56)

α=1

where we have used (1.56); multiplying from the left by the bra .〈n(0) , β| and using the orthonormality relations (1.48), we arrive to d

.

  cα 〈n(0) , β|Hˆ P |n(0) , α〉 − En(1) ΔEα,β = 0,

(1.57)

α=1

which a system of d homogeneous linear algebraic equations for the coefficients c. As the above system of equations is homogeneous, it will have a solution different from zero, if and only if the determinant of the system is null; the roots (1) of the resulting equation will be the corrections to the energy .En . If those roots are all different, the spectrum is non-degenerated and the original levels are split; it is said that the degeneracy has been left; the lifting of the degeneracy may be total or partial. Once the energy first-order corrections has been calculated and substitute in the system (1.48), the c coefficients can be determined solving the equations system, and hence, the zero-order adequate eigenfunctions are known. The procedure can be described in a somehow general form, but in concrete cases it is easier to reproduce the main steps that lead to the equation system (1.57) and solve it, therefore, to illustrate all the process we present an example.

1.3.1 Example: The Three-Dimensional Isotropic Harmonic Oscillator with an xy Perturbation The energy eigenfunctions of the stationary Schrödinger equation, for a particle 2 of mass .μ, in a three-dimensional spherical symmetric potential .V (r) = μω 2 r of angular frequency .ω are, in spherical coordinates,     μω 2 l+ 1 μω 2 r Ylm (θ, φ); r Lk 2 ϕk,l,m (r, θ, φ) = Nk,l r l exp − h¯ 2h¯

.

(1.58)

1.3 Discrete Degenerated Spectrum

17

the normalization constant .Nk,l is

Nk,l

.

# $ $ μ3 ω3  μω l k! 2k+2l+2 % , = 2h¯ (2k + 2l + 1)!! π h¯ 3

(1.59)

Lba (ζ ) are the generalized Laguerre polynomials [15–18], and .Ylm (θ, φ) are the spherical harmonics [15–18]. The corresponding energy eigenvalues are

.

  3 . = h¯ ω 2lk + l + 2

Ek,l

.

(1.60)

The quantum number k is a non-negative integer, and for .n = 2k + l even, .l = 0, 2, 4, . . . , n − 2, n, and for .n = 2k + l odd, .l = 1, 3, 5, . . . , n − 2, n; the quantum number m is an integer given by .−l ≤ m ≤ l. Thus, the degeneracy of level n is d=

.

(n + 1) (n + 2) . 2

(1.61)

We consider now the perturbation potential Hˆ P (r, θ, φ) = ηxy,

.

(1.62)

where .η is a constant that has the dimensions of energy over length squared. In spherical coordinates, the perturbative potential can be written as Hˆ P (r, θ, φ) = ηr 2 sin2 (θ ) sin (φ) cos (φ) .

.

(1.63)

The lower energy level of the three-dimensional isotropic harmonic oscillator is the one with .k = 0, l = 0, m = 0, which corresponds to .n = 2k + l = 0, which is The second energy level non-degenerated; the energy of that state is .E0,0 = 3hω/2. ¯ is when .k = 0, l = 1, m = 0, ±1, which corresponds to .n = 1, and that has a 3 degeneration; the energy of those three levels is .E0,1 = 5hω/2. We will calculate, in ¯ this last case, the first-order correction to the energy of each state, and the zero-order functions. We have to evaluate the matrix in Eq. (1.57), ⎞ 〈0, 1, −1|H 〉P |0, 1, −1〉 〈0, 1, −1|H 〉P |0, 1, 0〉 〈0, 1, −1|H 〉P |0, 1, 1〉 .M = ⎝ 〈0, 1, 0|H 〉P |0, 1, −1〉 〈0, 1, 0|H 〉P |0, 1, 0〉 〈0, 1, 0|H 〉P |0, 1, 1〉 ⎠ , 〈0, 1, 1|H 〉P |0, 1, −1〉 〈0, 1, 1|H 〉P |0, 1, 0〉 〈0, 1, 1|H 〉P |0, 1, 1〉 (1.64) ⎛

18

1 Standard Time-Independent Perturbation Theory

where the ket .|k, l, m〉 stands for the wave function .ϕk,l,m (r, θ, φ), the corresponding bra for the complex conjugated, and .

  〈k, l, m|H 〉P k , , l , , m, = & & ∞ & π dr r 2 dθ sin (θ ) 0

0

2π

0

∗ dφ ϕk,l,m (r, θ, φ)Hˆ P ϕk , ,l , ,m, (r, θ, φ);

(1.65)

calculating all the integrals, we get ⎛ ⎞ 0 0 −1 iηh¯ ⎝ .M = 0 0 0 ⎠. 2μω 10 0

(1.66)

' ( ηh¯ ηh¯ The eigenvalues of matrix .M are . 0, − 2μω , and the corresponding eigen, 2μω vectors are (0,1,0), (i,0,1), and (.−i,0,1). Thus, the energy levels to first order are E2,1 =

.

5hω ¯ , 2

E2,2 =

ηh¯ 5hω ¯ − , 2 2μω

E2,3 =

ηh¯ 5hω ¯ + , 2 2μω

(1.67)

and the corresponding zero-order functions are .

|0, 1, 0〉 ,

1 √ (i |0, 1, −1〉 + |0, 1, 1〉) , 2

1 √ (−i |0, 1, −1〉 + |0, 1, 1〉) . 2 (1.68)

Note that as soon as we have the above correct zero-order wave functions, the firstorder corrections to the energy levels can be calculated using the non-degenerated perturbation formula Eq. (1.15); indeed, when we introduce those wave vectors in ηh¯ ηh¯ expression (1.65), we obtain 0, .− 2μω , respectively. , and . 2μω

1.3.2 Example: The Stark Effect in the Hydrogen Atom The Stark effect is the shifting and splitting of the spectral lines of molecules and atoms when they are placed in an external electric field. The interaction between a molecule or atom with the external electric field .E is given by Vˆint = −E · μ,

.

(1.69)

where .μ is the dipole moment of the charge distribution. We will find, to first order, the modification to the energy levels of the first degenerated state of the hydrogen atom, when it is placed in an external electric

1.3 Discrete Degenerated Spectrum

19

field. The orthonormal wave functions of the hydrogen atom are    r   2r l 2r 2l+1 .ϕn,l,m (r, θ, φ) = Nn,l exp − Ln−l−1 Ylm (θ, φ), na na na where the normalization constant is  Nn,l =

.



2 an

3

(1.70)

(n − l − 1)! , 2n(l + n)!

(1.71)

and a=

.

4π ε0 h¯ 2 , μe4

(1.72)

being .ε0 the vacuum electric permittivity and e the electron electric charge. The energy eigenvalues are En = −

.

e4 μ

1 . n2

32π 2 ε02 h¯ 2

(1.73)

The principal quantum number n is .n = 1, 2, 3, . . . , the azimuthal quantum number l is .l = 0, 1, 2, . . . , n − 1, and the magnetic quantum number m is .m = −l, −l + 1, . . . , l − 1, l. The high degree of degeneracy in the hydrogen atom is accidental [22] and is given by .d = n2 . The first state, .n = 1, l = 0, m = 0, is not degenerated; the second state, .n = 2, has a degeneration .d = 4. As we said previously, we will calculate the first-order corrections to the energy of the states with .n = 2, which are .{|2, 0, 0〉 , |2, 1, 1〉 , |2, 1, 0〉 , |2, 1, −1〉}; the matrix corresponding to Eq. (1.57) is ⎛

0 12π ε0 h¯ 2 E ⎜ ⎜0 .M = − ⎝1 eμ 0

0 0 0 0

1 0 0 0

⎞ 0 0⎟ ⎟. 0⎠

(1.74)

0

' ( ε0 Eh¯ 2 ε0 Eh¯ 2 , 0, 0, 12π eμ and the eigenvecThe eigenvalues of this matrix are . − 12π eμ tors are .(−1, 0, 1, 0), .(0, 1, 0, 0), .(0, 0, 0, 1), and .(1, 0, 1, 0); then, the energies to first order are (1)

E2,0,0 = − .

(1)

E2,1,0 = −

e4 μ 128π 2 ε02 h¯ 2 e4 μ 128π 2 ε02 h¯

−

, 2

12π ε0 Eh¯ 2 , eμ

(1)

E2,1,1 = − (1)

E2,1,−1 = −

e4 μ 128π 2 ε02 h¯ 2 e4 μ 128π 2 ε02 h¯ 2

, +

12π ε0 Eh¯ 2 ; eμ (1.75)

20

1 Standard Time-Independent Perturbation Theory

and the good eigenvectors to zero order are .

1 √ (|2, 0, 0〉 + |2, 1, 0〉) , 2

|2, 1, 1〉 ,

|2, 1, −1〉 ,

1 √ (|2, 0, 0〉 − |2, 1, 0〉) . 2 (1.76)

The note we make at the end of the previous example, it is also valid in this case. If we use the correct zero-order vectors above in the expression for the nondegenerated event, Eq. (1.15), we obtain the first-order corrections to the energy levels, as can be easily verified calculating the corresponding integrals.

References 1. D.I. Blokhintsev, Quantum Mechanics (Springer, Holland, 1964) 2. A. Messiah, Quantum Mechanics (North-Holland Publishing Company, Amsterdam, 1991) 3. S. Gasiorowicz, Quantum Physics, 2nd edn. (Wiley, New York, 1995) 4. N. Zettili, Quantum Mechanics. Concepts and Applications, 2nd edn. (Wiley, Chichester, 2009) 5. D.J. Griffiths, D.F. Schroeter, Introduction to Quantum Mechanics (Cambridge University Press, Cambridge, 2018) 6. E. Merzbacher, Quantum Mechanics, 3rd edn. (Wiley, Hoboken, 1998) 7. P.M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34(1), 57–64 (1929) 8. E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28(6), 1049–1070 (1926) 9. E. Schrödinger, Quantisierung als eigenwertproblem. Annalen der Physik 80, 437–490 (1926) 10. B. Simon, Fifty years of eigenvalue perturbation theory. Bull. Am. Math. Soc. 24(2), 303–319 (1991) 11. M. Znojil, A quick perturbative method for Schrödinger equations. J. Phys. A: Math. Gen. 30(24), 8771–8783 (1997) 12. W. Hai, M. Feng, X. Zhu, L. Shi, K. Gao, X. Fang, Alternative quantum perturbation theory without divergences. Phys. Rev. A 61(5), 052105 (2000) 13. Z. Yun-Hui, H. Wen-Hua, Z. Cheng-Lin, L. Xiao-Bing, Boundedness and convergence of perturbed corrections for helium-like ions in ground states. Chin. Phys. B 17(5), 1720–1728 (2008) 14. B.C. Hall, Quantum Theory for Mathematicians (Springer, New York, 2013) 15. G.B. Arfken, H.J. Weber, F.E. Harris, Mathematical Methods for Physicists (Elsevier, New York, 2012) 16. M. Abramowitz, I.A. Stegun, R.H. Romer, Handbook of mathematical functions with formulas, graphs, and mathematical tables. Am. J. Phys. 56(10), 958–958 (1988) 17. K. Oldham, J. Myland, J. Spanier, An Atlas of Functions (Springer, New York, 2009) 18. National Institute of Standards, NIST Handbook of Mathematical Functions (Cambridge University Press, New York, 2010) 19. S. Flügge, Practical Quantum Mechanics (Classics in Mathematics) (Springer, Berlin, 1998) 20. Wolfram Research, Inc., Mathematica, Version 13.2. Champaign, 2022 21. Maplesoft, a division of Waterloo Maple Inc. Maple 22. H.V. McIntosh, On accidental degeneracy in classical and quantum mechanics. Am. J. Phys. 27(9), 620–625 (1959)

Chapter 2

Standard Time-Dependent Perturbation Theory

Abstract We present the standard time-dependent perturbation theory in this chapter. The method of variation of constants and the Dyson series are demonstrated. We analyze some cases in which certain characteristics of the perturbative potential are assumed, such as a constant potential in time, a potential that acts during a finite time, and a monochromatic potential. The fundamental application of quantum perturbation theory, which involves calculating the transition probabilities between levels, is briefly introduced.

2.1 Introduction As mentioned in the introduction of the previous chapter, the Schrödinger equation serves as the central equation in non-relativistic quantum mechanics. According to one of the postulates of quantum mechanics, all the information about a system is contained in the wave function, the solution of this equation [1, 2]. Consequently, finding its solution is a fundamental objective of quantum theory. In Chap. 1, we introduced the time-independent perturbation theory, which offers a method for approximating solutions to the stationary Schrödinger equation. However, solving the exact solutions of the time-dependent Schrödinger equation is a highly challenging task and can only be accomplished in rare cases. Nonetheless, given its significance, it necessitates a solution, even if it requires the use of approximations and numerical methods. One widely utilized and valuable method in such cases is the standard time-dependent perturbation theory. Within the theory of time-dependent perturbations, two equivalent methods exist: the variation of constants method and the Dyson series method. The variation of constants method, developed by Dirac during his studies on the quantum theory of radiation [3–6], involves expressing the solution to the time-independent unperturbed problem in terms of the energy eigenfunctions and allowing the expansion coefficients to vary with time. On the other hand, the Dyson series method [7]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6_2

21

22

2 Standard Time-Dependent Perturbation Theory

formulates the Schrödinger equation in the interaction picture and relies on the time propagator. The time-dependent perturbation theory is applicable when a Hamiltonian can be divided into two parts Hˆ = Hˆ 0 + λVˆ (t),

.

where .Hˆ 0 represents the non-perturbed Hamiltonian, assumed to have known solutions (i.e., known eigenvalues and eigenvectors). The second part of the Hamiltonian, .Vˆ (t), is time dependent and considered “small” compared to .H0 , thus earning the term “perturbation” due to its minor effects on the system. To indicate the magnitude of the terms in the method’s expansion, it is customary to express ˆ (t) using a dimensionless positive real parameter, .λ, which is much smaller than .V one. As previously stated, the task involves solving the perturbed Schrödinger equation for the Hamiltonian Hˆ = Hˆ 0 + λVˆ (t),

.

knowing the complete analytic solution of the unperturbed equation       Hˆ 0 n(0) = En(0) n(0) ,

.

where n indexes the eigenvalues, which can represent a set of variables comprising discrete variables, continuous variables, or a combination of both. Both operators, ˆ 0 and .Vˆ (t), are assumed to be Hermitian. .H Given that the perturbation parameter .λ is small and the Schrödinger equation is linear, it is reasonable to assume that the solution to the perturbed problem will not deviate significantly from the solution to the unperturbed problem. The chapter is organized as follows: Sect. 2.2 presents the method of variation of constants. We first examine the case when the spectrum is discrete and subsequently analyze scenarios with mixed spectra, involving both discrete and continuous components. The chapter briefly covers cases of finite time perturbation, constant perturbation, and monochromatic perturbation. Section 2.3 discusses the Dyson series method, revisiting the case of a constant perturbation in light of this method. Finally, Sect. 2.4 applies the theory of time-dependent perturbations to the calculation of transition probabilities between different states of a quantum system. As an example, transition probabilities are determined for both constant and monochromatic perturbations.

2.2 Method of Variation of Constants

23

2.2 Method of Variation of Constants 2.2.1 Time-Dependent Perturbation Theory When the Spectrum Is Discrete We consider a time-dependent Hamiltonian that is the sum of two terms Hˆ (t) = Hˆ 0 + λVˆ (t),

.

(2.1)

where the Hamiltonian .Hˆ 0 is time independent and describes completely the unperturbed system, .Vˆ (t) is the perturbation potential, which we will regard as time dependent and small with respect to .Hˆ , and .λ is a dimensionless parameter that will be useful to keep track of the different orders of the perturbation. Both operators, ˆ 0 and .Vˆ (t), are supposed to be Hermitian. .H The complete analytic solution of the unperturbed equation is       Hˆ 0 n(0) = En(0) n(0) ,

.

(2.2)

where n indexes the eigenvalues and can represent a set of variables, which in turn can be discrete variables, continuous variables, or variables that are a combination of discrete and continuous. However, to fix ideas, we will consider that .n = 0, 1, 2, 3, . . . .   The wave functions . n(0) are assumed to be orthonormal and to form a complete set; i.e.,  .

   m(0) n(0) = δm,n ,

n, m = 0, 1, 2, . . . ,

(2.3)

and Iˆ =

.

∞      (0) (0)  n . n

(2.4)

n=0

It is well known that the time-dependent solution of the Schrödinger equation corresponding to the non-perturbed Hamiltonian .Hˆ 0 is written as

.

|ϕ(t)〉 =

∞  n=0



(0)   En  cn exp −i t n(0) , h¯

(2.5)

where .cn , n = 0, 1, 2, . . . , are complex coefficients, whose modulus squared gives us the probability of finding the system in state n, when making a measurement [1, 2,

24

2 Standard Time-Dependent Perturbation Theory

8–10]. The method of variation of the constants consists precisely in making these constants depend on time; that is, they cease to be constant and now change over time. Thus, we suggest as solution of the complete perturbed Schrödinger equation i h¯

.

 ∂ |ψ(t)〉 = Hˆ |ψ(t)〉 = Hˆ 0 + λVˆ (t) |ψ(t)〉 , ∂t

(2.6)

this one .

|ψ(t)〉 =

∞  n=0



(0)   En  cn (t) exp −i t n(0) . h¯

(2.7)

Substituting this proposed solution into the Schrödinger equation (2.6), we get ∞  .

n=0

 

∞    dcn (t) i (0) i (0)  (0)  (0)  (0) cn (t) exp − En t En n exp − En t n + i h¯ = dt h¯ h¯ n=0



 ∞      i i   cn (t) exp − En(0) t Hˆ 0 n(0) + λ cn (t) exp − En(0) t Vˆ (t) n(0) , h¯ h¯ n=0 n=0 (2.8)

∞ 



  where the linearity of the Schrödinger equation has been applied; as .n(0) are eigenfunctions of .Hˆ 0 with eigenvalue .En(0) , we cast the last equation as

 

∞     dcn (t) i i   cn (t) exp − En(0) t En(0) n(0) + i h¯ exp − En(0) t n(0) = dt h¯ h¯ n=0 n=0

∞  .



 ∞      i i   cn (t) exp − En(0) t En(0) n(0) + λ cn (t) exp − En(0) t Vˆ (t) n(0) ; h¯ h¯ n=0 n=0 (2.9)

∞ 

collecting terms and simplifying, we arrive to i h¯

.

∞  dcn (t) n=0

dt





∞     i i   cn (t) exp − En(0) t Vˆ (t) n(0) . exp − En(0) t n(0) = λ h¯ h¯ n=0 (2.10)

Now, we multiply  scalarly by the left the previous equation by an arbitrary  eigenfunction . m(0) , and we obtain

2.2 Method of Variation of Constants

25

 ∞    i dcn (t)   exp − En(0) t n(0) = i h¯ m(0)  h¯ dt

.

n=0

 ∞    i (0)  λ m  cn (t) exp − En t Vˆ (t) n(0) ; h¯ n=0 

(0) 

(2.11)

  introducing . m(0)  inside the sums, we get ∞  dcn (t)

i h¯

.

n=0

dt



  i  exp − En(0) t m(0) n(0) = h¯ λ

    i   cn (t) exp − En(0) t m(0)  Vˆ (t) n(0) . h¯ n=0

∞ 

(2.12)

  We use now the fact that the set . n(0) is orthonormal, Eq. (2.3), to write i h¯

.

∞  dcn (t) n=0

λ

dt

 i (0) exp − En t δm,n = h¯

    i   cn (t) exp − En(0) t m(0)  Vˆ (t) n(0) ; h¯ n=0

∞ 

(2.13)

by the Kronecker delta properties [11], this last equation reduces to



 ∞  i (0) i (0)  (0)  ˆ  (0)  dcm (t) exp − Em t = λ . cn (t) exp − En t m  V (t) n .i h ¯ dt h¯ h¯ n=0 (2.14)     The quantities . m(0)  Vˆ (t) n(0) are the matrix elements of the perturbative potential in the energy representation, and they will be denoted as       Vm,n (t) = m(0)  Vˆ (t) n(0) ,

.

m, n = 0, 1, 2, . . . , ∞,

(2.15)

so, the formula (2.14) can be copied as i h¯

.

 i (0) dcm (t) exp − Em t = h¯ dt

 ∞  i (0) λ cn (t) exp − En t Vm,n (t), m = 0, 1, 2, . . . , ∞, h¯ n=0

(2.16)

26

2 Standard Time-Dependent Perturbation Theory

or canceling the exponentials, ∞

i h¯

.

   dcm (t) Vm,n (t) exp iωm,n t cn (t), =λ dt

m = 0, 1, 2, . . . , ∞,

(2.17)

n=0

where ωm,n =

.

0 − E0 Em n , h¯

m, n = 0, 1, 2, . . . , ∞.

(2.18)

Until now, we have not made any approximations at all, everything is exact. The ordinary differential equations system (2.17) is completely equivalent to the Schrödinger equation (2.6) of the perturbed system: equivalent in everything, even in difficulty; if the Schrödinger equation is difficult to solve, so will be the system (2.17). To solve an equation or a system of differential equations, it is necessary to have a set of corresponding initial conditions. It is common for the initial state of the system to be one of the eigenfunctions of the unperturbed Hamiltonian, let us say  (0) .|ψ (t = 0)〉 = k 0 , with .k0 any fixed value in the spectrum; in that case, the system (2.17) must satisfy the initial conditions .cn (0) = δk0 ,n , n = 0,  1, 2, . . . , ∞,  (0) implying that originally the system was initially in the state .k0 and that had (0) an energy .Ek0 . Note that if .Vˆ (t) = 0 in the system (2.17), all the .cn (t), n = 0, 1, 2, . . . , are constant, as it corresponds to the non-perturbed time-independent Hamiltonian .Hˆ 0 . We will go now to the approximations: Let us assume that the perturbation is “small,” in a sense that will be clarified in the process; we propose that

cm (t) =

∞ 

.

(s) λs cm (t),

(2.19)

s=0 (s) where .cm (t) will be the perturbative solution of order s. Substituting the proposal (2.19) in the exact system (2.17), we arrive to

i h¯

∞ 

.

s=0

∞

∞

    s (s) dcm (t) =λ Vm,n (t) exp iωm,n t λ cn (t), dt (s)

λs

n=0

=

∞ 

s=0

  Vm,n (t) exp iωm,n t

n=0

=

∞  s=0

 λ

s

∞  s=0

∞  n=0

λs+1 cn(s) (t),

   (s−1) Vm,n (t) exp iωm,n t cn (t) ,

2.2 Method of Variation of Constants

27

m = 0, 1, 2, . . . , ∞;

(2.20)

as the power of .λ is linearly independent, we get ∞

  dcm (t)  Vm,n (t) exp iωm,n t cn(s−1) (t), = dt (s)

i h¯

.

m, s = 0, 1, 2, . . . , ∞.

n=0

(2.21) Writing explicitly the three first orders, (0)

dcm (t) =0, . dt ∞ (1)   dcm (t)  i h¯ Vm,n (t) exp iωm,n t cn(0) (t), . = dt i h¯

.

(2.22a) (2.22b)

n=0

i h¯

(2) dcm (t)

dt

=

∞ 

  Vm,n (t) exp iωm,n t cn(1) (t),

(2.22c)

n=0

m = 0, 1, 2, . . . , ∞.    If we consider that the system is initially in the state .k0(0) , the first equation, the one to zero order, is easily solved as (0) cm (t) = δk0 ,m ,

.

m = 0, 1, 2, . . . , ∞,

(2.23)

which, as expected, is the solution of the unperturbed system. Now, we substitute the zero-order solution in Eq. (2.22b), to obtain ∞

  dcm (t)  Vm,n (t) exp iωm,n t δk0 ,n = dt n=0   =Vm,k0 (t) exp iωm,k0 t , (1)

i h¯

.

which has the solution    1 t (1) .cm (t) = Vm,k0 (τ ) exp iωm,k0 τ dτ, i h¯ 0

m = 0, 1, 2, . . . , ∞. (1)

(2.24)

(2.25)

And the procedure can continue; once we know .cm (t), it can be substituted in (2.22c), which, when solved, gives us the second-order solution, and so on. Usually second-order calculations get terribly complicated; furthermore, in most applications, the first-order correction is sufficient. Therefore, Eq. (2.25) is one of the most important of this perturbative method.

28

2 Standard Time-Dependent Perturbation Theory (s)

Note that once you have all the coefficients .cm , to a certain order s (as we said, usually the first), they can be substituted back into Eq. (2.7) and the approximate time-dependent wave function of the perturbed system obtained. Since one of the basic principles of quantum mechanics reads: “The state of any physical system is specified, at each time t, by a state vector in a Hilbert space; the wave function contains (and serves as the basis to extract) all the needed information about the system” [1, 10], once the wave function is obtained, we have all the information about the system. One aspect that is very important, and to which we will return later in this chapter, is the transition probabilities between the different states of the system. When the Hamiltonian of the system is independent of time, it is enough to solve the stationary Schrodinger equation, since the system is stationary; that is, if the system is at a given moment in state .|n〉, the system will stay there forever. However, a time-dependent disturbance will induce transitions between the different possible states, and the method we are studying allows us to calculate the transition probabilities between them. To go further, we must make some assumptions about the perturbative potential. First, in the next subsection, we will assume that the disturbance lasts for a finite time.

2.2.1.1

Finite Time Perturbation

We assume that the perturbation is of the form  V (x, t) =

.

V (x, t) /= 0,

0 ≤ t ≤ T,

0,

t < 0 or t > T ;

(2.26)

that is, the perturbation lasts only from 0 to T . Figure 2.1 shows a potential that is zero up to a certain time, then for a finite time interval it is some function in the spatial dimension, and then it returns to zero. Substituting this potential in (2.25), we obtain for .t ≥ T , (1) .cm

1 = i h¯



T

0

  Vm,k0 (τ ) exp iωm,k0 τ dτ,

m /= k0 ,

(2.27)

where we have changed the integration from 0 to T due to the fact that .t ≥ T , and where we have removed the temporal dependence on the coefficients for the same reason. As .V (x, t) is zero outside the interval .[0, T ], we can extend the integration from .−∞ to .∞, as follows: (1) cm =

.

Thus,

1 i h¯



∞

−∞

  Vm,k0 (τ ) exp iωm,k0 τ dτ,

m /= k0 .

(2.28)

2.2 Method of Variation of Constants

29

Fig. 2.1 Potential that is zero up to a certain time, then for a finite time interval it is some function in the spatial dimension, and then it returns to zero

(1) cm =

.

  2π ˜ Vm,k0 ωm,k0 , i h¯

(2.29)

where .V˜m,n (ω) is the Fourier transform of the matrix elements of the perturbative potential .V (x, t); i.e., ˜m,n (ω) = 1 .V 2π



∞

−∞

Vm,n (t) exp (iωt) .

(2.30)

Note that in (2.29), the Fourier transform of the perturbative potential matrix elements is evaluated at the resonant frequencies .ωm,k0 . Example As a concrete example, consider a particle of mass .μ in an infinite one-dimensional well of width a (Fig. 2.2); that is, we must solve the stationary one-dimensional Schrödinger equation for the potential

30

2 Standard Time-Dependent Perturbation Theory

Fig. 2.2 Infinite one-dimensional well of width a

⎧ ⎪ ⎪ ⎨∞, .V (x) = 0, ⎪ ⎪ ⎩∞,

x a.

The stationary solutions, the eigenfunctions of the Hamiltonian, are [2, 8, 10, 12–14]  ϕn (x) =

.

 π nx  2 sin , a a

(2.32)

and the energies, the eigenenergies of the Hamiltonian, are En =

.

π 2 h¯ 2 2 n . 2a 2 μ

(2.33)

Let us now think that we have a perturbation given by ⎧ ⎪ ⎪ ⎨0, x  α(t)x −1 , .V (x, t) = ⎪ a ⎪ ⎩ 0,

t a,

where .α(t) is an arbitrary function of time (Fig. 2.3). The matrix elements of the perturbation, Eq. (2.15), are

(2.34)

2.2 Method of Variation of Constants

31

Fig. 2.3 Finite time perturbation, Eq. (2.34), with .α(t) = α0 exp (−t/τ )

V2j,2k (t) =

.

⎧ ⎨ 2a2 π

⎩0, 

V2j +1,2k+1 (t) =

jk

(j 2 −k 2 )2

α(t), t ∈ [0, T ] t∈ / [0, T ]

(2.35a)

,.

2a (2j +1)(2k+1) α(t), π 2 (j −k)2 (j +k+1)2

t ∈ [0, T ]

0,

t∈ / [0, T ]

,.

(2.35b)

V2j +1,2k (t) =0, .

(2.35c)

V2j,2k+1 (t) =0.

(2.35d)

To proceed to concrete examples, where we can evaluate transition probabilities, we must specify the function .α(t); we will consider that

 t α(t) = α0 exp − , τ

.

(2.36)

being .τ a parameter that measures duration of the perturbation, and .α0 is a constant that gives us the intensity of the disturbance and that guarantees that the units of the perturbation are correct; the units of .τ are seconds and those of .α0 are Joule over meter. To avoid unnecessary complications and focus ideas, we will also assume that the initial state of the system was the ground state, that is, .k0 = 1. Then, substituting all the elements in the integral (2.27), we get

32

2 Standard Time-Dependent Perturbation Theory (1)

c2m =0, .

.

(1)

c2m+1

(2.37a)  2m + 1 τ aα0 T = 2 1 − exp iω2m+1,1 T − , 2 2 τ π h¯ 2m (m + 1) τ ω2m+1,1 + i (2.37b) 

where 2π 2 h¯ 2 π 2 h¯ 2 π 2 h¯ 2 2 = m (m + 1) . − + 1) (2m 2a 2 μ 2a 2 μ a2μ (2.38)  (0)   (0)    to an arbitrary state . m , is The transition probability, from the initial state . 1 ω2m+1,1 = E2m+1 − E1 =

.

given by .P1→m = |cm |2 ; hence, P1→2m =0, .

(2.39a)

.

P1→2m+1 =

(2m + 1)2 τ2 2 +1 π 4 h¯ 2 2m4 (m + 1)4 τ 2 ω2m+1,1

 

   T T exp − cosh − cos ω2m+1,1 T . τ τ a 2 α02

(2.39b)

Thus, the transition probability to the two first excited states is P1→2 =0, .

(2.40a)

P1→3

(2.40b)

.

2.2.1.2

 



  9a 2 α02 τ2 T T = cosh − cos ω3,1 T . exp − 2 +1 τ τ 32π 4 h¯ 2 τ 2 ω3,1 Constant Perturbation

A constant perturbation in time, not necessarily constant in space, V (x, t) = V0 (x),

.

(2.41)

leads us in (2.25) to (1) cm (t) =

.

1 V0;m,k0 i h¯



t 0

  exp iωm,k0 τ dτ,

(2.42)

where .V0 (x) is the value of the constant perturbation and can depend on the space variables, and .V0;m,k0 are their corresponding matrix elements in the energy representation. Carrying out the integral, we get

2.2 Method of Variation of Constants

33

Fig. 2.4 Plot of .sin2 (ωt/2) /ω2 as a function of .ω for a fixed time t

(1) .cm (t)

  V0;m,k0 1 − exp iωm,k0 t = , ωm,k0 h¯

(2.43)

and the probability for the system to make a transition from the state .k0 to a final state m is Pk0 →m =

.

2 4V0;m,k 0 2 h¯ 2 ωm,k 0

2

sin

ωm,k0 t 2

 .

(2.44)

Remark that similar to the case in which the disturbance acts for a finite time, the frequencies used to evaluate the transition probabilities are the resonant frequencies. In the attached figure, we show how the part that depends on the frequency .ω behaves. This function has a maximum at .ω = 0, where it has .t 2 /4 as height, and it has a width proportional to t; since the greatest amount of area is below the central peak, it is clear that the transition probability will be proportional to t. This also means that the probability of a transition to a certain final state is higher when .ωfinal,initial ≈ 0 or .Efinal ≈ Einitial (Fig. 2.4).

2.2.2 Time-Dependent Perturbation Theory When the Spectrum Is Discrete and Continuous We will consider now that the non-perturbed system has discrete and continuous spectrum; i.e., the complete analytic solution of the unperturbed equation is

34

2 Standard Time-Dependent Perturbation Theory

      Hˆ 0 n(0) = En(0) n(0) , n = 0, 1, 2, . . . .       Hˆ 0 α (0) = E (α)(0) α (0) , α ∈ R. .

(2.45a) (2.45b)

The eigenfunctions are assumed to be orthonormal and to form a complete set; i.e.,    m(0) n(0) = δm,n , n = 0, 1, 2, . . . , m = 0, 1, 2, . . . , .       α '(0) α (0) = δ α ' − α , α ' ∈ R, α ∈ R, .     n(0) α (0) = 0, n = 0, 1, 2, . . . , α ∈ R, 

.

(2.46a) (2.46b) (2.46c)

and Iˆ =

.

∞       (0) (0)  n + n n=0

∞ −∞

    (0)  α (0)  dα. α

(2.47)

The Schrödinger equation corresponding to the complete problem, the perturbed one, is i h¯

.

 ∂ |ψ(t)〉 = Hˆ |ψ(t)〉 = Hˆ 0 + λVˆ (t) |ψ(t)〉 , ∂t

(2.48)

and, as in the previous case, the main goal is to solve it. We proceed as in the discrete case; we write the time-dependent solution of the unperturbed problem using the eigenvalues and eigenfunctions of .Hˆ 0 as follows:

.

|ϕ(t)〉 =





  ∞  i i   cα exp − E (0) (α)t α (0) dα, cn exp − En(0) t n(0) + h¯ h¯ −∞ n=0 (2.49)

∞ 

where the complex numbers .cn , n = 0, 1, 2, . . . , and .cα , α ∈ R, do not depend on time. The method of variation of constants consists in letting the coefficients no longer be constant but depend on time; so, we write 

i (0)  (0)  . |ψ(t)〉 = cn (t) exp − En t n h¯ n=0 

 ∞  i  cα (t) exp − E (0) (α)t α (0) dα + h¯ −∞ ∞ 

as the solution to the Schrödinger equation (2.48).

(2.50)

2.2 Method of Variation of Constants

35

Inserting Eq. (2.50) into Eq. (2.48) leads us to i h¯

.



 ∞ E (0) (α)  (0)  En(0)  (0)  dcα (t) t n t α exp −i exp −i +i h¯ dα h¯ h¯ dt dt −∞

∞ (0)    E  n = cn (t)Vˆ (t) exp −i t n(0) h¯ n=0

 ∞ E (0) (α)  (0)  ˆ t α . + dαcα (t)V (t) exp −i (2.51) h¯ −∞

∞  dcn (t) n=0



  We make the left inner product by . α '(0) , use the orthonormality conditions (2.46), and at the end interchange .α and .α ' , to obtain

∞ (0) dcα (t)  E (0) (α) − En t Vα,n (t) = cn (t) exp i .i h ¯ h¯ dt n=0

 ∞ E (0) (α) − E (0) (α ' ) + t Vα,α ' (t)dα ' , cα ' (t) exp i h¯ −∞

α ∈ R, (2.52)

where       Vα,n (t) = α '(0)  Vˆ (t) n(0)

.

(2.53)

are the matrix elements of .Vˆ (t) between the discrete spectrum and the continuous, and      '(0)  ˆ .Vα,α ' (t) = α (2.54)  V (t) α (0) are the matrix elements between states of the continuous. We consider again that initially the complete system was in one of the eigenstates of the non-perturbed  (0)  Hamiltonian, let us say the .k0 ; so the initial condition is .|ψ(t = 0)〉 = k and this implies that cn (t = 0) = δn,k0 ,

.

cα (t = 0) = 0.

(2.55)

Everything is analytical and exact, and no approximation has been made. The set of equations (2.52) is totally equivalent to the Schrödinger equation (2.48). With the established initial conditions (2.55), the system (2.52) gives us

36

2 Standard Time-Dependent Perturbation Theory



∞ (0) (1) E (0) (α) − En dcα (t)  .i h δn,k0 exp i t Vα,n (t) = ¯ h¯ dt n=0



= exp i

E (0) (α) − Ek(0) 0 h¯

t Vα,k0 (t),

(2.56)

which has as a solution (1) .cα (t)

1 = i h¯



t 0

Vα,k0 (τ ) exp i

E (0) (α) − Ek(0) 0 h¯

τ dτ.

(2.57)

Nothing more can be done without knowing the explicit form of the perturbative potential .Vˆ ; to have a better understanding and fix some of the ideas, we will proceed to analyze what happens when the perturbative potential satisfies certain very general conditions.

2.2.2.1

Monochromatic Perturbation

Consider a monochromatic potential; it means a potential that oscillates in time with just one frequency. So, the potential is V (t) = Vr cos (ωt) + Vi sin (ωt) ,

.

(2.58)

and of which we show a generic form in the following figure (Fig. 2.5):

Fig. 2.5 Generic form of the monochromatic perturbative potential. Displayed as a function of time for fixed .ω, .Vr , and .Vi

2.2 Method of Variation of Constants

37

The monochromatic potential can be written as  V (t) = Re V exp (iωt) = V exp (iωt) + V ∗ exp (−iωt) ,

(2.59)

.

where V = Vr + iVi .

(2.60)

.

The necessary matrix elements of the perturbation can be put as ∗ exp (−iωt) Vˆα,n (t) = Vα,n exp (iωt) + Vα,n

(2.61)

.

and, substituted in (2.57), take us to (1) .cα (t)

1 = i h¯



t

0

Vα,k0 = i h¯ +



∗ Vα,k0 exp (iωt) + Vα,k 0





t

exp i

(0)

E (0) (α) − Ek0 + hω ¯ h¯

0

 ∗ Vα,k 0 i h¯



t

exp (−iωt) exp i

exp i

h¯

τ dτ

τ dτ

(0)

E (0) (α) − Ek0 − hω ¯ h¯

0

E (0) (α) − Ek(0) 0

(2.62)

τ dτ.

The integrals can be easily calculated, to bring us to (1) .cα (t)

=

+

Vα,k0



1 − exp i

(0)

E (0) (α) − Ek0 + hω ¯ ∗ Vα,k 0 (0)

E (0) (α) − Ek0 − hω ¯







1 − exp i



(0)

E (0) (α) − Ek0 + h¯ ω h¯

t



(0)

E (0) (α) − Ek0 − h¯ ω h¯

t

.

(2.63)

(0)

As in general, .ω > 0, .E (0) (α) > 0, and .Ek0 < 0, it is evident that (0) (0) E (0) (α) − Ek(0) + hω ¯ ≥ E (α) − Ek0 − h¯ ω, 0

.

and the first term is small relative to the second, especially at resonant frequencies (0) when .E (0) (α)−Ek0 − hω ¯ ≈ 0, where the second term becomes very big; neglecting then the first term, we have

38

2 Standard Time-Dependent Perturbation Theory

Fig. 2.6 The function .sin (ηt) /η for .t = 104

    Vα,k 2 1  (1) 2 0 . cα (t) =   h¯ 2 E (0) (α)−E (0) −hω2  ¯  k0

 (0)

2   E (α) − Ek(0) − hω ¯   0 t  . 1 − exp i   h¯ (2.64)

Since it is a transition from a state of the discrete spectrum to a state of the continuousspectrum, we must calculate the probability of transition per unit of time  from state .k (0) to the interval of the continuum .α, α + dα; i.e.,

pk0 →α dα =

.

 2   d cα(1) (t) dt

dα

(0) (0) E (α) − Ek0 − hω 2 ¯ 2  1  Vα,k0 t . = sin h¯ h¯ E (0) (α) − Ek(0) − hω ¯ 0 (2.65) In Fig. 2.6, we plot .sin (ηt) /η as function of .η for t very big, and it can be clearly inferred that when .t → ∞ [11], 1 .

E (0) (α) − Ek(0) − h¯ ω 0

sin



(0)

E (0) (α)−Ek0 −hω ¯ h¯

t

  (0) → π δ E (0) (α) − Ek0 − hω ; ¯ (2.66)

  therefore, the probability of transition per unit time from the unperturbed state .k (0) ,  belonging to the discrete spectrum, to the unperturbed state .α (0) , in the continuous

2.2 Method of Variation of Constants

39

part of the spectrum, is to first order pk0 →α dα =

.

