Quantum Optics Applications 9781774690512, 9781774692363

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Quantum Optics Applications

Quantum Optics Applications

Nelson Bolivar

www.arclerpress.com

Quantum Optics Applications Nelson Bolivar

Arcler Press 224 Shoreacres Road Burlington, ON L7L 2H2 Canada www.arclerpress.com Email: [email protected] HERRN(GLWLRQ2 ISBN:(HERRN)

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ABOUT THE AUTHOR

Nelson Bolivar is currently a Physics Professor in the Physics Department at the Universidad Central de Venezuela, where he has been teaching since 2007. His interests include quantum field theory applied in condensed matter. He obtained his PhD in physics from the Universite de Lorraine (France) in 2014 in a joint PhD with the Universidad Central de Venezuela. His BSc in physics is from the Universidad Central de Venezuela.

TABLE OF CONTENTS

List of Figures ........................................................................................................xi List of Abbreviations ............................................................................................xv Preface........................................................................ .................................. ....xvii Chapter 1

Fundamentals of Quantum Optics ............................................................ 1 1.1. Introduction ........................................................................................ 2 1.2. Development of Quantum Optics (QO) and the Photon Theory of Light ................................................................................. 3 1.3. Key Findings ....................................................................................... 7 1.4. Fundamental Concepts ....................................................................... 7 1.5. Coincidence Correlation ..................................................................... 9 1.6. Quantum Entanglement ...................................................................... 9 1.7. Quantum Teleportation ..................................................................... 10 1.8. Quantum Information Processing...................................................... 11 1.9. Quantum Communication ................................................................ 12 1.10. Quantum Electronics ...................................................................... 13 1.11. Quantum Optics (QO) and Many-Worlds Interpretation (MWI) ...... 14 References ............................................................................................... 22

Chapter 2

From Quantum Optics to Quantum Technologies .................................. 31 2.1. Introduction ...................................................................................... 32 2.2. Photons ............................................................................................ 36 2.3. Cavity QED ...................................................................................... 42 2.4. Trapped Ions ..................................................................................... 43 2.5. Optomechanics: Quantum Optics (QO) with Mechanical Oscillators...................................................................................... 44 2.6. Quantum Technologies ..................................................................... 49 2.7. Summary and Outlook...................................................................... 63 References ............................................................................................... 65

Chapter 3

Integration of Quantum Optics and Photonics ........................................ 79 3.1. Introduction ...................................................................................... 80 3.2. Plasmonics and Nanophotonics: Controlling Optical Fields and Propagation on the Nanoscale ....................................... 84 3.3. Coherent Electromagnetic Fields: Attosecond Time Scales and X-Ray Photon Energies ............................................................ 88 3.4. Optomechanical Interactions: From Single-Molecule Mechanics to the Macroscopic Quantum States ............................. 91 3.5. Seeing Beyond the Diffraction Limit and New Imaging Modalities .... 95 3.6. Creating and Controlling Quantum Coherence with Light................. 97 3.7. Controlling Molecules with Light and Light with Molecules ............ 100 3.8. Observing the Universe: Optics and Photonics for Astronomy and Astrophysics .......................................................................... 103 References ............................................................................................. 106

Chapter 4

Laser Spectroscopy and Quantum Optics ............................................. 115 4.1. Introduction .................................................................................... 116 4.2. Laser Sources Along with Other Trading Tools ................................. 116 4.3. Ultrasensitive Spectroscopy ............................................................ 118 4.4. Atomic Hydrogen’s Accuracy: Precision Spectroscopy and Spectral Resolution ...................................................................... 118 4.5. Optical Frequency Metrology ......................................................... 121 4.6. Manipulating Matter by Using Light ................................................ 123 4.7. Laser Cooling and Trapping............................................................. 123 4.8. From Laser Spectroscopy Techniques to Quantum Optical Techniques ................................................................................... 125 4.9. Recent Trends and Advances ........................................................... 126 References ............................................................................................. 134

Chapter 5

Quantum-Mechanical Concepts in Optical Coherence ......................... 143 5.1. Introduction .................................................................................... 144 5.2. History............................................................................................ 146 5.3. The Quantum Optic’s Present Status................................................ 153 5.4. Optical Frequency Comb Techniques and Laser-Based Precision Spectroscopy ............................................. 154 References ............................................................................................. 164

viii

Chapter 6

Mode Converter Linear Optical Devices ............................................... 171 6.1. Introduction .................................................................................... 172 6.2. The Mode Converter Basis Set ......................................................... 172 6.3. Derivation of Example Results......................................................... 177 References ............................................................................................. 181