 2  2π  − h ω dα. Vα,k0  δ E (0) (α) − Ek(0) ¯ 0 h¯

(2.67) (0)

It is very important to remark that .pk0 →α dα /= 0 if and only if .E (0) (α) − Ek0 = h¯ ω. The transitions have a resonant character; the quantum system behaves as if it were made up of a set of harmonic oscillators with frequencies equal to the Bohr frequency.

Example: Probability of ionization of a hydrogen atom In this example, we will calculate the probability of ionization of a hydrogen atom when it interacts with an oscillating electric field .E(t) = E0 sin (ωt), and that initially is in its ground state. We will assume a dipole interaction; so, the interacting potential is given by V (t) = −er · E(t) = −er · E0 sin (ωt) =

.

 e r · E0 exp (−iωt) − exp (iωt) , 2i (2.68)

that it is clearly a harmonic disturbance and that is why we present it as an example. 4 Initially the atom is in its ground states, which has an energy .E0 = − μe e2 2 , (4π ϵ0 ) 2h¯ where .μe is the rest mass of the electron, e is its electric charge, and .ε0 is the permittivity of free space; the ground state wave function is ϕ1,0,0 (r) = √

.

1 3/2

π a0

exp (−r/a0 ) ,

(2.69)

2

being .a0 = 4πe2ϵμ0 h¯ the Bohr radius [2, 8, 9, 14]. e Once the atom is ionized, the electron will be a free particle, and all its energy 2 2 will be kinetic, of value .Ekinetic = h¯2μke ; the free electron wave function will be [2, 8, 9, 14] ϕk (r) = (2π)−3/2 exp (ik · r) .

.

(2.70)

We already have all the elements to apply directly the formula (2.67). The matrix elements of the perturbation are Vk,100 (t) = 〈ϕk | V (t) |100〉 =

.

e 〈ϕk | r · E0 |100〉 ; 2i

(2.71)

if we denote the spherical coordinates polar angles of .E0 by A and B, we obtain

40

2 Standard Time-Dependent Perturbation Theory

√ 7/2 4 2a0 eE0 cos(A) k  2  . .Vk,100 (t) = − π a0 k 2 + 1 3

(2.72)

Substituting in Eq. (2.67), we arrive to the expression pk0 →k dk =

.

  64a07 e2 E20 cos2 (A) k2  2  δ E (0) (k) − Ek(0) − hω dk, ¯ 0 π h¯ a0 k 2 + 1 6 (2.73)

which gives us the probability of transition from the base state to only one of the states of the continuum, represented by k; hence, we have to sum over all the final states, that is, we have to integrate over the three-dimensional .k; finally, the probability transition rate of ionization .p0→ion can be written as p0→ion = 4π ϵ0

.

3/2

256a03 E20  ω0 6 ω −1 , 3h¯ ω ω0

(2.74)

4

where .ω0 = μe e2 3 stands for the frequency of the base state. (4π ϵ0 ) 2h¯ There is an aspect that this formula indicates to us and that is important to point out; if the frequency of the oscillating electric field is equal to .ω0 , the probability of ionization is zero; also, if the frequency .ω is less than .ω0 , there will be no ionization; and finally, ionization will occur only when .ω > ω0 . In Fig. 2.7, we depict the function (2.74) in terms of .ω. It has a maximum at √ 2 27 3a03 E0 .4ω0 /3, with value .4π ϵ0 , and for long times, it goes to zero. 16h¯

2.3 Method of Dyson Series Naturally, we start again with the Schrödinger equation for the complete problem, the perturbed problem. For ease of reading, we reproduce it here i h¯

.

 ∂ |ψ(t)〉 = Hˆ (t) |ψ(t)〉 = Hˆ 0 + λVˆ (t) |ψ(t)〉 . ∂t

(2.75)

The solution to this equation can be written formally in terms of the propagator Uˆ (t, t0 ) [2, 8–10, 15] as

.

.

|ψ(t)〉 = Uˆ (t, t0 ) |ψ(t0 )〉 ,

(2.76)

where .|ψ(t0 )〉 is the initial condition at time .t0 . We can transform this equation to the interaction picture [2, 8–10, 15], by means of the time-dependent unitary transformation

2.3 Method of Dyson Series

41

/

Fig. 2.7 The probability transition rate of ionization of an hydrogen atom, initially in its ground state

  Tˆ = exp it Hˆ 0 /h¯ .

.

(2.77)

In this description, the Schrödinger equation becomes i h¯

.

∂ |ψ(t)〉int = Hˆ int (t) |ψ(t)〉int , ∂t

(2.78)

where |ψ(t)〉int = Tˆ |ψ(t)〉 ,

(2.79)

Hˆ int (t) = λTˆ Vˆ (t)Tˆ † .

(2.80)

.

and .

With all these elements, we can express the solution as .

  |ψ(t)〉int = exp it Hˆ 0 /h¯ |ψ(t)〉 = Uˆ int (t, t0 ) |ψ(t)〉int ,

(2.81)

where Uˆ int (t, t0 ) = Tˆ Uˆ (t, t0 ) Tˆ † .

.

(2.82)

42

2 Standard Time-Dependent Perturbation Theory

Inserting (2.81) in the Schrödinger equation in the interaction picture, Eq. (2.78), we get an equation for the propagator, i h¯

.

Uˆ int (t, t0 ) = Vˆint (t)Uˆ int (t, t0 ) . dt

(2.83)

The formal solution to this last equation, with the obvious initial condition Uˆ int (t0 , t0 ) = Iˆ, is the integral equation

.

i Uˆ int (t, t0 ) = Iˆ − h¯



t

.

Vˆint (τ )Uˆ int (τ, t0 ) dτ ;

(2.84)

t0

of course, this integral equation can be solved exactly and analytically in very few cases. We are now in a position to start making approximations to obtain useful solutions to different problems, taking advantage of the assumption that the perturbative potential is, in some sense, small. As we did in the method of variation of the constants, we take advantage of the knowledge we have of the initial conditions, to use them as a zero-order approximation; so, the zero-order approximation will be .Uˆ int (t0 , t0 ) = Iˆ, which we replace in Eq. (2.84) to get i (1) Uˆ int (t, t0 ) = Iˆ − h¯



t

.

Vˆint (t1 )dt1 .

(2.85)

t0

Next, we follow an iterative procedure; we substitute (2.74) into (2.84) to obtain i (2) Uˆ int (t, t0 ) = Iˆ + h¯



t

.

t0

   i 2 t t1 ˆ Vˆint (t1 )dt1 + − Vint (t1 )Vˆint (t2 )dt1 dt2 . h¯ t0 t 0 (2.86)

We can substitute this expression again in (2.84) to arrive to the third-order correction, and so forth; doing this way, we arrive to the Dyson series [7]   

 i 2 t t1 ˆ i t ˆ (n) Vint (t1 )dt1 + − Vint (t1 )Vˆint (t2 )dt1 dt2 + . . . Uˆ int (t, t0 ) = Iˆ + h¯ t0 h¯ t0 t 0   

 tn−1 i n t t1 ... + − Vˆint (t1 )Vˆint (t2 ) . . . Vˆint (tn )dt1 dt2 . . . dtn . h¯ t0 t0 t 0 (2.87)

.

Summarizing, the wave function to order n will be .

     (n) (n) ψ (t) = exp −it Hˆ 0 /h¯ Uˆ int (t, t0 ) |ψ(t0 )〉 .

(2.88)

2.3 Method of Dyson Series

43

2.3.1 Constant Perturbation Under these circumstances, the Hamiltonian .Hˆ is time independent and the formal solution to Eq. (2.75) can be stated as    i ˆ ˆ |ψ(t)〉 H0 + λV0 t |ψ(0)〉 , = exp − . h¯

(2.89)

where .Vˆ0 is the constant, in time, perturbation potential, and we have .t0 = 0;  made  therefore, if the initial condition is the .Hˆ 0 eigenstate k, .|ψ(0)〉 = k (0) , we have .

    i  (0)  |ψ(t)〉 = exp − Ek + λVˆ0 t k (0) . h¯

(2.90)

We solve the same problem applying the Dyson series. From (2.85),  t  i i t ˆ (1) dt1 Vint (t1 )dt1 = Iˆ − Vˆint,0 Uˆ int (t, 0) =Iˆ − h¯ 0 h¯ 0 i =Iˆ − t Vˆint,0 ; h¯

.

(2.91)

using (2.87), ˆ (2) (t, 0) .U int



2 ˆint,0  t  t1 i V i dt1 dt2 =Iˆ − Vˆint,0 t + − h¯ h¯ 0 0

2  t2 i Vˆint,0 i ˆ ˆ ; =I − Vint,0 t + − 2 h¯ h¯

(2.92)

and so on, in such a way that we can infer that ˆ (n) (t, 0) .U int

k n  1 i Vˆint,0 t ; = − k! h¯

(2.93)

k=0

and taking the limit when .n → ∞,

k

∞  ˆint,0 ˆint,0 1 i V i V ˆ int (t, 0) = t = exp − t . .U − k! h¯ h¯

(2.94)

k=0

In Eq. (2.88), we have the wave function resulting from the Dyson series, then

44

2 Standard Time-Dependent Perturbation Theory



 ˆint,0  i V ˆ 0 /h¯ exp − . |ψ(t)〉 = exp −it H t k (0) h¯

  i Vˆint,0  (0)  (0) = exp −itEk /h¯ exp − t k h¯     i  (0)  Ek + λVˆ0 t k (0) , = exp − h¯ 



(2.95)

which matches the result found in (2.90), as expected. That these two results are the same is nothing extraordinary or wonderful, it is simply the verification of what we know must happen, and it serves to verify that the calculations are well done.

2.4 Transition Probabilities The theory of time-dependent perturbations naturally finds one of its fields of application in the calculation of transition probabilities between the levels of a quantum system. We will easily convince ourselves that this is true if we remember that the probability of passing from a state .|ψ(t0 )〉 at time .t = t0 to a state .|ψ(t)〉 at time t is given by .|〈ψ(t0 )|ψ(t)〉|2 [1, Section 1–3, page 23], and we realize that what the theory gives us is precisely the value of the wave function at time t; thus, for example, in the case of a system with a discrete spectrum, studied in Sect. 2.2.1, we will have that to first order .

  2 2        ψ(t)k (0)  = cn(1) (t) ;

(2.96)

in other words,  the  probability for that system to undergo a transition from the original state .k (0) to the state .n(0) is to first order,  2   (1) Pk→n = cn(1) (t) .

.

(2.97)

In the case of the interaction picture, we have .

  |ψ(t)〉 = exp −it Hˆ 0 Uˆ int |ψ(t0 )〉 ;

(2.98)

hence, the probability of a transition from the initial unperturbed state .|ψ(t0 )〉 to the final unperturbed state .|ψ(t)〉 at time t is  2   P|ψ(t0 )〉→|ψ(t)〉 = 〈ψ(t)| Uˆ int (t, t0 ) |ψ(t0 )〉 .

.

(2.99)

2.4 Transition Probabilities

45

Then, from  the  the  Dyson series,  first-order probability for the system to go from the state .k (0) to the state .n(0) is      i t  (0)    2 (1) n Vˆ (τ )|k (0) exp iωn,k τ dτ  . Pk→n = − h¯ 0

.

(2.100)

Let us apply these concepts to some of the examples previously developed in this chapter. We start with the constant potential.

2.4.1 Constant Perturbation We are going to calculate the probability of transition from the unperturbed state .k to the unperturbed state .m, when the perturbation is constant in time; first, we will use the formulas found using the variation of constants method, and second, we will use the formulas derived from the Dyson series. Using the method of variation of constants, we calculated in Eq. (2.43) (1) .cm (t)

  V0;m,k0 1 − exp iωm,k0 t , = ωm,k0 h¯

 (0)   and hence, the transition to other  (0)  probability from the non-perturbed state . k is to first order non-perturbed state .m (1) .P k→n

 2  

   V0;m,k 2  2   4 V0;m,k   (1) 2 2 ωm,k t   exp iωm,k t − 1 = . sin = cm (t) = 2 2 2 2 h¯ 2 ωm,k h¯ ωm,k (2.101)

In the case of the Dyson series, we established expression (2.100), which in the actual situation reduces to (1) .P k→n

   t  i   2  exp iωn,k τ dτ  = − V0;m,k h¯ 0   V0;m,k 2     exp iωm,k t − 12 = 2 2 h¯ ωm,k 2 

 4 V0;m,k  2 ωm,k t , = sin 2 2 h¯ 2 ωm,k

(2.102)

which is, of course, the same result as the one obtained with the method of variation of constants.

46

2 Standard Time-Dependent Perturbation Theory

The behavior of a system acted upon by a time-constant perturbation is extremely interesting and sheds light on many atomic phenomena. However, it is not the objective of our work to delve into the physical systems in which perturbative methods are applied, but rather to explain their essence in order to compare them. The interested reader will undoubtedly find many books in which he will be able to study these aspects in detail and in depth; as a reference we can cite [2, 8–10, 14, 15].

References 1. P. Roman, Advanced Quantum Theory (Addison Wesley, Reading, 1965) 2. A. Messiah, Quantum Mechanics (North-Holland Publishing Company, Amsterdam, 1991) 3. P.A.M. Dirac, On the theory of quantum mechanics. Proc. R. Soc. Lond. Ser. A, Contain. Pap. Math. Phys. Charact. 112(762), 661–677 (1926) 4. P.A.M. Dirac, The quantum theory of the emission and absorption of radiation. Proc. R. Soc. Lond. Ser. A 114(767), 243–265 (1927) 5. P.A.M. Dirac, The quantum theory of dispersion. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Charact. 114(769), 710–728 (1927) 6. P.W. Langhoff, S.T. Epstein, M. Karplus, Aspects of time-dependent perturbation theory. Rev. Modern Phys. 44(3), 602–644 (1972) 7. F.J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75(3), 486–502 (1949) 8. E. Merzbacher, Quantum Mechanics, 3rd edn. (Wiley, Hoboken, 1998) 9. S. Gasiorowicz, Quantum Physics, 2nd edn. (Wiley, New York, 1995) 10. N. Zettili, Quantum Mechanics. Concepts and Applications, 2nd edn. (Wiley, Chichester, 2009) 11. G.B. Arfken, H.J. Weber, F.E. Harris, Mathematical Methods for Physicists (Elsevier, New York, 2012) 12. G. Auletta, The Quantum Mechanics Conundrum (Springer Nature Switzerland AG, Cham, 2019) 13. S. Weinberg, The Quantum Theory of Fields, vol. 1. (Cambridge University Press, New York, 1995) 14. D.J. Griffiths, D.F. Schroeter, Introduction to Quantum Mechanics (Cambridge University Press, Cambridge, 2018) 15. J.J. Sakurai, Modern Quantum Mechanics – 2. edición (Addison Wesley, Boston, 2011)

Chapter 3

The Matrix Perturbation Method

Abstract The Matrix Perturbation Method is presented as a novel technique for obtaining approximate solutions to the Schrödinger equation in non-relativistic quantum mechanics. The key element of this method is the introduction of the matrix .M, defined in Eq. (3.56), which enables the transformation of the Taylor series for the wave function into a series of powers of the matrix .M. This alternative representation, in terms of products of the operators .Hˆ0 and .Hˆ p , proves to be more manageable. The Matrix Perturbation Method allows for the simultaneous determination of energy and wave function corrections within a single solution. In Sect. 3.1, we provide a comprehensive explanation of the method for the first-order correction. Subsequently, in Sect. 3.2, we estimate the second-order correction. In Sect. 3.3, we study the generalization to any order. The introduction of the normalization constant is covered in Sect. 3.4. Finally, in Sect. 3.5, we establish the relationship between the matrix perturbation method and the Dyson series. By applying our formalism, we are able to rewrite the Dyson series in matrix form.

3.1 Introduction In the previous two chapters, we discussed the significance of developing approximate methods for solving the Schrödinger equation in non-relativistic quantum mechanics. In this chapter, we propose an alternative perturbative approach known as the Matrix Perturbation Method [1–4]. This novel scheme, utilizing triangular matrices, offers an elegant and straightforward solution to the time-dependent Schrödinger equation. Unlike the standard perturbation theory, where wave function and energy corrections are calculated separately, the Matrix Method encapsulates both corrections within a single expression [1–4]. Furthermore, this method can be applied even when finding the eigenstates of the unperturbed Hamiltonian is not feasible, but a unitary evolution operator is known. Notably, the approximate solutions obtained through the Matrix Method exhibit not only conventional stationary terms but also time-dependent factors that provide insight into the temporal evolution of the corrections. Moreover, the general expression for computing these corrections applies uniformly to both degenerate and non-degenerate Hamiltonians [1–4]. It © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6_3

47

48

3 The Matrix Perturbation Method

is worth highlighting that the Matrix Method yields perturbative solutions with a suitable normalization constant, ensuring their normalization at all perturbative orders, which is a significant limitation of the standard perturbation theory as it often leads to divergent approximations for large times. Additionally, the Matrix Method enables the generation of a matrix form dual Dyson series, applicable to both weak and strong perturbations [5]. Another important feature is its extension to the Lindblad master equation [3], which will be explored in Chap. 6. Thus, the Matrix Method offers attractive characteristics absent in conventional treatments of perturbation theory. The organization of this chapter is as follows: In Sect. 3.2, we introduce the Matrix Method, which allows us to obtain a unified solution encompassing energy and wave function corrections. We present the necessary corrections to generate the Dyson series and express it in terms of a matrix series. Section 3.2.1 covers the first-order matrix perturbation method, while Sect. 3.2.2 addresses the secondorder method. Section 3.2.3 presents the generalization to any order. In Sect. 3.3, we derive the general expression for computing the normalization constant at any order. Finally, in Sect. 3.4, we utilize our formalism to rewrite the Dyson series in matrix form. It is very important to point out that from now on we will use a system of units in which .h¯ = 1; not only this is a common practice in quantum mechanics, but also the complexity of the expressions that will be presented in the next chapters makes it necessary.

3.2 Matrix Approach to the Perturbation Theory As in Chap. 1, we have a time-independent Hamiltonian, which can be separated in two parts, as follows: Hˆ = Hˆ 0 + λHˆ p ,

.

(3.1)

where .Hˆ 0 is called non-perturbed Hamiltonian, and it is usually assumed to have known solutions, i.e., its eigenvalues and eigenvectors are known. The second part of the Hamiltonian, .Hˆ p , is supposed to be small compared to .H0 ; thus .Hˆ p is called the “perturbation,” because its effect in the energy spectrum and in the eigenfunctions will be small. To keep track of the “size” of the terms that appear in the development of this method, it is usual to write .Hˆ p in terms of a dimensionless positive real parameter .λ, which is considered very small compared to one. As we already said, we want to solve the perturbed eigenvalue problem   Hˆ |n〉 = Hˆ 0 + λHˆ P |n〉 = En |n〉

.

knowing the complete analytic solution of the unperturbed equation

(3.2)

3.2 Matrix Approach to the Perturbation Theory

      Hˆ 0 n(0) = En(0) n(0) ,

.

49

(3.3)

where n indexes the eigenvalues and can represent a set of variables, which in turn can be discrete variables, continuous variables, or variables that are a combination of ˆ 0 and .Hˆ P , are supposed to be Hermitian. discrete and continuous. Both .H  operators,  The wave functions . n(0) are assumed to be orthonormal and to form a complete set; i.e., .

   m(0) n(0) = δm,n

(3.4)

and Iˆ =

.

 

  n(0) n(0) .

(3.5)

n

The Matrix Perturbation Method is a time-independent perturbative approach [1– 4, 6] based on the Taylor expansion of the evolution operator of the time-dependent ˆ Schrödinger equation .|ψ(t)〉 = e−i H t |ψ(0)〉.

3.2.1 First-Order Correction We find an approximated solution to the complete time-dependent Schrödinger equation; i.e., we will solve approximately the equation i

.

  ∂ |ψ(t)〉 = Hˆ |ψ(t)〉 = Hˆ 0 + λHˆ P |ψ(t)〉. ∂t

(3.6)

As we suppose that the Hamiltonian .Hˆ is time independent, the formal solution to this time-dependent Schrödinger equation is  |ψ(t)〉 = exp −it (Hˆ 0 + λHˆ P ) |ψ(0)〉,

.

(3.7)

where .|ψ(0)〉 is the initial condition. Expanding the exponential in its Taylor series, the previous solution can be written as |ψ(t)〉 =

.

∞ n

(−it)n  ˆ H0 + λHˆ P |ψ(0)〉. n! n=0

Explicitly, the parts of the operators of the first non-trivial terms are

(3.8)

50

.

3 The Matrix Perturbation Method

 0 Hˆ 0 + λHˆ P = Iˆ  1 Hˆ 0 + λHˆ P = Hˆ 0 + λHˆ P ,  2 Hˆ 0 + λHˆ P = Hˆ 02 + λHˆ 0 Hˆ P + λHˆ P Hˆ 0 + λ2 Hˆ P2 ,  3   Hˆ 0 + λHˆ P = Hˆ0 3 + λ Hˆ0 2 · HˆP + HˆP · Hˆ0 2 + Hˆ0 · HˆP · Hˆ0   + λ2 Hˆ0 · HˆP 2 + HˆP 2 · Hˆ0 + HˆP · Hˆ0 · HˆP + λ3 HˆP 3 ,  4 Hˆ 0 + λHˆ P = Hˆ0 4   + λ Hˆ0 3 · HˆP + HˆP · Hˆ0 3 + Hˆ0 · HˆP · Hˆ0 2 + Hˆ0 2 · HˆP · Hˆ0  + λ2 Hˆ0 2 · HˆP 2 + HˆP 2 · Hˆ0 2 + Hˆ0 · HˆP 2 · Hˆ0 + HˆP · Hˆ0 2 · HˆP  +Hˆ0 · HˆP · Hˆ0 · HˆP + HˆP · Hˆ0 · HˆP · Hˆ0   + λ3 Hˆ0 · HˆP 3 + HˆP 3 · Hˆ0 + HˆP · Hˆ0 · HˆP 2 + HˆP 2 · Hˆ0 · HˆP + λ4 HˆP 4 , .. ..

If we keep only the terms to first order in .λ, we have .

 0 Hˆ 0 + λHˆ P =Iˆ  1 Hˆ 0 + λHˆ P =Hˆ 0 + λHˆ P ,  2   Hˆ 0 + λHˆ P ≈Hˆ 02 + λ Hˆ 0 Hˆ P + Hˆ P Hˆ 0 ,  3   Hˆ 0 + λHˆ P ≈Hˆ0 3 + λ Hˆ0 2 · HˆP + HˆP · Hˆ0 2 + Hˆ0 · HˆP · Hˆ0 ,  4 Hˆ 0 + λHˆ P ≈Hˆ0 4 +   λ Hˆ0 3 · HˆP + HˆP · Hˆ0 3 + Hˆ0 · HˆP · Hˆ0 2 + Hˆ0 2 · HˆP · Hˆ0 , .. . n−1 n 

Hˆ 0 + λHˆ P ≈Hˆ0 n + λ Hˆ 0n−1−k Hˆ P Hˆ 0k . k=0

3.2 Matrix Approach to the Perturbation Theory

51

Rearranging the terms, to first order in .λ, and substituting them in (3.8), we have

|ψ(t)〉 ≈

∞

(−it)n

.

n!

n=0

Hˆ 0n

+λ

∞ n−1

(−it)n

n!

n=1

 Hˆ 0n−1−k Hˆ P Hˆ 0k

|ψ(0)〉.

(3.9)

k=0

The key ingredient of the Matrix Perturbation Method is the matrix  M=

.

Hˆ 0 Hˆ P 0 Hˆ 0

 ,

(3.10)

whose diagonal elements are all equal to the unperturbed part of the Hamiltonian, Hˆ 0 , and the superior triangle is given by the perturbation, .Hˆ P . Note how the different powers of this matrix are

.



M

.

0

M1 M2 M3 M4

 10 = , 01   Hˆ0 Hˆ P , = 0 Hˆ0   2 Hˆ0 Hˆ 0 Hˆ P + Hˆ 0 Hˆ P , = 0 Hˆ0 2   3 Hˆ0 Hˆ0 2 HˆP + HˆP Hˆ0 2 + Hˆ0 HˆP Hˆ0 , = 0 Hˆ0 3   3 2 Hˆ0 4 Hˆ0 HˆP + HˆP Hˆ0 3 + Hˆ0 HˆP Hˆ0 2 + Hˆ0 HˆP Hˆ0 = , 0 Hˆ0 4

.. .   n n−1 n−1−k ˆ Hˆ P Hˆ 0k Hˆ0 k=0 H0 . Mn = 0 Hˆ0 n If we denote the element .j, k of matrix .M as .Mj,k , it is very easy to get convinced that the following relations are satisfied: M1,2 = Hˆ P ,

.

M21,2 = Hˆ 0 Hˆ P + Hˆ P Hˆ 0 , M31,2 = Hˆ0 2 HˆP + Hˆ0 HˆP Hˆ0 + HˆP Hˆ0 2 , 3 2 M41,2 = Hˆ0 HˆP + HˆP Hˆ0 3 + Hˆ0 HˆP Hˆ0 2 + Hˆ0 ,

52

3 The Matrix Perturbation Method

.. . Mn1,2 =

n−1

Hˆ 0n−1−k Hˆ P Hˆ 0k .

k=0

Thus, Eq. (3.9) can be written as

∞

(−it)n

|ψ(t)〉 ≈

.

n!

n=0

Hˆ 0n + λ

∞

(−it)n Mn1,2



n!

n=1

|ψ(0)〉;

(3.11)

because .M0 = I (where .I is the .2 × 2 identity matrix), we trivially have that .M01,2 = 0, and we can start the second sum above from zero and cast the above equation as  |ψ(t)〉 ≈

.

∞

(−it Hˆ 0 )n

n!

n=0

n  ∞ 

(−itM)1,2 |ψ(0)〉 ; +λ n!

(3.12)

n=0

hence, identifying the Taylor development of the exponential function, we obtain    ˆ |ψ(t)〉 ≈ e−i H0 t + λ e−iMt

.

1,2

 |ψ(0)〉

  ˆ =e−i H0 t |ψ(0)〉 + λ e−iMt

1,2

|ψ(0)〉.

(3.13)

Note that in the last expression, we have separated the approximate solution in two parts; the first part is the solution of the non-perturbed system, which is well known, and the second part is the first-order correction to the wave function. We now show how the correction to first order may be calculated; for this, we rewrite Eq. (3.13) as |ψ(t)〉 ≈ |ψ (0) (t)〉 + λ(||ψ p 〉)1,2 ,

.

(3.14)

  ˆ where .ψ (0) (t) = e−i H0 t |ψ(0)〉 is, as we already said, the non-perturbed solution or zero-order solution, and where we have defined the matrix wave function  ||ψ p 〉 =

.

 |ψ1,1 〉 |ψ1,2 〉 . |ψ2,1 〉 |ψ2,2 〉

(3.15)

Deriving (3.13) and (3.14) with respect to time and equating them, we arrive to the equation   ˆ Hˆ 0 e−i H0 t |ψ(0)〉 + λ Me−itM |ψ(0)〉

.

1,2

=i

∂ ∂ (0) |ψ (t)〉 + iλ (||ψ p 〉)1,2 . ∂t ∂t (3.16)

3.2 Matrix Approach to the Perturbation Theory

53

As the powers of .λ are linearly independent, the coefficients of those powers in each side of the equation must be equal. For .λ0 = 1, we have   ∂ Hˆ 0 exp −i Hˆ 0 t |ψ(0)〉 = i |ψ (0) (t)〉, ∂t

.

(3.17)

which is the unperturbed time-dependent Schrödinger equation, with solution .

     (0) ψ (t) = exp −i Hˆ 0 t |ψ(0)〉 ,

(3.18)

and that in fact is the zero-order solution. For .λ, we have i

.

  ∂  p  ||ψ 〉 1,2 = Me−itM |ψ(0)〉 . 1,2 ∂t

(3.19)

We have to solve the previous equation or equivalently the system i

.

∂ ∂t



|ψ1,1 〉 |ψ1,2 〉 |ψ2,1 〉 |ψ2,2 〉



= Me−itM



 |ψ(0)〉 0 ; 0 |ψ(0)〉

(3.20)

integrating this equation, we have ||ψ 〉 = e p

.

−itM



 |ψ(0)〉 0 , 0 |ψ(0)〉

(3.21)

which is obviously equivalent to the differential equation i

.

∂ ||ψ p 〉 = M||ψ p 〉, ∂t

(3.22)

with the initial condition  ||ψ p (0)〉 =

.

 |ψ(0)〉 0 . 0 |ψ(0)〉

(3.23)

The needed solution is associated with the second column of matrix .||ψ p 〉, and thus we write     ∂ |ψ1,2 〉 |ψ1,2 〉 =M . (3.24) .i |ψ2,2 〉 ∂t |ψ2,2 〉 As .M, Eq. (3.10), is a tridiagonal matrix, the system may be directly integrated. We show it by writing explicitly the two equations

54

3 The Matrix Perturbation Method

i

.

∂ |ψ1,2 〉 = Hˆ 0 |ψ1,2 〉 + Hˆ P |ψ2,2 〉 ∂t

(3.25)

∂ |ψ2,2 〉 = Hˆ 0 |ψ2,2 〉, ∂t

(3.26)

and i

.

with the initial condition  .

 0 . |ψ(0)〉

(3.27)

Because we know the solution for .Hˆ 0 , Eq. (3.26) is solved trivially, ˆ

|ψ2,2 〉 = e−it H0 |ψ(0)〉,

.

(3.28)

and its substitution in (3.25) allows us to write i

.

∂ ˆ |ψ1,2 〉 = Hˆ 0 |ψ1,2 〉 + Hˆ P e−i H0 t |ψ(0)〉. ∂t

(3.29)

ˆ

Making the transformation .|ψ1,2 〉 = e−i H0 t |φ1,2 (x)〉 in the above equation, we arrive to i

.

∂ ˆ ˆ |φ1,2 〉 = ei H0 t Hˆ P e−i H0 t |ψ(0)〉, ∂t

(3.30)

which can be easily integrated to give 

t

|φ1,2 〉 = −i

.

ˆ

ˆ

ei H0 t1 Hˆ P e−i H0 t1 dt1 |ψ(0)〉,

(3.31)

0

and by transforming back, the correction to first order is obtained ˆ

|ψ1,2 〉 = −ie−i H0 t

.



t

ˆ

ˆ

ei H0 t1 Hˆ P e−i H0 t1 dt1 |ψ(0)〉.

(3.32)

0

Up to here, we have produced a first-order correction for the wave function with no assumptions on Hamiltonian degeneracy, therefore making this first-order correction valid also for degenerate Hamiltonians. We can show additionally that it is possible to write t   ˆ p −it Hˆ 0 1 −i 0 dt1 HP (t1 ) |ψ(0)〉, .||ψ 〉 = e (3.33) 0 1

3.2 Matrix Approach to the Perturbation Theory

55

where ˆ ˆ Hˆ P (t1 ) = ei H0 t1 Hˆ P e−i H0 t1 .

.

(3.34)

Summarizing, the wave function to first order is .

       (1)  ψ (t) = ψ (0) (t) + λ ψ1,2 =e

−it Hˆ 0

ˆ

|ψ(0)〉 + λ (−i) e−i H0 t



t

ˆ ˆ ei H0 t1 Hˆ P e−i H0 t1 dt1 |ψ(0)〉.

0

(3.35)

3.2.2 Second-Order Correction We now proceed to find the solution up to second order; that is, we must find the second-order correction to the Schrödinger equation i

.

  ∂ |ψ(t)〉 = Hˆ |ψ(t)〉 = Hˆ 0 + λHˆ P |ψ(t)〉, ∂t

(3.36)

whose solution is  |ψ(t)〉 = exp −it (Hˆ 0 + λHˆ P ) |ψ(0)〉,

.

(3.37)

where .|ψ(0)〉 is the initial condition. Expanding the exponential in its Taylor series, the previous solution can be written as |ψ(t)〉 =

.

∞ n

(−it)n  ˆ H0 + λHˆ P |ψ(0)〉. n! n=0

Explicitly, the parts of the operators of the first non-trivial terms are .

 0 Hˆ 0 + λHˆ P = Iˆ  1 Hˆ 0 + λHˆ P = Hˆ 0 + λHˆ P ,  2 Hˆ 0 + λHˆ P = Hˆ 02 + λHˆ 0 Hˆ P + λHˆ P Hˆ 0 + λ2 Hˆ P2 ,  3   Hˆ 0 + λHˆ P = Hˆ0 3 + λ Hˆ0 2 · HˆP + HˆP · Hˆ0 2 + Hˆ0 · HˆP · Hˆ0   + λ2 Hˆ0 · HˆP 2 + HˆP 2 · Hˆ0 + HˆP · Hˆ0 · HˆP + λ3 HˆP 3 ,

(3.38)

56

3 The Matrix Perturbation Method

 4 Hˆ 0 + λHˆ P = Hˆ0 4   + λ Hˆ0 3 · HˆP + HˆP · Hˆ0 3 + Hˆ0 · HˆP · Hˆ0 2 + Hˆ0 2 · HˆP · Hˆ0  + λ2 Hˆ0 2 · HˆP 2 + HˆP 2 · Hˆ0 2 + Hˆ0 · HˆP 2 · Hˆ0  +HˆP · Hˆ0 2 · HˆP + Hˆ0 · HˆP · Hˆ0 · HˆP + HˆP · Hˆ0 · HˆP · Hˆ0   + λ3 Hˆ0 · HˆP 3 + HˆP 3 · Hˆ0 + HˆP · Hˆ0 · HˆP 2 + HˆP 2 · Hˆ0 · HˆP + λ4 HˆP 4 , .. .. If we keep only the terms to second order in .λ, we have .

 0 Hˆ 0 + λHˆ P = Iˆ  1 Hˆ 0 + λHˆ P = Hˆ 0 + λHˆ P ,  2 Hˆ 0 + λHˆ P = Hˆ 02 + λHˆ 0 Hˆ P + λHˆ P Hˆ 0 + λ2 Hˆ P2 ,   3  Hˆ 0 + λHˆ P ≈ Hˆ0 3 + λ Hˆ0 2 · HˆP + HˆP · Hˆ0 2 + Hˆ0 · HˆP · Hˆ0   + λ2 Hˆ0 · HˆP 2 + HˆP 2 · Hˆ0 + HˆP · Hˆ0 · HˆP ,   4  4 Hˆ 0 +λHˆ P ≈ Hˆ0 + λ Hˆ0 3 · HˆP +HˆP · Hˆ0 3 +Hˆ0 · HˆP · Hˆ0 2 +Hˆ0 2 · HˆP · Hˆ0  + λ2 Hˆ0 2 · HˆP 2 + HˆP 2 · Hˆ0 2 + Hˆ0 · HˆP 2 · Hˆ0 + HˆP · Hˆ0 2 · HˆP  +Hˆ0 · HˆP · Hˆ0 · HˆP + HˆP · Hˆ0 · HˆP · Hˆ0 , .. . n−1 n 

Hˆ 0 + λHˆ P ≈ Hˆ 0n + λ Hˆ 0n−1−k Hˆ P Hˆ 0k k=0

+ λ2

n−1

n−k

n−k−j −1 ˆ ˆ j ˆ ˆ k−1 Hˆ 0 HP H 0 HP H 0 .

k=1 j =0

Rearranging the terms, to second order in .λ, and substituting them in (3.8), we have

3.2 Matrix Approach to the Perturbation Theory

|ψ(t)〉 ≈

∞

(−it)n

.

n=0

+ λ2

n! ∞

n=2

Hˆ 0n |ψ (0)〉 + λ

∞ n−1

(−it)n

n=1

n−1 n−k (−it)n

n!

57

n!

Hˆ 0n−1−k Hˆ P Hˆ 0k |ψ (0)〉

k=0

n−k−j −1 ˆ ˆ j ˆ ˆ k−1 Hˆ 0 HP H0 HP H0 |ψ (0)〉 .

(3.39)

k=1 j =0

In analogy with the first-order correction, we define now the matrix ⎛

Hˆ 0 ⎝ .M = 0 0

Hˆ P Hˆ 0 0

⎞ 0 Hˆ P ⎠ , Hˆ 0

(3.40)

whose diagonal elements are all equal to the unperturbed part of the Hamiltonian, Hˆ 0 , and the superior sub-diagonal is given by the perturbation, .Hˆ P . As in the first-order case, we can analyze the first powers of the matrix .M, which are

.

⎛

M0

.

M1

M2

M3

⎞ 100 =⎝0 1 0⎠, 001 ⎛ ⎞ Hˆ 0 Hˆ P 0 = ⎝ 0 Hˆ 0 Hˆ P ⎠ , 0 0 Hˆ 0 ⎛ 2 ⎞ Hˆ0 Hˆ0 HˆP + HˆP Hˆ0 HˆP 2 =⎝ 0 Hˆ0 2 Hˆ0 HˆP + HˆP Hˆ0 ⎠ , 0 0 Hˆ0 2 ⎞ ⎛ 3 Hˆ0 Hˆ0 2 HˆP + HˆP Hˆ0 2 + Hˆ0 HˆP Hˆ0 Hˆ0 HˆP 2 + HˆP 2 Hˆ0 + HˆP Hˆ0 HˆP =⎝ 0 Hˆ0 3 Hˆ0 2 HˆP + HˆP Hˆ0 2 + Hˆ0 HˆP Hˆ0 ⎠ , 0 0 Hˆ0 3 .. .;

and then, it is very easy to see that M11,3 =0,

.

M21,3 =Hˆ P2 , M31,3 =Hˆ 0 Hˆ P2 + Hˆ P Hˆ 0 Hˆ P + Hˆ P2 Hˆ 0 , M41,3 =Hˆ0 2 HˆP 2 + HˆP 2 Hˆ0 2 + Hˆ0 HˆP 2 Hˆ0 + HˆP Hˆ0 2 HˆP + Hˆ0 HˆP Hˆ0 HˆP

58

3 The Matrix Perturbation Method

+ HˆP Hˆ0 HˆP Hˆ0 .. . Mn1,3 =

n−k n−1



n−k−j −1 ˆ ˆ j ˆ Hˆ 0 HP H0 HP (Hˆ 0 )k−1 ;

k=1 j =0

thus, we can write Eq. (3.39) as

|ψ(t)〉 ≈ e

.

−i Hˆ 0 t

  + λ e−iMt

+λ

1,2

2



∞

(−it)n n=2

Mn1,3

n!

|ψ(0)〉,

(3.41)

and using that .M01,3 = M1,3 = 0, we obtain in a completely analogous way to the case of the first-order perturbation "   ˆ |ψ(t)〉 ≈ e−i H0 t + λ e−iMt

.

1,2

  + λ2 e−iMt

#

1,3

|ψ(0)〉.

(3.42)

Inserting now the matrix, ⎛

⎞ |ψ1,1 〉 |ψ1,2 〉 |ψ1,3 〉 p .||ψ 〉 = ⎝ |ψ2,1 〉 |ψ2,2 〉 |ψ2,3 〉 ⎠ , |ψ3,1 〉 |ψ3,2 〉 |ψ3,3 〉

(3.43)

where the first-order correction .|ψ1,2 〉 and the second-order correction .|ψ1,3 〉 are included. Expanding .|ψ〉(t) to second order in .λ, we get |ψ(t)〉 ≈ |ψ (0) (t)〉 + λ||ψ p 〉1,2 + λ2 ||ψ p 〉1,3 .

(3.44)

.

Following the same procedure as in the first-order correction case, we derive equations (3.42) and (3.44) with respect to time and equate the corresponding equations to obtain ∂ ∂ (0) ∂ |ψ (t)〉 + iλ ||ψ p 〉1,2 + λ2 ||ψ p 〉1,3 ∂t ∂t ∂t     ˆ −i H t −itM = Hˆ 0 e 0 |ψ(0)〉 + λ Me |ψ(0)〉 + λ2 Me−itM |ψ(0)〉

i

.

1,2

1,3

. (3.45)

Equating powers of .λ2 , we can establish that i

.

  ∂ ||ψ p 〉1,3 = Me−iMt |ψ(0)〉 , 1,3 ∂t

(3.46)

3.2 Matrix Approach to the Perturbation Theory

59

or equivalently i

.

∂ ||ψ p 〉 = M||ψ p 〉, ∂t

(3.47)

with the initial condition ⎛

⎞ |ψ(0)〉 0 0 p .||ψ (0)〉 = ⎝ 0 |ψ(0)〉 0 ⎠ . 0 0 |ψ(0)〉

(3.48)

Equation (3.47) is similar to (3.22), and thus we can again proceed as in the firstorder case, choosing the third column in both sides of the equation and getting the differential equation system ⎞ ⎛ ⎞ ⎛ |ψ1,3 〉 |ψ 〉 ∂ ⎝ 1,3 ⎠ .i |ψ2,3 〉 = M ⎝ |ψ2,3 〉 ⎠ , ∂t |ψ3,3 〉 |ψ3,3 〉

(3.49)

with the initial condition ⎛

⎞ 0 .⎝ 0 ⎠. |ψ(0)〉

(3.50)

The correction we are looking for is then 2 −i Hˆ 0 t

|ψ1,3 〉 = (−i) e

.

 t 0

t1

ˆ ˆ ˆ ˆ ei H0 t1 Hˆ P e−i H0 t1 ei H0 t2 Hˆ P e−i H0 t2 dt2 dt1 |ψ(0)〉,

0

(3.51) which can be written in the compact form ⎛

1 −i p −it Hˆ 0 ⎝ .||ψ 〉 = e 0 0

t 0

dt1 Hˆ P (t1 ) (−i)2 1 0

t 0

t ⎞ dt1 01 dt2 Hˆ P (t1 )Hˆ P (t2 ) t ⎠ |ψ(0)〉. −i 0 dt1 Hˆ P (t1 ) 1 (3.52)

Summarizing, the wave function to second order is .

         (2)  ψ (t) = ψ (0) (t) + λ ψ1,2 + λ2 ψ1,3  t ˆ ˆ ˆ ˆ = e−it H0 |ψ(0)〉 + λ (−i) e−i H0 t ei H0 t1 Hˆ P e−i H0 t1 dt1 |ψ(0)〉 0

60

3 The Matrix Perturbation Method 2 −i Hˆ 0 t

+ λ (−i) e 2

 t 0

t1

ˆ ˆ ˆ ˆ ei H0 t1 Hˆ P e−i H0 t1 ei H0 t2 Hˆ P e−i H0 t2 dt2 dt1 |ψ(0)〉.

0

(3.53)

3.2.3 Higher Order Corrections In this subsection, we generalize our method to higher orders. We propose the perturbation series |ψ(t)〉 ≈ |ψ

.

(0)

(t)〉 +

m



λ ||ψ 〉1,m+1 = e n

p

n=1

−it Hˆ 0

+

m

   −itM λ e |ψ(0)〉 n

n=1

(3.54) with ⎞ |ψ1,1 〉 · · · |ψ1,m+1 〉 ⎟ ⎜ .. .. p .. .||ψ 〉 = ⎝ ⎠, . . . |ψm+1,1 〉 · · · |ψm+1,m+1 〉 ⎛

where m is the correction order and ⎛ Hˆ 0 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 .M = ⎜ . ⎜ . ⎜ . ⎜ . ⎜ . ⎝ .

⎞ Hˆ P 0 · · · · · · 0 ⎟ Hˆ 0 Hˆ P 0 · · · 0 ⎟ ⎟ . ⎟ 0 Hˆ 0 Hˆ P . . 0 ⎟ ⎟ . . . . . . . . .. ⎟ ; . . . . . ⎟ ⎟ .. .. .. ˆ ˆ ⎟ . . . H0 HP ⎠ 0 0 0 · · · · · · Hˆ 0

(3.55)

(3.56)

i.e., the matrix has .Hˆ 0 in the main diagonal and .Hˆ p in the superior sub-diagonal, and all the other elements are zero. Following the method for the first- and second-order corrections, we deduce the following system of differential equations: ∂ ||ψ p 〉1,m+1 = M||ψ p 〉1,m+1 , ∂t

(3.57)

⎞ ⎛ ⎞ |ψ1,m+1 〉 |ψ1,m+1 〉 ∂ ⎜ ⎟ ⎜ ⎟ .. .. .i ⎝ ⎠ = M⎝ ⎠, . . ∂t |ψm+1,m+1 〉 |ψm+1,m+1 〉

(3.58)

i

.

or ⎛

3.3 Normalization Constant

61

with the initial condition ⎛ .

⎞

0 .. .

⎜ ⎝

⎟ ⎠,

(3.59)

|ψ(0)〉 whose solution is |ψ1,k+1 〉 = (−i)k e−iH0 t

 t

t1





t2

...

.

0

0

0

tk−1

eiH0 t1 HP e−iH0 t1 eiH0 t2 HP e−iH0 t2 . . .

0

. . . eiH0 tk HP e−iH0 tk |ψ(0)〉 dtk . . . dt3 dt2 dt1 ,

k = 1, 2, 3, . . . . (3.60)

This time-ordered series, restricted to the interval .[0, t], is the fundamental piece to calculate the different correction terms; furthermore, we should point out that the relationship (3.60) is the mathematical representation of the Dyson series [7–9]. Although this latter expression only applies for weak perturbations, its strong perturbation counterpart (.λ → ∞) can be derived in a straightforward way interchanging the unperturbed Hamiltonian .Hˆ0 with the perturbation .HˆP and rescaling the time as .τ = λt. This duality on the Matrix Method gives us the possibility to analyze the solution of a quantum system in both regimes of the perturbation parameter.

3.3 Normalization Constant In the previous section, we showed that the approximated solution of the Schrödinger equation can bewritten as a power series of the perturbation parameter .λ, along with the element .ψ1,k+1 of the perturbed matrix. It is appropriate to mention that the expression (3.54) is not normalized and it is convenient to get a normalization factor .Nk that preserves its norm at any order. Then, let us define the next normalized solution ⎞ ⎛ k  

   (0) + . |Ψ(t)〉 = Nk ⎝ψ λj ψ1,j +1 ⎠ , (3.61) j =1

where the corresponding value of .Nk may be easily determined by the normalization condition .〈Ψ(t)|Ψ(t)〉 = 1 and can be expressed as follows:

62

3 The Matrix Perturbation Method

Table 3.1 The different terms of the double summation contained in the second term of Eq. (3.62), when m and j run from 1 to k 1

m/j

&  ψ1,2 |ψ1,2 &  3 .λ ψ1,3 |ψ1,2 &  4 .λ ψ1,4 |ψ1,2 .. .. .. .. &  1+k ψ .λ 1,k+1 |ψ1,2 2 .λ

1 2 3 .. .. .. .. .k

2

3

3 .λ

4 .λ

&  ψ1,2 |ψ1,3 &  4 .λ ψ1,3 |ψ1,3 &  5 .λ ψ1,4 |ψ1,3 .. .. .. .. &  2+k ψ .λ 1,k+1 |ψ1,3

.λ

5

6 .λ

& & &

ψ1,2 |ψ1,4 ψ1,3 |ψ1,4 ψ1,4 |ψ1,4

.. . .



.. . .



.. . .



.. . .

.. .. .. ..

.

&  3+k ψ .λ 1,k+1 |ψ1,4

..

.

.. . .

k

&  ψ1,2 |ψ1,k+1 &  2+k ψ |ψ .λ 1,3 1,k+1 &  3+k ψ |ψ .λ 1,4 1,k+1 .. .. .. .. &  2k ψ .λ 1,k+1 |ψ1,k+1 .λ

1+k

1

Nk = '

.

1+2

k  j =1



&

λj Re ψ (0) |ψ1,j +1 +

k 

λl+j

l,j =1

&  ψ1,l+1 |ψ1,j +1

,

(3.62)

for the real part of z. The first contribution is due to the where& .Re (z) stands  term . ψ (0) |ψ (0) , whereas the second one arises from two single finite sums, one referent & (0) to the  inner product of the zero-order term with the j &th-order correction  . ψ |ψ1,j +1 , and the other respect to its complex conjugate . ψ1,j +1 |ψ (0) ; as a consequence, a purely real contribution is obtained of the sum of both over all k. The last part of the above equation is merely handled if we run m and j from 1 to k, such as presented in Table 3.1. It is easy to see that the double summation in (3.62) can be split in two parts: one where .m = j and which contains all diagonal terms and &the other for the offdiagonal terms represented in a double sum of the real part of . ψ1,m+1 |ψ1,j +1 , i.e., k

.

k & 

&  λm+j ψ1,m+1 |ψ1,j +1 = λ2n ψ1,n+1 |ψ1,n+1

m,j =1

n=1

+2

k k−1



&  λn+m Re ψ1,n+1 |ψ1,m+1 ;

n=1 m=n+1 k>1

(3.63) applying the change of variable .m = p − n and replacing in Eq. (3.62), we arrive to ⎡ .Nk = ⎣1 + 2

k

j =1

j

λ Re

 ψ

(0)

|ψ1,j +1



+

k

n=1

&  λ2n ψ1,n+1 |ψ1,n+1

3.4 Connection with the Standard Time-Independent Perturbation Theory

+2

k−1

63

⎤− 1

n+k

⎥ λ Re ψ1,n+1 |ψ1,p−n+1 ⎦ &

p

2

,

(3.64)

n=1 p=2n+1 k>1

which is the normalization constant for the approximated analytical solution of the Schrödinger equation defined in Eq. (3.61). In principle, the inclusion of the factor .Nk in our calculations can give a fairly good approximation to the solution without convergence difficulties. An important remark on the proposed normalization procedure is that we have not invoked the usual intermediate normalization used &  in the standard perturbation theory, i.e., the imposition . ψ (0) |ψ1,n+1 = 0 for all .λ. Such a condition does not apply in our case, because the inner product of the zero-order correction with the first two correction terms is different from zero; in particular, these complex inner products have non-zero imaginary parts which should not be neglected if the called intermediate normalization is applied; for this reason, we have adopted other procedure to obtain the factor .Nk , which ensures real values at any power of .λ.

3.4 Connection with the Standard Time-Independent Perturbation Theory In this section, we will derive some of the formulas of the standard time-independent perturbation theory from the NMPM. We start with the two main formulas to first order: the one for the energy and the other for the wave function. Although these expressions were deduced in Chap. 1, to facilitate reading and understanding, we reproduce them below; indeed, for the first-order correction to energy, we have      ΔEn(1) = n(0) Hˆ P n(0) ,

.

(3.65)

being .Hˆ P the perturbation part of the complete Hamiltonian, and for the first-order correction of the wave function, we have    (1) . n =

& (0)   (0)    m Hˆ P n  (0) . m (0) (0) m (m/=n) En − Em

(3.66)

Let us write the formula that defines the first-order correction of the wave function, particularized to the case in which the initial condition is   one of the eigenstates of the unperturbed Hamiltonian .Hˆ 0 , that is, .|ψ(0)〉 = m(0) ,   −i Hˆ 0 t . ψ1,2 = −ie



t 0

  ˆ ˆ  ei H0 t1 Hˆ P e−i H0 t1 dt1 m(0) .

(3.67)

64

3 The Matrix Perturbation Method

 & As the set . k (0)  is orthonormal, we can use the closure relation [10–12] that   (0)  & (0)  ˆ k , and inserting it in the expression expresses the unit  operator as .I = k k above for .ψ1,2 , we get  t    

   ˆ ˆ  (0) (0)  −i Hˆ 0 t  k e . ψ1,2 = −i ei H0 t1 Hˆ P e−i H0 t1 dt1 m(0) . k k

(3.68)

0

  As the vectors .k (0) are eigenfunctions of the non-perturbed Hamiltonian .Hˆ 0 , with     (0) ˆ 0 k (0) = E (0) k (0) , and also, for any welleigenvalues .Ek , we have that .H k     (0)  (0)  k ; then, it is very easy to calculate behaved function, .f Hˆ 0 k (0) = f E k

that       ˆ ˆ ˆ  (0) (0) (0)  (0) . k  − ie−i H0 t ei H0 t1 Hˆ P e−i H0 t1 m(0) = − i exp −itEk + it1 Ek − Em      k (0)  Hˆ P m(0) , which can be integrated without any difficulty to give    ⎧  (0) (0)  exp −itEm −exp −itEk & (0)   ⎪ ⎨ k  Hˆ P m(0) , k /= m, (0) (0) E −E m . k   ⎪ (0) & (0)  ˆ  (0)  ⎩−it exp −itEm m HP m , k = m.

Thus, the first-order correction to the wave function can be written as     (0) (0)     

exp −itEm − exp −itEk   (0)  ˆ  (0)  (0) H k · k . ψ1,2 =  m P (0) (0) Em − Ek k, k/=m           (0) m(0)  Hˆ P m(0) · m(0) ; (3.69) − it exp −itEm splitting the sum over k and defining      &k (0)  Hˆ P m(0)     .

 (1)  (0) (0) k , . n (t) = exp −itEk  (0) (0) Em − Ek k, k/=m

(3.70)

we get .

        ψ1,2 = − n(1) (t) + exp −itE (0) · n(1) (0) m           (0) m(0)  Hˆ P m(0) · m(0) . − it exp −itEm

(3.71)

3.4 Connection with the Standard Time-Independent Perturbation Theory

65

  Note that .n(1) (0) is the expression that the traditional time-independent perturbative method offers as a first-order correction to the wave function. Now, we proceed to calculate the first-order correction to the energy. We take the complete wave function to first order .

        (1) = ψ (0) + λ ψ1,2 , ψ

(3.72)

     (0) = exp −itEk(0) |m〉 ψ

(3.73)

where .

is the non-perturbed solution, and we calculate the mean value of the complete Hamiltonian            &    (1)  ˆ  (1) = ψ (0)  + λ ψ1,2  Hˆ 0 + λHˆ P ψ (0) + λ ψ1,2 . . ψ  H ψ (3.74) Expanding the right-hand side of (3.74), we arrive to .

          ψ (1)  Hˆ ψ (1) = ψ (0)  Hˆ 0 ψ (0) &        &  &  + λ ψ1,1  Hˆ P ψ1,1 + ψ1,2  Hˆ 0 ψ1,1 + ψ1,1  Hˆ 0 ψ1,2 &        &  &  + λ2 ψ1,2  Hˆ P ψ1,1 + ψ1,2  Hˆ 0 ψ1,2 + ψ1,1  Hˆ P ψ1,2   &  (3.75) + λ3 ψ1,2  Hˆ P ψ1,2 .

As we are looking for the correction to the energy to first order, we discard the terms in .λ2 and .λ3 , and we can write         (1)  ˆ  (1) ≈ ψ (0)  Hˆ 0 ψ (0) . ψ  H ψ &        &  &  (3.76) + λ ψ1,1  Hˆ P ψ1,1 + ψ1,2  Hˆ 0 ψ1,1 + ψ1,1  Hˆ 0 ψ1,2 . The non-perturbed Hamiltonian .Hˆ 0 is Hermitian, so it is obvious that .

    &  &  ψ1,2  Hˆ 0 ψ1,1 = ψ1,1  Hˆ 0 ψ1,2 ∗ ,

(3.77)

where the asterisk represents complex conjugation; it is straightforward to calculate that        & &   ˆ 0 ψ1,1 = itE (0) m(0)  Hˆ P m(0) = − ψ1,2  Hˆ 0 ψ1,1 , . ψ1,2  H (3.78) k

66

3 The Matrix Perturbation Method

as also the perturbation Hamiltonian .Hˆ P is Hermitian, and then .

        &      ψ (1)  Hˆ ψ (1) ≈ ψ (0)  Hˆ 0 ψ (0) + λ ψ1,1  Hˆ P ψ1,1 ;

furthermore, on the one hand we have    (0) (0)  ˆ  (0) =E , . ψ  H0 ψ

(3.79)

(3.80)

k

and on the other hand, we have that      &    ψ1,1  Hˆ P ψ1,1 = m(0)  Hˆ P m(0) ;

(3.81)

          (0) ψ (1)  Hˆ ψ (1) ≈ Ek + m(0)  Hˆ P m(0) ;

(3.82)

.

hence, finally, .

and we have recovered the formula (3.65) of the standard time-independent perturbation theory.

3.5 The Dyson Series in the Matrix Method In Sect. 2.3 we have the Dyson series, and we saw that the wave functions of the perturbed problem are written as #  "  t ˆ |ψ(t)〉 = e−i H0 t Tˆ exp −iλ dt1 Vˆ (t1 ) |ψ(0)〉,

.

(3.83)

0

where .Tˆ is the time-ordered operator; i.e., if we have the time-dependent operators ˆ and .B(t), ˆ then A(t)

.

ˆ 2 )] = ˆ 1 )B(t .Tˆ [A(t



ˆ 1 ) if t2 > t1 , ˆ 2 )A(t B(t ˆ ˆ A(t1 )B(t2 ) if t1 > t2 .

(3.84)

On the other hand, from Eq. (3.54), we can write

|ψ(t)〉 = e

.

−i Hˆ 0 t

+

∞

n=1

  λn e−iMt

(1,n+1)

 |ψ(0)〉,

so comparing Eq. (3.83) with Eq. (3.84), we derive the formula

(3.85)

References

67

#  "  t ∞   ˆ n dt1 Vˆ (t1 ) = ei H0 t λ e−iMt Tˆ exp −iλ

.

0

n=0

(1,n+1)

,

(3.86)

which offers a matrix expansion for the Dyson operator and that links our matrix method with the Dyson series.

References 1. J. Martinez-Carranza, F. Soto-Eguibar, H. Moya-Cessa, Alternative analysis to perturbation theory in quantum mechanics. Eur. Phys. J. D 66(1) (2012). https://doi.org/10.1140/epjd/ e2011-20654-5 2. J. Martinez-Carranza, H.M. Moya-Cessa, F. Soto-Eguibar, La teoría de perturbaciones en la mecánica cuántica (Editorial Académica Española, 2012) 3. B.M. Villegas-Martinez, F. Soto-Eguibar, H.M. Moya-Cessa, Application of perturbation theory to a master equation. Adv. Math. Phys. 2016, 1–7 (2016) 4. B.M. Villegas-Martinez, H.M. Moya-Cessa, F. Soto-Eguibar, Normalization corrections to perturbation theory based on a matrix method. J. Mod. Opt. 65(8), 978–986 (2017) 5. R. Chen, Z. Xu, L. Sun, Finite-difference scheme to solve Schrödinger equations. Phys. Rev. E 47, 3799–3802 (1993) 6. B.M. Villegas-Martinez, H.M. Moya-Cessa, F. Soto-Eguibar, Exact and approximated solutions for the harmonic and anharmonic repulsive oscillators: matrix method. Euro. Phys. J. D 74(7) (2020). https://doi.org/10.1140/epjd/e2020-10128-2 7. F.J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75(3), 486–502 (1949) 8. A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publications, Mineola, 2003) 9. G.W. Johnson, M.L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus (Oxford Mathematical Monographs) (Oxford University Press, Oxford, 2002) 10. A. Messiah, Quantum Mechanics (North-Holland Publishing Company, Amsterdam, 1991) 11. N. Zettili, Quantum Mechanics. Concepts and Applications, 2nd edn. (John Wiley and Sons, Inc., Hoboken, 2009) 12. E. Merzbacher, Quantum Mechanics, 3rd edn. (John Wiley and Sons, Inc., Hoboken, 1998)

Chapter 4

Examples of the Matrix Perturbation Method

Abstract In this chapter we present several examples of the Matrix Perturbation Method. First, we analyze the harmonic oscillator perturbed by a linear anharmonic term; this example has the advantage that it can be solved exactly and allows a very good evaluation of the Matrix Perturbation Method. Second, we treat the quadratic potential perturbed by a cubic term; we calculate the corrections to second order, and we consider a number state and a coherent state as initial conditions. Finally, we study the repulsive quadratic potential with a linear perturbation; again this system has an exact solution which permits a good evaluation of the Matrix Perturbation Method.

4.1 Introduction In the previous chapter the Matrix Perturbation Method was introduced. In this chapter, we present several examples of the utilization of the method, and we compare the obtained results with the exact ones, when they are available, and with those earn using the traditional perturbation methods. In Sect. 4.2, we present a harmonic oscillator perturbed by a linear anharmonic potential; this example has the advantage that an exact analytic solution is well known. We obtain an exact analytic solution by means of operator techniques, and we compare the results with those up to second order obtained by using the Matrix Perturbation Method. In Sect. 4.3, we analyze the case of a quadratic potential plus a cubic term; the corrections up to second order are presented and analyzed. Finally, in Sect. 4.4, the case of the repulsive quadratic potential is studied; we solve that potential exactly, and after we add a perturbative linear term. This new system is also solved exactly, and also the second-order corrections by means of the Matrix Perturbation Method are calculated. Both results are compared, showing a very good concordance of the perturbative results with the exact ones.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6_4

69

70

4 Examples of the Matrix Perturbation Method

4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term Let us study the case of the harmonic oscillator disturbed by a linear potential. This quantum system is described by the following Schrödinger equation i

.

d|ψ(t)〉 = dt



 pˆ 2 ω2 2 + xˆ + λxˆ |ψ (t)〉, 2 2

(4.1)

∂ and .xˆ = x are the momentum and position operators, whereas where .pˆ = −i ∂x the quantity .λ is a dimensionless scale parameter which quantifies the perturbation strength of the linear anharmonic term; without loss of generality, the mass .μ of the oscillator is set equal to 1. We also remind the reader that we are using a system of units in which .h¯ = 1. From a classical point of view, the above timedependent Schrödinger equation gives a physical description of a particle with charge q oscillating harmonically with angular frequency .ω, located in a weak electric field of strength .ε [1–5]. This example, although simple and well known, has the advantage of having an exact solution, and then it is possible to clearly evaluate the validity of the different perturbative methods. We will start by just finding the exact solution, using an operator formalism.

4.2.1 The Exact Solution Standard differential equation techniques have been used with great success to get an exact solution of this system, especially when a time dependence is involved in the anharmonic part [6–9]; here, we adopt an operator formalism to solve Eq. (4.1) exactly. The momentum and position operators can be expressed in terms of the well-known raising and lowering operators .aˆ † and .aˆ [2, 10–12] as  1  † aˆ + aˆ , . xˆ = √ 2ω    ω † pˆ =i aˆ − aˆ . 2

(4.2a)

.

(4.2b)

These ladder operators satisfy the commutation relations [1–5] .

  a, ˆ aˆ † = 1,

  n, ˆ aˆ † = aˆ † ,



n, ˆ aˆ = −a, ˆ

(4.3)

being .nˆ = aˆ † aˆ the number operator; then, Eq. (4.1) turns into the equivalent form

4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term

71

    1 d|ψ(t)〉 λ  † aˆ + aˆ |ψ (t)〉. = ω nˆ + .i +√ dt 2 2ω

(4.4)

To simplify the above Schrödinger equation and reach an exactly solvable form, it is convenient to perform the transformation .

|φ (t)〉 = Dˆ

 √

λ 2ω3

 |ψ (t)〉 ,

(4.5)

where   ˆ D(β) = exp β aˆ † − β ∗ aˆ

(4.6)

.

ˆ is the Glauber displacement operator [13–17]. The displacement operator .D(β) is † unitary and acts as a displacement upon the amplitudes .aˆ and .aˆ , as [15, 17] ˆ D(β) aˆ Dˆ † (β) =aˆ − β, .

(4.7a)

ˆ D(β) aˆ † Dˆ † (β) =aˆ † − β ∗ .

(4.7b)

.

So, Eq. (4.4) is transformed into    λ2 1 d |φ (t)〉 |φ (t)〉 , − = ω nˆ + .i 2 dt 2ω2

(4.8) 2

λ which is nothing else than the harmonic oscillator displaced by a quantity . 2ω 2 , and with a quantized energy

  λ2 1 − .En = ω n + , 2 2ω2

n = 0, 1, 2, . . . .

(4.9)

If we now integrate the resulting expression with respect to time and then transform it back to .|ψ (t)〉, one obtains       

 t λ λ λ2 † ˆ ˆ |ψ (0)〉 . exp −iωt nˆ D √ ω− 2 . |ψ (t)〉 = exp −i D √ 2 ω 2ω3 2ω3 (4.10) ˆ In order to this

 simplify

 expression, one can insert the identity operator .I = exp iωt nˆ exp −iωt nˆ into the previous equation as follows:       

 it λ λ λ2 † ˆ ˆ exp −iωt nˆ D √ ω− 2 . |ψ (t)〉 = exp − D √ 2 ω 2ω3 2ω3



 × exp iωt nˆ exp −iωt nˆ |ψ (0)〉 . (4.11)

72

4 Examples of the Matrix Perturbation Method

Using the Hadamard formula [18–21]        3  2  ˆ ˆ −δ Aˆ ˆ A, ˆ Bˆ + δ A, ˆ A, ˆ A, ˆ Bˆ ˆ Bˆ + δ A, + ..., eδ A Be = Bˆ + δ A, 2! 3! (4.12)

.

it is very easy to prove that

 . exp −iωt n ˆ Dˆ





λ

√ 2ω3

 exp iωt nˆ = Dˆ

 √

λ 2ω3

 exp (−iωt) ,

(4.13)

and with the aid of the displacement operator property [15]   ∗ αβ − α ∗ β ˆ ˆ ˆ .D(α)D(β) = exp D(α + β), 2

(4.14)

and the fact that .Dˆ † (β) = Dˆ (−β) [15–17, 22], Eq. (4.11) is simplified to .

 |ψ (t)〉 = exp [γ1 (t)] Dˆ (ξ1 (t)) exp −iωt nˆ |ψ (0)〉 ,

(4.15)

   λ2 λ2 i t ω − 2 + 3 sin(ωt) , .γ1 (t) = − 2 ω ω

(4.16)

λ exp(−iωt) − 1 . ξ1 (t) = √ 3 2ω

(4.17)

where

and .

4.2.1.1

A Coherent State as Initial State

We consider now the special case when the initial state is a coherent state .|α〉 [15– 17, 22]. Substituting this initial condition in (4.15), we get .

 |ψ (t)〉 = exp [γ1 (t)] Dˆ (ξ1 (t)) exp −iωt nˆ |α〉

(4.18)

and as [15–17, 22] .

 exp γ nˆ |α〉 = |exp (γ ) α〉 ,

(4.19)

expression (4.18) can be written in the form .

|ψ (t)〉 = exp [γ1 (t)] Dˆ (ξ1 (t)) |α (t)〉 ,

(4.20)

4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term

73

being α (t) = exp (−iωt) α.

(4.21)

.

We use now the definition of a coherent state as a displacement of the vacuum, i.e., |α〉 = Dˆ (α) |0〉, to cast (4.20) as

.

.

|ψ (t)〉 = exp [γ1 (t)] Dˆ (ξ1 (t)) Dˆ (α (t)) |0〉 .

(4.22)

Applying again the formula (4.14), performing some algebra and using the alternative definition of a coherent state as an eigenfunction of the down operator .a, ˆ we arrive to .

|ψ (t)〉 = exp [γ2 (t)] |ξ2 (t)〉 ,

(4.23)

where γ2 (t) = √

.





1 − exp (iωt) α (t) + exp (−iωt) α ∗ (t)

λ

8ω3   i   3 2 2 ωt ω + λ − − λ sin , (ωt) 2ω3

(4.24)

and ξ2 (t) = α (t) + √

.

λ 2ω3



exp (−iωt) − 1 .

(4.25)

Finally, assuming that the amplitude of the initial coherent state .α is real, and using the well-known coordinate representation of a coherent state [15–17, 22]

.

〈x|α〉 =

 ω 1/4 π



|α|2 √ ωx 2 α2 + 2ωαx − − exp − 2 2 2

 ,

(4.26)

we can write, in the coordinate representation, the exact solution of the Schrödinger equation (4.1) as ψ (x, t) =

.

 ω 1/4 π

exp [γ3 (x, t)] ,

(4.27)

where γ3 (x, t) = f0 (x, t) + f1 (x, t)α + f2 (t)α 2 ,

.

(4.28)

74

4 Examples of the Matrix Perturbation Method

being

λ2 3 − 2iωt − 4 exp (−iωt) + exp (−2iωt) .f0 (x, t) = − 4ω3

  λx exp (−iωt) − 1 1 2 − iω t − ix + ,. 2 ω √

2 exp (−iωt) λ − λ cos(tω) + xω2 ,. f1 (x, t) = ω3/2

1 f2 (t) = − 1 + exp (−2iωt) . 2

(4.29a) (4.29b) (4.29c)

It is a simple exercise to verify, by direct substitution, that the wave function (4.27) is indeed a solution of the Schrödinger equation (4.1); it is also very easy to check that the initial condition is satisfied and that the wave function is normalized. In Fig. 4.1, we plot in two dimensions, the square of the wave function .ψ(x, t) as function of x, for four times: .t = 0.0, .t = 0.5, .t = 1.0, and .t = 1.5; we

Fig. 4.1 Two dimensional plot of .|ψ(x, t)|2 , from the solution (4.27), as function of x for (a) = 0.0, (b) .t = 0.5, (c) .t = 1.0, and (d) .t = 1.5. The parameters utilized are .ω = 10, .λ = 0.01, and .α = 4

.t

4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term

75

Fig. 4.2 Three-dimensional plot of .|ψ(x, t)|2 , from the solution (4.27), as function of x and t; the parameters utilized are .ω = 10, .λ = 0.01, and .α = 4. The time evolution of the initial coherent state is clearly exposed

have chosen, completely arbitrarily, the parameters .ω = 10, .λ = 0.01, and .α = 4. We get an oscillating Gaussian, initially moving to the left. In Fig. 4.2, the 3D plot shows .|ψ(x, t)|2 as function of x and t, for the same parameters as in the previous figure; there the oscillations of the coherent state are clear.

4.2.2 The Approximated Perturbative Solution Let us proceed to solve the same quantum system, but now through our perturbative treatment, the Matrix Perturbation Method. In this case,   ˆ 0 = ω nˆ + 1 .H (4.30) 2 denotes the unperturbed harmonic oscillator, and  1  † aˆ + aˆ HˆP = xˆ = √ 2ω

.

the small perturbation, quantified by .λ.

(4.31)

76

4 Examples of the Matrix Perturbation Method

4.2.2.1

Zero-Order Term

The zero-order term is just the harmonic oscillator solution, .

     (0) = exp −i Hˆ 0 t |ψ (0)〉 ψ    1 |ψ (0)〉 . = exp −iωt nˆ + 2

(4.32)

  As .exp −i Hˆ 0 t is a unitary operator, it is clear that the normalization is conserved;   therefore, if the initial condition is normalized, so will be .ψ (0) . To go further, the initial state has to be specified; next we analyze the cases when the initial condition is a number state and when it is a coherent state. A Number State |m〉 as Initial State If the initial state is a number state .|m〉, we have trivially .

     1  (0) |m〉 . = exp −iωt m + ψ 2

(4.33)

The coordinate representation of the Fock states .|n〉 is [2, 3, 11, 12]  

√  1  ω 1/4 ωx 2 Hn ωx , ϕn (x) = 〈x|n〉 = √ exp − 2 2n n! π

.

m = 0, 1, 2, . . . , (4.34)

where .Hn (ζ ) are the Hermite polynomials [23, 24]; then, the coordinate representation of the zero-order wave function is    ψ (0) (x) = x ψ1,1       ω 1/4

√  1 ωx 2 1 Hm ωx , exp − = exp −iωt m + √ 2 2 2m m! π (4.35)

.

which are nothing more than the well-known of the harmonic   eigenfunctions (0) oscillator, with a temporal phase which is .exp −itEm .

A Coherent State |α〉 as Initial State If the initial state is a coherent state .|α〉, we can use (4.19) and get

4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term

    ωt  (0) |α (t)〉 , = exp −i . ψ 2

77

(4.36)

and in the coordinates representation, we have [3, 12] ψ

.

(0)

(x, t) =

 ω 1/4 π



 |α(t)|2 √ ωt α 2 (t) ωx 2 − + 2ωα(t)x − − exp −i , 2 2 2 2 (4.37)

where .α(t) = exp(−iωt)α.

4.2.2.2

First-Order Term

The first-order correction is from (3.60) with .k = 1,      1  nˆ + . ψ1,2 = − i exp −iωt 2 t 0

    †   1 1 aˆ + aˆ dt1 |ψ(0)〉 . exp iωt1 nˆ + exp −iωt1 nˆ + √ 2 2 2ω (4.38)

By means of some simple algebraic operations, we can write   

  i ωt  exp −iωt nˆ . ψ1,2 = − √ exp −i 2 2ω  t  



 exp iωt1 nˆ aˆ † + aˆ exp −iωt1 nˆ dt1 |Ψ(0)〉 .

(4.39)

0

Using the transformation rules [15, 17] eiωt nˆ aˆ † e−iωt nˆ = eiωt aˆ † , .

(4.40a)

eiωt nˆ ae ˆ −iωt nˆ = e−iωt a, ˆ

(4.40b)

.

derived from the Hadamard formula (4.12), and integrating with respect to .t1 , we find         2 ωt −iωt n+ ˆ 12 e aˆ † eiωt/2 + ae . ψ1,2 = −i sin ˆ −iωt/2 |ψ(0)〉 . (4.41) 3 2 ω

78

4 Examples of the Matrix Perturbation Method

Inserting the identity operator, written as .Iˆ = eiωt nˆ e−iωt nˆ , just before .|ψ(0)〉, and using again (4.40), we can cast the formula as   . ψ1,2 = −i



   2 ωt e−iωt aˆ † + aˆ e−iωt nˆ |ψ(0)〉 . sin 3 2 ω

(4.42)

To further analyze the problem, we need to specify some initial condition; first, we analyze when said state is a number state, and then, when it is a coherent state. A Number State |m〉 as Initial State We consider as initial state .|ψ(0)〉 a number state .|m〉; in this case, the first-order correction to the wave function becomes      √   √ 2 ωt −imωt −iωt  |m |m e e m − 1〉 . ψ1,2 = −i sin m + 1 + 1〉 + . 2 ω3 (4.43) If we use the coordinate representation of the eigenfunctions of the harmonic oscillator, Eq. (4.34), we get  ψ1,2 (x) = −i

.

    √ √ ωt 2 −imωt −iωt e m ϕ (x) . sin m + 1 ϕ (x) + e m+1 m−1 2 ω3 (4.44)

The complete wave function to first order is .

        (1) = ψ (0) + λ ψ1,2 ψ    1 |m〉 = exp −iωt m + 2      √ √ 2 ωt −imωt −iωt |m |m e e m − 1〉 , m + 1 + 1〉 + sin − iλ 2 ω3 (4.45)

which is not normalized. We can use the formula (3.62) to find the normalization constant, N1 = 

1

.

1 + λ2 2(2m+1) ω3

sin2

ωt  ; 2

(4.46)

4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term

thus, the normalized wave function to first order is        (1)  . Ψ = N1 ψ (0) + λ ψ1,2 ;

79

(4.47)

  it is simple to verify that .Ψ (1)  = 1. The energy mean value .E¯ can be calculated as       E¯ = ψ (1)  Hˆ ψ (1)             = ψ (0)  + λ ψ1,2  Hˆ 0 + λHˆ P ψ (0) + λ ψ1,2 ;

.

(4.48)

using (4.30), (4.31), (4.45), and after some algebra, we obtain  

2 1 (2m + 1)2 ¯ .E = ω m + exp (−iωt) exp (iωt) − 1 ; − λ2 2 2 4ω since we are to first order, we discard the terms in .λ2 , and we can write   1 ¯ .E ≈ ω m + . 2

(4.49)

(4.50)

Hence, the energy correction to first order is zero, as with the standard timeindependent perturbation theory. A Coherent State |α〉 as Initial State When the initial states is a coherent state .|α〉, we can use (4.14) to write   . ψ1,2 = −i



   2 ωt −iωt † |α(t)〉 . sin a ˆ + a ˆ e 2 ω3

(4.51)

We know, by definition, that .aˆ |α〉 = α |α〉, and we also know that [15] aˆ |α〉 = α |α〉 , .   α∗ ∂ † |α〉 ; aˆ |α〉 = + ∂α 2 .

(4.52a) (4.52b)

thus,   . ψ1,2 = −i



    2 ωt α ∗ (t) ∂ −iωt sin e + + α(t) |α(t)〉 . 2 ∂α(t) 2 ω3

(4.53)

80

4 Examples of the Matrix Perturbation Method

Going now to the configuration representation, we get  ψ1,2 (x, t) = −i

.

  √  2 ωt −i ωt 2 e sin 2ωx − α(t) + α ψ (0) (x, t), 2 ω3

(4.54)

where .α(t) = exp(−iωt)α, and .ψ (0) (x, t) is given in Eq. (4.37).

4.2.2.3

Second-Order Term

The second-order term can be calculated by using (3.60) with .k = 2,   †   t    t1   1 aˆ + aˆ 1 2  dt1 dt2 exp iωt1 nˆ + . ψ1,3 = (−i) exp −iωt nˆ + √ 2 2 2ω 0

0

     †     1 1 1 aˆ + aˆ |ψ(0)〉 . exp −iωt2 nˆ + exp iωt2 nˆ + exp −iωt1 nˆ + √ 2 2 2 2ω (4.55) Performing the same steps as in the first-order correction, we arrive to ωt

.

−i 2   ψ1,3 = − e 2ω3

    ωt 1 − iωt − e−iωt + 2 sin2 e−iωt aˆ †2 + 2aˆ † aˆ + eiωt aˆ 2 2

× e−iωt nˆ |ψ(0)〉 .

(4.56)

A Number State |m〉 as Initial State In this case .|ψ(0)〉 = |m〉, and applying the operators in (4.56), we get   −iωt m+ 21

   −iωt 2 ωt |m〉 1 − iωt − e + 4m sin 2 2ω3    2 ωt e−iωt (m + 1)(m + 2) |m + 2〉 +2 sin 2   (4.57) +eiωt m(m − 1) |m − 2〉 .

  e . ψ1,3 = −

A Coherent State |α〉 as Initial State We consider now a coherent state as initial state, and we obtain

4.2 The Harmonic Oscillator Perturbed by a Linear Anharmonic Term

81

      1 1 −i ωt  −iωt −i ωt 2 ωt  2 2 e 1 − iωt − e +e . ψ1,3 = − sin 2 ω3 2     ∂ ∂ α ∗ (t) α ∗ (t) 2 −iωt iωt 2 |α(t)〉 . α(t) + e α (t) +2 e + + 2 ∂α(t) 2 ∂α(t) (4.58) In the coordinate representation, we get   ψ1,3 (x, t) = −3 + iωt + eitω + 2e−itω + e−itω 2x 2 ω − 1 [cos(ωt) − 1]      3 √ tω + 2ω xe−2itω −1 + eitω α(t) + 8 sin4 α 2 (t) ψ (0) (x, t), 2 (4.59)

.

where .α(t) = exp(−iωt)α, and .ψ (0) (x, t) is given in Eq. (4.37).

4.2.2.4

The Perturbative Solution Up to Second Order

Finally, we have an analytic expression for the perturbative solution up to second order, ψ (2) (x, t) = ψ (0) (x, t) + λψ1,2 (x, t) + λ2 ψ1,3 (x, t),

.

(4.60)

where .ψ (0) (x, t) is given in Eq. (4.37), .ψ1,2 (x, t) is given by Eq. (4.54), and .ψ1,3 (x, t) is given by Eq. (4.59). We now proceed to compare this approximated solution, obtained with the Matrix Perturbation Method, with the exact solution (4.27).

4.2.3 Comparison Between the Exact and the Approximated Solutions 4.2.3.1

The Exact and the Approximated Solutions for ω = 1, λ = 0.01, and α = 4 at Different Times

We can see in Fig. 4.3 that for such a small value of .λ = 0.01 the coincidence of the exact and the approximated solutions is excellent. We also notice that the convergence regime does not seem to depend on the amplitude .α of the initial coherent state.