Chapter 7

Fundamentals of Nonlinear Quantum Optics ........................................ 185 7.1. Introduction .................................................................................... 186 7.2. Second-Order Effects ...................................................................... 188 7.3. Third-Order Effects .......................................................................... 189 7.4. Parametric Processes ....................................................................... 191 7.5. Electro-Optics ................................................................................. 192 7.6. Magneto-Optics .............................................................................. 192 7.7. Photorefractivity ............................................................................. 193 7.8. Current Research in Nonlinear Quantum Optics (QO) .................... 194 References ............................................................................................. 196

Chapter 8

Recent Advances in Quantum Optics.................................................... 201 8.1. Introduction .................................................................................... 202 8.2. Quantum Measurements................................................................. 202 8.3. Two-Photon Interferometry.............................................................. 205 8.4. Nonlocal Cancellation of Dispersion .............................................. 209 8.5. Dynamic Phase of Electromagnetic Field ........................................ 210 8.6. Quantum Cryptography .................................................................. 210 8.7. Quantum Computing ...................................................................... 214 References ............................................................................................. 216 INDEX ................................................................................................... 221

ix

LIST OF FIGURES Figure 1.1. Setup for the quantum optics experiments showing the source of polarizationentangled photons, the ULWBS, and the detection system Figure 1.2. Electromagnetic propagation of light waves Figure 1.3. Quantum optics experiment Figure 1.4. Illustration of quantum entanglement Figure 1.5. Quantum teleportation phenomena Figure 1.6. Quantum information processing Figure 1.7. Illustration of quantum communication Figure 1.8. New components in quantum electronics Figure 1.9. Optical illustration of many world interpretation Figure 1.10. Illustration of diverging timelines via many world interpretations Figure 2.1. Applications of the quantum technology Figure 2.2. Creation of photons in an atom Figure 2.3. A superposition of Fock states |1i and 0i is formed inside a cavity. The mirror oscillates around diverse equilibrium separations corresponding to every Fock state, thus producing a superposition whose components comprise discrete coherent states at    

      this superposition could be certified through probing the coherence among the cavity     Figure 2.4. Illustration of a quantum simulator Figure 2.5. Applications of quantum computing Figure 2.6. “Fusion gates” realize a parity measurement through a single polarizing beam splitter followed through measurement. They could be utilized to combine entangled multiphoton states into bigger cluster states, resources for the universal quantum computing through single-qubit measurement alone Figure 2.7. Spatially detached qubits (here ions stuck in separate cavities) could be entangled through interference on a beam splitter and the emission of photons followed through photon detection. The deletion of which-path information on the beam splitter takes to the entanglement Figure 3.1. Diagram illustration of the 2D scanning mechanism utilizing quantum optics

Figure 3.2. Illustration of probable integration of numerous new nanophotonic devices, based on plasmonic response and photonic crystals, for a high-performance communications application Figure 3.3. Probing and generating highly restrained plasmons on a graphene surface utilizing a plasmonic excitation of a suitably shaped metal tip. The grapheme plasmon is tunable through carrier density and restricted to below than 1/100 of the free-space wavelength of the light Figure 3.4. Schematic representation of HHG (high-harmonic generation) procedure during a solo optical cycle of an ultrashort laser pulse (top). The spectrum of radiation generated for excitation with femtosecond laser pulses of numerous wavelengths (bottom) and correlation trace presenting attosecond (as) pulse duration (middle) Figure 3.5. Novel structural analysis on an enzyme-linked to sleeping sickness through means of x-ray diffraction with pulses from the LCLS XFEL. This exhibits the “diffract before destroy” principle Figure 3.6. Optical fields with orbital angular momentum could be utilized to apply exactly controlled torque to dielectric particles. Through pairing molecules to this article, one could extend former investigations of the effect of force on individual molecules to inspect the effect of torque Figure 3.7. Combination of the “laser tweezers” method to apply specific forces to individual molecules with details of the consequences of the force utilizing singlemolecule spectroscopy through FRET Figure 3.8. Cavity optomechanics inquiries are presently being pursued on structures from individual atoms to macroscopic masses, covering more than 10 orders of magnitude in frequency and 20 orders of magnitude in mass Figure 3.9. Structure of periodic rings in neurons exposed by super-resolution STORM microscopy Figure 3.10. Optical lattices formed by interfering laser beams allow precise tuning of interactions among atoms, allowing the formation of new states of quantum simulators and matter Figure 3.11. NV centers in the diamond give a solid-state material structure with suitable optical access to a quantum structure having extended spin coherence times. In isotopically pure carbon, spin coherence times of a fraction of a second had been perceived at room temperature Figure 3.12. Photosystem in which energy is composed in a light-harvesting complex and transmitted to the reaction center. The latest research shows the role of quantum entanglement and coherence in the energy transfer procedure Figure 3.13. Photo driven unidirectional rotation has been revealed in molecular motor founded on chiral molecules Figure 3.14. Production of a 3D woodpile photonic crystal structure utilizing photopolymerization with a 532 nm laser. The nonlinearity improves resolution and xii