82

4 Examples of the Matrix Perturbation Method

Fig. 4.3 Comparison between the exact and the approximated solutions, Eqs. (4.27) and (4.60), respectively. The square absolute value of both functions is plotted as function of x, for (a) .t = 0.0, (b) .t = 1.0, (c) .t = 10.0, and (d) .t = 100.0. The parameters utilized are .ω = 1, .λ = 0.01, and .α = 4

4.2.3.2

The Exact and the Approximated Solutions for ω = 1, λ = 0.01, and α = 10 at Different Times

We present now the numerical results for a greater value of .α; we go from .α = 4 to .α = 10, in order to show that even in those circumstances the approximation provided by the matrix method is good. We also let time run to analyze what happens with the temporal evolution; the results are clear, as long as the smallness requirement of .λ is preserved, the approximate solution is good. All these aspects are presented in Fig. 4.4.

4.2.3.3

The Exact and the Approximated Solutions for ω = 1, λ = 0.1, and α = 10 at Different Times

It stands to reason that if the value of lambda is not really small, the approximation provided by the matrix method will fail. This is shown in Fig. 4.5; as time passes, the exact solution and the approximate solution begin to differ fundamentally.

4.3 The Harmonic Oscillator Plus a Cubic Potential

83

Fig. 4.4 Comparison between the exact and the approximated solutions, Eqs. (4.27) and (4.60), respectively. The square absolute value of both functions is plotted as function of x, for (a) .t = 0.0, (b) .t = 1.0, (c) .t = 10.0, and (d) .t = 100.0. The parameters utilized are .ω = 1, .λ = 0.01, and .α = 10

4.2.3.4

Does the Coincidence Regime Depend on the Initial Condition?

It seems that indeed the coincidence of the exact and the approximated solutions depends in the value .α of the initial coherent state (Fig. 4.6).

4.3 The Harmonic Oscillator Plus a Cubic Potential Several examples have been presented in the articles where the Matrix Perturbation Method has been developed. In [25], we present a harmonic oscillator perturbed by a quadratic term. In [26], we present the method for the Lindblad master equation and it is applied to the problem of a lossy cavity filled with a Kerr medium; finally, in [27], we show the case of binary waveguide array. In this section, in honor of the clarity and completeness, we present the case of a harmonic oscillator perturbed by a cubic potential, .x 3 .

84

4 Examples of the Matrix Perturbation Method

Fig. 4.5 Comparison between the exact and the approximated solutions, Eqs. (4.27) and (4.60), respectively. The square absolute value of both functions is plotted as function of x, for (a) .t = 0.0, (b) .t = 1.0, (c) .t = 10.0, and (d) .t = 100.0. The parameters utilized are .ω = 1, .λ = 0.1, and .α = 10

The Hamiltonian of a harmonic oscillator perturbed by a cubic term is ω2 2 1 xˆ + λxˆ 3 . Hˆ = pˆ 2 + 2 2

.

(4.61)

We consider that .λ ⪡ 1 is the perturbation parameter, that 1 ω2 2 xˆ Hˆ 0 = pˆ 2 + 2 2

.

(4.62)

is the non-perturbed Hamiltonian, and that Hˆ p = xˆ 3

.

is the perturbation.

(4.63)

4.3 The Harmonic Oscillator Plus a Cubic Potential

85

Fig. 4.6 Comparison between the exact and the approximated solutions, Eqs. (4.27) and (4.60), respectively. The square absolute value of both functions is plotted as function of x, for (a) .α = 1, (b) .α = 4, (c) .α = 10, and (d) .α = 20, always at .t = 10.0. The parameters utilized are .ω = 1 and .λ = 0.1

4.3.1 First Order The correction to first order is given by expression (3.60) with .k = 1,   −i Hˆ 0 t . ψ1,2 = −ie

t

ˆ

ˆ

ei H0 t1 xˆ 3 e−i H0 t1 dt1 |ψ(0)〉 ;

(4.64)

0

we write the position operator in terms of the creation and annihilation operators,  xˆ = √1 aˆ † + aˆ , use that

.

2ω

ˆ

ˆ

ei H0 t aˆ † e−i H0 t = aˆ † eiωt ,

.

ˆ

ˆ

ei H0 t ae ˆ −i H0 t = ae ˆ −iωt

which is derived from the Hadamard formula (4.12), to arrive to .

         ψ1,2 = √ 1 9 1 − eitω aˆ + 1 − e3itω aˆ 3 − 9 1 − e−itω aˆ †2 aˆ 6 2 ω5/2

(4.65)

86

4 Examples of the Matrix Perturbation Method

         + 9 1 − eitω aˆ † aˆ 2 − 9 1 − e−itω aˆ † − 1 − e−3itω aˆ †3 exp −it Hˆ 0 |ψ(0)〉 . (4.66) The complete solution to first order is then .

        (1) = ψ (0) + λ ψ1,2 , ψ

(4.67)

  being .ψ (0) the solution of the unperturbed harmonic oscillator, Eq. (4.32).

4.3.1.1

A Number State |m〉 as Initial State

To go further, we must choose an initial state; in order to compare with the standard time-independent perturbation theory, first we choose an eigenstate of the harmonic oscillator, i.e., a number state, .|ψ(0)〉 = |n〉. In this case, the solution to first order is        1 itω i 1 e− 2 itω − e 2 . ψ1,2 = − √ exp −iωt m + 2 6 2 ω5/2  itω   × e 2 eitω + e2itω + 1 m(m − 1)(m − 2) |m − 3〉 √ √ 1 m m |m − 1〉 + 9e− 2 itω (m + 1) m + 1 |m + 1〉    5 (m + 1)(m + 2)(m + 3) |m + 3〉 . +e− 2 itω eitω + e2itω + 1 +9e

itω 2

(4.68) The first-order solution is not normalized, and we have affirmed that one of the advantages of the Matrix Perturbation Method is that at each step the normalization constant can be easily found using the general expression, (3.62). Thus, we proceed to find the normalization constant for the wave function to first order when the initial state is a number state,      λ2 (2n + 1)  2 2 3ωt n N1 = 1 + + n + 6 sin 2 18ω5 1   −   2 ωt . −9 n2 + n + 1 sin2 2

.

(4.69)

The complete normalized solution to first order is then .

        (1) = N1 ψ (0) + λ ψ1,2 , Ψ

(4.70)

4.3 The Harmonic Oscillator Plus a Cubic Potential

87

  being .ψ (0) the solution of the unperturbed harmonic oscillator, Eq. (4.32).

4.3.1.2

A Coherent State |α〉 as Initial State

To emphasize the possibilities of the Matrix perturbation Method, we considered now a coherent state .|ψ(0)〉 = |α〉, as initial state, and we get .

        (1) = N1 ψ (0) + λ ψ1,2 Ψ   ωt  = N1 e−i 2 αe−iωt      ωt   αe−i 2 ωt  −iωt 2 −3iωt − iωt 2 α e + λN1 √ sin − 1 − 18ie αe 2 6 2 ω5/2

−iωt n ∞   −i ωt |α|2 ! αe e 2 1 − e3iωt e− 2 n(n − 1)(n − 2) |n〉 + λN1 √ √ 6α 3 2 ω5/2 n! n=0   ∞ −iωt n   2 ! αe 3i ωt − |α|2 e n n + α 2 e−iωt |n〉 , − λN1 √ sin √ 2 α 2 ω5/2 n! n=0

(4.71) with the normalization constant   λ2  8 α cos(6ωt) + 9α 4 α 2 + 2 cos(4ωt) N1 = 1 − 5 36ω     − 2α 8 − 11α 6 − 45α 4 − 99α 2 − 6 cos(3ωt) − 63α 4 α 2 − 3 cos(2ωt)

.

  − 1 2 − 9 7α 4 + 58α 2 + α 6 + 9 cos(ωt) + α 8 + 163α 6 + 612α 4 + 738α 2 + 87 . (4.72) We can write this wave function in the coordinate representation x, i.e., ψ (1) (x, t) = N1

.



   x|ψ (0) + λ x|ψ1,2 ,

(4.73)

where by definition .

and to first order we get

|x〉 =

1 π

x2

e 2 e− 1/4

aˆ 2† 2

√    2x ,

(4.74)

88

ψ

.

4 Examples of the Matrix Perturbation Method

(1)

  √ x2 α N1 ωt −iωt −2iωt − − 1+e + 2xαe (x, t) = 1/4 exp −i 2 2 2 π       1 1+λ √ α 3 e−3iωt − 1 + 9α e−iωt − 1 6 2 ω5/2   3   √  √ + e−3iωt − 1 2x − αe−iωt 2x − αe−iωt − 3 e−3iωt − 1   √  2x − αe−iωt + 9α 2 e−2iωt 1 − eiωt 2    √  − 9α 1 − e−iωt 2x − αe−iωt + 9αe−iωt 1 − e−iωt   √   −iωt −iωt −9 1−e 2x − αe .

(4.75)

4.3.2 Second Order The second-order term is given again by (3.60), now with .k = 2,   2 −i Hˆ 0 t . ψ1,3 = (−i) e

 t 0

t1

ˆ

ˆ

ˆ

ˆ

ei H0 t1 xˆ 3 e−i H0 t1 ei H0 t2 xˆ 3 e−i H0 t2 dt2 dt1 |ψ(0)〉 ,

0

(4.76) where .Hˆ 0 is given by (4.62), and .xˆ = first-order case .

  ψ1,3 = −

√1 2ω

†  aˆ + aˆ ; proceeding similarly to the

1  g0 + g1 a 2 + g2 a 4 + g3 a 6 + g4 a † a + g5 a †3 a + g6 a †5 a 288ω5

+g7 a †2 a 2 + g8 a †4 a 2 + g9 a † a 3 + g10 a †3 a 3 + g11 a †2 a 4  ˆ +g12 a † a 5 + g13 a †2 + g14 a †4 + g15 a †6 e−it H0 |ψ(0)〉 , where g0 = − 396itω − 324e−itω − 24e−3itω + 348   g1 =108e−itω 3e2itω − 2e3itω − 1

.

2    eitω + e2itω + 1 g2 = − 36 −1 + eitω 2  g3 = − 2 −1 + e3itω

(4.77)

4.3 The Harmonic Oscillator Plus a Cubic Potential

89

  g4 =36 −6itω − 9e−itω − 9eitω − 2e−3itω + 20  2   11eitω + 2e2itω + 2 g5 = − 18e−3itω −1 + eitω g6 =36e−itω − 9e−4itω − 27   g7 =36 15itω − 18eitω − e−3itω + 19 g8 = − 36eitω − 18e−2itω + 54  2   11eitω + 2e2itω + 2 g9 = − 18e−itω −1 + eitω  2 g10 = − 4e−3itω −1 + e3itω g11 =54e2itω − 36e3itω − 18 g12 =36e3itω − 27e4itω − 9   g13 =108 3e−itω − e−3itω − 2   g14 =36 e−itω + e−3itω − e−4itω − 1  2 g15 = − 2e−6itω −1 + e3itω .

4.3.3 Initial Condition Equal to a Number State ˆ As the number states are eigenfunctions √ of the non-perturbed Hamiltonian .H0 , also √ aˆ |m〉 = m |m − 1〉, and .aˆ † |m〉 = m + 1 |m + 1〉, it is to evaluate (4.77) when .|ψ(0)〉 = |m〉, and we get .



1



−iωt m+ 2   e {g0 . ψ1,3 = − 288ω5 √ √ + m − 1 m [g1 + g9 (m − 2) + g11 (m − 3)(m − 2)] |m − 2〉 √ √ √ √ + m − 3 m − 2 m − 1 m [g2 + g12 (m − 4)] |m − 4〉 √ √ √ √ √ √ +g3 m − 5 m − 4 m − 3 m − 2 m − 1 m|m − 6〉

+m [g4 + g7 (m − 1) + g10 (m − 2)(m − 1)] |m〉 √ √ + m + 1 m + 2 [mg5 + g8 m(m − 1) + g13 ] |m + 2〉 √ √ √ √ + m + 1 m + 2 m + 3 m + 4 [g6 m + g14 ] |m + 4〉

90

4 Examples of the Matrix Perturbation Method

 √ √ √ √ √ √ +g15 m + 1 m + 2 m + 3 m + 4 m + 5 m + 6|m + 6〉 . (4.78) The results for the wave function and for the energy coincide exactly with the ones obtained by the traditional time-independent perturbative method [2, 3, 11, 12, 28]. To verify all the details of the calculations involved in this section, we invite the reader to examine the Appendix 2 of reference [27].

4.4 The Repulsive Quadratic Potential Plus a Linear Term 4.4.1 Exact Solution of the Repulsive Quadratic Potential The Hamiltonian of the repulsive harmonic oscillator is  1 2 pˆ − ω2 xˆ 2 , Hˆ = 2

.

(4.79)

d where .pˆ = −i dx and .xˆ = x are the momentum and position operators, respectively, and .ω is a strictly positive parameter that represents the strength of the repulsive potential. As the Hamiltonian is time independent, the formal solution of the corresponding time-dependent Schrödinger equation is

.

  i  |ψ (t)〉 = exp − pˆ 2 − ω2 xˆ 2 t |ψ (0)〉 , 2

(4.80)

being .|ψ (0)〉 the initial state. Introducing the harmonic oscillator ladder operators [2, 3, 11, 12]  1  † aˆ + aˆ , .x ˆ=√ 2ω

 pˆ = i

 ω † aˆ − aˆ , 2

the formal solution reads as    π ωt i π 2 |ψ(0)〉 . . |ψ(t)〉 = exp e 2 aˆ − e−i 2 aˆ †2 2

(4.81)

(4.82)

  ˆ ) = exp 1 ξ ∗ aˆ 2 − ξ aˆ †2 We identify the propagator with the squeeze operator .S(ξ 2 [15, 17, 29–31] with a time-dependent squeeze amplitude .r = ωt and a squeeze phase equal to .φ = −π/2. Thus, it is found that (4.80) can be written with a squeeze operator with a purely imaginary parameter .ξ = −iωt, i.e., .

|ψ(t)〉 = Sˆ (−iωt) |ψ(0)〉 .

(4.83)

4.4 The Repulsive Quadratic Potential Plus a Linear Term

91

If the system starts in a coherent state .|ψ(0)〉 = |β〉, it evolves into a squeezed coherent state [15–17] .

|−iωt, β〉 = Sˆ (−iωt) |β〉 .

(4.84)

Using the known representation of the squeezed coherent states in terms of the Hermite polynomials (expression (3.134), page 92 of the book of Vogel et al. [15])   ∞    n/2  β 1 ν ν∗ 2 ! 1 1 2 |β| − β , Hn √ . |ξ, β〉 = √ exp − √ μ 2 μ 2μν n! 2μ n=0

(4.85) where μ = cosh |ξ |,

.

ν = exp (iφ) sinh |ξ |,

(4.86)

we arrive to   |β|2 1 i 2 exp − . |−iωt, β〉 = √ + tanh(ωt)β 2 2 cosh (ωt)  " n/2  ∞ ! i 1 i β |n〉 , − tanh (ωt) Hn √ 2 sinh (2ωt) n!

(4.87)

n=0

where .|n〉 are the eigenfunctions of the harmonic oscillator. We can get the coordinate representation of the state ψβ (x, t) = 〈x|−iωt, β〉

.

using the explicit representation of the Hermite–Gauss functions and the Mehler formula for the product of two Hermite polynomials [24, 32], ψβ (x, t) =

.

 ω 1/4 π

  √  |β|2 σ ∗ (t)  2 2ω β 1 2 + x− β + ωx exp − , √ 2 σ (t) 2σ (t) σ (t) (4.88)

where, for the sake of simplicity, we have introduced σ (t) = cosh (ωt) + i sinh (ωt) .

.

(4.89)

This wave function is, of course, solution of the Schrödinger equation with the Hamiltonian (4.79); it is normalized at all times and, as must reduce, when .t = 0, it reduces to the one of the initial coherent state.

92

4 Examples of the Matrix Perturbation Method

Fig. 4.7 Probability density of the repulsive harmonic oscillator with a coherent state as initial state. Eq. (4.88). The parameters are .ω = 1 and .β = 1, 6. We observe that the solution is a Gaussian function that is repelled to the positive part of the X axis and evolves following a parabolic trajectory. (a) .β = 1. (b) .β = 6

In Fig. 4.7, we plot the probability density for the repulsive harmonic oscillator for .ω = 1 and two values of .β, 1 and 6, when the initial state is a coherent state of amplitude .β. It can be appreciated that the solution of the repulsive oscillator for an initial coherent state exhibits a curved amplitude squeezed behavior in the positive X axis direction and that the squeezing curvature is proportional to the amplitude .β of the initial coherent state. The solution is always a Gaussian curve that is repelled from the origin in the positive direction of the X axis and evolves following a parabolic trajectory. From all the previous development, it is easy to see that the exact solution of the repulsive oscillator is ψβ (x, t) + eiη ψ−β (x, t) ψcat (x, t) =    , 2 −2|β| 2 1+e cos η

.

(4.90)

when it is initially prepared in a Schrödinger cat state .

|β〉 + eiη |−β〉 |ψ(0)〉 =   , 2 2 1 + e−2|β| cos η

(4.91)

where .η is an arbitrary phase. Figure 4.8 shows the result associated with a cat state with .β =6, .ω = 1, and .η = 0. We can observe that the same parabolic pattern repeats itself in both X directions. On the other hand, one can notice that the degree of curvature of the solution for .β = 6 is more remarkable than for .β = 1 (coherent state); these results suggest that it is possible to control the degree of squeezing by increasing or decreasing the value of .β. The above argument not only applies to the case of a coherent state but also

4.4 The Repulsive Quadratic Potential Plus a Linear Term

93

Fig. 4.8 When the initial state is a cat case, Eq. (4.90), with .ω = 1, .β = 6.0, and .η = 0

for the coherent superposition; this is clearly reflected in Fig. 4.8, but with the main difference that the squeeze curvature starts to divide in two curves with opposite directions when .β takes large enough values. At first sight, one would expect the quantum interference effects characteristic of Schrödinger cat states; however, this is not the case since what we really have is a superposition of squeezed coherent states. The situation could change if one evaluates the system under the initial condition

.

|ψ(0)〉 =

Sˆ † (−iωt) |β〉 + eiφ Sˆ † (−iωt) |−β〉 ,    2 2 1 + cos(φ)e−2|β|

(4.92)

when the interference is recovered.

4.4.2 Exact Solution to the Quadratic Repulsive Potential Plus a Linear Term In an analogous procedure as the one of Sect. 4.4.1, we have also explored the situation of the quadratic repulsive potential with a linear term. The Schrödinger equation associated with this system reads as

94

4 Examples of the Matrix Perturbation Method

i

.

 d 1 2 |Ψ(t)〉 = pˆ − ω2 xˆ 2 + λxˆ |Ψ(t)〉 , dt 2

(4.93)

where the perturbation strength of the linear term is quantified by the dimensionless scale parameter .λ; furthermore, we have used the notation .Ψ instead of .ψ to distinguish the solutions of this system from those obtained without the linear term. Now, using the ladder operators (4.81), we may rewrite the previous Schrödinger equation as    d ω 2 λ  †2 † |Ψ(t)〉 = − |Ψ(t)〉 . aˆ + aˆ .i aˆ + aˆ +√ dt 2 2ω

(4.94)

The second term transformation  on the  right side is fully removed by applying the  λ † † ∗ ˆ ˆ √ |Ψ(t)〉, where .D(α) = exp α aˆ − α aˆ is the Glauber .|φ(t)〉 = D 3 2ω

ˆ displacement operator [13, 14]. Indeed, the operator .D(α) acts as a displacement † † † ˆ ˆ ˆ ˆ = aˆ † + α ∗ ; then, upon .aˆ and .aˆ as .D (α)aˆ D(α) = aˆ + α and .D (α)aˆ † D(α) Eq. (4.94) becomes   d ω 2 λ2 †2 |φ(t)〉 |φ(t)〉 , .i = − aˆ + aˆ + dt 2 2ω2

(4.95)

2

λ which leads to the repulsive oscillator shifted by a factor . 2ω 2 . Integrating (4.95) with respect to t and then transforming back to .|Ψ(t)〉 gives us

      λ λ λ2 † ˆ ˆ ˆ |Ψ(0)〉 . S (−iωt) D √ . |Ψ(t)〉 = exp −i t D √ 2ω2 2ω3 2ω3

(4.96)

Inserting the identity operator .Iˆ = Sˆ † (−iωt) Sˆ (−iωt) before the initial state, using the relations ξ † Sˆ (ξ ) aˆ Sˆ † (ξ ) = aˆ cosh (|ξ |) + aˆ sinh (|ξ |) , . |ξ |

(4.97a)

ξ∗ Sˆ (ξ ) aˆ † Sˆ † (ξ ) = aˆ † cosh (|ξ |) + aˆ sinh (|ξ |) , |ξ |

(4.97b)

.

and the displacement operator identity .

we can write

 ∗  ηγ − η∗ γ ˆ D (η + γ ) , Dˆ (η) Dˆ (γ ) = exp 2

(4.98)

4.4 The Repulsive Quadratic Potential Plus a Linear Term

95

    2 λ λ ˆ . |Ψ(t)〉 = exp i [sinh(ωt) − ωt] D √ [1 − σ (t)] Sˆ (−iωt) |Ψ(0)〉 . 2ω3 2ω3 (4.99) If we consider now the particular case when the initial state is a coherent state, let ˆ |0〉, we can write the exact solution as say .|β〉 = D(β) .

    2   Ψβ (t) = exp i λ [sinh(ωt) − ωt] Dˆ √ λ [1 − σ (t)] Sˆ (−iωt) D(β) ˆ |0〉. 2ω3 2ω3 (4.100)

ˆ |0〉 is the squeezed coherent state .|−iωt, β〉 [15, 17], which The state .Sˆ (−iωt) D(β) can also be written as [15] ˆ ' )Sˆ (−iωt) |0〉 |−iωt, β〉 = D(β

(4.101)

β ' = cosh(ωt)β + i sinh(ωt)β ∗ ;

(4.102)

.

with .

thus,  2      λ λ  ˆ ˆ ' )Sˆ (−iωt) |0〉. . Ψβ (t) = exp i [sinh(ωt) − ωt] D √ [1 − σ (t)] D(β 2ω3 2ω3 (4.103) Using again the fact that the product of two displacement operators is again a displacement operator, with another parameter and multiplied by an exponential, expression (4.98), we arrive to .

  Ψβ (t) = exp [f1 (t)] |−iωt, f2 (t)〉 ,

(4.104)

where f1 (t) =

.

iλ2 iλ {Im(β) [cosh(ωt) − 1] + Re(β) sinh(ωt)} , [sinh(ωt) − ωt] − √ 2ω3 2ω3/2 (4.105)

λ ∗ σ (t) − 1 , f2 (t) = β + √ 2ω

.

(4.106)

and .|ξ, β〉 is the standard notation for the coherent squeezed state with parameters β and .ξ ; the real and imaginary parts of a complex number z are denoted by .Re(z) and .Im (z), respectively.

.

96

4 Examples of the Matrix Perturbation Method

Fig. 4.9 Probability density of the repulsive harmonic oscillator with a linear term and a coherent state as initial state, Eq. (4.107). The parameters are .ω = 1, .λ = 1, and .β = 1, 6. We observe that the solution is again a Gaussian function that is repelled to the positive part of the X axis and evolves following a parabolic trajectory; however, in this case the repulsion is diminished by the linear term. (a) .β = 1. (b) .β = 6

Considering again the representation of the coherent squeezed states in terms of the Hermite polynomials, Eq. (4.85), and after a lot of algebra, we arrive to the final expression λ2 σ ∗ (t) − 1 λ x − i 2t .Ψβ (x, t) = ψβ (x, t) exp σ (t) ω 2ω  ⎧ √ ⎫ ⎨ 8βλω3/2 − 2λ2 [cosh(ωt) − 1] + iλ2 sinh(ωt) ⎬ × exp , ⎩ ⎭ 2ω3 σ (t) 

(4.107) where .ψβ (x, t) is the solution (4.88) of the repulsive oscillator defined by the Hamiltonian (4.79). This wave function is solution of the Schrödinger equation (4.93), it is normalized at all times, and when .t = 0, it reduces to the one of the initial coherent state. In Fig. 4.9, we plot the probability density for the perturbed repulsive harmonic oscillator, i.e., the repulsive quadratic potential plus the linear term, for .ω = 1, .λ = 1 and two values of .β, 1 and 6, when the initial state is a coherent state of amplitude .β. We observe a similar behavior to the case when there is no perturbation, but now the repulsion is attenuated by the linear term. Now, if initially .|Ψ(0)〉 = |Cat〉, one can easily check that ΨCat (x, t) =

.

Ψβ (x, t) + eiφ Ψ−β (x, t)  . 2 2 + 2 cos(φ)e−2|β|

(4.108)

4.4 The Repulsive Quadratic Potential Plus a Linear Term

97

4.4.3 Perturbative Solution Let us proceed to find the approximated solution of the linear anharmonic repulsive oscillator through the

 Matrix Perturbation Method. To do so, we

assume  that ˆ 0 = − ω aˆ 2 + aˆ †2 is the unperturbed part and that .Hˆ p = √1 aˆ + aˆ † is the .H 2 2ω perturbation, as is written in Eq. (4.94).

4.4.3.1

Zero-Order Correction

According to the main result of the Matrix Perturbation Method, expression (3.60), the zero-order correction is     ˆ 0 |ψ (0)〉 , . ψ1,1 = exp −it H (4.109)   that in this case as .exp −it Hˆ 0 → Sˆ (−iωt) reduces to .

  ψ1,1 = Sˆ (−iωt) |ψ (0)〉 .

(4.110)

This result is obvious, since it is just the exact solution of the repulsive quadratic potential.

4.4.3.2

First-Order Correction

Using now (3.60) with .n = 1, .Hˆ p = .

√1 2ω

 aˆ + aˆ † and identifying

  exp −it Hˆ 0 → Sˆ (−iωt) ,

the first-order correction is   i ˆ . ψ1,2 = − √ S(−iωt) 2ω

t

  ˆ Sˆ † (−iωt1 ) aˆ + aˆ † S(−iωt 1 )dt1 |ψ(0)〉 .

0

(4.111) From (4.97a), it is clear that Sˆ † (−iωt) aˆ Sˆ (−iωt) = cosh (ωt) aˆ + i sinh (ωt) aˆ † , .

(4.112a)

Sˆ † (−iωt) aˆ † Sˆ (−iωt) = −i sinh (ωt) aˆ + cosh (ωt) aˆ † ,

(4.112b)

.

98

4 Examples of the Matrix Perturbation Method

and we get .

  ψ1,2 = −i √1 Sˆ (−iωt) 2ω

 t  σ ∗ (t1 ) aˆ + σ (t1 ) aˆ † dt1 ,

(4.113)

0

whereas was defined in (4.89), .σ (t) = cosh (ωt) + i sinh (ωt). Performing the integration, we obtain .

    ψ1,2 = Sˆ (−iωt) ζ (t) aˆ − ζ ∗ (t) aˆ † |ψ (0)〉 ,

(4.114)

where we have introduced ζ (t) =

.

1 − σ (t) . √ 2ω3

(4.115)

For some initial states, it will be more convenient to move the squeezing operator to act directly in the initial ket; to do this, we insert the identity operator written as ˆ † (−iωt) Sˆ (−iωt) and we get .Iˆ = S .

   ψ1,2 = ζ (t) Sˆ (−iωt) aˆ Sˆ † (−iωt) Sˆ (−iωt)  −ζ ∗ (t) Sˆ (−iωt) aˆ † Sˆ † (−iωt) Sˆ (−iωt) |ψ (0)〉 ;

(4.116)

taking the adjoint of (4.112), Sˆ (−iωt) aˆ Sˆ † (−iωt) = cosh (ωt) aˆ − i sinh (ωt) aˆ † , .

(4.117a)

Sˆ (−iωt) aˆ † Sˆ † (−iωt) = i sinh (ωt) aˆ + cosh (ωt) aˆ † ,

(4.117b)

.

and substituting in (4.116), we obtain .

    ψ1,2 = ζ (t) aˆ † − ζ ∗ (t) aˆ Sˆ (−iωt) |ψ (0)〉 .

(4.118)

Note that in (4.114) and (4.118) the ladder operators are interchanged. 4.4.3.3

Second-Order Correction

Following the same steps that in the first-order correction, the second-order correction is  t1  t     1 ˆ  ˆ dt1 dt2 Sˆ † (−iωt1 ) aˆ + aˆ † S(−iωt . ψ1,3 = − S(−iωt) 1) 2ω 0 0   ˆ Sˆ † (−iωt2 ) aˆ + aˆ † S(−iωt (4.119) 2 ) |ψ(0)〉 .

4.4 The Repulsive Quadratic Potential Plus a Linear Term

99

Using the transformations (4.112) and performing the integration, we obtain .

   ψ1,3 = 1 − 1 − iωt + σ (t) − 1 cosh (2ωt) Sˆ (−iωt) |ψ (0)〉 2 2 2ω3  1 ˆ 1 ∗ + S (−iωt) 1 − σ (t) − i sinh (2ωt) aˆ †2 2 2ω3 + [−1 + 2 cosh (ωt) − cosh (2ωt)] aˆ † aˆ   1 + 1 − σ (t) + i sinh (2ωt) aˆ 2 |ψ (0)〉 . 2

(4.120)

As for first-order correction, it will be advantageous for some initial conditions to have the squeeze operator acting directly to the initial state. We do that exactly as in the first-order case, inserting the identity operator as .Iˆ = Sˆ † (−iωt) Sˆ (−iωt) and using Eqs. (4.117), .

   1 1 ψ1,3 = 1 − − iωt + σ (t) − cosh (2ωt) 2 2 2ω3  1 + 1 − σ (t) + i sinh (2ωt) aˆ †2 2 + [−1 + 2 cosh (ωt) − cosh (2ωt)] aˆ † aˆ   1 ∗ + 1 − σ (t) − i sinh (2ωt) aˆ 2 Sˆ (−iωt) |ψ (0)〉 . 2

4.4.3.4

(4.121)

The Solution to Second Order

Of course, the approximated solution to second order is .

4.4.3.5

        ψap = ψ1,1 + λ ψ1,2 + λ2 ψ1,3 .

(4.122)

A Coherent State as Initial State

If the initial state is a coherent state .|β〉, we can write .|β〉 = Dˆ (β) |0〉, and using the definition of a coherent squeezed state as .|ξ, η〉 = Sˆ (ξ ) Dˆ (η) |0〉, we have   ψ1,1 = |−iωt, β〉 ,

(4.123)

    ψ1,2 = ζ (t) aˆ † − ζ ∗ (t) aˆ |−iωt, β〉

(4.124)

.

.

100

4 Examples of the Matrix Perturbation Method

and    1 1 1  − − iωt + σ (t) − cosh (2ωt) . ψ1,3 = 2 2 2ω3  1 + 1 − σ (t) + i sinh (2ωt) aˆ †2 2 + [−1 + 2 cosh (ωt) − cosh (2ωt)] aˆ † aˆ   1 + 1 − σ ∗ (t) − i sinh (2ωt) aˆ 2 |−iωt, β〉 . 2

(4.125)

We can obtain the coordinate representation of this wave functions. First, the coordinate representation of a coherent squeezed state .|ξ, η〉 is ψη,ξ = 〈x|ξ, η〉 = √

.



 ω 1/4 1 μ−ν π

 √ |η|2 ν ∗ η2 ωx 2 η2 νωx 2 2ωηx + − + + − × exp − , 2 2μ 2 2μ (μ − ν) μ−ν μ−ν (4.126) where μ = cosh |ξ |,

.

 ν = exp iφξ sinh |ξ |,

(4.127)

being .|ξ | the magnitude and .φξ the phase of the complex number .ξ , respectively. Thus, the zero-order correction is ψ0 (x, t) = 〈x|−iωt, β〉

.

  √  |β|2 σ ∗ (t)  2 2ωβx 1  ω 1/4 2 + − β + ωx exp − =√ . 2 σ (t) 2σ (t) σ (t) π (4.128)

To obtain the coordinate representation of the first and second-order corrections, we need the coordinate representation of the ladder operators; it is very well known that  aˆ →

.

  1 d ω x+ , 2 ω dx

 aˆ † →

  1 d ω x− . 2 ω dx

(4.129)

4.4 The Repulsive Quadratic Potential Plus a Linear Term

101

Using these operators and (4.128) in (4.124) and (4.125), we find    ψ1 (x, t) = x ψ1,2     √ √ √  ωt 1 − i ωx sinh2 (ωt) ψ0 (x, t) , 2β + ωx sinh2 2 =√ 2 ω3 σ (t) (4.130)

.

and  √    1 1 2 2 6β ψ2 (x, t) = x ψ1,3 = 2ωβx − 2iωt + 4ωx − 1 + 6 4ω3 1 + i sinh (2ωt)     √ √ − 4 2β 2 + ωx 2 + 2 2ωβx − 1 cosh (ωt) + 4i ωx 2 + 2ωβx + 1 sinh (ωt)   √ + −i + 2ωt − 2iωx 2 − 2i 2ωβx sinh (2ωt)    √ + 2β 2 + 2 2ωβx − 3 cosh (2ωt) ψ0 (x, t) . (4.131)

.

Summarizing, when the initial state is a coherent state .|β〉, the approximated solution in the coordinate representation provided by the Matrix Perturbation Method is ψap (x, t) = ψ0 (x, t) + λψ1 (x, t) + λ2 ψ2 (x, t) .

.

4.4.3.6

(4.132)

The Normalized Solution

In all previous developments we consciously avoided introducing the normalization constant, calculated in a general way in Chap. 3; the idea was to expose the Matrix Perturbation Method as clearly as possible. However, of course the normalization constant is important, and below we present the complete results taking it into account. Restricting ourselves to the case when the initial condition is a coherent state, we take .|ψ(0)〉 = |β〉, and proceeding in a completely analogous way to the case in which we do not consider the normalization constant, the perturbative solution up to second-order is  (2)    Ψ (t)      (0) + λ ψ1,2 + λ2 ψ1,3 = . ψ (2) Nβ (t)    β∗ d ∗ |−iωt, β〉 + = |−iωt, β〉 + λ ζ (t)β − ζ (t) dβ 2

102

4 Examples of the Matrix Perturbation Method

 |ζ (t)|2 i ζ 2 (t)β 2 + λ2 − − − sinh(ωt)] + [ωt 2 2 2ω3     d β∗ ζ ∗2 (t) d 2 β ∗2 ∗ d |−iωt, β〉, + β − |ζ (t)|2 β + + + dβ 2 2 dβ 4 dβ 2 (4.133) with the corresponding normalization constant −2 (2) Nβ (t)

 .



2 β sinh(ωt)[ωt − sinh(ωt)] ω9   λ4  4 2 2 2 2 + − 12β + 8ω t + 12 2β − 3 cosh(ωt) 17 + 12β 32ω6     + 4 7 − 4β 4 cosh(2ωt) − 12 2β 2 + 1 cosh(3ωt)   + 4β 4 + 12β 2 + 3 cosh(4ωt) =1+λ

3

−16ωt sinh(ωt)] ;

(4.134)

the subscript .β indicates the initial coherent state, whereas the superscript (2) denotes the order of the approximation. The solution in the coordinate representation is (2)

(2)

Ψβ (x, t) =

.

Nβ (t) 4ω3 [1 + i sinh(2ωt)]

[d0 + d1 sinh(ωt) + d2 cosh(ωt)

+d3 sinh(2ωt) + d4 cosh(2ωt)] ψβ (x, t),

(4.135)

where    √ d0 =4ω3 + 2 2ω3 βλ + λ2 6β 2 − 2iωt + 4ωx 2 + 6 2ωβx − 1 , .      √ d1 =λ −4i 2ω3 β − 4iω2 x + λ2 4i + 4iωx 2 + 4i 2ωβx , .      √ d2 =λ −4 2ω3 β − 4ω2 x + λ2 4 − 8β 2 − 4ωx 2 − 8 2ωβx , .    √ d3 =4iω3 + 2i 2ω3 βλ + λ2 2ωt − 2iωx 2 − 2i 2ωβx − i , .      √ d4 =λ 2 2ω3 β + 4ω2 x + λ2 2β 2 + 2 2ωβx − 3 .

.

(4.136a) (4.136b) (4.136c) (4.136d) (4.136e)

4.4 The Repulsive Quadratic Potential Plus a Linear Term

4.4.3.7

103

A Cat State as Initial State

If the initial state is a cat state, .

|Ψ(0)〉 = 

|β〉 + eiφ |−β〉 2 2 + 2e−2|β|

,

(4.137)

cos(φ)

the approximated solution is given by   (2) (2) (2) (2) ΨCat (x, t) = NCat (t) Ψβ (x, t) + eiφ Ψ−β (x, t)

.

(4.138)

with normalizing constant (2) NCat (t)    1 λ4 4 4 4 2 2 2 3|ζ (t)| +2Re(β ζ (t))−12Re(β ζ (t))+ 6 [ωt− sinh(ωt)] = 2+ 2 ω  3   λ 2 2 × 1+e−2|β| cosh(φ) +4 [ωt − sinh(ωt)] Re(βζ (t))e−2|β| sin(φ) ω    −1/2 4 2 2 2 2 2 −2|β|2 + 2λ |ζ (t)| |β| 3|ζ (t)| − 2Re(β ζ (t)) 1 − e cosh(φ) .

.

(4.139)

4.4.4 Comparison of the Exact and the Perturbative Solutions In order to test the accuracy and the validity of the above perturbative solution, we compare our normalized approximated solution (to second order) Eq. (4.135), with the exact analytic expression (4.107) of Sect. 4.4.2. At a fixed time, we plot the probability densities of each one of the solutions, the exact and the approximated, as functions of the position x. In Fig. 4.10, we consider the case when .ω = 1, β = 1, λ = 0.05; in the left side of the figure, we can see that from .t = 0 to .t = 2 it is almost impossible to distinguish the perturbative solution from the exact one; analyzing the right side of the figure, we conclude that for times after .t = 3 is no longer a good solution. The same behavior is present in Fig. 4.11, where .ω = 1, β = 1, λ = 0.1; but, as expected, the time where the exact and perturbative solutions become different is shorter, as the parameter .λ is larger. Also, if we increase the amplitude of the initial coherent state, .β, the interval of time in which the perturbative solutions is adequate is diminished. In Fig. 4.12, we present the case when .ω = 1, β = 6, λ = 0.05.

104

4 Examples of the Matrix Perturbation Method

Fig. 4.10 Probability densities of the exact (black dotted line) and perturbative (red dashed line) solutions, when .ω = 1, β = 1, λ = 0.05 at two fixed times, .t = 2 and .t = 3, as functions of the position x. (a) .t = 2. (b) .t = 3

Fig. 4.11 Probability densities of the exact (black dotted line) and perturbative (red dashed line) solutions, when .ω = 1, β = 1, λ = 0.1 at two fixed times, .t = 2 and .t = 3, as functions of the position x. (a) .t = 2. (b) .t = 3

Fig. 4.12 Probability densities of the exact (black dotted line) and perturbative (red dashed line) solutions, when .ω = 1, β = 6, λ = 0.05 at two fixed times, .t = 1 and .t = 2, as functions of the position x. (a) .t = 1. (b) .t = 2

References

105

References 1. J.J. Sakurai, Modern Quantum Mechanics - 2. edición (Addison Wesley, Boston, 2011) 2. E. Merzbacher, Quantum Mechanics, 3rd edn. (John Wiley and Sons, Inc., Hoboken, 1998) 3. A. Messiah, Quantum Mechanics (North-Holland Publishing Company, Amsterdam, 1991) 4. J.S. Townsend, A Modern Approach to Quantum Mechanics - 2. edición (University Science Books, 2012) 5. B.C. Hall, Quantum Theory for Mathematicians (Springer, New York, 2013) 6. P. Carruthers, M.M. Nieto, Coherent states and the forced quantum oscillator. Am. J. Phys. 33(7), 537–544 (1965) 7. T.-J. Li, Exact wave functions and coherent states for a forced damped harmonic oscillator. Rep. Math. Phys. 62(2), 157–165 (2008) 8. V.V. Dodonov, V.I. Man’ko, Coherent states and the resonance of a quantum damped oscillator. Phys. Rev. A 20(2), 550–560 (1979) 9. D. Velasco-Martinez, V.G. Ibarra-Sierra, J.C. Sandoval-Santana, J.L. Cardoso, A. Kunold, Unitary approach to the quantum forced harmonic oscillator (2014). arXiv:1409.0236 [quantph]. https://doi.org/10.48550/arXiv.1409.0236 10. G. Messinis, A.D. Ahmed, Innovation, technology diffusion and poverty traps: the role of valuable skills. Working Paper, Victoria University, Melbourne, Australia (2010). https://vuir. vu.edu.au/id/eprint/15934 11. S. Gasiorowicz, Quantum Physics, 2nd edn. (John Wiley and Sons, Inc., Hoboken, 1995) 12. N. Zettili, Quantum Mechanics. Concepts and Applications, 2nd edn. (John Wiley and Sons, Inc., Hoboken, 2009) 13. R.J. Glauber, Coherent and incoherent states of the radiation field. Phys. Rev. A 131(6), 2766– 2788 (1963) 14. R.J. Glauber, The quantum theory of optical coherence. Phys. Rev. 130(6), 2529–2539 (1963) 15. W. Vogel, Quantum Optics (Wiley-VCH, Weinheim, 2006) 16. J.C. Garrison, Quantum Optics (Oxford University Press, Oxford, 2008) 17. C. Gerry, P. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005) 18. R.R. Puri, Mathematical Methods of Quantum Optics (Springer, Berlin, 2001) 19. B. Hall, Lie Groups, Lie Algebras, and Representations (Springer-Verlag GmbH, Berlin, 2015) 20. W.H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons, Hoboken, 1973) 21. F. Soto-Eguibar, H.M. Moya-Cessa, Introduction to Quantum Optics (Rinton Press, Princeton, 2011) 22. M. Fox, Quantum Optics: An Introduction (Oxford University Press, Oxford, 2006) 23. G.B. Arfken, H.J. Weber, F.E. Harris, Mathematical Methods for Physicists (Elsevier, Amsterdam, 2012) 24. M. Abramowitz, I.A. Stegun, R.H. Romer, Handbook of mathematical functions with formulas, graphs, and mathematical tables. Am. J. Phys. 56(10), 958–958 (1988) 25. J. Martinez-Carranza, F. Soto-Eguibar, H. Moya-Cessa, Alternative analysis to perturbation theory in quantum mechanics. Euro. Phys. J. D 66(1) (2012). https://doi.org/10.1140/epjd/ e2011-20654-5 26. B.M. Villegas-Martinez, F. Soto-Eguibar, H.M. Moya-Cessa, Application of perturbation theory to a master equation. Adv. Math. Phys. 2016, 1–7 (2016) 27. B.M. Villegas-Martinez, H.M. Moya-Cessa, F. Soto-Eguibar, Normalization corrections to perturbation theory based on a matrix method. J. Mod. Opt. 65(8), 978–986 (2017) 28. L. Landau, Zur theorie der energieubertragung II. Physik. Z. Sowjet 2, 46–50 (1932) 29. G. Auletta, The Quantum Mechanics Conundrum (Springer International Publishing, Berlin, 2019)

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30. B. Mehmani, A. Aiello, Visualizing the quantum interaction picture in phase space. Euro. J. Phys. 33(5), 1367–1381 (2012) 31. A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publications, Mineola, 2003) 32. W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer London, Limited, London, 2013)

Chapter 5

Applications of the Matrix Perturbation Method

Abstract This chapter provides an explicit expression for the second-order perturbative solution of a single trapped ion at the high intensity regime. Unlike other perturbative schemes, where the ion-laser dynamics have been explored using unitary transformations and the Lamb-Dicke regime, this analysis relies instead on a direct perturbation method that may be implemented in a simple manner and works especially well without resorting to additional approximations. Based on a matrix method and a final normalization of the perturbed solutions, the second-order perturbative analysis supplies the probability to find the ion in its excited state; the resulting perturbative solution renders a high accuracy, comparable to the one based on the small rotation approximation.

5.1 Introduction This chapter presents some important and interesting applications of the Matrix Perturbation Method. The first application explores the interaction between a trapped ion and a laser beam. Second, we analyze the perturbative solution to the Rabi mode. Finally, we study the exact and perturbative solutions of a waveguide array. Trapped ions interacting with laser beams represent elementary quantum optical systems that have gained considerable attention in quantum information, both experimentally and theoretically, due to their potential for realizing quantum computation. These systems are essential for tasks such as preparing non-classical states of the ion’s vibrational motion [1–8], reconstructing quasi-probability distribution functions [9], producing robust quantum gates, and preparing entangled Bell states in quantum computers [10–12], among many others. Although the interaction of a single ion qubit with laser light is a fundamental model for studying such systems, the trapped ion Hamiltonian leads to a Schrödinger equation that is extremely difficult to solve exactly. However, the dynamics of the system can be described theoretically using suitable approximations. For example, the Lamb-Dicke approximation [13] assumes that the ion’s motion is confined to a region much smaller than the laser wavelength. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6_5

107

108

5 Applications of the Matrix Perturbation Method

vibrational rotating wave approximation (RWA) [14] neglects the counter-rotating terms, and the weak excitation regime [15, 16] assumes that the Rabi frequency, proportional to the intensity of the laser field, is much smaller than the vibrational frequency of the ion. However, the weak excitation regime overlooks the case of an intense laser field, which is now important for faster quantum gates [17–19]. In the framework of perturbation techniques [20, 20–29], only a few works have proposed perturbative solutions in the strong excitation regime, each with its own constraints. For instance, Aniello et al. [30] developed a perturbative procedure on the ion-trap Hamiltonian that provides approximate solutions of the system’s evolution operator. However, this approach only works in the LambDicke regime and requires a set of unitary transformations to obtain a diagonal balanced Hamiltonian. This method has also been extended to a system of N equal ions, where the new perturbative parameter is no longer proportional to the laser intensity [30, 31]. Alternatively, Aniello has analyzed the laser-driven trapped ion Hamiltonian through a large and non-trivial perturbative decomposition of the evolution operator in a generalized Magnus expansion [32]. However, this method also focuses on the Lamb-Dicke regime. Despite these theoretical attempts, none of the available perturbation treatments provides a reliable solution in the strong field regime without imposing an auxiliary unitary transformation or a second approximation to the full ion-laser Hamiltonian. Consequently, there is a need for an appropriate theoretical approach to obtain reliable solutions of the Schrödinger equation for a single trapped ion, valid for large laser intensities. In fact, any of these single techniques may yield highly misleading results regarding the exact system dynamics behavior when operating outside the Lamb-Dicke approximation. Therefore, it is necessary to develop an appropriate theoretical approach to obtain reliable solutions of the Schrödinger equation for a single trapped ion, valid for large laser intensities. This prompts us to explore a straightforward alternative to bridge this knowledge gap–the Matrix Perturbation Method. Depending on the free choice of which part of the system represents the unperturbed part and the perturbation, this method generates a dual Dyson series in matrix form. This offers the possibility to analyze the solution of a quantum system in both regimes of the perturbative parameter, a duality that is very convenient for studying the single trapped ion system in the high intensity regime. Additionally, unlike existing literature, the Matrix Perturbation Method is capable of providing normalized solutions at any correction order–a property that distinguishes our approach. Obtaining a general formulation of a normalization constant is not easy in the other methods [33–38]. Furthermore, we explore the Rabi model, which is closely related to the ion-trap system and has been demonstrated to be equivalent under a unitary transformation [8]. Thus, the results obtained in Sect. 5.2 are adapted to this case, and the secondorder solutions are presented. Finally, we introduce the case of a binary waveguide array, which involves optical analogues of quantum systems. For instance, the system of a charged particle hopping on an infinite linear chain driven by an electric field can be mimicked by the light propagation in a binary waveguide array [39, 40]. This optical structure

5.2 Trapped Ion Hamiltonian

109

has also demonstrated its potential to emulate other typical phenomena of quantum mechanics, such as optical Bloch oscillations [41], discrete spatial solitons [42], quantum walks [43], discrete Fourier transforms [44], and parity-time symmetry [45], among others. This chapter is organized as follows: In Sect. 5.2, we solve the Schrödinger equation perturbatively for a single trapped ion up to the second order. We assess the validity of our results by comparing them with the reported solution based on the small rotation approximation. Since the ion-laser interaction Hamiltonian is similar to the Rabi Hamiltonian [8], we also outline the second-order solution for that problem in Sect. 5.3. In Sect. 5.4, we present a binary waveguide array. We obtain the exact analytic solution and demonstrate the Matrix Perturbation Method corrections up to the second order. In all three applications presented in this chapter, the Matrix Perturbation Method successfully compares with the exact results or with the results previously obtained by other methods, whether perturbative or otherwise.

5.2 Trapped Ion Hamiltonian We now consider a simplified model of a single trapped ion interacting with a classical laser field; the Hamiltonian of this system is [8, 16, 46–50]   δ ˆ (5.1) Hˆ ion = ν nˆ + σˆ z + Ω σˆ + D(iη) + σˆ − Dˆ † (iη) , 2    where .Dˆ (iη) = exp iη aˆ + aˆ † is the Glauber displacement operator [51–56], .aˆ † (.a) ˆ is the ion’s vibrational creation (annihilation) operator, .nˆ = aˆ † aˆ is the number operator, .η is the Lamb-Dicke parameter [57–61], .ν is the vibrational frequency of the ion, .δ = νκ is the laser-ion detuning, .Ω is the Rabi frequency, and .σˆ + = | ↑〉〈↓ | and .σˆ − = | ↓〉〈↑ | are the atomic raising and lowering operators (Pauli matrices) expressed in terms of the excited .|↑〉 = (1, 0) and ground .|↓〉 = (0, 1) states of the two-level ion. Note that the ion-laser coupling strength is the Rabi frequency. We consider that the laser-ion detuning .δ is a multiple integer of the vibrational frequency of the and  ion. The  atomic raising   lowering operators satisfy the commutation relations . σˆ + , σˆ − = σˆ z and . σˆ z , σˆ ± = ±2σˆ ± . The dynamics of a single trapped ion can be studied by solving the timedependent Schrödinger equation .

i

.

d |Ψ(t)〉ion = Hˆ ion |Ψ(t)〉ion . dt

(5.2)

We are interested in solving perturbatively the Schrödinger equation through the Matrix Perturbation Method. Indeed, our perturbative scheme provides us the great flexibility to choose the unperturbed and perturbed parts of the full Hamiltonian (5.1); using this freedom, we can get normalized perturbative solutions for the cases

110

5 Applications of the Matrix Perturbation Method

when the amplitude .Ω of the laser is very small compared with the vibrational frequency of the ion, .Ω ⪡ ν, and vice versa, i.e., when .Ω ⪢ ν. These two approximations correspond to the weak and strong laser intensity regimes, respectively. In addition, the ion-trap system is formally equivalent to the quantum Rabi model, when we consider a certain unitary transformation .Tˆ [8]; therefore, we can perform the transformation .|φ(t)〉Rabi = Tˆ |Ψ(t)〉ion and also get perturbative solutions of the quantum Rabi model for the weak and strong coupling regimes; this will be done in Sect. 5.3.

5.2.1 High Intensity Regime 5.2.1.1

First-Order Correction

Let us begin our perturbative analysis by solving the high intensity case .(Ω ⪢ ν). In such a scenario, we must consider that κ Hˆ p = nˆ + σˆ z 2

.

(5.3)

is the perturbation with perturbative parameter .λ = ν/Ω, whereas ˆ Hˆ 0 = σˆ + D(iη) + σˆ − Dˆ † (iη)

.

(5.4)

plays the role of the unperturbed part. If we re-scale time as .τ = Ωt and set .k = 1 into Eqs. (3.60), (3.61), and (3.62), we get the approximate solution to first order .



 |Ψ(τ )〉ion ≈ N (1) (τ ) ψ (0) + λ ψ (1) ,

(5.5)

where .



ˆ (0) = e−i H0 τ |ψ(0)〉 , . ψ τ

 κ  ˆ ˆ (1) −i Hˆ 0 τ ei H0 τ1 nˆ + σˆ z e−i H0 τ1 dτ1 |ψ(0)〉 , . = −ie ψ 2



N (1) (τ )

−2

0

= 1 + 2λ Re







ψ (0) ψ (1) + λ2 ψ (1) ψ (1) ,

(5.6a) (5.6b)

(5.6c)

being .Re (ζ ) the real part of the complex number .ζ . The integral in Eq. (5.6b) requires to calculate the product of exponential operators with .nˆ + κ2 σˆ z . To do this, we first expand in Taylor series the exponential ˆ operator .ei H0 τ1 and split the series in even and odd powers of .Hˆ 0 ,

5.2 Trapped Ion Hamiltonian

ˆ

ei H0 τ1 =

.

111

∞  (−1)n τ 2n 1

n=0

(2n)!

Hˆ 02n + i

∞  (−1)n τ 2n+1 1

n=0

(2n + 1)!

Hˆ 02n+1 ;

(5.7)

one can easily check that .Hˆ 02n = 1ˆ and .Hˆ 02n+1 = Hˆ 0 , then equation Eq. (5.7) becomes ˆ

ei H0 τ1 = cos(τ1 ) + i sin(τ1 )Hˆ 0   ˆ = cos(τ1 ) + i sin(τ1 ) σˆ + D(iη) + σˆ − Dˆ † (iη) .

.

(5.8)

It is also possible to show that    κ  i ˆ ˆ ei H0 τ1 nˆ + σˆ z e−i H0 τ1 = nˆ cos2 (τ1 ) + sin(2τ1 ) Hˆ 0 , nˆ + sin2 (τ1 ) Hˆ 0 nˆ Hˆ 0 2 2 κ κ (5.9) + cos(2τ1 )σˆ z + i sin(2τ1 )Hˆ 0 σˆ z . 2 2     ˆ nˆ = a, ˆ we obtain that As . aˆ † , nˆ = −aˆ † , and . a, .

.

      ˆ Hˆ 0 , nˆ =σˆ + D(iη), nˆ + σˆ − Dˆ † (iη), nˆ     ˆ ˆ ˆ + σˆ − Dˆ † (iη) nˆ − D(iη) nˆ Dˆ † (iη) . nˆ − Dˆ † (iη)nˆ D(iη) =σˆ + D(iη) (5.10)

Using the Hadamard formula (4.12), the expression (5.10) is simplified to .

       ˆ Hˆ 0 , nˆ = − η σˆ + D(iη) + σˆ − Dˆ † (iη) η + i aˆ − aˆ † σˆ z .

(5.11)

Another equation that it is not difficult to prove is     ˆ 0 D(iη) nˆ Dˆ † (iη) 2 † = n+η ˆ σˆ z , +iη a ˆ − a ˆ Hˆ 0 nˆ Hˆ 0 = ˆ 0 Dˆ † (iη)nˆ D(iη)

.

(5.12)

and when this result is substituted in Eq. (5.9), together with Eq. (5.11), yields to  κ  ˆ ˆ ei H0 τ1 nˆ + σˆ z e−i H0 τ1 = nˆ 2     κ iη  + ˆ σˆ D(iη) + σˆ − Dˆ † (iη) η + i aˆ − aˆ † σˆ z − σˆ z sin(2τ1 ) − 2 η     κ + cos (2τ1 ) σˆ z + η η + i aˆ − aˆ † σˆ z sin2 (τ1 ) , (5.13) 2

.

that can be easily integrated to give

112

5 Applications of the Matrix Perturbation Method

τ .

0

   τ    κ  ˆ ˆ ei H0 τ1 nˆ + σˆ z e−i H0 t1 τ1 = 2nˆ + η η + i aˆ − aˆ † σˆ z 2 2     η κ † η + i aˆ − aˆ σˆ z − σˆ z sin (2τ ) − 4 η     iη  + ˆ κ σˆ D(iη) + σˆ − Dˆ † (iη) η + i aˆ − aˆ † σˆ z − σˆ z sin2 (τ ). − η 2 (5.14)

Substituting the above expression in Eq. (5.6b), and after some algebra, the firstorder term is obtained,     κ 

 η (1) =− i cos(τ ) nτ ˆ + [τ − tan(τ )] η+i aˆ − aˆ † σˆ z + tan (τ )σˆ z |ψ(0)〉 . ψ 2 2       η η+i aˆ − aˆ † σˆ z |ψ(0)〉 . − τ sin(τ ) σˆ + Dˆ (iη)+ σˆ − Dˆ † (iη) n+ ˆ 2 (5.15) Now, the normalization constant .N (1) (τ ) of Eq. (5.6c) is obtained doing the inner (1)  product of . ψ with itself, and once it is calculated and being substituted in Eq. (5.5), give us .

|Ψ(τ )〉ion ≈ N (1) (τ ) cos(τ )       λκ η tan (τ ) σˆ z |ψ(0)〉 × 1 − iλτ nˆ − iλ [τ − tan(τ )] η + i aˆ − aˆ † σˆ z − i 2 2   − iN (1) (τ ) sin(τ ) σˆ + Dˆ (iη) + σˆ − Dˆ † (iη)     η2 ητ  † |ψ(0)〉 . aˆ − aˆ σˆ z − iλτ nˆ + × 1+λ 2 2

(5.16)

The above expression is the first-order approximated solution of Eq. (5.2), and the normalization constant is .

 λ2 κ 2  −2  λ2 η 2  2 sin2 (τ ) η + 1 sin2 (τ ) + τ [τ − sin(2τ )] + N (1) (τ ) =1 + 4 4 + λ2 τ 2 〈ψ(0)| nˆ 2 |ψ(0)〉  λ2 η 2  2 sin (τ ) + τ [τ − sin(2τ )] 4       × 〈ψ(0)| 2nˆ − aˆ 2 + aˆ †2 + 2iη aˆ − aˆ † σˆ z |ψ(0)〉 +

5.2 Trapped Ion Hamiltonian

113

λ2 τ η [2τ − sin(2τ )] 4        × 〈ψ(0)| 2ηnˆ + i 2 aˆ − aˆ † nˆ − aˆ + aˆ † σˆ z |ψ(0)〉

+

λ2 κ sin (2τ ) 2      η × 〈ψ(0)| nτ ˆ + [τ − tan(τ )] η + i aˆ − aˆ † σˆ z σˆ z |ψ(0)〉 . 2 (5.17) +

Initial State |ψ(0)〉 = |n〉 |g〉 Let us consider as initial state .|ψ(0)〉 = |n〉 |g〉, which represents n vibrational quanta and the ion in the ground internal state .|g〉; with this initial state the solution to first order is .

(1) |Ψ(τ )〉ion,|n,g〉 ≈ N|n,g〉 (τ ) cos(τ )   η2 λκ × 1 − iλτ n − iλ [τ − tan(τ )] + i tan(τ ) |n〉 |g〉 2 2 √  √ λη (1) N|n,g〉 (τ ) cos(τ ) [τ − tan(τ )] n |n − 1〉 − n + 1 |n + 1〉 |g〉 − 2    η2 (1) |iη; n〉 |e〉 − iN|n,g〉 (τ ) sin(τ ) 1 − iλτ n + 2 √  √ λητ (1) (5.18) N|n,g〉 (τ ) sin(τ ) n |iη; n − 1〉 − n + 1 |iη; n + 1〉 |e〉 , +i 2

where .|α; m〉 ≡ Dˆ (α) |m〉 is a displaced number state [51–56], whereas the (1) normalization constant .N|n,g〉 (τ ) is given by  .

(1)

N|n,g〉 (τ )

−2

=1

     λ2  2  2 4τ n η + 2n + 2 η2 η2 + 2n + 1 + κ 2 sin2 (τ ) 8   λ2 κ    sin(2τ ) 2nτ + η2 [τ − tan(τ )] . +2τ η2 η2 + 4n + 1 [τ − sin(2τ )] − 4 (5.19)

+

114

5 Applications of the Matrix Perturbation Method

Initial State |ψ(0)〉 = |n〉 |e〉 If we suppose an initial condition with the ion in the excited state, .|ψ(0)〉 = |n〉 |e〉, we get .

(1) |Ψ(τ )〉ion,|n,e〉 ≈ N|n,e〉 (τ ) cos(τ )   η2 λκ × 1 − iλτ n − iλ [τ − tan(τ )] − i tan(τ ) |n〉 |e〉 2 2 √  √ λη (1) + N|n,e〉 cos(τ ) [τ −tan(τ )] n |n − 1〉 − n + 1 |n + 1〉 |e〉 2    η2 (1) |−iη; n〉 |g〉 − iN|n,e〉 sin(τ ) 1−iλτ n + 2  √ √ λτ η (1) −i N|n,e〉 sin(τ ) n |−iη; n−1〉− n + 1 |−iη; n+1〉 |g〉 , 2 (5.20)

where the normalization constant is .

      λ2 κ (1) −2 (1) −2 sin(2τ ) 2nτ + η2 [τ − tan(τ )] . N|n,e〉 = N|n,g〉 + 2

(5.21)

Initial State |ψ(0)〉 = |iα〉 |e〉 As a third initial condition, we assume .|ψ(0)〉 = |iα〉 |e〉, which means an initial vibrational coherent state for the field and the excited state for the ion; then Eq. (5.16) becomes .

with

(1) |Ψ(τ )〉ion,|iα,e〉 = N|iα,e〉 (τ ) cos(τ )   λη λκ tan(τ ) |iα〉 |e〉 × 1+i (α − η) [τ − tan(τ )] − i 2 2   ∂ λ (1) − i N|iα,e〉 (τ ) cos(τ ) {2ατ − η [τ − tan(τ )]} + α |iα〉 |e〉 ∂α 2   ∂ λτ (1) N + α − η |i (α − η)〉 |g〉 − (τ ) (2α − η) sin(τ ) 2 |iα,e〉 ∂ (α − η)   αηλτ (1) |i (α − η)〉 |g〉 , − iN|iα,e〉 (τ ) sin(τ ) 1 − i (5.22) 2

5.2 Trapped Ion Hamiltonian

.

115

−2  (1) =1 N|iα,e〉 (τ )    λ2 κ 2     λ2  + αη (α−η) η−4ατ 2 +2ατ 2 α α 2 +1 −η η2 +1 sin2 (τ ) 2 4     λ2  2  η (2α−η)2 +1 cos(2τ )−2τ η (2α − η) η2 +2α (α − η)+1 sin(2τ ) − 8  λ2 η 2     λ2 κ + sin(2τ ) 2α 2 τ − (2α − η) η [τ − tan(τ )] + η2 + 1 2τ 2 + 1 , 4 8 (5.23) +

where we have used the coherent states properties [51–55, 62] aˆ |α〉 = α |α〉 , .   α∗ ∂ ∂ † ∗ aˆ |α〉 = + α |α〉 , . + ∂α α ∂α ∗ .

Dˆ † (α) aˆ † Dˆ (α) = aˆ † + α ∗ .

(5.24a) (5.24b) (5.24c)

For simplicity, we have taken .α as a real number, but all calculation can be done with an .α complex.

5.2.1.2

Second-Order Correction

Let us now turn to get the second-order perturbative solution by using again the general solution Eqs. (3.60), (3.61), and (3.62), but now running .k = 2; we obtain the equation .





 |Ψ(τ )〉ion ≈ N (2) (τ ) ψ (0) + λ ψ (1) + λ2 ψ (2) ,

(5.25)

where .



ˆ (2) = − e−i H0 τ ψ

 κ  ˆ ˆ ei H0 τ1 nˆ + σˆ z e−i H0 τ1 2 0 τ1  κ  ˆ ˆ ei H0 τ2 nˆ + σˆ z e−i H0 τ2 dτ2 dτ1 |ψ(0)〉 , 2 0 τ

(5.26)

and  .

N (2) (τ )

−2

 



 =1 + λ2 2Re ψ (1) ψ (2) + ψ (1) ψ (1) 



+ 2λ3 Re ψ (1) ψ (2) + λ4 ψ (2) ψ (2) .

(5.27)

116

5 Applications of the Matrix Perturbation Method

We insert Eq. (5.13) into (5.26), and after integration, one gets

.

 

e−i Hˆ 0 τ    (2) = − 4 sin4 (τ ) + sin2 (2τ ) Oˆ 7 + Oˆ 5 − Oˆ 2 + Oˆ 8 sin(2τ ) ψ 4   + 4Oˆ 4 + 2Oˆ 6 − Oˆ 3 sin2 (τ )    +τ −2Oˆ 8 − 2Oˆ 5 + 2 cos(2τ )Oˆ 2 + sin(2τ ) Oˆ 3 − 2Oˆ 6   −τ 2 2Oˆ 1 + Oˆ 3 |ψ(0)〉 ,

(5.28)

with    η  Oˆ 1 =nˆ 2 + nˆ η + i aˆ − aˆ † σˆ z , . (5.29a) 2   η4  η2  2 η2 † + − ˆ ˆ ˆ ˆ ˆ aˆ + aˆ †2 − − O2 =i σ D (iη) + σ D (iη) − η2 nˆ + 4 4 4          2 κ η κ ηκ η aˆ − aˆ † nˆ + η2 σˆ z + nˆ + +i aˆ − aˆ † σˆ z σˆ z , . −i 2 2 4 4 (5.29b)

.

      η2  Oˆ 3 = (5.29c) 4nˆ − aˆ 2 + aˆ †2 + η2 + 1 + iη aˆ − aˆ † nˆ + η2 σˆ z , . 2     η κ (5.29d) Oˆ 4 = nˆ η + i aˆ − aˆ † σˆ z − σˆ z , . 4 η 2      η2  κ † † + − ˆ ˆ ˆ ˆ ˆ ˆ η + i aˆ − aˆ σˆ z − σˆ z O5 =O2 + σ D (iη) + σ D (iη) i 2 η       η     η2 κ † † † −i η + i aˆ − aˆ σˆ z − σˆ z η + i aˆ − aˆ σˆ z − aˆ + aˆ σˆ z , . 4 η 2 (5.29e)      κ Oˆ 6 = (5.29f) 2nˆ + η η + i aˆ − aˆ † σˆ z σˆ z , . 4      κ Oˆ 7 = (5.29g) κ σˆ z − η η + i aˆ − aˆ † σˆ z σˆ z , . 8   Oˆ 8 = − 2i σˆ+ Dˆ (iη) + σˆ− Dˆ † (iη) Oˆ 7 . (5.29h) ˆ

Applying the unperturbed evolution operator .e−i H0 τ and after some algebraic manipulation, we arrive to

5.2 Trapped Ion Hamiltonian

117



 τ (2) = − cos(τ ) . ψ 8      × 4τ nˆ 2 + η2 [τ − tan(τ )] 6nˆ + η2 + 1 − η2 [τ − tan(τ )] aˆ 2 + aˆ †2       η2 σˆ+ Dˆ (iη)+ σˆ− Dˆ † (iη) 2nˆ + η2 + 1 cos(τ ) τ −tan(τ ) 1+τ 2 8       η2 σˆ+ Dˆ (iη) + σˆ− Dˆ † (iη) aˆ 2 + aˆ †2 + i cos(τ ) τ − tan(τ ) 1 + τ 2 8      κη cos(τ ) [τ − tan(τ )] σˆ+ Dˆ (iη) − σˆ− Dˆ † (iη) η + i aˆ − aˆ † σˆ z −i 4       3 η cos(τ ) τ − tan(τ ) 1 + τ 2 σˆ+ Dˆ (iη) − σˆ− Dˆ † (iη) aˆ − aˆ † − 4       η σˆ+ Dˆ (iη) − σˆ− Dˆ † (iη) aˆ + aˆ † − cos(τ ) τ − tan(τ ) 1 − τ 2 4      ητ −i cos(τ ) [τ − tan(τ )] aˆ 2nˆ + η2 − 1 − aˆ † 2nˆ + η2 + 1 σˆ z 4   κ2 − i cos(τ ) [τ − tan(τ )] σˆ+ Dˆ (iη) + σˆ− Dˆ † (iη) 8    ητ 2 sin(τ ) σˆ+ Dˆ (iη) − σˆ− Dˆ † (iη) aˆ − aˆ † nˆ + 2     τ2 + i sin(τ ) σˆ+ Dˆ (iη) + σˆ− Dˆ † (iη) nˆ nˆ + η2 2   κ − 4nˆ σˆ z + κ τ sin(τ ) |ψ(0)〉 . (5.30) 8 −i

Once the second-order term has been calculated, the normalization constant and the complete solution can be obtained using Eqs. (5.27) and (5.25). Initial State |ψ(0)〉 = |iα〉 |e〉 For practical purpose, let us consider as initial condition the field in a coherent state and the ion in the excited state, i.e., .|ψ(0)〉 = |iα〉 |e〉; the solution to second order is given by .

|Ψ(τ )〉ion,|iα,e〉     λη κ (2) = N|iα,e〉 (τ ) cos(τ ) 1 + i (α − η) [τ − tan(τ )] − tan(τ ) 2 η

118

5 Applications of the Matrix Perturbation Method

 ∂ ∂2 2 + 2α −λ +α ∂α ∂ 2α   ∂ λ (2) |iα〉 |e〉 +α − i N|iα,e〉 {cos(τ ) [(2α − η) τ +η tan(τ )]− 2iλF2 (α, τ )} 2 ∂α    λ (2) ∂ +α−η + − N|iα,e〉 (τ ) [(2α − η) τ sin(τ ) + 2iλF5 (α, τ )] 2 ∂ (α − η)   ∂ ∂2 (2) − iλ2 N|iα,e〉 (τ )F6 (α, τ ) 2 + (α − η)2 + 2 (α − η) ∂ (α − η) ∂ (α − η)     αηλτ (2) (5.31) − λ2 F4 (α, τ ) |i (α − η)〉 |g〉 , − iN|iα,e〉 (τ ) sin(τ ) 1 − i 2 

2

(2) (2) N|iα,e〉 (τ )F1 (α, τ ) |iα〉 |e〉 − λ2 N|iα,e〉 (τ )F3 (α, τ )

where −2  τ 2 λ2  2 (2) (τ ) =1− αη η + 1 sin2 (τ ) N|iα,e〉 4     4 2 2 + λ F1 (α, τ )+F2 (α, τ )+2 F32 (α, τ )+F62 (α, τ ) +F42 (α, τ ) + F52 (α, τ )   + α 2 λ4 2 [F1 (α, τ ) + 2F3 (α, τ )] F3 (α, τ ) + F22 (α, τ )

 .

+ α 3 λ4 [2F2 (α, τ ) + αF3 (α, τ )] F3 (α, τ ) + 2αλ4 [F1 (α, τ ) + 2F3 (α, τ )] F2 (α, τ ) + 2 (α − η) λ4 [2F6 (α, τ ) − F4 (α, τ )] F5 (α, τ )   + (α − η)2 λ4 F52 (α, τ ) − 2 [F4 (α, τ ) − 2F6 (α, τ )] F6 (α, τ ) + (α − η)3 λ4 [2F5 (α, τ ) + (α − η) F6 (α, τ )] F6 (α, τ ) ,

(5.32)

with      κ2  τη 2 2 .F1 (α, τ ) = cos (τ ) [τ − tan (τ )] α η + η + 1 (η − 2α) + tan (τ ) , . 8 η (5.33a)     τ F2 (α, τ ) = cos (τ ) 2ατ + η 3ηα − 2α 2 − η2 + 1 4 [τ − tan (τ )] + 2ακ tan (τ )} , .   τ F3 (α, τ ) = cos (τ ) 4τ α 2 + η (η − 4α) [τ − tan (τ )] , . 8  η F4 (α, τ ) = cos (τ ) τ 2 [αη (α + η) + 3α − η] tan (τ ) 8

(5.33b) (5.33c)

5.2 Trapped Ion Hamiltonian



119



  κ2 − 2 (ακ + η − α) + η α + 2 + 1 [τ − tan (τ )] , . (5.33d) η  cos (τ )  F5 (α, τ ) = η (αη+κ +1) [τ −tan (τ )]−(2α−η) (αη+1) τ 2 tan (τ ) , . 4 (5.33e)     cos (τ ) 2 (5.33f) τ 4α (η − α) − η2 tan (τ ) + η2 [τ − tan (τ )] . F6 (α, τ ) = 8 2

5.2.1.3

Comparison of the Perturbative Solution with the Small Rotation Approximation Solution

In order to verify the validity and accuracy of our perturbative solution, we calculate the probability to find the ion in its excited state .Pe (τ ) = |〈e|Ψ(τ )〉|2 and compare it with the expression Pe (τ )SRA

.

    1 η 2 2  high  = sin τ χ 1 + exp −2 α − 2 2       η 2 high high cos τ 2 − χ sin 2τ χ − α− , 2

(5.34)

where .χ high = −λ2 η2 /2 in the case of high intensity regime. The probability .Pe (τ )SRA was calculated using the small rotation approximation (SRA) by ZuñigaSegundo et al. in reference [63] and is established in Eqs. (16) and (18) of their article. Also, it should be emphasized that does not exist other solution to compare, since ours is the first perturbative solution in this regime, i.e., without consider the Lamb-Dicke approximation, which implies supposing that .η ⪡ 1. Using Eq. (5.31) to calculate .Pe (τ ) yields the following expression  2    (2) Pe (τ ) = N|iα,e〉 (τ ) − λ2 cos (τ ) 2α 2 F3 (α, τ ) + αF2 (α, τ ) + 2F1 (α, τ )   + λ4 F12 (α, τ ) + F22 (α, τ ) + 2F32 (α, τ ) + cos2 (τ )     + λ2 G21 (α, τ ) + α 2 + 1 G22 (α, τ ) − 2αG1 (α, τ ) G2 (α, τ )  + 2λ4 α 3 F2 (α, τ ) F3 (α, τ )

.

2    (2) F22 (α, τ ) + 2F3 (α, τ ) [F1 (α, τ ) + 2F3 (α, τ )] + α 2 λ4 N|iα,e〉 (τ ) 2    (2) 2F2 (α, τ ) [F1 (α, τ ) + 2F3 (α, τ )] + α 3 F32 (α, τ ) , + αλ4 N|iα,e〉 (τ ) (5.35)

120

5 Applications of the Matrix Perturbation Method

Fig. 5.1 Plot of the probability to find the ion in its excited state as a function of .τ from our perturbative solution, Eq. (5.35), and from the small rotation approximation, Eq. (5.34); the black dotted line represents the small rotation approximation solution and the red dashed line denotes the perturbative solution. The parameters used are .κ = 0, .η = 0.1, .α = 4, and .λ = 0.1, 0.2, 0.3, and .0.4. (a) Parameters .α = 4.0, η = 0.1, κ = 0.0, λ = 0.1. (b) Parameters .α = 4.0, η = 0.1, κ = 0.0, λ = 0.2. (c) Parameters .α = 4.0, η = 0.1, κ = 0.0, λ = 0.3. (d) Parameters .α = 4.0, η = 0.1, κ = 0.0, λ = 0.4

where G1 (α, τ ) =

  κ η cos (τ ) (α − η) [τ − tan (τ )] − tan (τ ) , . 2 η

(5.36a)

G2 (α, τ ) =

cos (τ ) {2ατ − η [τ − tan (τ )]} . 2

(5.36b)

.

The .Pe (τ ) obtained in our approach, Eq. (5.35), and that from the small rotation approximation, Eq. (5.34), are plotted in Fig. 5.1, for several values of the perturbative parameter .λ. It is clear that the perturbative results, indicated by the red dashed line, are sufficiently accurate to reproduce the small rotation approximation solution, denoted by the black dotted line, provided the condition .λτ ⪡ 1 holds. Otherwise, it is logical to expect that when .λτ ⪢ 1 a substantial difference will arise between the perturbative solution and the small rotation approximation solution.

5.3 Perturbative Solution for the Rabi Model

121

In this section, we have shown that by using the Matrix Perturbation Method allowed us to produce an appropriate perturbative treatment of a single trapped ion interacting with a laser field in the high intensity regime. In fact, two features make the procedure presented here particularly attractive; the first one is that the Matrix Perturbation Method makes the problem amenable to be solved directly without falling in the use of auxiliary unitary transformations or the Lamb-Dicke approximation. The second appealing aspect lies in the fact that the perturbative solution is capable of reproducing, with high accuracy and self-consistency, the already known results of the small rotation approximation. Albeit in a technical level the normalized perturbative solutions appear too long and complicated, such solutions generate reliable results; this is a noticeable advantage over others perturbative approaches, with rather cumbersome algebra and based on special assumptions that produce too complicated solutions that could lead to imprecise results. We believe that our work could pave the way to address a more complicated, but physically important, generalization, which is the study of the perturbative solution of a system of N equal ions. Indeed, the case of many ions is a very promising scenario, which could be useful to study the trapped ion quantum logic operation outside the Lamb-Dicke regime.

5.3 Perturbative Solution for the Rabi Model As we already mentioned in the Introduction, it has been shown in [8] that the iontrap system is formally equivalent to the quantum Rabi model when we consider the unitary transformation   1  1  Tˆ = √ Dˆ † (iη/2) + Dˆ (iη/2) Iˆ + √ Dˆ † (iη/2) − Dˆ (iη/2) σˆ z 2 2 2 2   1 (5.37) + √ σˆ + Dˆ (iη/2) − σˆ − Dˆ † (iη/2) , 2

.

and we take .ν → ω, .Ω → ω20 and . ην 2 → g; when this transformation is done, we get the transformed Schrödinger equation i

.

d |φ(t)〉Rabi = Hˆ Rabi |φ(t)〉Rabi , dt

(5.38)

where    ωκ  +  ω0 σˆ z + ig aˆ − aˆ † σˆ + + σˆ − − σˆ + σˆ − Hˆ Rabi = Tˆ Hˆ ion Tˆ † = ωnˆ + 2 2 ηg . + (5.39) 2

.

122

5 Applications of the Matrix Perturbation Method

Hence, we can perform the transformation .|φ(t)〉Rabi = Tˆ |Ψ(t)〉ion and also get perturbative solutions of the quantum Rabi model for the weak .(g/ω) ⪡ 1 and strong .(g/ω) ⪢ 1 coupling regime. As the low intensity regime case has been considered extensively [1–8], we focus now on the high intensity regime. Taking advantage of this equivalence and applying .Tˆ to expression (5.31), one gets 

.

  η cos(τ ) + iλ G1 (α, τ ) − G2 (α, τ ) 2   2 η η − λ2 F1 (α, τ ) + F2 (α, τ ) + F3 (α, τ ) |−〉 |γ 〉 4 2    η η2 (2) 2 + N|iα,e〉 (τ ) iλ F4 (α, τ ) + F5 (α, τ ) − F6 (α, τ ) − i sin (τ ) 2 4    η − λ G3 (α, τ ) − G4 (α, τ ) |+〉 |γ 〉 2   ∂  (2) + γ |−〉 |γ 〉 + N|iα,e〉 (τ ) λ2 [F2 (α, τ ) + ηF3 (α, τ )] + iλG2 (α, τ ) ∂γ   ∂  (2) + γ |+〉 |γ 〉 − N|iα,e〉 (τ ) λG4 (α, τ ) + iλ2 [F5 (α, τ ) − ηF6 (α, τ )] ∂γ  2  ∂ ∂ (2) + γ 2 |γ 〉 + λ2 N|iα,e〉 (τ ) [F3 (α, τ ) |−〉 − iF6 (α, τ ) |+〉] 2 + 2γ ∂γ ∂ γ (5.40)

|φ(τ )〉Rabi ≈

(2) −N|iα,e〉 (τ )

which is the second-order perturbative solution for the Rabi model and where |±〉 = √1 [|g〉 ± |e〉], .γ = i (α − η/2), .G3 (α, τ ) = τ αη 2 sin (τ ), and .G4 (α, τ ) = 2 τ − η) sin (2α (τ ). 2

.