also allows 3D patterning. The structure could be tuned by the laser power, as revealed in structures (a) and (b) Figure 3.15. Preparation of a hybrid CMOS imaging sensor. This design gives the optimized read-out incorporated circuit in CMOS technology with the choice of most appropriate detector material for the precise application. The image to the right displays a wafer-to-wafer hybridization with 150 M interconnections Figure 4.1. Two-photon doppler-free hydrogen spectrum of the 1S-2S resonance, noticed through exciting an atomic(cold) beam by using a standing optical wave at 243 nm. The linewidth and frequency ratio are less than a single part in a trillion Figure 4.2. The advancement in achieving accuracy in the field of optical spectroscopy of hydrogen atom in the 20th century Figure 4.3. The resonance fluorescence of an ion is shown. In part (a) of the figure, the spectrum is shown that is evaluated in a heterodyne experiment. At low intensities, the reemitted fluorescence radiation is monochromatic. In the laboratory air, the light beam’s phase fluctuations limit the linewidth. In a Twiss and Hanbury-Brown experiment, they investigated the same radiation (b) Figure 4.4. Rb Rydberg atom’s micromaser setup. We can control the velocity of the atoms in the atomic beam through the velocity subgroup of an atom, simply by exciting them. Field ionization selectively detects the atoms in the lower and upper maser levels Figure 5.1. Optical coherence tomography’s schematic illustration Figure 5.2. Quantum theory applications Figure 5.3. Three experimental setups for Glauber’s correlation function’s measurement of light that is being emitted by CdSe/CdS DRs and their clusters using an ICCD camera Figure 5.4. Double slit interference experiment Figure 5.5. Laser-induced spectroscopy’s illustration Figure 5.6. This setup was formed for determining the precision in atomic hydrogen’s 1s-2s transition. At 486 nm, a stabilized narrow-band laser is frequency-doubled at 243 nm for inducing a two-photon doppler-free transition in hydrogen that is cryogenically cooled. We can achieve longer interaction times by using delayed irradiation to restrict the detection to the slow tail and also through pulsing the hydrogen beam. Through an optical frequency comb generator, the blue laser’s frequency is measured (in the cesium atomic clock terms) Figure 5.7. In precision spectroscopy, the development in relative accuracy is shown. In both, microwave and optical-based systems, 10–15 level has been achieved Figure 5.8. Femtosecond radiation’s frequency and time representations. The laser light’s electrical field generally advances under the pulse envelope. Down to frequency 0, we can extrapolate the frequency comb, and then usually there is an off-set fCEO Figure 6.1. Demonstration of 4 representative optical device configurations Figure 8.1. Evaluation of the polarization of a couple of obscure photons together with    ! "#!  $ xiii

Figure 8.2. Probability regarding the measured polarization of a couple of photons will #       ! "#!  $ %  of quantum mechanics is represented by the red line, while the maximum correlation in any classical theory is corresponded by the black line where the information cannot be disseminated swiftly than the speed of light Figure 8.3. As marked by single-photon detectors D1,’ D1, D2,’ D2, a two-photon interferometer showing nonclassical correlations within output ports selected by photons &1 &2. Phase shifts 1 and 2 are proposed into the way through each interferometer Figure 8.4. From an ultraviolet laser, individual photons being break up into two secondary photons meanwhile conserving momentum and energy in the process Figure 8.5. Outcomes are illustrated for an experiment executed by Ray Chiao’s group, a two-photon interferometer, at the University of California at Berkeley. Interference in the two-photon coincidence counting rate and not in the single-photon rates is demonstrated. The units conformed to the number of events of every type gained in a 10-s interval Figure 8.6. Scattering of two classical optical pulses diffused by a source and propagating in two scattered media. The diffusion encountered by one photon can be abolished by that encountered by a faraway photon in quantum optics Figure 8.7. Two coordinate frames pivoted by 45° and utilized for the calculation of the linear polarization of single photons Figure 8.8. Block diagram of a functional system for quantum cryptography, in addition with a feedback loop, to reimburse for time-dependent variations in the phase of polarization of single photons disseminating in an optical fiber Figure 8.9. A capture from APL, the quantum cryptography laboratory. On the lefthand side of the optical table, there shows a prototype system based on a single-photon polarization. On the right-hand side, laser beams and other equipment shown are part of a two-photon interferometer experiment Figure 8.10. Communication of secure information from computer 1 (i.e., top) to computer 2 (i.e., bottom). The encrypted message was transferred through an open transmission line with the decoded message