5.4 The Binary Waveguide Array We consider now light propagation in a binary waveguide array [39, 40]. The linear behavior of light propagation over this kind of waveguide arrangement is governed by the infinite system of ordinary differential equations i

.

dEn (z) = ω (−1)n En (z) + α [En+1 (z) + En−1 (z)] , dz n = −∞, . . . , −2, −1, 0, 1, 2, . . . , ∞,

(5.41)

where .En (z) represents the amplitude of light field confined in the n-th waveguide, z the longitudinal propagation distance, .2ω the mismatch propagation constant, and

5.4 The Binary Waveguide Array

123

α the hopping rate between two adjacent waveguides. Physically, equation (5.41) describes the effective evanescent field coupling between the nearest-neighbor waveguide interactions. As this system is linear, it is enough to consider as initial conditions only one of the waveguides with a field different from zero; i.e., we will assume that the initial conditions for the system Eq. (5.41) is

.

En (z) = δn,m ,

n = −∞, . . . , −2, −1, 0, 1, 2, . . . , ∞,

.

(5.42)

where .δ is the Kronecker delta [64], and m is an arbitrary fixed integer that indicates which guide has a non-null field; in what follows, we will usually take that number m equal to zero, indicating that only the “central”is excited. It has been shown [65] that this system Eq. (5.41) can be associated with the Schrödinger type equation i

.

d |ψ(z)〉 = Hˆ |ψ(z)〉 , dz

(5.43)

where   Hˆ = ω (−1)nˆ + α Vˆ + Vˆ † ,

.

(5.44)

being .(−1)nˆ the parity operator, and .Vˆ and .Vˆ † peculiar ladder operators, called Susskind-Glogower operators, defined as [66–68]

.

Vˆ =

∞ 

|n〉 〈n + 1| ,

Vˆ † =

n=−∞

∞ 

|n + 1〉 〈n| ,

(5.45)

n=−∞

acting over the vector space generated by the complete and orthonormal set {|n〉 ; n = −∞, . . . , ∞}. The .|n〉 represents the classical analogue of Fock states, and the previous operators, down and up, act over them as

.

.

Vˆ |n〉 = |n − 1〉 ,

Vˆ † |n〉 = |n + 1〉 ;

(5.46)

something similar to the annihilation and creation operators in quantum optics, but without the square root term that characterizes them. Note that if the solution of the Schrödinger type equation, Eq. (5.43), is written in terms of the waveguide number basis [39] as

.

|ψ(z)〉 =

∞ 

En (z) |n〉 ,

(5.47)

n=−∞

and if this proposal is substituted into Eq. (5.43), the infinite system given by Eq. (5.41) is recovered. Hence, if we find the solution .|ψ(z)〉 of the Schrödinger type equation (5.43), we also have the field in the binary waveguide array as

124

5 Applications of the Matrix Perturbation Method

En (z) = 〈n|ψ(z)〉 ,

n = −∞, . . . , −2, −1, 0, 1, 2, . . . , ∞.

.

(5.48)

We have a few words to say about the conditions of the equation of the Schrödinger type equation (5.43); if the initial condition (5.42) is chosen for the array, then the equivalent initial condition for Eq. (5.43) is .

|ψ (z = 0)〉 = |m〉 .

(5.49)

5.4.1 Exact Solution It has been proven [69] that the exact solution of the system (5.41) is 1 .En (z) = π

π



 sin[Ω(φ)z] dφ, cos (nφ) cos[Ω(φ)z] − i[2α cos φ + (−1) ω] Ω(φ) n

0

n = −∞, . . . , −2, −1, 0, 1, 2, . . . , ∞.

(5.50)

with Ω(φ) =



.

ω2 + 4α 2 cos2 φ.

(5.51)

5.4.2 Small Rotation Approximated Solution Despite the above equation represents the exact solution for the amplitude of the light field, .En = 〈n|ψ(z)〉, a substantial alternative approximated solution is derived in [69]; under the condition .α ⪡ ω, and performing the unitary transformation    α nˆ ˆ † ˆ ˆ V + V .R = exp , the following solution is found, (−1) 2ω En (z) = (−1)

.

 Jr

n(n−1) 2

 2 2  s ω +α z (−1) i exp −i (−1) ω r=−∞ s=−∞ ∞ 

∞ 

sr r

   α α2 α z Js Jn+2r+s , ω ω ω

(5.52)

where .Jn (ζ ) are the Bessel functions of the first kind of order n [52]; in fact, the unitary transformation .Rˆ constitutes the small rotation approximation [70].

5.4 The Binary Waveguide Array

125

5.4.3 Matrix Perturbative Solution We will solve now, approximately, the Schrödinger type equation Eq. (5.43), with the formalism presented in Chap. 3. First, we need to regard the variable z as the time; this is intuitively reasonable, if we want to describe the optical field propagation on the waveguide array in evolutionary terms. Besides, as we already said, since the problem is linear, for simplicity and convenience, we consider the initial condition .|ψ(0)〉 = |m〉, which corresponds to a single excitation in the m-th guide. Considering that .ω (−1)nˆ is the unperturbed part of the Hamiltonian (5.44), ˆ + Vˆ † is the perturbation, and .α is the perturbation parameter, we can write the .V fundamental equations (3.60) of Chap. 3 as |ψ1,k+1 〉 = (−i)k e−iω(−1)

.

nˆ z





z z1

zk−1

... 0

0

nˆ z 1

eiω(−1)

  nˆ Vˆ + Vˆ † e−iω(−1) z1

0

  nˆ Vˆ + Vˆ † e−iω(−1) z2 e   nˆ nˆ . . . eiω(−1) zk Vˆ + Vˆ † e−iω(−1) zk |m〉 dzk−1 . . . dz2 dz1 , iω(−1)nˆ z2

k = 1, 2, 3, . . . .

(5.53)

Since the normalization constant in each of the steps depends only on the wave functions, we do not need to do any adaptation for this particular case; however, to facilitate the reading of the calculations that we will make later, we reproduce here the formula (3.62) of Chap. 3, 1

Nk = 

.

1+2

k  j =1

k     λj Re ψ1,1 |ψ1,j +1 + λl+j ψ1,l+1 |ψ1,j +1



, k=1, 2, 3, . . . .

l,j =1

(5.54)

5.4.3.1

Zero-Order Perturbative Solution

The zero-order solution is trivial, since for .α = 0 the Schrödinger type equation reduces to   d ψ1,1 (z) = ω (−1)nˆ ψ1,1 (z) , (5.55) .i dz  with the initial condition . ψ1,1 (0) = |m〉; thus,

126

5 Applications of the Matrix Perturbation Method

.

   ψ1,1 (z) = exp −iωz (−1)nˆ |m〉   = exp −iωz (−1)m |m〉 .

(5.56)

This wave function is already normalized. The corresponding fields .En (z) in the waveguides are given by .〈n|ψ1,1 (z)〉; i.e.,   En (z) = exp −iω(−1)n z En (0),

n = −∞, . . . , −2, −1, 0, 1, 2, . . . , ∞, (5.57)

.

where .En (0) is the initial field in guide n; if, as supposed, only the guide m is excited, we have  exp [−iω(−1)m z] , n = m, .En (z) = (5.58) 0, n /= m. A logical and expected result, since only the m guide has been excited and there is no interaction between the guides. This result is also obvious if one notices that at zero order, the system of equations (5.41) reduces to i

.

dEn (z) = ω (−1)n En (z), dz

n = −∞, . . . , −2, −1, 0, 1, 2, . . . , ∞,

(5.59)

which together with the assumed initial condition, clearly has as a solution (5.58).

5.4.3.2

First-Order Perturbative Solution

We will go through the calculations in this subsection in enough detail to serve as an illustration of the techniques to calculate the wave functions and the normalization factors. The calculations of the second and third-order corrections are completely analogous but more cumbersome and do not add anything new to the understanding of the method, and therefore, we will briefly present the calculations and state the results. All the calculations can be made with any computer algebra system, as Mathematica, Maple, or Mathcad, among many others; in our case, we use Mathematica and Mathcad.  The first-order correction, . ψ1,2 , requires the use of Eq. (5.53) with .k = 1 to give  −iω(−1)nˆ z . ψ1,2 = −ie

z eiω(−1) 0

nˆ z 1



 nˆ Vˆ + Vˆ † e−iω(−1) z1 |m〉 dz1 .

(5.60)

5.4 The Binary Waveguide Array

127

The simplest way to calculate the previous expression is the direct one; indeed, we first have that e−iω(−1)

.

nˆ z 1

|m〉 = e−iω(−1)

mz 1

|m〉 ,

(5.61)

and then     ˆ + Vˆ † e−iω(−1)nˆ z1 |m〉 = Vˆ + Vˆ † e−iω(−1)m z1 |m〉 . V   m = e−iω(−1) z1 Vˆ + Vˆ † |m〉 = e−iω(−1)

mz 1

(|m − 1〉 + |m + 1〉) ,

(5.62)

where in the last line we have used the property (5.46) of the Susskind-Glogower operators. The next step is easy eiω(−1)

.

nˆ z 1

  nˆ m nˆ Vˆ + Vˆ † e−iω(−1) z1 |m〉 = e−iω(−1) z1 eiω(−1) z1 (|m + 1〉 + |m − 1〉) = e−2iω(−1)

mz 1

(|m + 1〉 + |m − 1〉) . (5.63)

The integral over .z1 is trivially done, and z e

.

iω(−1)nˆ z1



 m   i(−1)m −1 + e−2iω(−1) z † −iω(−1)nˆ z1 ˆ ˆ |m〉 dz1 = V +V e 2ω

0

(|m + 1〉 + |m − 1〉) .

(5.64) nˆ

All that remains is to apply the first operator (from left to right), .e−iω(−1) z , to the result obtained, .

 m   i(−1)m −1 + e−2iω(−1) z ψ1,2 = − ie−iω(−1)nˆ z (|m + 1〉 + |m − 1〉) 2ω  m   (−1)m −1 + e−2iω(−1) z  −iω(−1)m+1 z m−1 |m + 1〉+e−iω(−1) z |m − 1〉 e = 2ω  m  m (−1) −1 + e−2iω(−1) z iω(−1)m z e = (5.65) (|m + 1〉 + |m − 1〉) . 2ω

Finally, using the fact that the sinus is an odd function, we get .

 ψ1,2 = −i sin (ωz) (|m + 1〉 + |m − 1〉) . ω

(5.66)

128

5 Applications of the Matrix Perturbation Method

Then, the complete wave function to first order is .



  (1) ψ (z) = ψ1,1 + α ψ1,2   sin (ωz) = exp −iωz (−1)m |m〉 − iα (|m + 1〉 + |m − 1〉) . ω

(5.67)

However, this wave function is not normalized, and one of the advantages of our method is that it provides normalization at each step. So, we now proceed to calculate the normalization constant; we use Eq. (5.54) to obtain     N1−2 = 1 + 2αRe ψ1,1 |ψ1,2 + α 2 ψ1,2 |ψ1,2 .

.

(5.68)

We have     sin (ωz) m 〈m| × −i . ψ1,1 |ψ1,2 = exp iωz (−1) (|m + 1〉 + |m − 1〉) = 0, ω (5.69) as the set .{|n〉 ; n = −∞, . . . , ∞} is orthonormal, and 

.



     sin (ωz) sin (ωz) ψ1,2 |ψ1,2 = i (〈m + 1| + 〈m − 1|) −i (|m + 1〉 + |m − 1〉) ω ω =

2 sin2 (ωz) ; ω2

(5.70)

thus, N1 = 

1

.

1+

2α 2 sin2 (ωz) ω2

.

(5.71)

Henceforth, for purposes of this application, we will denote the normalized wave function with a capital psi, such that the normalized first-order solution is  

  sin (ωz) (1) m . Ψ (z) = N1 exp −iωz (−1) |m〉 − iα (|m + 1〉 + |m − 1〉) . ω (5.72) We now turn our attention to the calculation of the electric field in the binary waveguide arrangement. As .En(1) (z) = 〈n|Ψ (1) (z)〉, we find directly that      sin (ωz)  δn,m+1 + δn,m−1 . En(1) (z) = N1 exp −iωz (−1)m δn,m − iα ω (5.73)

.

5.4 The Binary Waveguide Array

5.4.3.3

129

Second-Order Perturbative Solution

The second-order correction, from Eq. (3.61), is .



   (2) Ψ (z) = N2 ψ1,1 + α ψ1,2 + α 2 ψ1,3 ;

(5.74)

  we have already calculated, two  in the previous  subsections, . ψ1,1 and . ψ1,2 , so, we need to calculate . ψ1,3 and .N2 . For . ψ1,3 , we use Eq. (5.53) with .k = 2; i.e.,  −iω(−1)nˆ z . ψ1,3 = −e

z z1 eiω(−1) 0

e

iω(−1)nˆ z2

nˆ z 1



 nˆ Vˆ + Vˆ † e−iω(−1) z1

0

  nˆ Vˆ + Vˆ † e−iω(−1) z2 |m〉 dz2 dz1 .

(5.75)

Again, the easiest method is the direct method, just as in the first-order case. The direct method also has the advantage of being very easily implemented in a computer algebra system. As we explained before, the second and third-order calculations are cumbersome and do not contribute much to the understanding of the problem and the method, so we present directly the results obtained using Mathematica [71]. We get     (−1)m sin (ωz)−ωz exp −iωz (−1)m . ψ1,3 = i (|m + 2〉 + 2 |m〉 + |m − 2〉) , 2ω2 (5.76) and N2−2 = 1 +

.

 3α 4  2 2 1 + 2ω z − 2ωz sin(2ωz) − cos(2ωz) ; 4ω4

(5.77)

such that the wave function normalized to second order is 

  sin (ωz) (2) . Ψ (z) = N2 exp −iωz (−1)m |m〉 − iα (|m + 1〉 + |m − 1〉) ω     m m sin − ωz exp −iωz (ωz) (−1) (−1) +iα 2 (|m + 2〉 + 2 |m〉 + |m − 2〉) . 2ω2 (5.78) The electric field in each of the waveguides of the equivalent array can be calculated (1) using the expression .En (z) = 〈n|Ψ (1) (z)〉, we find directly that

130

(2) .En (z)

5 Applications of the Matrix Perturbation Method

=N2



  exp −iωz (−1)m

   sin (ωz) − ωz exp −iωz (−1)m  δn,m + iα ω2  sin (ωz)  δn,m+1 + δn,m−1 − iα ω     m  sin (ωz) − ωz exp −iωz (−1)m  (−1) 2 + iα δ . + δ n,m+2 n,m−2 2ω2 (5.79) 2 (−1)

5.4.3.4

m

Third-Order Perturbative Solution

For the next order correction, we have 

    (3) . Ψ (z) = N3 ψ1,1 + α ψ1,2 + α 2 ψ1,3 + α 3 ψ1,4 ,

(5.80)

 and we need to calculate . ψ1,4 and .N3 . From (5.53) with .k = 3, we can write  −iω(−1)nˆ z . ψ1,4 = − e

z

z1 dz1

0

z2 dz3 eiω(−1)

dz2 0

nˆ z 1

  nˆ Vˆ + Vˆ † e−iω(−1) z1

0

    nˆ nˆ nˆ nˆ eiω(−1) z2 Vˆ + Vˆ † e−iω(−1) z2 eiω(−1) z3 Vˆ + Vˆ † e−iω(−1) z3 |m〉 , (5.81) and after a long, but direct, algebra calculation, we get  ψ1,3 = i sin (ωz) − ωz cos (ωz) (|m + 3〉 + 3 |m + 1〉 + 3 |m − 1〉 + |m − 3〉) . 2ω3 (5.82) For the normalization factor, we arrive to .

N3−2 =1 +

.

+

 3α 4  2 2 2ω z + 2ωz sin + 3 cos − 3 (2ωz) (2ωz) 4ω4

5α 6 [sin (2ωz) − ωz cos (2ωz)]2 . ω6

(5.83)

5.4 The Binary Waveguide Array

131

The normalized wave function is 

  sin (ωz) (3) . Ψ (z) = N3 exp −iωz (−1)m |m〉 − iα (|m + 1〉 + |m − 1〉) ω    m sin (ωz) − ωz exp −iωz (−1)m 2 (−1) +iα (|m + 2〉 + 2 |m〉 + |m − 2〉) 2ω2  3 sin (ωz) − ωz cos (ωz) iα (|m + 3〉 + 3 |m + 1〉 + 3 |m − 1〉 + |m − 3〉) . 2ω3 (5.84) Finally, we can evaluate the electric field in each guide, finding  En(3) (z) =N3

.

   exp −iωz (−1)m

    sin (ωz) − ωz exp −iωz (−1)m +iα δn,m ω2  3   3α sin (ωz)  +i δn,m+1 + δn,m−1 [sin (ωz) − ωz cos (ωz)] − α ω 2ω3    m  sin (ωz) − ωz exp −iωz (−1)m  2 (−1) + iα δn,m+2 + δn,m−2 2ω2    α3 + i 3 [sin (ωz) − ωz cos (ωz)] δn,m+3 + δn,m−3 . (5.85) 2ω 2 (−1)

m

5.4.4 Comparison of the Perturbative Solution with the Exact Solution and with the Small Rotation Solution In order to illustrate the high degree of accuracy that can be obtained with the thirdorder correction for the amplitude of the electrical field in each of the guides in the binary array, the comparison of the perturbative result (5.85), with the exact solution (5.50), and with the small rotation solution (5.52), is given in Figs. 5.2 and 5.3, using 2 .ω = 0.9. In these figures, we present the intensity distribution .I (z) = |En (z)| for the first four guides, considering two values of the perturbation parameter, .α = 0.1 and .α = 0.3. It is noteworthy that for .α = 0.1 the approximate solution converges to the exact one uniformly even over large propagation distance. On the other hand, for .α = 0.3 both solutions are very similar only for short distances, but we still obtain a good approximation; in fact, the real measure of the perturbation is the product .αz, in same fashion that previous examples on this chapter but with t instead of z.

132

5 Applications of the Matrix Perturbation Method

Fig. 5.2 Field intensity .I (z) = |En (z)|2 versus propagation distance z, using the exact solution (solid black line), the third-order solution (red dashed line) and the small rotation method solution (blue dotted line), with .α = 0.1 and .ω = 0.9, for the first four guides. (a) Guide 0. (b) Guide 1. (c) Guide 2. (d) Guide 3

Moreover, for the previous two values of .α, it becomes evident the improvement provided by the third-order correction with respect to the reported results using the small rotation method. Thus, the assessment of higher-order terms can give us a reliable solution into the system described here. The perturbative solutions of the equations that describe the binary waveguide array, obtained applying the Matrix Perturbation Method, are highly accurate. Furthermore, it is shown that the third-order approximated solution matches exactly the known exact solution, not only for small values of the perturbative parameter .α, but also for large values; in fact, the real measure of the perturbation is the product .αz. On the other hand, it becomes evident the improvement of this method with respect to the reported results using the small rotation method. Therefore, the assessment of higher-order terms can give us a reliable solution into the system described here.

References

133

Fig. 5.3 Field intensity .I (z) = |En (z)|2 versus propagation distance z, using the exact solution (solid black line), third-order solution (red dashed line) and the small rotation method solution (blue dotted line), with .α = 0.3 and .ω = 0.9, for the first four guides. (a) Guide 0. (b) Guide 1. (c) Guide 2. (d) Guide 3

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Chapter 6

The Matrix Perturbation Method for the Lindblad Master Equation

Abstract This chapter presents the Matrix Perturbation Method for the Lindblad master equation, which provides both first- and second-order corrections; to ensure generality, the method is extended to higher orders. The perturbation approach is applied to tackle a particular problem related to a lossy cavity containing a Kerr medium; and the estimation of second-order corrections is performed, and a comparison is made with the known exact analytic solution. This comparative analysis is based on calculating essential quantities such as the Husimi Q-function, the average number of photons, and the distance between density matrices.

6.1 Introduction The dynamics of quantum mechanical systems interacting with an environment is of paramount importance across various branches of physics, including cosmology [1–4], quantum information [5], quantum optics [6], and condensed matter [7], among others [8–10]. Referred to as open quantum systems, these systems exhibit dynamics that elude full description through unitary evolution, which results from their interaction with the environment. The significant role of the environment becomes evident as it impacts irreversible information loss and gives rise to dissipative processes stemming from interactions with an external reservoir [11, 12]. Such interactions deeply affect the overall dynamics of the physical systems, leading to the emergence of mixed states. This implies that the states of the systems are no longer purely determined, but rather represent incomplete knowledge about the system, in contrast to pure states where maximal knowledge is available. In the framework of quantum mechanics, a mixed state is mathematically represented by a density operator or state matrix denoted as .ρ [13]. Particularly, for an open quantum system with Markovian evolution, the state matrix .ρ obeys a master equation, and the most general mathematically valid form of this equation is known as the Lindblad equation [6, 7, 14, 15]. However, obtaining exact solutions to these master equations remains a challenging task [16–18]. As a result, perturbative methods are often employed to address these complexities. Numerous perturbative approaches have been developed © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6_6

137

138

6 The Matrix Perturbation Method for the Lindblad Master Equation

to tackle specific problems modeled by the Lindblad master equation. Some of these problems include the dynamics of a two-level nonlinear quantum system, a singlemode field in a lossy cavity, a two-level atom coupled to a Bose-mode environment, and a single atom coupled to a mode of a lossy cavity [19–21]. Furthermore, studies have demonstrated that even in the presence of decoherence, quasiprobability distribution functions can still be reconstructed in scenarios involving atom–field interactions [22, 23] or laser-trapped ion systems [24]. Hence, the purpose of this chapter is to demonstrate how the matrix perturbation method can be extended to mixed states and applied to treat the Lindblad master equation perturbatively. This perturbative analysis provides a straightforward and effective means of determining the mth-order correction. The chapter is organized into two parts. In the first part, the perturbative method for the Lindblad master equation is developed using super-operator techniques in terms of matrices, with a focus on insights into the first- and second-order corrections, and the extension of this methodology to higher order corrections. In the second part, the method is validated through its application to a specific problem involving a lossy cavity filled with a Kerr medium, a problem for which an exact analytical solution exists. The approximate solution derived from the proposed method is compared with the exact result, and their accuracy is evaluated using the Husimi Q-function, the average photon number, and the measure of the distance between their density matrices.

6.2 Lindblad Master Equation The Lindblad master equation, which describes the interaction between a given system and its environment at zero temperature, is given by Breuer and Petruccione [6], Lindblad [14], Caldeira and Leggett [7], Soto-Eguibar and Moya-Cessa [15] .

  dρ = Sˆ +λLˆ ρ; dt

(6.1)

in this equation, .ρ is the density matrix; the super-operator .Sˆ is (we have set .h¯ = 1) .

  Sˆ ρ = −i Hˆ , ρ ,

(6.2)

ˆ which being .Hˆ the time-independent interaction Hamiltonian; the super-operator .L, describes the interaction of the system with the environment, is given by .

  ˆ = γ 2aρa † − a † aρ − ρa † a , Lρ

(6.3)

6.2 Lindblad Master Equation

139

with a and .a † the usual creation and annihilation operators and .γ the rate at which the system loses energy; and finally, .λ is a perturbation parameter (effectively, the perturbation parameter is .λγ as we consider small .γ ’s). The formal solution of the master equation is 

ρ(t) = e

.

 Sˆ +λLˆ t

(6.4)

ρ(0),

where .ρ(0) is the density matrix of the initial state of the system.

6.2.1 First-Order Correction The first-order correction to the non-perturbed solution of the master equation is obtained by expanding the exponential in Eq. (6.4) using the Taylor series and retaining only the terms up to first order in .λ,  ρ(t) ≈

∞ n  t

.

n=0

n!

ˆn

S +λ

∞ n  n−1  t n=1

n!

 S Lˆ S ˆm

ˆ n−m−1

ρ(0).

(6.5)

m=0

In order to simplify the above expression [25], it becomes necessary to define a triangular .2 × 2 array of super-operators, where the diagonal elements are given by the non-perturbed system and the superior triangle contains the perturbation, Mρ =

.

Sˆ 0

Lˆ ρ. Sˆ

(6.6)

Equation (6.5) can then be written as   ˆ ρ(t) ≈ eS t + λ(eMt )1,2 ρ(0),

.

(6.7)

where .(eMt )1,2 stands for the element (1,2) of the matrix .eMt . Notice that the expression (6.7) conveniently decomposes the first-order approximation into two parts, one involves only the system and the other tells us how the environment affects the system. Consequently, the density matrix can be split into one part concerning the non-perturbed system and a small contribution in terms of .λ referent to firstorder perturbation. As a result, Eq. (6.7) can be written as follows:   ρ(t) ≈ ρ (0) (t) + λ ρ P

.

1,2

where the .2 × 2 perturbed density matrix is given by

,

(6.8)

140

6 The Matrix Perturbation Method for the Lindblad Master Equation

ρ1,1 ρ1,2 . .ρ = ρ2,1 ρ2,2 P

(6.9)

Deriving (6.7) and (6.8) with respect to time and equating the .λ terms, one obtains the differential equation .

  d P (ρ )1,2 = MeMt ρ(0) 1,2 dt

(6.10)

or the equivalent systems of differential equations .

d dt



ρ(0) 0 ρ1,1 ρ1,2 = MeMt . ρ2,1 ρ2,2 0 ρ(0)

(6.11)

This system of equations can be readily solved, and it becomes evident that the differential equation .

d P ρ = Mρ P dt

(6.12)

is also satisfied. Performing a matrix multiplication in the above expression, it can be demonstrated that the first-order solution is related to the second column of the perturbed density matrix, given by .

d dt





ρ1,2 ρ = M 1,2 . ρ2,2 ρ2,2

(6.13)

ˆ

Doing the transformation .φ1,2 = e−St ρ1,2 , one yields .

d ˆ ˆ Sˆ t φ1,2 = e− S t Le ρ(0), dt

(6.14)

and after integrating the above expression with respect to time and transforming back the resulting expression to .ρ1,2 , one obtains ⎛ ˆ

ρ1,2 = eS t ⎝

.

t

⎞ ˆ

ˆ

ˆ S t1 dt 1 ⎠ ρ(0), e− S t1 Le

0

which is the first-order correction term.

(6.15)

6.2 Lindblad Master Equation

141

6.2.2 Second-Order Correction The second-order correction to the non-perturbed solution of the master equation can be obtained by considering the terms in .λ2 in the Taylor series expansion (6.4). Subsequently, the following expression emerges: ρ(t) ≈

∞ n  t

.

n!

n=0

+ λ2

Sˆ n ρ(0) + λ

∞ n  n−1  t n=1

n−1 n−m−1 ∞ n    t n=2

n!

m=1

n!

Sˆ m Lˆ Sˆ n−m−1 ρ(0)

m=0

Sˆ n−m−j −1 Lˆ Sˆ j Lˆ Sˆ m−1 ρ(0).

(6.16)

j =0

In this case, the dimension of the super-operators matrix is increased accordingly, ⎛

Sˆ .Mρ = ⎝ 0 0

⎞ 0 Lˆ ⎠ ρ. Sˆ

Lˆ Sˆ 0

(6.17)

In the previous section, it was demonstrated that the element .(1, 2) of .M n provided all the terms of the first-order correction. For the second order, a similar situation arises: all the relevant information resides in the element .(1, 3) of the new M raised to the power n. Indeed, one obtains   ˆ ρ(t) ≈ eS t + λ(eMt )1,2 + λ2 (eMt )1,3 ρ(0).

(6.18)

.

By following analogous steps as in the first-order case, the density matrix can be expressed as   ρ(t) ≈ ρ (0) (t) + λ ρ (P )

.

1,2

  + λ2 ρ (P )

1,3

(6.19)

,

where the solution of .ρ1,3 will be associated with the third column of the perturbed density matrix represented as ⎞ ⎛ ⎞ ⎛ ρ1,3 ρ d ⎝ 1,3 ⎠ . ρ2,3 = M ⎝ρ2,3 ⎠ . dt ρ3,3 ρ3,3

(6.20)

ˆ

Solving the system of equations through the transformations .φ1,3 = e−St ρ1,3 and ˆ −St .φ2,3 = e ρ2,3 yields the second-order correction term

142

6 The Matrix Perturbation Method for the Lindblad Master Equation

ρ1,3 = e

.

Sˆ t

t t1 0

ˆ

ˆ

ˆ

ˆ

ˆ S t1 e− S t2 Le ˆ S t2 dt2 dt1 ρ(0). e− S t1 Le

(6.21)

0

6.2.3 Higher Orders The generalization of the method for higher order corrections can be directly derived from the outcomes of the first-order and second-order corrections. By following the same steps that led to the expression (6.21), one can define the density matrix for the mth-order correction as   m m       ˆ (0) n P n Mt St .ρ (t) + λ ρ ≈ e + λ e (6.22) ρ(0), 1,n+1

n=1

1,n+1

n=1

where the perturbed density matrix is given by ⎞ ρ1,1 · · · ρ1,m+1 ⎟ ⎜ .. P .. .ρ = ⎝ ... ⎠. . . ρm+1,1 · · · ρm+1,m+1 ⎛

(6.23)

Considering the above, the semi-infinite super-operators array is defined as follows: ⎛ˆ S ⎜0 ⎜ ⎜. ⎜. .M = ⎜ . ⎜. ⎜ .. ⎝

Lˆ · · · 0 Sˆ Lˆ · · · . 0 Sˆ . . . 0 0 .. . 0 .. 0 0

⎞ 0 0⎟ ⎟ .. ⎟ .⎟ ⎟, ⎟ ˆ L⎟ ⎠ Sˆ

(6.24)

meanwhile, one arrives at a system of differential equations for the mth order, which is determined by the product of matrix M with the last column of the perturbed density matrix ⎛

⎞ ⎛ ⎞ ρ1,m+1 ρ1,m+1 d ⎜ ⎟ ⎜ ⎟ .. .. . ⎝ ⎠=M⎝ ⎠ . . dt ρm+1,m+1 ρm+1,m+1 with the initial condition

(6.25)

6.3 Lossy Cavity Filled with a Kerr Medium

143

⎛

⎞ 0 ⎜ . ⎟ .⎝ . ⎠. .

(6.26)

ρ(0) Finally, the calculus of the mth-order correction terms can be expressed as

ρ1,m+1 = e .

Sˆ t

t m −1

t t1 0

0 ˆ

ˆ

ˆ

ˆ

ˆ

ˆ S t1 e− S t2 Le ˆ S t2 e− S t1 Le

....

(6.27)

0 ˆ

ˆ S tm dtm . . . dt2 dt1 ρ(0). . . . e− S tm Le

6.3 Lossy Cavity Filled with a Kerr Medium 6.3.1 Exact Solution To demonstrate the accuracy and capability of the method, the perturbative solution of the master equation is obtained for a Kerr medium filling an optical cavity with losses. The exact analytic solution for the master equation in this scenario is given by Arévalo-Aguilar et al. [26] ⎡

ˆ

ˆ

ρexa (τ ) = eS τ eLτ

.

⎤ ˆ −2τ (λ+i R) 1 − e  Jˆ⎦ ρ(0), exp ⎣  2 λ + i Rˆ

(6.28)

where .τ = χ t, .λ = γ /χ , and the parameter .λ represents the ratio between the cavity decay and the Kerr medium constant. The exact solution (6.28) serves as a benchmark for evaluating the accuracy of the perturbative solution given by the ˆ .Jˆ, .L, ˆ and .Rˆ are defined as Matrix Perturbation Method. The super-operators .S, .

  Sˆ ρ = −i a †2 a 2 , ρ , .

(6.29a)

Jˆρ = 2λaρ a † , .

(6.29b)

ˆ = −λ(a † aρ + ρa † a), . Lρ

(6.29c)

ˆ = a † aρ − ρa † a Rρ

(6.29d)

and satisfy the commutation relations .

  ˆ Jˆ ρ = 2i Rˆ Jˆρ, S,

  ˆ Jˆ ρ = 2λJˆρ, L,

  ˆ Jˆ ρ = 0. R,

(6.30)

144

6 The Matrix Perturbation Method for the Lindblad Master Equation

If one assumes a coherent state, i.e., .ρ(0) = |α〉 〈α|, as the initial state of the system, we get ⎡

ρexa (τ ) = e

.

ˆ Sˆ τ Lτ

e

⎤ ˆ −2τ (λ+i R) 1 − e  Jˆ⎦ |α〉 〈α| . exp ⎣  2 λ + i Rˆ

(6.31)

In order to determine the impact of the first super-operator on the coherent state, we expand the exponential in a Taylor series ⎡

  ⎤   −2τ λ+i Rˆ ˆ −2τ (λ+i R) 1 − e 1 1 − e   Jˆ⎦ |α〉 〈α| = |α〉 〈α| + . exp ⎣ Jˆ |α〉 〈α| 2 λ + i Rˆ 2 λ + i Rˆ   ⎤k ⎡ −2τ λ+i Rˆ ∞  1 ⎣1 − e Jˆ⎦ |α〉 〈α| ; + 2k k! λ + i Rˆ k=2

(6.32) note that Jˆ |α〉 〈α| = 2λ |α|2 |α〉 〈α| ,  2 Jˆ2 |α〉 〈α| = 2λ |α|2 |α〉 〈α| .

.. .  k Jˆk |α〉 〈α| = 2λ |α|2 |α〉 〈α| ,

(6.33)

and then, it is straightforward to determine from the previous calculations that ⎛ .

⎝1 − e

 ⎞ −2τ λ+i Rˆ

λ + i Rˆ

e

−|α|2

⎛ ⎝1 − e

∞  (α)n (α ∗ )m √ n!m! n=m=0



1 − e−2τ (λ+i(n−m)) λ + i(n − m)

 |n〉 〈m| ,

  ⎞2 −2τ λ+i Rˆ

λ + i Rˆ

e

⎠ |α〉 〈α| =

−|α|2

⎠ |α〉 〈α| =

∞  (α)n (α ∗ )m √ n!m! n=m=0



1 − e−2τ (λ+i(n−m)) λ + i(n − m)

2 |n〉 〈m| ,

6.3 Lossy Cavity Filled with a Kerr Medium

.. . ⎛

145

  ⎞k −2τ λ+i Rˆ

⎝1 − e λ + i Rˆ e

−|α|2

⎠ |α〉 〈α| =

∞  (α)n (α ∗ )m √ n!m! n=m=0



1 − e−2τ (λ+i(n−m)) λ + i(n − m)

k |n〉 〈m| .

(6.34)

Hence, the expression (6.32) is rewritten as ⎡

⎤ ˆ −2τ (λ+i R) 1 − e   Jˆ⎦ |α〉 〈α| = . exp ⎣ 2 λ + i Rˆ   ∞ −2τ (λ+i(n−m))  (α)n (α ∗ )m −|α|2 2 1−e |n〉 〈m| . =e exp λ |α| √ λ + i(n − m) n!m! n=m=0 (6.35) In order to apply the remaining exponentials that involve the super-operators .Lˆ and ˆ on the Fock states, it is essential to express these super-operators, which include .S † .a a, in terms of the number operator; this process leads to ˆ

eLτ |n〉 〈m| = e−λ(n+m)τ |n〉 〈m| ,

.

ˆ

eS τ |n〉 〈m| = e−i[n(n−1)−m(m−1)]τ ,

(6.36)

where   Lˆ |n〉 〈m| = −λ a † a |n〉 〈m| + |n〉 〈m| a † a = −λ(n + m) |n〉 〈m| ,   ˆ |n〉 〈m| = −i [n(n − 1) − m(m − 1)] |n〉 〈m| . Sˆ |n〉 〈m| = −i nˆ 2 − n,

.

(6.37)

Using the aforementioned calculations into Eq. (6.31), one yields ρexa (τ ) = e−|α|

2

∞  ∞ 

.

Cn,m |n〉 〈m| ,

(6.38)

n=0 m=0

where α n α ∗m −i[n(n−1)−m(m−1)]τ Cn,m = √ Dn,m , e n!m!

.

(6.39)

146

6 The Matrix Perturbation Method for the Lindblad Master Equation

with 

Dn,m = e

.

−λ(n+m)τ

 1 − e−2τ (λ+i(n−m)) exp λ |α| . λ + i(n − m) 2

(6.40)

The exact density matrix shows how the initial coherent state structure is lost due to the dissipation of energy generated by the cavity walls and the quadratic terms associated with the nonlinear medium.

6.3.2 Perturbative Solution 6.3.2.1

First-Order Correction

The approximate solution for the cavity problem is derived by employing the ˆ expression for the mth-order correction and considering that the super-operator .Lρ ˆ Keeping this in mind is defined as the sum of the super-operators .Jˆρ and .Lρ. and using the commutation relations (6.29), the initial step in the process entails obtaining the first-order correction term as follows: ⎡ ˆ ρ1,2 = eS τ ⎣

τ

.

⎡ ˆ = eS τ ⎣

⎡

τ

ˆ

ˆ

ˆ

e− S τ1 JˆeS τ1 dτ 1 +

0

 =e

ˆ

ˆ S τ1 dτ 1 ⎦ ρ(0) e− S τ1 Le

0

ˆ = eSτ ⎣Jˆ

Sˆ τ

⎤

τ 0

τ

ˆ

e−2i Rτ1 dτ 1 + Lˆ

0 ˆ

1 − e−2i Rτ 2i Rˆ



τ 0

⎤ ˆ

ˆ

ˆ S τ1 dτ 1 ⎦ ρ(0) e− S τ1 Le ⎤

dτ 1 ⎦ ρ(0)



ˆ ρ(0). Jˆ + Lτ

(6.41)

In this case, the existing commutation relation between super-operators .Jˆ and .Rˆ ˆ allows for situating .Jˆ on either the left side or the right side of .R.

6.3.2.2

Second-Order Correction

The subsequent step involves the calculation of the second-order term. This procedure entails resorting again to expression (6.27) and using the result obtained from the first-order term in the following manner:

6.3 Lossy Cavity Filled with a Kerr Medium

ρ1,3 = e

.

Sˆ τ

τ τ1 0

= eS τ 0 ˆ



ˆ Jˆe−2i Rτ1 + Lˆ

  ˆ Jˆe−2i Rτ2 + Lˆ dτ2 dτ1 ρ(0)

0

τ τ1

= eS τ 0

    ˆ ˆ ˆ ˆ e− S τ1 Jˆ + Lˆ eS τ1 e− S τ2 Jˆ + Lˆ eS τ2 dτ2 dτ1 ρ(0)

0

τ τ1

ˆ

147



 ˆ ˆ ˆ 1 ˆ −2i Rτ Jˆ2 e−2i R(τ1 +τ2 ) + Lˆ Jˆe−2i Rτ2 + JˆLe + Lˆ 2 dτ2 dτ1 ρ(0).

0

(6.42) After performing the integration, one obtains ⎡ ρ1,3 = e

.

Sˆ τ

+



⎣1 2!

ˆ

1 − e−2i Rτ ˆ J 2i Rˆ



2

τe

2i Rˆ

ˆ −2i Rτ



2 ˆ + Lτ ˆ

1 − e−2i Rτ ˆ − J 2i Rˆ

 ˆ 1 − e−2i Rτ ˆ Lˆ 2 τ 2 J + 2! 2i Rˆ

 (6.43)

ρ(0).

Therefore, the full second-order perturbation solution for the optical cavity filled with a Kerr medium takes the following form:  ρ(t) ≈e

.

Sˆ τ

ρ(0) + λe

Sˆ τ

⎡ 2 Sˆ τ

+λ e

⎣1 2! 

2 Sˆ τ

+λ e

ˆ Lτ





ˆ

1 − e−2i Rτ 2i Rˆ ˆ

1 − e−2i Rτ ˆ J 2i Rˆ ˆ



2 Sˆ τ

ˆ ρ(0) + λ e Jˆ + Lτ

2

1 − e−2i Rτ ˆ J 2i Rˆ





+

2 2i Rˆ

 ˆ

τ e−2i Rτ −

 Lˆ 2 τ 2 ρ(0) 2! ⎤ ˆ

1 − e−2i Rτ ˆ ⎦ J ρ(0) 2i Rˆ

 (6.44)

ρ(0).

The difference between Eq. (6.44) and Eq. (6.28) is that for the approximated solution .λ has been considered as a perturbation parameter. It is noteworthy that a significant portion of the perturbative

solution contains the term involving the product of super-operators . term acts on a coherent state, it yields

ˆ

1−e−2i Rτ 2i Rˆ

Jˆ . Notably, when this

148

 .

6 The Matrix Perturbation Method for the Lindblad Master Equation

   ˆ ˆ −2i Rτ 1 − e 1 − e−2i Rτ ˆ |α〉 〈α| J |α〉 〈α| = 2 |α|2 2i Rˆ 2i Rˆ 2 −|α|2

= 2 |α| e

∞  (α)p (α ∗ )q √ p!q! p,q=0



1 − e−2i(p−q)τ 2i(p − q)

 |p〉 〈q| , (6.45)

as a result, by considering the initial state as a coherent state, analogous to the exact solution case, and using the aforementioned result, the derivation of the following approximate density matrix is obtained

.

ρapro (τ ) = e−|α|

2

∞ ∞  

Fn,m |n〉 〈m| ,

(6.46)

n=0 m=0

where  α n α ∗m −i[n(n−1)−m(m−1)]τ  e 1 + Hn,m , Fn,m = √ n!m!

.

(6.47)

and Hn,m =

.

2 λ2 τ 2  n + m − 2 |α|2 e−i(n−m)τ sinc [(n − m)τ ] 2 − λ(n + m)τ + 2λ |α|2 τ e−i(n−m)τ sinc [(n − m)τ ]   2i(n−m)τ + 2i(n − m)τ 2 2 −2i(n−m)τ 1 − e − λ |α| e . (n − m)2

(6.48)

6.4 Comparison Between the Exact and the Approximated Solution In a straightforward and direct manner, one can visualize the evolution of a cavityKerr system in phase space by calculating a quasi-probability function. The Husimi Q-function stands as the simplest basic among all quasi-probability functions, and it is precisely defined as the expectation value of the density matrix in a coherent base [27–30]. Thus the Q-function is given by the following expressions: Q(β) =

.

1 〈β| ρ(t) |β〉 , π

(6.49)

6.4 Comparison Between the Exact and the Approximated Solution

149

being .|β〉 a coherent state. Consequently, when the exact density matrix is used to evaluate the Q-function together with the orthogonality condition, .δm,n = 〈n|m〉, it yields the following solution: ∞

Qexa (β) =

.

∞

1 −|α|2 +|β|2    (αβ ∗ )n (α ∗ β)l −λ(n+l)τ e e Dn,l , π n!l!

(6.50)

n=0 l=0

with 

Dn,l = e

.

−i[n(n−1)−l(l−1)]τ

1 − e−2τ [λ+i(n−l)] exp λ |α| λ + i(n − l)



2

.

(6.51)

For the approximated density matrix, the result is Qapro (β) =

.

∞ ∞  1 −|α|2 +|β|2    (αβ ∗ )n (α ∗ β)l −i[n(n−1)−l(l−1)]τ  e e 1 + Gn,l , π n!l! n=0 l=0 (6.52)

with Gn,l =

.

2 λ2 τ 2  n + l − 2 |α|2 e−i(n−l)τ sinc [(n − l)τ ] 2 − λ(n + l)τ + 2λ |α|2 τ e−i(n−l)τ sinc [(n − l)τ ] + 4τ λ2 |α|2 e−i(n−l)τ

e−i(n−l)τ − sinc [(n − l)τ ] . 2i (n − l)

(6.53)

Figure 6.1 illustrates the time evolution of the Husimi function for .λ = 0.05 and α = 2. The left side displays the outcomes of the exact solution, whereas the right side showcases the results for the second-order correction solution. It is worth noting that, at short times, both solutions produce similar results; however, as time increases, some differences begin to emerge between the two solutions. Figure 6.2 illustrates the time evolution of the Husimi function for .λ = 0.13 and .α = 2. The graph for .τ = 0 has been omitted since it is straightforward to observe that, at that time, the exact and approximated solutions coincide. Notably, as the value of lambda increases, the time range during which the exact and approximated solutions remain similar becomes smaller. This behavior can be easily understood by recognizing that the real perturbation parameter is .λτ and not just .λ. Furthermore, the density matrices .ρexa (τ ) and .ρapro (τ ) can be used to calculate the mean photon number, which is a relevant physical quantity of the Kerr lossy cavity. By doing so, one can obtain information about how closely both solutions are when one measures this observable, offering a second alternative to test the accuracy of the perturbative approximation in comparison to the exact solution. Therefore, the procedure to determine the average number of photons for the exact solution is .

150

6 The Matrix Perturbation Method for the Lindblad Master Equation

Fig. 6.1 The evolution of the Husimi Q-function, .Q(q, p), with .α = 2 and .λ = 0.05. The left column displays the exact solution, while the right column shows the approximated perturbative solution. (a) .τ = 0. (b) .τ = 0. (c) .τ = 1. (d) .τ = 1. (e) .τ = 2. (f) .τ = 4. (g) .τ = 4. (h) .τ = 4

6.4 Comparison Between the Exact and the Approximated Solution

151

Fig. 6.2 Evolution of the Husimi Q-function, .Q(q, p), with .α = 2 and .λ = 0.13. The left column is the exact solution and the right one is the approximated perturbative solution. (a) .τ = 1. (b) .τ = 1. (c) .τ = 2. (d) .τ = 2. (e) .τ = 4. (f) .τ = 4

.

〈N 〉exa =

∞ 

〈j | N ρexa (τ ) |j 〉

j =0

= e−|α|

2

∞  n,m,j =0

Cn,m 〈j | N |n〉 〈m | j 〉

152

6 The Matrix Perturbation Method for the Lindblad Master Equation

= e−|α|

2 e−2λτ

∞  |α|2j e−2j λτ ; (j − 1)!

(6.54)

j =1

by applying the change of variable .s = j −1 and replacing in Eq. (6.54), one obtains .

〈N〉exa = |α|2 e−2λτ .

(6.55)

Similarly, one can repeat the procedure for the second-order approximated solution, resulting in the following expression:  .

〈N〉apro = |α|

2

 (−2λτ )2 . 1 − 2λτ + 2!

(6.56)

These results are presented in Fig. 6.3. The chosen parameter values are .α = 2, and λ = 0.05, 0.07, 0.09, 0.11, and 0.13. The solid lines represent the exact solution, whereas the dotted lines display the results of the perturbative solution. As expected, the perturbation solutions are in good agreement with the exact solution at different values of .λ, but only for short times. This is associated with the requirement that .λτ ⪡ 1 must be satisfied to ensure the accuracy of the approximation, as depicted in Fig. 6.3. Finally, as another measure of proximity for the solutions, the distance between the exact density matrix and the approximated density matrix is evaluated [10, 31– 33]. In this case, the geometrical measure of the distance between two density matrices is given by

.

Fig. 6.3 The mean photon number .〈N 〉 versus time .τ , with .α = 2 and .λ = 0.05, 0.07, 0.09, 0.11, and 0.13. The solid lines are for the exact solution, while the dotted lines are for the perturbative solution

6.4 Comparison Between the Exact and the Approximated Solution

Fig. 6.4 Plot of .F = Tr (ρ1 ρ2 ) +

1 2

  2 − Trρ12 − Trρ22 versus .τ for different values of .λ

‖ρ1 − ρ2 ‖2 =

.

153

1 Tr (ρ1 − ρ2 )2 2

= 1 − F,

(6.57)

  where .F = Tr (ρ1 ρ2 ) + 12 2 − Trρ12 − Trρ22 is a parameter that evaluates the closeness of .ρ1 and .ρ2 . Both matrices will be similar if .F = 1 or completely different if .F = 0. Therefore, to determine the degree of similarity between the density matrices .ρexa (τ ) and .ρ(τ )apro , a numerical calculation of F as a function of time is essential. This computation is depicted in Fig. 6.4. Figure 6.4 illustrates the numerical evaluation of the parameter F with respect to .τ , considering .α = 2 and varying the values of .λ. The plot demonstrates that the density matrices exhibit strong similarity at short times; these results coincide and are in agreement with the temporal behavior of the Q-function and the average number of photons. Based on these findings, it can be concluded that the Matrix Perturbation Method provides favorable perturbative solutions for the studied scenario of a lossy cavity filled with a Kerr medium. It is important to note that as time grows, the perturbative results start to differ from the exact solution, which is expected due to the real measure of the perturbation being determined by the product .λτ , as previously mentioned. A similar logical behavior is also observed in the case of the Husimi function and the parameter F , where a good agreement is found for short times, and differences arise as time increases. The second-order result proves to be sufficient for accurately reproducing the exact solution for this specific system. However, in the study of more complex systems, higher order contributions may become relevant.

154

6 The Matrix Perturbation Method for the Lindblad Master Equation

References 1. W.H. Zurek, Complexity, Entropy and the Physics of Information (Westview Press, Boulder, 1990) 2. W.H. Zurek, Complexity, Entropy and the Physics of Information (CRC Press, Boca Raton, 2018) 3. M. Gell-Mann, J.B. Hartle, Classical equations for quantum systems, arXiv:gr-qc/9210010v2 (1992) 4. M. Gell-Mann, J.B. Hartle, Classical equations for quantum systems. Phys. Rev. D 47(8), 3345–3382 (1993) 5. W.G. Unruh, Maintaining coherence in quantum computers. Phys. Rev. A 51(2), 992–997 (1995) 6. H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007) 7. A.O. Caldeira, A.J. Leggett, Quantum tunnelling in a dissipative system. Ann. Phys. 149(2), 374–456 (1983) 8. M.O. Scully, M. Suhail Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997) 9. R Alicki, The quantum open system as a model of the heat engine. J. Phys. Math. Gen. 12(5), L103–L107 (1979) 10. M.A. Nielsen, M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, 10th edn. (Cambridge University Press, Cambridge, 2010) 11. D.F. Walls, M.J. Collet, G.J. Milburn, Analysis of a quantum measurement. Phys. Rev. D 32(12), 3208–3215 (1985) 12. B.L. Hu, J.P. Paz, S. Sinha, Minisuperspace as a quantum open system, in Directions in General Relativity (Cambridge University Press, Cambridge, 1993), pp. 145–165 13. U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993) 14. G. Lindblad, On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119–130 (1976) 15. F. Soto-Eguibar, H.M. Moya-Cessa, Introduction to Quantum Optics (Rinton Press, Princeton, 2011) 16. A.B. Klimov, J.L. Romero, An algebraic solution of Lindblad-type master equations. J. Opt. B Quant. Semiclassical Opt. 5(3), S316–S321 (2003) 17. T. Prosen, Third quantization: a general method to solve master equations for quadratic open fermi systems. New J. Phys. 10(4), 043026 (2008) 18. L.M. Arévalo-Aguilar, H. Moya-Cessa, Solution to the master equation for a quantized cavity mode. Quant. Semiclassical Opt. J. Euro. Opt. Soc. Part B 10(5), 671–674 (1998) 19. Z.-J. Zhang, D.-G. Jiang, W. Wang, Perturbation theory for open two-level nonlinear quantum systems. Commun. Theor. Phys. 56(1), 67–70 (2011) 20. X.X. Yi, C. Li, J.C. Su, Perturbative expansion for the master equation and its applications. Phys. Rev. A 62(1), 013819 (2000) 21. J.I. Kim, M.C. Nemes, A.F.R. de Toledo Piza, H.E. Borges, Perturbative expansion for coherence loss. Phys. Rev. Lett. 77(2), 207–210 (1996) 22. H. Moya-Cessa, J.A. Roversi, S.M. Dutra, A. Vidiella-Barranco, Recovering coherence from decoherence: a method of quantum-state reconstruction. Phys. Rev. A 60(5), 4029–4033 (1999) 23. H. Moya-Cessa, S.M. Dutra, J.A. Roversi, A. Vidiella-barranco, Quantum state reconstruction in the presence of dissipation. J. Mod. Opt. 46(4), 555–558 (1999) 24. H. Moya-Cessa, F. Soto-Eguibar, J.M. Vargas-Martínez, R. Juárez-Amaro, A. Zúñiga-Segundo, Ion-laser interactions: the most complete solution. Phys. Rep. 513, 229–261 (2012) 25. J. Martinez-Carranza, F. Soto-Eguibar, H. Moya-Cessa, Alternative analysis to perturbation theory in quantum mechanics. Euro. Phys. J. D 66(1) (2012). https://doi.org/10.1140/epjd/ e2011-20654-5

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26. L.M. Arévalo-Aguilar, R. Juárez/Amaro, J.M. Vargas-Martínez, O. Aguilar-Loreto, H. MoyaCessa, Solution of master equations for the anharmonic oscillator interacting with a heat bath and for parametric down conversion process. Appl. Math. Inform. Sci. 2(1), 43–49 (2008) 27. W.P. Schleich, Quantum Optics in Phase Space (Wiley, Hoboken, 2001) 28. R. Loudon, Coherent states: applications in physics and mathematical physics. Optica Acta Int. J. Opt. 33(3), 219–219 (1986) 29. K. Husimi, Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn. 3rd Ser. 22(4), 264–314 (1940) 30. Y. Kano, A new phase-space distribution function in the statistical theory of the electromagnetic field. J. Math. Phys. 6(12), 1913–1915 (1965) 31. S.N. Filippov, V.I. Man’ko, Geometrical interpretation of the density matrix: mixed and entangled states. J. Russ. Laser Res. 29(6), 564 (2008) 32. V. Bargmann, On unitary ray representations of continuous groups. Ann. Math. 59(1), 1 (1954) 33. J. Anandan, A geometric approach to quantum mechanics. Found. Phys. 21(11), 1265–1284 (1991)

Chapter 7

Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

Abstract In this concluding chapter, we extend the Matrix Perturbation Method to address time-dependent Hamiltonians. Utilizing unitary transformations, we factorize specific time dependencies, enabling perturbative solutions. We explore two cases: a particle with pulsating mass in a linear potential and a particle with exponentially increasing mass in a linear potential. Additionally, we present illustrative examples of time-varying potentials. This approach expands the method’s applicability, offering a powerful tool for understanding and solving diverse quantum mechanical systems.

7.1 Introduction The Matrix Perturbation Method has demonstrated its efficacy in solving stationary systems, offering valuable solutions in various models adhering to this constraint. However, its application as a methodology for addressing time-dependent system scenarios, without resorting to standard time-dependent perturbation theory, has been largely unexplored. In this concluding chapter, we bridge this gap by presenting an alternative approach that complements the Matrix Perturbation Method, enabling perturbative solutions for specific types of time-dependent Hamiltonians. The core idea behind this technique involves the implementation of two consecutive time-dependent unitary transformations, utilizing operators associated with the algebra su(1,1). These operators, generated by the elements .xˆ 2 , .pˆ 2 , and .xˆ pˆ + pˆ x, ˆ satisfy the following commutation relations .

    xˆ 2 , pˆ 2 = 2i xˆ pˆ + pˆ xˆ ,

  xˆ pˆ + pˆ x, ˆ xˆ 2 = −4i xˆ 2 ,



 xˆ pˆ + pˆ x, ˆ pˆ 2 = 4i pˆ 2 . (7.1)

By employing two transformations, the aim is to render the Hamiltonian under investigation time independent and amenable to the subsequent application of the Matrix Perturbation Method, thereby obtaining perturbative solutions. This approach proves viable for solving the Schrödinger equation in scenarios governed © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6_7

157

158

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

by the following time-dependent Hamiltonians     ∂|ψ(t)〉 pˆ 2 + λV x, ˆ μ(t) |ψ(t)〉, . .i = ∂t 2μ2 (t)   2 √  ∂|ψ(t)〉 pˆ 2 i + λμ (t)V m0 xμ(t) ˆ |ψ(t)〉. = ∂t 2m0

(7.2a) (7.2b)

In Eq. (7.2a), the Schrödinger equation describes the dynamics of a free particle with a time-varying mass, denoted as .μ(t), and subjected to a time-dependent potential  V. xμ(t) ˆ . This formulation characterizes the restricted class of Hamiltonian systems determined by the linear time dependence of the free particle’s mass .μ(t), as explained in subsequent subsections. In contrast, Eq. (7.2b) represents the second Hamiltonian structure, describing a free particle with a constant mass .m0 ; in this case, the temporal dependence is solely concentrated in the potential term, and the class of solvable Hamiltonians allowed includes functions .μ(t) that are polynomials of degree 2 or 1 with respect to t. This chapter provides a detailed explanation of both situations, illustrating the successful elimination of time dependence through the proposed unitary transformations.

7.2 First Case Let us begin by deriving the solution to the Schrödinger equation for the class of time-dependent Hamiltonians described by Eq. (7.2a) i

.

    ∂|ψ(t)〉 pˆ 2 + λV xμ(t) ˆ |ψ1 (t)〉. = ∂t 2μ2 (t)

(7.3)

The initial step in the analysis involves applying a unitary transformation to eliminate .μ(t) from the quadratic term in .pˆ and make the potential time independent. This is accomplished by changing the variable .|ψ(t)〉 = Uˆ1 (t)|φ(t)〉, where   ln |α(t)| (xˆ pˆ + pˆ x) ˆ ; Uˆ 1 (t) = exp i 2

.

(7.4)

by using Hadamard’s lemma, (4.12), it is straightforward to show that the .Uˆ 1 (t) transformation results in the following changes in the coordinate and momentum operators Uˆ 1† (t)pˆ 2 Uˆ 1 (t) = α 2 (t)pˆ 2 ,

.

xˆ Uˆ 1† (t)xˆ Uˆ 1 (t) = . α(t)

(7.5)

7.2 First Case

159

Subsequently, applying the unitary transformation to the wave function .|ψ(t)〉 on both sides of Eq. (7.3), the Schrödinger equation for the transformed wave function .|φ(t)〉 can be expressed as    pˆ 2 μ(t) α(t) ˙ ∂|φ(t)〉 2 = (xˆ pˆ + pˆ x) ˆ + λV xˆ |φ(t)〉. α (t) + .i ∂t 2α(t) α(t) 2μ2 (t) Choosing .α(t) in such a way that it satisfies .μ(t) = mass of the particle, one yields to

√

(7.6)

m0 α(t), where .m0 is the

 2  √  pˆ ∂|φ(t)〉 α(t) ˙ = (xˆ pˆ + pˆ x) ˆ + λV .i + m0 xˆ |φ(t)〉. ∂t 2m0 2α(t)

(7.7)

Notice that the aforementioned transformation results in a Hamiltonian that concentrates the time dependence in the new mixed terms involving .xˆ and .p. ˆ Additionally, the removal of time dependence on the potential is guaranteed at this stage, regardless of its complexity at any ˆ This assurance is based on

 power n of .x. ˆ xˆ n = −2inxˆ n , which yields .Uˆ 1† (t)xˆ n Uˆ 1 (t) = the commutation relation. xˆ pˆ + pˆ x, xˆ n α(t)−n .   Subsequently, it becomes imperative to eradicate the time-dependent . xˆ pˆ + pˆ xˆ term. This can be achieved by introducing a second unitary transformation, denoted as .|φ(t)〉 = Uˆ 2 (t)|χ (t)〉, where now   ˙ m0 α(t) xˆ 2 , Uˆ 2 (t) = exp −i 2α(t)

.

(7.8)

which when applied to Eq. (7.7) gives i

.

 2  √  pˆ ¨ m0 α(t) ∂|χ (t)〉 xˆ 2 + λV = − m0 xˆ |χ (t)〉; 2m0 2α(t) ∂t

(7.9)

thus, this second transformation yields a Hamiltonian that governs the evolution of .|χ (t)〉 and exhibits characteristics of a harmonic oscillator. Specifically, this ¨ occurs when the term .− α(t) α(t) , acting as a time-dependent frequency, is equal to

¨ 2 a real constant frequency denoted as .Ω20 . Alternatively, when . α(t) α(t) = Ω0 , the system represents a repulsive oscillator. In both cases, there is an additional timeindependent perturbative potential. These conditions, of course, impose a constraint that can be expressed as an auxiliary equation in the form of .α(t) ¨ ± Ω20 α(t) = 0, which corresponds to the classical equation of motion for a time-dependent harmonic or repulsive harmonic oscillator, and the general solutions of these equations can be expressed as linear combinations of functions of the form .exp (Ω0 t), .exp (−Ω0 t) or .sin (Ω0 t), .cos (Ω0 t) with linear terms in t. Therefore, these general solutions explicitly define the kind of mass functions .μ(t) and subsequently the class of admissible time-dependent Hamiltonians that can be solved. As will be

160

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

extensively discussed in this chapter, this information greatly aids in eliminating time dependence from numerous widely recognized Hamiltonians. Furthermore, it is noteworthy to emphasize a significant aspect of the second transformation. In α(t) ˙ certain scenarios, there is a possibility where the exponential term . α(t) equals a constant. This condition implies that only the first transformation is necessary to remove the time dependence from the free particle term and the potential under consideration. Consequently, the second transformation becomes time independent and is responsible for eliminating the term involving the product of position and momentum operators, which arise from the application of the first transformation. In summary, the two unitary transformations implemented here enable the derivation of a time-independent Hamiltonian. In this case, the original Hamiltonian of the free particle with potential transforms into a new Hamiltonian that simplifies to the familiar equation representing either a harmonic or a repulsive oscillator, depending on the value of the frequency constraint. Furthermore, these oscillator representations can even be obtained in the absence of the time-dependent potential term. This feature is advantageous since permits the equivalence of a free particle with the harmonic or repulsive oscillator in a distinct manner, as reported in [1–3]. Hence, the aforementioned transformations enable the elimination of time dependence, leading to the application of standard time-independent quantum mechanics to address the original time-dependent problem (7.2a). Consequently, the approach presented here allows for the determination of the solution of the Schrödinger equation for both systems,

√  m0 Ω20 2 pˆ 2 + m0 xˆ |χ1 (t)〉, . xˆ + λV 2m0 2

√  m0 Ω20 2 ∂|χ2 (t)〉 pˆ 2 i − m0 xˆ |χ2 (t)〉, = xˆ + λV ∂t 2m0 2

∂|χ1 (t)〉 .i = ∂t



(7.10a)

(7.10b)

where the feasibility of obtaining an exact solution is naturally dependent on the complexity of the potential being studied. In this case, the exact solutions are achievable for linear and quadratic potentials. However, numerical methods or perturbative approaches become essential for obtaining solutions for potentials beyond the third order. In our specific context, the Matrix Perturbation Method can be employed to effectively address such problems. Let us thoroughly examine the efficacy of the method by closely analyzing two straightforward illustrative examples, allowing a direct comparison between their exact solutions and the corresponding perturbative counterparts.

7.2 First Case

161

7.2.1 A Particle with Strongly Pulsating Mass Moving in a Linear Potential 7.2.1.1

Exact Solution

Let us consider a specific case of physical interest, where a particle has a mass that pulsates strongly with a constant frequency .δ. The system is subject to a timedependent linear potential and is described by the Hamiltonian [4, 5] Hˆ (t) =

.

√ pˆ 2 ˆ + λ m cos (δt) x. 2 2m cos (δt)

(7.11)

In order to solve the Schrödinger equation associated with .Hˆ (t), given by √ d i dt |ψ(t)〉 = Hˆ (t)|ψ(t)〉, we identify .μ(t) with . m cos (δt), and .cos (δt) with .α(t), in Eq. (7.11). According to the procedure outlined in Sect. 7.2, the first transformation to consider is of the form     i (7.12) .|ψ(t)〉 = exp ln | cos (δt) | xˆ pˆ + pˆ xˆ |φ(t)〉, 2

.

then the transformed Schrödinger equation becomes i

.

  2   √ pˆ δ d|φ(t)〉 = − tan (δt) xˆ pˆ + pˆ xˆ + λ m xˆ |φ(t)〉. dt 2m 2

(7.13)

The next step in the procedure entails the elimination of the term .xˆ pˆ + pˆ x, ˆ ˙ which is achieved by determining .− α(t) . In this particular case, it is found to be α(t) .δ tan (δt). This result is then linked to the second and final transformation explained in Sect. 7.2. Here, the transformation to be applied is   mδ 2 tan (δt) xˆ |χ (t)〉, .|φ(t)〉 = exp i 2

(7.14)

¨ 2 and the frequency constraint for this system is determined to be .− α(t) α(t) = δ , resulting in a harmonic system. Consequently, the application of the second transformation yields the following Schrödinger equation

d|χ (t)〉 = .i dt



√ pˆ 2 mδ 2 2 + xˆ + λ m xˆ |χ (t)〉, 2m 2

(7.15)

describing a harmonic oscillator perturbed by a linear anharmonic potential. This specific system has already been addressed and solved in Chap. 4, providing both the exact and perturbative solutions in the coordinate representation. In this case, the exact solution when the initial condition is a coherent state .|χ (0)〉 = |β〉 takes the following form

162

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

 χex (x, t) = 〈x|χ (t) =

.

δm π

1/4 exp [γ3 (x, t)] ,

(7.16)

where γ3 (x, t) = f0 (x, t) + f1 (x, t)β + f2 (t)β 2 ,

.

(7.17)

being

  λ2 3 − 2iδt − 4 exp (−iδt) + exp (−2iδt) 1  .f0 (x, t) = − − iδ t − imx 2 3 2 4δ

 √ λ m x exp (−iδt) − 1 (7.18a) ,. + δ √ 

√ 2 exp (−iδt) λ − λ cos(tδ) + m xδ 2 f1 (x, t) = ,. (7.18b) δ 3/2  1

f2 (t) = − (7.18c) 1 + exp (−2iδt) , 2 and where the subscript ex means exact solution. Now, to get the original wave function .|ψ(t)〉, we have to apply the inverse transformations that have been established in this section, in combination with the initial states .|ψ(0)〉 = |φ(0)〉 = |χ (0)〉 = |β〉. Subsequently, this process leads to       i i 2 .ψex (x, t) = x| exp ln | cos (δt) | xˆ pˆ + pˆ xˆ exp mδ tan (δt) xˆ |χ (t) 2 2      d i = cos (δt) exp ln | cos (δt) |x exp mδ tan (δt) x 2 χex (x, t) ; dx 2 (7.19) 



using  thed known actionυ of the dilatation operator on a given function .f (x), f (x) = f (e x) [6], the full solution of the original system is obtained exp υx dx as    i 2 .ψex (x, t) = cos (δt) exp mδ sin (2δt) x χex (x cos (δt) , t) . (7.20) 4

.

√ It is worth noting that initially, an alternative option to using .μ(t) = √ m cos(δt) would have been to employ a time-dependent mass given by .μ(t) = m sin(δt). In such a situation, the Hamiltonian for a particle with a mass that oscillates rapidly pˆ 2 while moving in a linear potential can be expressed as .Hˆ (t) = 2 (δt) + 2m sin √ λ m sin(δt)x. ˆ With this information, it is clear that .α(t) = sin(δt), and the two subsequent unitary transformations to implement are of the form .|ψ(t)〉 =

7.2 First Case

163

 

 2 exp 2i ln | sin (δt) | xˆ pˆ + pˆ xˆ |φ(t)〉 and .|φ(t)〉 = exp − imδ 2 cot (δt) xˆ |χ (t)〉. By following the same procedure as the previous example, the final solution in the coordinate representation can be obtained    i 2 .ψex (x, t) = sin (δt) exp mδ sin (2δt) x χex (x sin (δt) , t) . 4

7.2.1.2

(7.21)

Perturbative Solution

The quantum system is now solved using the perturbative approach, and for the sake of simplicity, the equations are inverted, and the expressions of the operators .pˆ and .x ˆ are substituted in terms of the ladder operators .aˆ and .aˆ † into Eq. (7.15). By doing  so, one can identify and associate the unperturbed term as .Hˆ 0 = δ nˆ + 1/2 , and   the perturbation as .Hˆp = √1 aˆ † + aˆ , quantified by the parameter .λ. Following 2δ the same steps as reported in Chap. 4 and performing straightforward calculations, the analytic expression for the perturbative solution up to the second order in the coordinate representation is obtained   (2) χap (x, t) = Nβ (t) χ (0) (x, t) + λχ (1) (x, t) + λ2 χ (2) (x, t) ,

.

(7.22)

where the subscript ap stands for approximated, whereas the super-indexes denote the perturbative terms up to the k-th order and are provided by

|β(t)|2 √ δt β 2 (t) δmx 2 .χ exp −i − + 2δmβ(t)x − − ,. 2 2 2 2 (7.23a)     δt √ 2 δt χ (1) (x, t) = − i 3 sin 2δmx − β(t) + β χ (0) (x, t), . e−i 2 2 δ (7.23b)   1  χ (2) (x, t) = 3 −3 + iδt + eitδ + 2e−itδ + e−itδ 2mx 2 δ − 1 [cos(δt) − 1] 2δ     3  √ −2itδ itδ 4 tδ 2 2δmxe β (t) χ (0) (x, t), . −1 + e β(t)+8 sin + 2 (7.23c) √ −2  2 =1 − λ3 9/2 β sin (δt) [δt − sin (δt)] Nβ(2) (t) δ  4 λ  + 6 4 sin2 (δt/2) + δt [δt − 2 sin (δt)] 4δ 

(0)

δm (x, t) = π

1/4

164

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

 2λ4  4 cos + 1 sin4 (δt/2) β (2δt) δ6    2λ4   + 6 β 2 3 β 2 + 2 + 2 cos (δt) 2β 2 + 3 sin4 (δt/2) , δ

+

(7.23d) being .β(t) = β exp (−iδt). By repeating the same procedure as in the case of the exact solution, the complete solution for the system is obtained as ψap (x, t) =

.





 i 2 cos (δt) exp mδ sin (2δt) x χap (x cos (δt) t) . 4

(7.24)

Similar √ to the exact case, in the scenario √ of using a time-dependent mass given by μ(t) = m sin(δt) instead of .μ(t) = m cos(δt), one gets

.

ψap (x, t) =

.

7.2.1.3





 i 2 sin (δt) exp mδ sin (2δt) x χap (x sin (δt) t) . 4

(7.25)

Comparison Between the Exact and Perturbative Solutions

Figures 7.1 and 7.2 compare the accuracy and validity of the perturbative solution obtained above, (7.24), with the exact solution presented in Sect. 7.2.1.1, (7.20), by plotting the probability densities .|ψex,ap (x, t)|2 of each solution, as functions of the position x at four times (.t = 0, 10, 20, 30), and for two different values of the perturbative parameter .λ. Figure 7.1 displays the probability densities of the exact and of the approximated solutions against x for .λ = 0.01 and reveals that the perturbative solution, labeled by the blue dotted line, is highly effective, even for times as far out as .t = 30; in fact, for this value of .λ the solutions are indistinguishable. Figure 7.2 is the same situation as Fig. 7.1, but for .λ = 0.1; it can be observed that the perturbative solution still provides an accurate approximation of the exact solution, with small differences starting to emerge. Nonetheless, selecting a larger value for this parameter would likely result in more significant differences between the solutions; however, this phenomenon is completely predictable and normal, since perturbative solutions are suitable only for small values of .λ. Both graphics were obtained using the common set of parameter values .δ = 1, .m = 1, and .β = 2.

7.2 First Case

165

Fig. 7.1 The probability densities of the exact and perturbative solutions, denoted by the red dashed and the blue dotted lines, are depicted as functions of the position x at times .t = 0, 10, 20, 30 with .λ = 0.01. The common parameters .δ = 1, .m = 1, and .β = 2 were used for all calculations

Fig. 7.2 The probability densities of the exact and perturbative solutions, denoted by the red dashed and the blue dotted lines, are depicted as functions of the position x at times .t = 0, 10, 20, 30 with .λ = 0.1. The common parameters .δ = 1, .m = 1, and .β = 2 were used for all calculations

166

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

7.2.2 A Particle with Exponentially Increasing Mass in the Presence of a Linear Potential 7.2.2.1

Exact Solution

This section highlights an illustrative example in which the implementation of the pair of time-dependent unitary transformations gives rise to a repulsive oscillator. The specific case under investigation involves a particle exhibiting exponential mass growth while subjected to a linear potential. In this context, the objective is to solve the Schrödinger equation governing the system, which can be described as follows: i

.

  2 √ pˆ −2(at+b) ∂|ψ2 (t)〉 e = + λ m xe ˆ at+b |ψ2 (t)〉, 2m ∂t

(7.26)

where both, a and b, are real constants. Identifying for this specific case that .eat+b = α(t), the initial transformation that needs to be applied is of form  |ψ2 (t)〉 = exp

.

   i (at + b) xˆ pˆ + pˆ xˆ |φ2 (t)〉. 2

(7.27)

By implementing this transformation on the Schrödinger equation(7.26), it can be directly transformed into i

.

 2  √ pˆ a ∂|φ2 (t)〉 = + (xˆ pˆ + pˆ x) ˆ + λ m xˆ |φ2 (t)〉; ∂t 2m 2

(7.28)

˙ similar to the first example in Sect. 7.2.1, one can identify that . α(t) α(t) = a for the current problem. This prompts us to introduce a second unitary transformation in the form of  ma  xˆ 2 |χ2 (t)〉, (7.29) .|φ2 (t)〉 = exp −i 2

it should be noted that, in the specific case under consideration, the second ˙ transformation is independent of time due to the term . α(t) α(t) yields a constant value represented by the real parameter a. The primary objective of the first transformation is to eliminate the temporal dependence, while the second transformation is responsible for eliminating the mixed terms involving position and momentum operators, as discussed in the concluding remarks of Sect. 7.2.2.1. Furthermore, it should be mentioned that, in this particular scenario, the frequency constraints ¨ 2 dictate that . α(t) α(t) = a . Consequently, the system undergoes a transformation into a repulsive oscillator, accompanied by the inclusion of a linear anharmonicity term

7.2 First Case

167

∂|χ2 (t)〉 .i = ∂t



√ pˆ 2 m 2 2 − a xˆ + λ m xˆ |χ2 (t)〉, 2m 2

(7.30)

where the parameter a plays the role of the strength of the repulsive potential in the system, while the dimensionless scale parameter .λ measures the perturbation strength of the linear anharmonic term. It is worth noting that this system has been previously solved algebraically in an exact form in Chap. 4 for a constant unitary mass and under the initial condition of a coherent state. Hence, by integrating (7.30) with respect to time and then transforming it back to .|ψ2 (t)〉, one can readily derive the exact solution    ma    i .|ψ2 (t)〉 = exp xˆ 2 (at + b) xˆ pˆ + pˆ xˆ exp −i 2 2  2   λ λ (7.31) exp i 3 [sinh(at) − at] Dˆ √ [1 − σ (t)] 2a 2a 3    ma   ib  2 ˆ S (−iat) exp i xˆ exp − xˆ pˆ + pˆ xˆ |ψ2 (0)〉, 2 2 where the notation .σ (t) = cosh (at) + i sinh (at) has been introduced. Let us consider a specific scenario where the initial state is 

  ma   ib  .|ψ2 (0)〉 = exp xˆ pˆ + pˆ xˆ exp −i xˆ 2 |β〉, 2 2

(7.32)

being .|β〉 a coherent state. The exact solution of Eq. (7.31) in the coordinate representation, denoted as .ψex (x, t) = 〈x|ψ2 (t), can be achieved by using the solution derived in Chap. 4. Subsequently, through the application of appropriate algebraic manipulations, involving the rearrangement of terms, it is straightforward to cast the solution in the following form  ma 1/4

   ma  1 ψex (x, t) = √π exp e2(at+b) x 2 (at + b) exp −i 2 2 σ (t)

√  |β|2 2ma β at+b σ ∗ (t)  2 2(at+b) 2 exp − − x + xe β + mae 2 σ (t) 2σ (t)  ∗  σ (t) − 1 λ √ λ2 at+b exp mxe − i 2t σ (t) a 2a  ⎧ √ ⎫ ⎨ 8βλa 3/2 − 2λ2 [cosh(at) − 1] + iλ2 sinh(at) ⎬ exp (7.33) . ⎩ ⎭ 2a 3 σ (t)

.

168

7.2.2.2

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

Perturbative Solution

The perturbative solution is determined at this stage using a strategy that involves expressing the operators .xˆ and .pˆ in terms of the raising and lowering operators .aˆ and .a ˆ † in Eq. (7.30). This converts the Hamiltonian into a more manageable system that can be solved perturbatively using the Matrix Perturbation Method. Specifically, for this case, the unperturbed part is given by .Hˆ 0 = − a2 aˆ 2 + aˆ †2 , while the   perturbation is .Hˆ p = √1 aˆ + aˆ † . To obtain the perturbative solution up to the 2a second order with the same initial state as the exact case, the same steps as in Chap. 4 are followed. Afterward, the inverse two transformations are applied to recover the original state to finally get   exp 12 (at + b) Nβ(2) (t) √ 4a 3 σ (t) [1 + i sinh(2at)]

 ma 1/4 ψap (x, t) =

.

π

[g0 + g1 sinh(at) + g2 cosh(at) + g3 sinh(2at) + g4 cosh(2at)]   ma e2(at+b) x 2 exp −i 2

√  |β|2 2ma β at+b σ ∗ (t)  2 − + xe β + mae2(at+b) x 2 , exp − 2 σ (t) 2σ (t) (7.34) where  g0 =4a 3 + 2 2a 3 βλ   √ (7.35a) + λ2 6β 2 − 2iat + 4max 2 e2(at+b) + 6 2maβxeat+b − 1 , .   √ g1 = − 4iλ 2a 3 β + a 2 mxeat+b   √ (7.35b) + 4iλ2 amx 2 e2(at+b) + 2maβxeat+b + 1 , .   √ g2 = − 4λ 2a 3 β + a 2 mxeat+b   √ (7.35c) + 4λ2 1 − 2β 2 − max 2 e2(at+b) − 2 2maβxeat+b , .  g3 =4ia 3 + 2i 2a 3 βλ   √ (7.35d) + λ2 2at − 2imae2(at+b) x 2 − 2i 2maβxeat+b − i , .     √ √ g4 =2λ 2a 3 β + 2 ma 2 xeat+b + λ2 2β 2 + 2 2maβxeat+b − 3 ,

.

(7.35e)

7.2 First Case

169

with normalization constant   −2 2 (2) 3 β sinh(at)[at − sinh(at)] . N =1 + λ β (t) a9   λ4  4 2 2 2 2 17 + 12β − 12β + 8a t + 12 2β − 3 cosh(at) + 32a 6     +4 7 − 4β 4 cosh(2at) − 12 2β 2 + 1 cosh(3at)    + 4β 4 + 12β 2 + 3 cosh(4at) − 16at sinh(at) . (7.36)

7.2.2.3

Comparison Between the Exact and Perturbative Solutions

Figures 7.3 and 7.4 present an evaluation of the accuracy of the perturbative solution, given by Eq. (7.34), in comparison to the exact solution, denoted by Eq. (7.33). The probability densities .|ψex,ap (x, t)|2 for both solutions are plotted against the position variable x. This analysis is performed at four specific time points (.t = 0, 1, 2, 3) while considering two distinct values of the perturbative parameter .λ. Upon comparing the perturbative solution (represented by the blue dotted line) with the exact solution (represented by the red dashed line), it is evident that a good agreement is achieved for each time instance when a small value of the parameter .λ = 0.01 is employed, as depicted in Fig. 7.3. However, when the magnitude of the perturbative parameter is increased to .λ = 0.1, the approximate solution loses precision and is not expected to be highly accurate. Notably, it only displays noticeable similarities with the exact solution at early times, as observed in Fig. 7.4. Another instance of a noteworthy physical system that can be solved analytically and results in a repulsive oscillator upon the application of the two unitary transformations is represented by the Hamiltonian governing a particle subjected to a uniform gravitational field with linear damping [7, 8]. The corresponding Schrödinger equation for this system is expressed as follows:   2 √ d pˆ |ψ3 (t)〉 = exp (−γ t) − λ mxˆ exp (γ t/2) |ψ3 (t)〉, .i dt 2m

(7.37)

where the damping coefficient is denoted as .γ and .λ acts as a constant gravitational field. By examining the Hamiltonian in the Schrödinger equation, it becomes evident that

.α(t)  = exp (γ t/2). This observation √ arises from the potential in question, .V xμ(t) ˆ = xμ(t), ˆ where .μ(t) = − m exp (γ t/2) and .α(t) = exp (γ t/2). Similar to the initial example in this subsection, two unitary transformations are employed. The first transformation, dependent on time, eliminates the temporal dependence, while the second transformation, independent of time, eliminates the

170

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

Fig. 7.3 The probability densities of the exact and perturbative solutions are represented by the red dashed and blue dotted lines, respectively. These solutions are shown as functions of the position x at times .t = 0, 1, 2, 3, with a fixed value of .λ = 0.01. The calculations for both solutions were performed using the common parameters .m = 1, .a = 1, .b = 0.1, and .β = 2

Fig. 7.4 The probability densities of the exact and perturbative solutions are represented by the red dashed and blue dotted lines, respectively. These solutions are shown as functions of the position x at times .t = 0, 1, 2, 3, with a fixed value of .λ = 0.1. The calculations for both solutions were performed using the common parameters .m = 1, .a = 1, .b = 0.1, and .β = 2

7.2 First Case

171

xˆ pˆ + pˆ xˆ term resulting from the application of the first transformation. These transformations can be expressed as follows:

.