LIST OF ABBREVIATIONS

BEC

Bose-Einstein Condensation

BI CMOS

Backside-Illuminated CMOS

CARS

Coherent Anti-Stokes Raman Scattering

CPA

Chirped-Pulse Amplification

JCM

Jaynes-Cummings Model

KLM

Knill, Laflamme, and Milburn

LEO

Linear Electro-Optic Effect

MWI

Many-Worlds Interpretation

MZI

Mach-Zehnder Interferometer

NLO

Nonlinear Quantum Optical

NPI

National Photonics Initiative

NQIT

Networked Quantum Information Technologies

NV

Nitrogen-Vacancy

OFC

Optical Fiber Communication

PALM

Photo Activated Localization Microscopy

QED

Quantum Electrodynamics

QO

Quantum Optics

S&T

Science and Technology

SHG

Second Harmonic Generation

SPDC

Spontaneous Parametric Down-Conversion

STED

Stimulated Emission Depletion

STORM

Stochastic Optical Reconstruction Microscopy

XFEL

X-Ray Free-Electron Lasers

PREFACE

Quantum optics (QO) can be called a union of physical optics and quantum field theory. Presently, the field of quantum optics is undergoing an era of revolutionary changes. This area of science has evolved from older studies on the coherence characteristics of radiations (e.g., quantum statistical laser theories in the 1960s) to present areas of exploration encompassing the role of squeezed radiation field states and atomic coherence in extinguishing quantum noise in optical amplifiers and interferometry. On the contrary, counter-intuitive models such as single atom (micro) lasers and masers in addition to inversion free lasing are now experimental realities. Many of the above-mentioned techniques promise the advent of novel devices with a capability to possess sensitivity that surpasses the standard quantum limitations. Moreover, quantum optics also offers a strong probe to address the fundamental problems of quantum mechanics, e.g., hidden variables, complementarity, and other facets essential for the foundations of philosophy and quantum physics. The purpose of this book is to offer various exciting advances in the field of quantum optics. Currently, quantum optics is being applied in many applications which involve electronics, climate control, space science, medical science, and industrial sectors. This book put emphasis on basic concepts and applications of quantum optics and its associated fields, so as to facilitate the students and researchers to perform their research in this amazing field. There are eight chapters in the book. The first chapter deals with the introduction and fundamentals of quantum optics and photon theory. The chapter also deals with applications of Many World Interpretation in the field of quantum optics. Chapter 2 introduces the readers to the concepts of quantum technologies and applications of quantum optical phenomena in modern technologies. Chapter 3 deals with the applications of quantum optics in the field of photonics. The integration principles of these two technologies have been thoroughly discussed in the chapter. Laser technology is being applied in various industries due to its excellent features. The applications of quantum optics in the field of laser spectroscopy are briefly discussed in Chapter 4. The applications of quantum optics and quantum mechanics in optical coherence are discussed in Chapter 5. Historical

perspectives of quantum mechanics and quantum optics are also discussed in the chapter. Chapter 6 deals with applications of quantum optics in modeconverter linear optical devices. Fundamental and operating principles of mode converter devices are discussed in the chapter. Chapter 7 discusses the fundamentals of nonlinear quantum optics, and it is their applications. The detailed descriptions of second and third-order effects are discussed in the chapter, along with a brief introduction of magneto-optics and electro-optics. Finally, Chapter 8 discusses the recent developments in the field of quantum optics. Different modern applications of quantum optics are discussed in the chapter. The book can be used as a standard textbook for graduate students having some background knowledge in the field of quantum mechanics and physical optics. The book is equally beneficial for readers from diverse backgrounds, which may include students, scientists, teachers, and industrialists working in the field of modern technologies. —Author

xviii

Chapter

1

Fundamentals of Quantum Optics

CONTENTS 1.1. Introduction ........................................................................................ 2 1.2. Development of Quantum Optics (QO) and the Photon Theory of Light ................................................................................. 3 1.3. Key Findings ....................................................................................... 7 1.4. Fundamental Concepts ....................................................................... 7 1.5. Coincidence Correlation ..................................................................... 9 1.6. Quantum Entanglement ...................................................................... 9 1.7. Quantum Teleportation ..................................................................... 10 1.8. Quantum Information Processing...................................................... 11 1.9. Quantum Communication ................................................................ 12 1.10. Quantum Electronics ...................................................................... 13 1.11. Quantum Optics (QO) and Many-Worlds Interpretation (MWI) ...... 14 References ............................................................................................... 22

2

Quantum Optics Applications

1.1. INTRODUCTION The field of physics which deals with the study of the nature of light, its propagation in diverse media (such as a vacuum), and its interaction with diverse materials is called Optics. Quantum in this context is a package of energy emitted or absorbed in the form of minute packets called quanta (Vogel and Welsch, 2006; Perina et al., 2012). The amount of energy absorbed or released in a physical process is always a multiple of a definite unit of energy called a quantum, which is also identified as a photon. In physics, quantum optics (QO) is a field of research that deals with the application of quantum mechanics to phenomena, including light and its connections with matter (Figure 1.1) (Klauder and Sudarshan, 2006; Lambropoulos and Petrosyan, 2007).