   γt  |ψ3 (t)〉 = exp i xˆ pˆ + pˆ xˆ |φ3 (t)〉, 4  mγ  xˆ 2 |χ3 (t)〉. |φ3 (t)〉 = exp −i 4

.

(7.38)

By applying these transformations to Eq. (7.37), the resulting expression is obtained  2  √ d pˆ mω˜ 2 xˆ 2 .i |χ3 (t)〉 = − − λ mxˆ |χ3 (t)〉, dt 2m 2

(7.39)

with .ω˜ = γ /2. This leads to a system that has well-known exact and approximate solutions, as has been consistently observed throughout this chapter. Then if the Eq. (7.39) is integrated with respect to t and subsequently transformed back to the original system, as .|ψ3 (t)〉, while applying the initial condition .|ψ3 (0)〉 =

represented  2 |β〉. Consequently, the exact solution of this system in the coordinate exp −i mγ x ˆ 4 representation denoted as .ψex (x, t) = 〈x|ψ3 (t) renders to ψex (x, t) =

.

  1 mγ 2 γ t γ t e + x exp −i √ 2π 4 4 η(t)

√  mγ β γ t/2 η∗ (t)  2 |β|2 2 γt exp − − + xe 2β + mγ x e 2 η(t) 4η(t)   ∗  η (t) − 1 λ √ λ2 γ t/2 exp −2 m xe + i 2t η(t) γ γ     3/2 2 + 2λ [cosh(γ t/2) − 1] − iλ2 sinh(γ t/2) βλγ exp − , γ 3 η(t)/4 (7.40)  mγ 1/4

where .η(t) = cosh(γ t/2)+i sinh(γ t/2). In contrast, the perturbative solution of the system described by Eq. (7.37), considering second-order corrections, is expressed as follows:   mγ 1/4 1 mγ 2 γ t γ t (2) .ψap (x, t) =2N (t) e + x exp −i √ β 2π 4 4 γ 3 η(t) [1 + i sinh(γ t)] [h0 + h1 sinh(γ t/2) + h2 cosh(γ t/2) + h3 sinh(γ t) + h4 cosh(γ t)]

√  mγ β γ t/2 η∗ (t)  2 |β|2 exp − − + xe 2β + mγ x 2 eγ t , 2 η(t) 4η(t) (7.41)

172

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

where h0 =

.

γ3 − 2

  √ γ 3 βλ + λ2 6β 2 − iγ t + 2mγ x 2 eγ t + 6 mγ βxeγ t/2 − 1 , .



√ h1 =iλ 2 γ 3 β + γ 2 mxeγ t/2



(7.42a)   √ + 2iλ2 γ mx 2 eγ t + 2 mγ βxeγ t/2 + 2 , . (7.42b)

 2√ γ t/2 3 h2 =λ 2 γ β + γ mxe   γ √ + 4λ2 1 − 2β 2 − m x 2 eγ t − 2 mγ βxeγ t/2 , . 2   3 γ √ h3 =i − i γ 3 βλ + λ2 2γ t − imγ eγ t x 2 − 2i mγ βxeγ t/2 − i , . 2  h4 = − λ

γ 3β +

√



(7.42c)

(7.42d)

  √ mγ 2 xeγ t/2 + λ2 2β 2 + 2 mγ βxeγ t/2 − 3 , (7.42e)

with normalization constant  .

(2)

Nβ (t)

−2

16λ3 β =1−  sinh(γ t/2)[γ t − 2 sinh(γ t/2)] γ9 2λ4  + 6 17 + 12β 4 − 12β 2 + 2γ 2 t 2 − 8γ t sinh(γ t/2) γ     +12 2β 2 − 3 cosh(γ t/2) + 4 7 − 4β 4 cosh(γ t)      −12 2β 2 + 1 cosh(3γ t/2) + 4β 4 + 12β 2 + 3 cosh(2γ t) . (7.43)

Figure 7.5 illustrates a comparison between the probability densities of both solutions at specific time points (.t = 0, 4, 8, 12), considering a perturbative parameter value of .λ = 0.2. The plot reveals an excellent agreement between the second-order perturbative solution (depicted by the red dashed line) and the exact solution (represented by the blue dotted lines) during the initial stages; this is because the product of perturbation strength and time is significantly smaller than unity, satisfying the condition .λt ⪡ 1 which allows the perturbative solution works well. However, as previously indicated, a notable discrepancy arises between both solutions as time progresses beyond the point where the condition .λt ⪢ 1 is not satisfied anymore. This discrepancy is evident in the bottom panels, where the slope of the second-order solution does not match the exact result.

7.3 Second Case

173

Fig. 7.5 The probability densities of the exact and perturbative solutions are represented by the red dashed and blue dotted lines, respectively. These solutions are shown as functions of the position x at times .t = 0, 4, 8, 12, with a fixed value of .λ = 0.2. The calculations for both solutions were performed using the common parameters .m = 1, .γ = 0.5, and .β = 4

It is clear that the accuracy of the perturbative solution depends on the magnitude of the applied perturbation and the evolution time of the system. For larger perturbations, the approximate solution is effective for very short times, while for smaller perturbation values, the approximate solution is more accurate for longer times, always adhering to the constraint .λt ⪡ 1. This fact is intuitively obvious and is confirmed in the first example of this section. Furthermore, it is important to note that problem-specific characteristics play a crucial role. In the second problem discussed in this section, it is worth mentioning that if higher-order perturbative contributions were calculated while satisfying the condition .λt ⪡ 1, the perturbative solution in all four panels of Fig. 7.4 would exhibit a strong agreement with the exact result during the four-time points when .λ = 0.1.

7.3 Second Case Let us now focus on determining the solution for the class of the time-dependent Hamiltonian described by Eq. (7.2b) i

.

∂|ψ(t)〉 = ∂t





√  pˆ 2 + λμ2 (t)V m0 xˆ μ(t) |ψ(t)〉. 2m0

(7.44)

174

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

In this procedure, the central idea is to incorporate time dependence into the first component of the Hamiltonian that influences the evolution of the state vector .|ψ(t)〉, specifically the free particle term. This requirement is essential as it enables the factorization of the time dependence of the Hamiltonian through a time-dependent scale transformation, resulting in a time-independent Hamiltonian, always that the constraint of time frequency is set equal to a constant. To begin this process, the first transformation to be considered is .|ψ(t)〉 = Uˆ1 (t)|φ(t)〉, where   ln |α(t)| (xˆ pˆ + pˆ x) ˆ , Uˆ 1 (t) = exp −i 2

.

(7.45)

which is similar to the first case but with the opposite sign. Then, the transformed Schrödinger equation is i

.

∂|φ(t)〉 = ∂t



 

√ α(t) ˙ pˆ 2 2 − ( x ˆ p ˆ + p ˆ x) ˆ + λμ m x ˆ μ(t) α(t) |φ(t)〉, (t)V 0 2m0 α 2 (t) 2α(t) (7.46)

by choosing .μ(t) in a manner that allows for the factorization of .α 2 (t), specifically by setting .μ(t) = 1/α(t), this leads to i

.

 2  √  1 pˆ α(t)α(t) ˙ ∂|φ(t)〉 (xˆ pˆ + pˆ x) ˆ + λV = 2 m0 xˆ |φ(t)〉. − 2 ∂t α (t) 2m0

(7.47)

  To eliminate the time-dependent terms . xˆ pˆ + pˆ xˆ , it is necessary to apply a second time-dependent unitary transformation. This transformation can be expressed as .|φ(t)〉 = Uˆ 2 (t)|χ (t)〉, where now   m 0 α(t)α(t) ˙ xˆ 2 , Uˆ 2 (t) = exp i 2

.

(7.48)

after applying this transformation to Eq. (7.47), one renders to the new Schrödinger equation i

.

 2  √  1 pˆ m0 3 ∂|χ (t)〉 α (t)α(t) = 2 + ¨ xˆ 2 + λV m0 xˆ |χ (t)〉, 2 ∂t α (t) 2m0

(7.49)

and performing the time-dependent scale transformation !

t

τ=

.

0

lead us to

dt ' , α 2 (t ' )

(7.50)

7.3 Second Case

∂|χ (τ )〉 .i = ∂τ

175



 √  m0 3 pˆ 2 2 + ¨ xˆ + λV m0 xˆ |χ (τ )〉. α (t)α(t) 2m0 2

(7.51)

It is worth noting that in order to achieve a time-independent Hamiltonian, it is desirable that the term proportional to .xˆ 2 be equal to a real constant, which can be achieved by imposing the constraint .α 3 (t)α(t) ¨ = ±Ω20 . This restriction on the timedependent frequency leads to a system that characterizes either a harmonic oscillator when .α 3 (t)α(t) ¨ = Ω20 or a repulsive oscillator when .α 3 (t)α(t) ¨ = −Ω20 . This allows the study of solutions to the Schrödinger equation in both scenarios

√  m0 Ω20 2 pˆ 2 + m0 xˆ |χ1 (τ )〉, . xˆ + λV 2m0 2

√  m0 Ω20 2 ∂|χ2 (τ )〉 pˆ 2 i − m0 xˆ |χ2 (τ )〉. = xˆ + λV ∂τ 2m0 2

∂|χ1 (τ )〉 .i = ∂τ



(7.52a)

(7.52b)

These equations are analogous to the first case discussed in Sect. 7.2. However, it is important to note that in this specific case; the constraints lead to a differential t 2 a 2 +2ta 2 b+a 2 b2 ±Ω2

0 equation with a general solution given by .α(t) = ± , where a a and b are constants. Therefore, the aforementioned procedure is only applicable to a specific kind of function where .α(t) corresponds to a polynomial of degree 2 or 1 in terms of t. In order to gain a better understanding of the above restriction, it is useful to examine a straightforward example.

7.3.1 Example

 √ Let us consider the potential .V xμ(t) ˆ = xˆ (2at + b)−1/2 , being .α(t) = 2at + b, into Eq. (7.44). Then the corresponding Schrödinger equation to solve is √   2 ∂|ψ(t)〉 pˆ λ m .i xˆ |ψ(t)〉, = + ∂t 2m (2at + b)3/2

(7.53)

based on this information, the first transformation to be implemented is   √   i .|ψ(t)〉 = exp − ln | 2at + b| xˆ pˆ + pˆ xˆ |φ(t)〉, 2

(7.54)

this transformation leads to i

.

  2 √ ∂|φ(t)〉 1 pˆ a = − (xˆ pˆ + pˆ x) ˆ + λ mxˆ |φ(t)〉, ∂t 2at + b 2m 2

(7.55)

176

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

1 where the term . 2at+b has been factorized from the involved Hamiltonian. Since .α(t)α(t) ˙ = a, the second transformation to consider is



 ima 2 .|φ(t)〉 = exp xˆ |χ (t)〉, 2

(7.56)

after applying this transformation to Eq. (7.55), the resulting equation becomes  2  √ pˆ 1 ma 2 2 ∂|χ (t)〉 = − xˆ + λ m xˆ |χ (t)〉. .i ∂t 2at + b 2m 2

(7.57)

To find the exact solution of the original system, one can perform the time-dependent "t ' 1 scale transformation .τ = 0 2atdt' +b = 2a ln | 2at+b b |. Integrating the resulting expression over .τ and applying the previously established inverse transformations leads to the exact solution # #  # #   2  1 ## 2at + b ## 1 ## 2at + b ## λ 1 − sinh .|ψ(t)〉 = exp i ln ln 2 # b # 2 # b # 2a 3 (2at + b)1/4  #√ #   ima 2 # # exp −i ln # 2at + b# xˆ pˆ exp xˆ 2 # #    ima 2 λ i ## 2at + b ## ˆ ˆ exp − xˆ D √ [1 − σ (t)] S − ln # b # 2 2 2a 3   √   i ln | b| xˆ pˆ + pˆ xˆ |ψ(0)〉, (7.58) exp 2 where in this particular case .σ (t) = cosh

7.3.1.1



1 2

# # # #  # # 2at+b # # 1 + i sinh ln ln # 2at+b # # b 2 b # .

Exact Solution

Let us consider a specific scenario where the initial state is given by    ma  √   i xˆ 2 |β〉, |ψ(0)〉 = exp − ln | b| xˆ pˆ + pˆ xˆ exp i 2 2

.

(7.59)

with .|β〉 being a coherent state. The solution in the coordinate representation can be obtained by following a similar procedure as in Sect. 7.2.2.1 and following the steps presented in Chap. 4. After performing algebraic manipulations, the resulting solution is given as

7.3 Second Case

177

  1/4 ma 1 ma exp i x2 √ π (2at + b) 2 (2at + b) σ (t) √  |β|2 2ma β ma σ ∗ (t) + β2 + x2 x− exp − √ 2 2σ (t) 2at + b σ (t) 2at + b # # # #     ⎧ √ # # # 2at+b # ⎫ 1 2 ⎨ ⎬ 8βλa 3/2 − 2λ2 cosh 12 ln # 2at+b b # − 1 + iλ sinh 2 ln # b # exp ⎩ ⎭ 2a 3 σ (t) 

ψex (x, t) =

.

 exp

7.3.1.2

# # √  # 2at + b # λ mx λ2 σ ∗ (t) − 1 # . # − i 3 ln # √ σ (t) b # 4a a 2at + b

(7.60)

Perturbative Solution

Similar to the exact case, the repulsive system with a linear anharmonic term obtained in Eq. (7.57) can be addressed using the same procedure outlined in Sect. 7.2.1.1 and Chap. 4. In this specific instance, the solution of the original wave function in the coordinate representation, up to the second-order correction, takes the following form: 

ψap (x, t) =

.

1/4

(2)

Nβ (t) #   # √ # # 4a 3 σ (t) 1 + i sinh ln # 2at+b b # # # # #    1 ## 2at + b ## 1 ## 2at + b ## + q q0 + q1 sinh ln # ln cosh 2 2 b # 2 # b # # #   #  # # # 2at + b # # # + q4 cosh ln # 2at + b # + q3 sinh ln ## # # b b #   ma exp i x2 2 (2at + b) √  |β|2 2ma β ma σ ∗ (t) 2 2 exp − β + + x x− , √ 2 2σ (t) 2at + b σ (t) 2at + b (7.61) ma π (2at + b)

where  q0 =4a 3 + 2 2a 3 βλ % $  # # 2 # 2at + b # 4max 2ma 2 2 #+ +6 βx − 1 , . + λ 6β − i ln ## b # 2at + b 2at + b

.

(7.62a)

178

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

  2 3 q1 = − 4iλ 2a β + a

m x 2at + b



%  2ma amx 2 + 4iλ (7.62b) + βx + 1 , . 2at + b 2at + b   m 2 3 2a β + a q2 = − 4λ x 2at + b % $  2ma amx 2 2 2 + 4λ 1 − 2β − (7.62c) −2 βx , . 2at + b 2at + b  q3 =4ia 3 + 2i 2a 3 βλ % $ #  # # 2at + b # 2ma 2ima 2 2 #− + λ ln ## (7.62d) x − 2i βx − i , . b # 2at + b 2at + b % $    m 2ma 2 2 2 q4 =2λ 2a 3 β + 2 a x + λ 2β + 2 βx − 3 , 2at + b 2at + b (7.62e) $

2

with normalization constant −2 =1 Nβ(2) (t)  # #  # # # #    2 1 ## 2at + b ## 1 ## 2at + b ## 1 ## 2at + b ## 3 β sinh ln ln ln +λ − sinh 2 # b # 2 # b # 2 # b # a9 #  #  # 2at + b # 2 λ4 4 2 # # − 12β + 2 ln 17 + 12β + # b # 32a 6 # # #   #     # 2at + b # 1 ## 2at + b ## 2 4 # # + 12 2β − 3 cosh ln + 4 7 − 4β cosh ln # 2 # b # b # # #    3 ## 2at + b ## 2 ln − 12 2β + 1 cosh 2 # b # # #    # 2at + b # # + 4β 4 + 12β 2 + 3 cosh 2 ln ## b # # # # #   # 2at + b # 1 ## 2at + b ## # # − 8 ln # (7.63) ln sinh . b # 2 # b #

 .

7.3 Second Case

179

Fig. 7.6 The probability densities of the exact and perturbative solutions are represented by the red dashed and blue dotted lines, respectively. These solutions are shown as functions of the position x at times .t = 0, 1, 2, 3, with a fixed value of .λ = 0.1. The calculations for both solutions were performed using the common parameters .m = 1, .a = 1, .b = 1, and .β = 4

7.3.1.3

Comparison Between the Exact and Perturbative Solutions

Figure 7.6 presents a comparison between the probability densities of the secondorder perturbative solution (depicted by the red dashed line) and the exact solution (represented by the blue dotted lines) at specific time points (.t = 0, ; 1, ; 2, ; 3), with a perturbative parameter value of .λ = 0.1. As expected, the plot demonstrates an excellent agreement between the two solutions during the initial stages. This is because the product of the perturbation parameter and time, .λt, is significantly smaller than unity, satisfying the condition .λt ⪡ 1, which allows the perturbative solution to provide an excellent approximation to the exact one.

7.3.2 Other Examples The preceding examples highlight the role of the two unitary time-dependent transformations in rendering a newly transformed Hamiltonian time independent, allowing for the description of a harmonic or repulsive system with a perturbative term, and making it easy to apply the perturbative method when an exact solution is not feasible. It should be noted that the methodology described in the first case can be extended to other quantum systems characterized by time-dependent masses. Prominent examples include the most studied physical systems like the

180

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

harmonic oscillator with a mass growing with time or rapidly growing with time, a harmonic oscillator with a strongly vibrating mass, and the Caldirola-Kanai oscillator, to name just a few. Such systems have been studied previously in the literature to calculate their propagators and well-known approaches, such as the Lewis-Riesenfeld invariant operator method [9], Feynman path integral [10–13], Schwinger method [12–17], Green function [18, 19], or the initial value problem method [20], have been employed to solve their corresponding Schrodinger’s equations exactly. However, the first three methods may encounter mathematical challenges, which involve the application of the integrals of the motion to calculate the propagator in coordinate representation. Furthermore, some of these equations’ integrals are not straightforward to analyze. Even seemingly simple methods, such as the initial value problem method, require several steps, including solving the Heisenberg equation for momentum and position operators, converting the system’s propagator into an eigenvalue equation, and obtaining a two-point characteristic function before substituting it into the Schrödinger equation. Additionally, the effectiveness of these methods varies depending on the specific problem, with some techniques performing well for certain problems while others may not. Therefore, this subsection demonstrates that the aforementioned physical systems can be effectively simplified by applying the two unitary transformations, reducing them to a harmonic or repulsive oscillator combined with a potential. In this context, the introduction of a linear anharmonic potential serves as a mathematical convenience, simplifying or enabling the derivation of an exact solution. Only the exact solutions are provided in this chapter, while the perturbative solutions are left as exercises for the readers to familiarize themselves with the concepts presented in this chapter.

7.3.2.1

Harmonic Oscillator with a Quadratically Growing Mass

Let us examine a system that comprises a harmonic oscillator with a mass that quadratic increases over time, as mentioned in reference [21, 22]. Additionally, this system includes a linear anharmonicity term, which can be effectively described by the Schrödinger equation i

.

  √ d mω2 xˆ 2 pˆ 2 2 + + λ m + ϵt) x ˆ |ψ(t)〉 |ψ(t)〉 = + ϵt) (1 (1 dt 2 2m (1 + ϵt)2 (7.64)

such a scenario describes the case where the mass of the oscillator is increasing, with the parameter .ϵ being time independent and small compared to the unity [21, 22]. It is worth noting that the Hamiltonian in the Schrödinger equation can be obtained by

 2 2 2 √ considering the potential .V xμ(t) ˆ = xμ(t) ˆ + ω μ2λ(t)xˆ , with .μ(t) = m (1 + ϵt) and .α(t) = 1 + ϵt can be distinguished clearly in this case. The subsequent steps involve the use of two time-dependent unitary transformations, =  namely: .|ψ(t)〉 

  mϵ exp 2i ln | (1 + ϵt) | xˆ pˆ + pˆ xˆ |φ(t)〉 and .|φ(t)〉 = exp −i 2(1+ϵt) xˆ 2 |χ (t)〉.

7.3 Second Case

181

These transformations lead us to a time-independent harmonic oscillator, which is perturbed by a linear anharmonic potential, and that was solved in the first section d .i |χ (t)〉 = dt



√ pˆ 2 mω2 xˆ 2 + + λ mxˆ |χ (t)〉. 2m 2

(7.65)

The logical extension of the aforementioned result naturally leads to considering a generalization where the expression .α(t) = (1 + ϵt)n , with .n ≥ 1, replaces .(1 + ϵt). Within this context, the transformations take the following form: .|ψ(t)〉 =  

  mnϵ ˜ ˜ exp 2i ln | (1 + ϵt)n | xˆ pˆ + pˆ xˆ |φ(t)〉 and .|φ(t)〉 = exp −i 2(1+ϵt) xˆ 2 |χ˜ (t)〉. Applying these transformations leads to a significant change in the Schrödinger equation governing the system, resulting in the transformed equation d .i |χ(t)〉 ˜ = dt



   √ nϵ 2 (1 − n) mxˆ 2 pˆ 2 2 + ω + + λ mxˆ |χ˜ (t)〉. 2m 2 (1 + ϵt)2

(7.66)

It is important to note, however, that the new Hamiltonian in the Schrödinger 2 equation is no longer time independent due to the presence of the term . nϵ (1−n) . (1+ϵt)2 To obtain a transformed time-independent Hamiltonian, it is necessary to set .α(t) as a strictly linear function of t, which occurs when .n = 1. This result highlights the reason for the limitation in certain systems where the solution is constrained to .α(t) with only linear terms in t.   By implementing the transformation .|χ (t)〉 = exp i ω2λ√m pˆ |υ(t)〉 into Eq. (7.65) and subsequently integrating the resulting expression over time, and transformed it back to .|χ (t)〉, yields the following result in terms of position and momentum operators   2   it  2 λ t λ |χ( t)〉 = exp i 2 exp i 2 √ pˆ exp − pˆ + m2 ω2 xˆ 2 2m 2ω ω m  λ exp −i 2 √ pˆ |χ (0)〉. (7.67) ω m

.

It is worth mentioning that the solution presented above is composed of the product of three exponential operators: the first one (from left to right) is a displacement operator, which acts over the propagator related to the harmonic oscillator, followed by another displacement operator. Moreover, one can use results reported by Arevalo et al. [23, 24] to factorize the exponential operator of the harmonic oscillator as follows:       mω tan ωt it  2 2 2 2 2 . exp − pˆ + m ω xˆ pˆ 2 exp −i sin (ωt) xˆ 2 = exp −i 2m 2mω 2  ωt  tan 2 2 exp −i (7.68) pˆ , 2mω

182

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

and transforming back to the original representation .|ψ(t)〉, which in the coordinate representation is .ψ(x, t) = 〈x||ψ(t)〉, then one finds the exact formal solution    2 √ d mϵx 2 λ t exp ln |1 + ϵt|x exp −i .ψ(x, t) = 1 + ϵt exp i dx 2 (1 + ϵt) 2ω2 $ %  ωt     mω tan 2 d 2 d λ 2 exp exp i exp −i sin x √ (ωt) 2 2mω dx 2 ω2 m dx $ %     mϵ  tan ωt d2 λ d 2 exp i exp i x 2 ψ(x, 0). exp − √ 2mω dx 2 2 ω2 m dx (7.69) The next step is to evaluate the exact solution with a given initial condition; we   2 x 1 choose as an initial wave function a Gaussian .ψ(x, 0) = √ √ exp − 4κ 2 , κ 2π

with a width given by .κ. Consequently, the application of the first two exponential operators to the initial state yields    mϵ  x2 1 d λ x 2 exp − 2 = exp i . √ exp − 2 √ 2 4κ ω m dx κ 2π   2  d 1 1 − 2iκ 2 mϵ x λ  √ exp − 2 √ , exp − (7.70) 4 κ2 ω m dx κ 2π  d f (x) = and using the known action of the translation operator, .exp λ dx f (x + λ), acting on a given function .f (x) [6, 25], one arrives to    mϵ  d x2 1 λ exp i x 2 exp − 2 = . √ exp − 2 √ 2 4κ ω m dx κ 2π ⎡  2 ⎤  x − ω2λ√m 1 1 κ ⎥ ⎢ (7.71) √ √ √ exp ⎣− ⎦, 2 4ζ 2π 1 − 2iκ mϵ ζ one obtains a new Gaussian function which is now displaced a distance . ω2λ√m with 2

κ a modified width denoted by .ζ = 1−2iκ 2 mϵ . Substituting the above result into Eq. (7.69) and applying the product of the exponential operators of the harmonic oscillator on this Gaussian function, the following result is obtained ,  2 1 1 + ϵt λ t .ψ(x, t) = exp i 2 √ i 2ω κ 2π cos (ωt) + sin (ωt) 2mωζ

   d mϵx 2 λ d exp −i exp exp ln |1 + ϵt| x √ dx 2 (1 + ϵt) ω2 m dx

7.3 Second Case

183

⎧  ωt 

2 ⎫ imω 1 ⎨ ⎬ tan + λ 2 2 4ζ    x − exp − √ ⎩ cos (ωt) + i sin (ωt) ⎭ ω2 m 1 + 2iζ mω tan ωt 2 2mωζ ⎡ ⎢ exp ⎣

2 λ√ 4ζ ω2 m  ωt  imω 1 2 tan 2 + 4ζ 

⎤

   mω ωt ⎥ − tan x2 , ⎦ exp −i 4 2 2 4mω λ2

(7.72) where it has been the Blinder’smethod,  employed  which involvesapplying the   ∂2 ∂ −1/2 exp −x 2 /4z identity . ∂x = z z−1/2 exp −x 2 /4z , a method 2 ∂z which proves to be advantageous when applying the evolution operator to Gaussian wave packets such as the ones given in [6, 25, 26], followed by the procedure outlined in [23]. Subsequently, by applying the translation and dilatation operators, one yields the final solution , ψ(x, t) =

.

1 + ϵt √ κ 2π

1 cos (ωt) +

i 2mωζ

 2 λ t exp i 2 2ω sin (ωt)

⎡  2 λ√  mϵ  2 4ζ ω m ⎢ exp −i (1 + ϵt) x 2 exp ⎣ imω  ωt  2 tan 2 2 +  exp

−

imω 2

tan

cos (ωt) +

⎤ 1 4ζ

−

λ2 4ζ mω4

 ωt 

1 2 + 4ζ i 2mωζ sin (ωt)

$

1   × x (1 + ϵt) + 1− ω m 1 + 2iζ mω tan ωt 2 2     ωt λ mω tan x (1 + ϵt) + 2 √ . exp − i 2 2 ω m

7.3.2.2

⎥ ⎦

λ √ 2

% 2 

(7.73)

Forced Caldirola–Kanai Oscillator

Let us now determine the exact solution of the Schrödinger equation for a widely recognized quantum system, specifically, the quantum Caldirola-Kanai oscillator [27–32], incorporating an extra linear anharmonicity term i

.

  2 √ d pˆ mω2 xˆ 2 |ψ(t)〉 = exp (−σ t) + exp (σ t) + λ mxˆ exp (σ t/2) |ψ(t)〉, dt 2m 2 (7.74)

184

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

where .σ represents a real parameter with frequency dimension, which is commonly referred to as the damping parameter. The Hamiltonian associated with the Schrödinger equation above can be derived by considering the expression

√  2 2 √ .V x ˆ m exp (σ t/2) = xˆ m exp (σ t/2) + mω xˆ 2λexp(σ t) . In this scenario, it is necessary to apply the unitary transformations 

  σt  .|ψ(t)〉 = exp i xˆ pˆ + pˆ xˆ |φ(t)〉, . 4  mσ  |φ(t)〉 = exp −i xˆ 2 |χ (t)〉, 4

(7.75a) (7.75b)

to obtain   2 √ d pˆ mxˆ 2 2 .i |χ (t)〉 = + ω˜ + λ mxˆ |χ (t)〉, dt 2m 2

(7.76)

 where .ω˜ = ω2 − σ 2 /4 is a modified frequency. Following the same procedure and the same initial condition as used in the earlier example, the solution for the time-dependent Schrodinger equation for the forced Caldirola–Kanai oscillator in the coordinate representation reads as  2  

mσ   λ t 2 exp σ4t exp i 2ω˜ 2 exp −i 4 exp (σ t) x .ψ(x, t) =  √ 2 κ 2π cos (ωt) ˜ + 2i 1−imσ2 κ sin (ωt) ˜ ⎧ ⎪ ⎪ ⎪ ⎨ exp



⎪ imω˜ ⎪ ⎪ ⎩ 2 tan ⎛

exp ⎝−

mω˜

  2 λ 1−imσ κ 2 √ 4κ 2 ω˜ 2 m



imω˜ 2

ωt ˜ 2

× x exp ⎩  −i

+

 tan

cos (ωt) ˜ + ⎧ ⎨

exp





4κ

1−imσ κ 2 4κ 2

ωt ˜ 2



+

σt 2

4κ 2 mω˜ 4

⎪ ⎪ ⎪ ⎭

1−imσ κ 2 4κ 2

2i 1−imσ κ 2 mω˜ 4κ 2



−

⎫ ⎪  ⎪ ⎪ λ2 1 − imσ κ 2 ⎬

sin (ωt) ˜

  ⎫2 ⎞ √ ˜ ⎬ 2iλ mκ 2 tan ωt 2 ⎟     ⎠ +  ⎭ ωt ˜ ω˜ 1 + imκ 2 2ω˜ tan 2 − σ

2     mω˜ ωt ˜ σt λ tan x exp + 2√ . 2 2 2 ω˜ m

(7.77)

It is worth pointing out that one also can obtain the exact solution of the Schrodinger equation for the inverted Caldirola-Kanai oscillator [33, 34]

7.3 Second Case

185

 2  d pˆ mω2 xˆ 2 .i |ψ(t)〉 = exp (−σ t) − exp (σ t) |ψ(t)〉, dt 2m 2

(7.78)

by applying the same unitary transformations defined into Eq. (7.55), one gets  2  d pˆ mω˜ 2 xˆ 2 .i |χ (t)〉 = − |χ (t)〉, dt 2m 2

(7.79)

 where .ω˜ = ω2 + σ 2 /4 is now the modified frequency. Upon transforming this expression back to .|ψ(t)〉 and subsequently applying the Gaussian initial state as demonstrated in the two preceding examples, the exact solution of this system in the coordinate representation is then given by  σt 1 1 exp ψ(x, t) =  √ 2 4 2i 1−imσ κ κ 2π cosh (ωt) ˜ + mω˜ 4κ 2 sinh (ωt) ˜ $ 3

  4% ˜ − σ + imκ 2 4ω˜ 2 − σ 2 sinh (ωt) ˜ m exp (σ t) x 2 2ω˜ cosh (ωt)   . exp − 8mκ 2 ω˜ cosh (ωt) ˜ + 4 i + mκ 2 σ sinh (ωt) ˜ (7.80)

.

7.3.2.3

Particle with a Hyperbolic Growing Mass

Let us conclude this subsection with the study of a particle with a hyperbolic growing mass [35] with a linear potential term. The dynamic of such a system is described by the Schrödinger equation   d λ√ pˆ 2 2 .i mxˆ cosh (ϒt) |ψ(t)〉. |ψ(t)〉 = ϒ + dt ϒ 2m cosh2 (ϒt)

(7.81)

In this case, one can easily identify that .α(t) = cosh(ϒt) . Therefore the two timeϒ dependent unitary transformations to consider are of the form # #  i ## cosh (ϒt) ## .|ψ(t)〉 = exp ln ˆ |φ(t)〉, . #(xˆ pˆ + pˆ x) 2 # ϒ   mϒ 2 |φ(t)〉 = exp −i tanh (ϒt) xˆ |χ (t)〉, 2 

(7.82a) (7.82b)

and after applying them to Eq. (7.81), it yields a new Schrödinger equation describing the well-known system of a repulsive oscillator with a linear anharmonic term

186

7 Eliminating the Time Dependence for a Class of Time-Dependent Hamiltonians

 2  √ mϒ 2 2 pˆ d − xˆ + λ mxˆ |χ (t)〉, |χ (t)〉 = .i 2m 2 dt

(7.83)

by employing the two inverse transformations outlined in Eq. (7.82) to the aforementioned equation and subsequently multiplying both sides of the resulting expression by the bra .〈x|, one can obtain the exact solution of the original system in the coordinate representation  ψ(x, t) =

.

#  #  # cosh (ϒt) # d λ2 t cosh (ϒt) # #x exp ln exp −i # # dx ϒ ϒ 2ϒ 2    m λ d exp −i ϒ tanh (ϒt) x 2 exp − 2 √ 2 ϒ m dx     2  imϒ i ϒt d 2 exp sinh x exp tanh (ϒt) 2 dx 2 2 2mϒ 2    2  d λ i ϒt d exp ψ(x, 0). exp tanh √ 2 dx 2 2mϒ 2 ϒ 2 m dx

Taking as an initial state, .ψ(x, 0) = √  ψ(x, t) =

.

1 cosh (ϒt)  √ ϒ κ 2π

1 √

κ 2π

(7.84)

  x2 exp − 4κ 2 , then one finally obtains 1

cosh (ϒt) +



i 2mκ 2 ϒ

 λ2 t exp −i 2ϒ 2 sinh (ϒt)

    √ i  2 √ ϒt λ mxϒ + 2λ cosh tanh mxϒ − (ϒt) 2 2ϒ 3   ⎧ ⎡ ⎤  ϒt  1 ⎨ − i mϒ 2 tanh 2 4κ 2 ⎦    exp − ⎣ ⎩ 1 sinh (ϒt) cosh (ϒt) + 2i exp

mϒ



4κ 2

   √ 2iκ 2 λ m tanh ϒt x cosh (ϒt) 2 × +   2 3 ϒ ϒ − 2imκ ϒ tanh ϒt 2 ⎡ ⎤ 2  λ√ λ2 ⎥ 4κ 2 ϒ 2 m ⎢ − 2 4 ⎦. exp ⎣     1 4κ ϒ m − i mϒ tanh ϒt 4κ 2

2

2

(7.85)

References

187

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27. T. Osotchan, S. Pepore, P. Winotai, U. Robkob, Path integral for a harmonic oscillator with time-dependent mass and frequency. ScienceAsia 32, 173–179 (2006) 28. P. Caldirola, Forze non conservative nella meccanica quantistica. Il Nuovo Cimento 18(9), 393–400 (1941) 29. E. Kanai, On the quantization of the dissipative systems. Prog. Theor. Phys. 3(4), 440–442 (1948) 30. V. Manko, V. Dodonov, Invariants and the Evolution of Nonstationary Quantum Systems (Nova Science Publishers, Hauppauge, 1989) 31. H. Dekker, Classical and quantum mechanics of the damped harmonic oscillator. Phys. Rep. 80(1), 1–110 (1981) 32. P. Exner, Complex-potential description of the damped harmonic oscillator. J. Math. Phys. 24(5), 1129–1135 (1983) 33. S. Baskoutas, A. Jannussis, Quantum mechanics of the inverted Caldirola-Kanai oscillator. Il Nuovo Cimento B 107(3), 255–267 (1992) 34. S. Baskoutas, A. Jannussis, R. Mignani, Dissipative tunnelling of the inverted Caldirola-Kanai oscillator. J. Phys. A Math. General 27(6), 2189–2196 (1994) 35. I. Ramos-Prieto, A. Espinosa-Zuñiga, M. Fernández-Guasti, H.M. Moya-Cessa, Quantum harmonic oscillator with time-dependent mass. Mod. Phys. Lett. B 32(20), 1850235 (2018)

Index

B Binary waveguide array, 122 comparison of solutions, 131 electric field, 128 electric field to second order, 129 equations system, 122 exact solution, 124 first order solution, 126 initial condition, 123 matrix perturbative solution, 125 normalization constant, 128 normalized first-order solution, 128 normalized wave function to second order, 129 second order solution, 129 small rotation approximation, 124 third order electric field, 131 third order normalized wave function, 131 third order solution, 130 zero order solution, 125

F Forced Caldirola–Kanai oscillator, 183

D Dyson series method, 40 constant perturbation, 43 wavefunction to order n, 42

L Lindblad master equation, 138 Lossy cavity with a Kerr medium, 143 distance between density matrices, 152 exact solution, 143 initial copherent state, 144 mean photon number, 149 solutions comparation, 148

E Exact solution, 143

H Harmonic oscillator with a cubic term first order perturbative solution, 85 second order perturbative solution, 88 Harmonic oscillator with a linear term coherent state as initial state, 72 comparison between the exact and the approximated solutions, 81 exact solution, 70 first order perturbative solution, 77 perturbative solution, 75 perturbative solution up to second order, 81 second order perturbative solution, 80 zero order perturbative solution, 76 Harmonic oscillator with a quadratically growing mass, 180

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Soto-Eguibar et al., The Matrix Perturbation Method in Quantum Mechanics, https://doi.org/10.1007/978-3-031-48546-6

189

190 M Matrix Perturbation Method, 47 connection with the standard timeindependent pertubation theory, 63 Dyson series in the matrix method, 66 examples, 69 first order correction, 49 harmonic oscillator with a cubic term, 83 harmonic oscillator with a linear term, 70 higher order corrections, 60 Lindblad master equation, 137 first order correction, 139 higher order corrections, 142 second order correction, 141 lossy cavity with a Kerr medium fisrt order solution, 146 second order correction, 146 matrix .M, 51 normalization constant, 61 repulsive quadratic potential, 90 second order correction, 55 trapped ion, 110 wave function to first order, 55 wave function to second order, 59

P Particle with a hyperbolic growing mass, 185 Perturbative solution for the Rabi model, 121

Q Quadratic repulsive potential plus a linear term comparison of the exact and the perturbative solutions, 103 exact solution, 93 first order perturbative solution, 97 normalized perturbative solution, 101 perturbative solution, 97 perturbative solution up to second order, 99 second order perturbative solution, 98 zero order perturbative solution, 97

Index R Rayleigh-Schrödinger theory, 1 Repulsive quadratic potential exact solution, 90 T Time-dependent perturbation theory, 22 constant perturbation, 32, 45 discrete spectrum, 23 finite time perturbation, 28 hydrogen ionization probability transition rate, 40 infinite one-dimensional well, 29 mixed sprectrum, 33 monochromatic perturbation, 36 probability of ionization of a hydrogen atom, 39 transition probabilities, 44 variation of constants, 23 Time-independent perturbation theory, 1 discrete degenerated spectrum, 14 discrete non-degenerated spectrum, 3 energy up to second order, 7 first-order correction to the eigenvectors, 5 first-order correction to the energy, 5, 6 first-order correction to the wave function, 6 non-perturbed Hamiltonian, 2 one dimensional harmonic oscillator with a cubic perturbation, 8 perturbed eigenvalue problem, 2 second order correction to the energy, 6, 7 second order correction to the wavefunction, 7 three dimensional isotropic harmonic oscillator with a xy perturbation, 16 unperturbed equation, 2 wavefunction up to second order, 8 Trapped ion comparison of solutions, 119 first order solution, 110 high intensity regime, 110 second order solution, 115 Trapped ion Hamiltonian, 109