Figure 1.1. Setup for the quantum optics experiments showing the source of polarization-entangled photons, the ULWBS, and the detection system. * + %+  }   !   }   " %}   ~  „  Sudarshan, John R. Klauder, Dirac, Leonard Mandel, and Roy J. Glauber to gain a further detailed understanding of the statistics and photodetection of light %% >    !         in the 1950s and 1960s. This guided to the introduction of the coherent state as an idea that addressed distinctions among thermal light, laser light, exotic squeezed states, etc., as it became clear that light cannot be entirely described only by    "%   {    picture. In 1977, Kimble et al. visualized a single atom emitting one photon at a whilst, a further fascinating indication that light consists of photons. †   >           ~    states, like squeezed light was later discovered (López et al., 1996). Progress of ultrashort and short laser pulses established by Q modelocking and switching methods unlocked the method to the study of what became called ultrafast procedures. Mechanical forces of light on the matter were considered, and applications for solid-state research (e.g., Raman %  %! ~    %   ‡  clouds of atoms or even minor biological samples in an optical tweezer or optical trick by the laser beam. This, together with Sisyphus cooling and Doppler cooling, was the critical technology required to attain the celebrated Bose-Einstein condensation (BEC). Other outstanding outcomes are the demonstration of quantum logic gates, quantum teleportation, and quantum entanglement. In quantum information theory, the mark is of much notice, from QO, a subject which partially appeared, partially from theoretical computer science (Brookes et al., 1980). Today’s arenas of attention between QO researchers contain parametric oscillation, parametric down-conversion, even shorter light pulses, for information use of QO, Bose-Einstein condensates, manipulation of single  }   %%  }     %      #= >! called atom optics), coherent perfect absorbers, and much more. Under the   ‚  %   } %  !  %%        and engineering revolution, frequently go under the modern term photonics (Fischer et al., 2008). For work in QO numerous Nobel prizes have been given. These were awarded: ˆ

In 1997, William Daniel Phillips, Claude Cohen-Tannoudji, and Steven Chu;

Fundamentals of Quantum Optics

ˆ ˆ ˆ

7

In 2001, Carl Wieman, Eric Allin Cornell, and Wolfgang Ketterle; In 2005, John L. Hall, Roy J. Glauber and Theodor W. Hänsch; In 2012, David J. Wineland and Serge Haroche “for groundbreaking experimental approaches that allow measuring and manipulation of individual quantum systems.”

1.3. KEY FINDINGS QO or quantum physics as a whole believes that electromagnetic radiation travels in the form of both a particle and a wave at the same time. This phenomenon is known as wave-particle duality. The phenomena could be explained in such a way that the photons move in a stream of particles, but by a quantum wave function, the overall behavior of those particles is determined through which the probability of the particles being in a given location at a given time could be determined. Taking results from QED (quantum electrodynamics), it is also probable to understand QO in the form of the annihilation and creation of photons, #! %    %     approaches that are suitable in evaluating the behavior of light, though       % ! !  ~ %         debate (though most people view it as just a suitable mathematical model).

1.4. FUNDAMENTAL CONCEPTS Quantum theory says that light is not only taken as an electromagnetic wave but also it is taken as a “stream” of particles called photons that travel with c, the vacuum speed of light. These particles should not be considered as quantum mechanical particles defined by a wavefunction spread over a limited region but not considered as classical billiard balls, (Horodecki et al., 2009). Each particle has one quantum of energy that is equal to hf, where h is known as Planck’s constant and f is the called the frequency of the light. That energy owned by a single photon corresponds precisely to the transition among distinct energy levels in an atom (or other systems) that produced the photon; the absorption of a photon by a material is the reverse process. The presence of stimulated emission is also predicted from Einstein’s description of spontaneous emission, the principle upon which the laser lies. Though, the real creation of the maser (and laser) many years back was reliant on a process to produce a population inversion (Van Raamsdonk, 2010).

8

Quantum Optics Applications

Statistical mechanics is in the core of QO: For creation and annihilation of photons, light is defined in terms of field operators—i.e., in the language of QED. A commonly come across a state of the light field is the coherent state, as in 1960 introduced by E.C. George Sudarshan. This state, which can be applied to almost define the output of a single-frequency laser that is well above the laser threshold, shows Poissonian photon number data. Through particular nonlinear relations, a coherent state can be altered into a squeezed coherent state, by using a squeezing operator which can show super- or sub-Poissonian photon numbers. Such light is known as squeezed light. Further significant quantum features are connected to associations of photon statistics among different beams. For instance, the so-called ‘twin beams’ could be generated through spontaneous parametric downconversion (SPDC), where (preferably) every photon of one beam is related with a photon in the other beam (Pfaff et al., 2014). Atoms are taken as quantum mechanical oscillators with a distinct energy spectrum, with the transitions among the energy eigenstates being determined by the emission or absorption of light according to Einstein’s theory. In the case of solid-state matter, one applies the energy band models of solid-state physics. This is vital for understanding how light is perceived by solid-state devices, generally used in experiments. In recent years there has been a substantial appearance of quantumbased applications, which has primarily been taken about by scientists understanding how the quantum properties of nanoscale materials can be utilized and manipulated. QO is one of such subdivision of quantum technology, and in this chapter, we will give a brief outline of the several       ‚   Š[*   } 2006; Ren et al., 2017).

Figure 1.3. Quantum optics experiment. Source: https://www.azooptics.com/Article.aspx?ArticleID=1502.

Fundamentals of Quantum Optics

9

QO is mainly a field of physics that uses a mixture of quantum mechanics principles and semi-classical physics at the subatomic level to manipulate and examine that how photons of light interrelate with the matter and the phenomena which can be produced. This is the way you can describe QO in its comprehensive sense (Nielsen et al., 1998; Olmschenk et al., 2009). However, at the same time as some of the most fore standing applications is quantum computing and lasers, there has been numerous researches into the important principles of how photons work at this level, and this has assisted to realize several different phenomenon and subsets inside QO which contribute deeply to the understanding of the physical applications (Raimond et al., 2001).

1.5. COINCIDENCE CORRELATION Coincidence correlation is a field of QO that is utilized to detect if somebody is observing a single quantum system. This is realized by assuming that an isolated system can just emit one photon while observers (through a photodetector) collect a single photon from the emitter. If it is found that more than one detector perceives the source, rather than the probability, it is interpreted that is nor a one-photon system hence is not likely to be a single quantum system. It is an essential process that allows someone to regulate the existence of a single quantum system, i.e., it is not an application but a test, but it can only be used in conjunction with other quantum optic application areas (Terhal, 2002; Bengtsson and Œ!?~ ~, 2017). Quantum entanglement is an example of a coincidence correlation that is explained in detail. Coincidence correlation can be used to disprove or prove the correlations with a quantumly entangled network and will engage a mixture of photodetectors and optical polarizers to sieve quantum states and regulate if there is correspondence at both ends of the entangled pair (Karlsson and Bourennane, 1998; Œ!?~ ~ et al., 2001).

1.6. QUANTUM ENTANGLEMENT Quantum entanglement is a phenomenon that happens among quantum systems, where the components of every quantum system become correlated, i.e., the entire system becomes one quantum state rather than two separate quantum states. The realm where we can observe this phenomenon includes molecules, photons, atoms, and electrons. This spreads to long-range distances, and the dimension of one part of the quantum system allows the

10

Quantum Optics Applications

properties of the consistent particle in the quantum system to be exposed (Figure 1.4) (Pittman et al., 1995; Fickler et al., 2012).

Figure 1.4. Illustration of quantum entanglement. Source: https://en.wikipedia.org/wiki/Quantum_entanglement.

The properties that can be exposed at various ends of an entangled network are polarization, spin, momentum, and position. In several cases,       % }     >  %      superimposition with an unlimited value (Barrett et al., 2004; Sarovar et al., 2010). If one of these particles how ever is measured, it can give certain worth for the corresponding pair. In quantum computing applications, quantum entanglement is often applied.

1.7. QUANTUM TELEPORTATION Quantum teleportation is one more phenomenon that has interesting applications in quantum communications, along with, quantum computing and is thoroughly connected to quantum entanglement. Quantum teleportation is the procedure through which the information seized inside a quantum bit (qubit) can be moved from one place to another, without the quantum bit (qubit) itself being moved (Bouwmeester et al., 1997; Furusawa et al., 1998; Alicea et al., 2011).

Fundamentals of Quantum Optics

11

Qubit, also known as a quantum bit, is the unit quantum networks, particularly in quantum information processing applications, and can assume a superposition of 0 or 1 value or a 0 value and a 1 value. This means that quantum bits (qubits) can achieve quantum operations in more than one value simultaneously (Figure 1.5) (Karlsson et al., 1999; Lo, 2000; Riebe et al., 2004).

Figure 1.5. Quantum teleportation phenomena. * + %+ ?      {{    % £ # oscillations   {  {!†>  !  setup (Loudon and Knight, 1987; Aasi et al., 2013). An information     §–€    {!   % ! prepared in a big amplitude comprehensible state would become a coherent %%     %  —  ‚    physics of atoms and light, quantum information emphases the applications and properties of the qubit. A quantum bit or qubit is the quantum mechanical expansion of a conventional bit {0,1}. As a quantum state, the state of a qubit |ªi could be in a superposition of 0 and 1: |ªi = «|0i + ­|1i where; «and ­are complicated numbers satisfying |«|2 + |­|2 = 1. The number of complicated numbers required to explain the state of a quantum structure grows exponentially with its complication (for instance, the

From Quantum Optics to Quantum Technologies

33

numbers of its constituents), which rapidly makes it prohibitively hard to perform precise calculations of its behavior, mainly those emergent properties which cannot be easily guessed or approximated through human ingenuity. This fact leads Feynman to propose, in the initial 1980s, that the properties of complicated quantum systems should preferably be studied with collections of programmable and controllable quantum entities that interrelate with each other to “mimic” the behavior of the structure being studied (Shore and Knight, 1993; He et al., 2016; Somaschi et al., 2016). The counterpart to consistency in QO is the quantum mechanical superposition of the qubit. Extended across several qubits, superposition takes to entanglement, and information could be processed and represented in an intrinsically nonclassical method. Therefore, a quantum computer would be capable of computing in an intrinsically diverse method to standard classical computers. Quantum computing turn into the subject of concentrated study after Shor invented an effective quantum algorithm for factoring, trouble for which no effective classical algorithm is recognized despite centuries of study (Jaynes and Cummings, 1963; Klauder and Sudarshan, 1968). An additional application of quantum mechanics in the processing of information is found in sending hidden messages. It is well renowned that a one-time pad (a series of random numbers shared through two people) is the safest way to send a hidden message. In 1984, Brassard and Bennett invented a scheme to make a one-time pad among distant partners utilizing      %%   “}^®¯± and Shapiro, 1978). The present development of quantum information processing and quantum technology is based on our capability to control coherences of a huge quantum structure. In doing this, one of the key hurdles are the system’s uncontrollable interface with its environment. There had been comprehensive studies of decoherence mechanisms in the framework of nonMarkovian and Markovian open quantum trajectories and quantum systems. Knight worked on the characteristics and sources of quantum decoherence utilizing a quantum jump method with his colleague Martin Plenio (Plenio and Knight, 1998). He also worked on numerous ways to defend a quantum state from decoherence utilizing the concept of decoherence-free subspace and the phase control of the structure. A key to overwhelmed the decoherence  >   %   {   >         needs a large number of entangled resources. The theory of quantum

34

Quantum Optics Applications

    { %     %  =   quantum computation (Eberly et al., 1980; Rempe et al., 1987). Quantum entanglement              constituents in quantum technology. We say two quantum structures are tangled when the density operator of the entire system can’t be written as a weighted sum of the product states (Wineland et al., 1988):

(1)

where; is a density operator of the system; and Pn is a probability function of the system i = 1,2. Entanglement could be thought of as a distinctive type of correlation where it is probable for the states of local structures to transmit more uncertainty than the state of the global structure, something improbable in classical statistics. Entanglement, which is closely associated with nonlocality in the EPR (Einstein-Podolsky-Rosen) paradox, has had several uninterrupted applications. Ekert suggested an arrangement to share a one-time pad amongst the two users of a hidden message, utilizing the entangled nonlocal state of light     !%   _  {!    % # of quantum mechanics to information technology (Phoenix and Knight, 1988; Gea-Banacloche, 1990). ‚ { %>    %   quantum teleportation where the entanglement of the quantum network is      %   — =      quantum superposition conditions is greatly challenging because like the quantum state is subject to decoherence (Wigner, 1997; Deleglise et al., 2008). An alternate is teleporting the quantum state. Although the entangled quantum network is also subject to decoherence, it is in theory, probable to distill entanglement to remove a small quantity of highly entangled pairs from a huge quantity of weakly entangled pairs. A simple however powerful distillation protocol was developed using entanglement swapping. Entanglement could be described by utilizing the Schmidt decomposition. Searching for a measure of entanglement was one of the problems to characterize quantum entanglement, Peter Knight and his colleagues    {>    >  Š 2.1) (Deutsch, 1985; Lv et al., 2017).

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Figure 2.1. Applications of the quantum technology. Source: https://www.insidequantumtechnology.com/news/future-quantum-supremacy/.

Atom optics, which is a struggle to utilize quantum coherence of atomic motion, is an additional branch of QO. Certainly, a very early type of atom interferometry is the Ramsey interferometry founded on the quantum coherence in atomic interior states. Atomic clocks had been developed, and precision accelerometers and sensors had been inspected based on atom optics (Shor, 1999; Cirac and Zoller, 2012). In this review, we investigate how QO had developed into quantum technology, emphasizing the characteristics and roles of atoms, ions, photons, and mechanical oscillators. In Section II, we evaluate the quantum properties of light itself, concentrating on coherent states and photons. In *  ƒƒƒ}       §–€ % {  ‡  

   the study of the interaction of quantum atoms and light. In Section IV, we "   %%  }    #           } and a platform for the progress of quantum technologies. In Section V, we present mechanical oscillators, demonstrating an analog of quantum light in microscopic matter. Lastly, in Section VI, we examine how the application of these tools and ideas for the precise manipulation of quantum structures is giving rise to innovative technologies (Paspalakis and Knight, 1998; Bennett and Brassard, 2020).

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Quantum Optics Applications

2.2. PHOTONS The photons study is at the heart of QO. The concept of particles of light had gone by numerous stages. Although the quantization of energy was proposed to explain the blackbody radiation at the early stage of quantum mechanics, it was not obvious which energy was quantized as well as its results. Glauber defined the coherent state of photons as an eigenstate of the extinction operator and noticed that the coherent state is signified as a displaced vacuum in phase space (Shor, 1995; Beige et al., 2000). The coherent state is recognized to describe the coherence properties of a laser field and the photon number statistics (Figure 2.2).

Figure 2.2. Creation of photons in an atom. Source: https://commons.wikimedia.org/wiki/File:Fluorescent-lamp-atom.gif.

2.2.1. Nonclassicalities Characterization of the nonclassical behavior, which cannot be described through classical theories, was a significant problem of QO and photons were at the heart of this study. Sub-Poissonian and Antibunching statistics were investigated (Caves, 1981; Grosshans et al., 2003). However, a laser field was invented founded on a quantum-mechanical interaction amongst light fields and atoms, properties of the laser field could be known founded

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on the theory of constant light field in classical optics. In early 1980, light fields and Knight investigated signatures of nonclassical behaviors of light fields and a completely quantum-mechanical atom field interaction model (Einstein et al., 1935; Steane, 1999).   =  

          ²    photon statistics and its intensity correlation g(2)(³  }  g(2) +

(2)

where; is the intensity operator; T and “:” denoting time and normalordering of the operators, respectively. All over the chapter, an operator is represented by “ˆ.” Using the Cauchy inequality, it could be shown that the intensity correlation function is equal to or greater than unity if   > ?€  “ "% !{ #= Poissonian photon statistics of a particular photon state which was produced, conditioned on a single photon measurement of an identical beam from a parametric conversion procedure (Ekert, 1991; Bennett et al., 1996). As it  #{     =%   }   !}    #  !  ¯  } sub-Poissonian photon statistics with g(2) = 0. Another result of Cauchy’s disproportion on the classical intensity correlation function is g(2)(³) < g(2) +   !           =#         Mandel, Kimble, and Dagenais experimentally demonstrated anti bunching %          % #!      excited through a dye laser (Zukowski et al., 1993; Bose et al., 1999). In mechanics, dynamics of a structure are often studied in phase space formed of position and momentum more usually, two canonically conjugate variables, q, and p. At a given time, a physical structure is explained by a joint probability function P(p,q). All the statistical properties of the structure at that time could then be calculated utilizing a simple probability theory: There been a quantum joint probability P0, we would have the following:

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Quantum Optics Applications

Though, in quantum mechanics, the two canonically conjugate operators do not commute therefore and Eqn. (3) cannot be applied to quantum mechanics. Therefore, it is obvious that there is no joint probability in quantum mechanical phase space. Seeing this, Wigner derived a probability-like function W(p,q) which fulfills the marginality in probability theory; and where P(p) and P(q) are the marginal probability functions in q and p which are well described in quantum mechanics. It was the future found that the Wigner function is the distinctive function that fulfills the marginality in quantum mechanics. Later then, a few quasiprobability functions had been proposed by various researchers (Ekert and Knight, 1995; Bose et al., 1998). Although the Wigner function shares the properties of the probability  }  % # #! ƒ%}  { { values at certain points of the phase space, not like a probability function, and negativity of a Wigner function is recognized to be a nonclassical behavior}   { "% !   {  with a single atom. Due to the over extensiveness of the coherent state, any density operator could be represented as a total of diagonal matrix elements in the coherent state basis (Vedral et al., 1997; Lee et al., 2000).

2.2.2. Parametric Down-Conversion One significant nonclassical condition in QO is a squeezed condition whose quadrature noise is decreased under the vacuum limit. In specific, a squeezed vacuum is generated through the SPDC (spontaneous parametric downconversion). As early as 1970, Burnham and Weinberg (1970) proposed to generate optical photon pairs utilizing the µ(2) nonlinear optical procedure. Later then, the SPDC has certainly been one of the most frequently utilized processes in QO, not only to produce squeezed states but also, for instance, to produce single photons and entangled photons (Hong and Mandel, 1986; Knight and Milonni, 1980). The SPDC transforms a pump photon into two daughter photons whose propagation and frequency direction is determined thru the principles of energy and momentum conservation, which takes the two daughter photons associated with energy and momentum. Considering the probability of the number of daughter photons produced decreases exponentially, the quantum state of the daughter photons is signified by:

(3)

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39

which is the two modes squeezed form with the squeezing parameter ¶. When ¶ · }        >?   #   "!   EPR state which was under examination by Einstein, Rosen, and Podolsky (Kimble et al., 1977; Reid, 1989). Though such a state is unphysical since the mean energy of the state is endless. Even yet not maximal, it is obvious that the two modes squeezed state is tangled; however when it de coheres the entanglement has vanished, and the state becomes mixed (Simon et al., 2000). To characterize quantum correlations of the mixed state, standards for quantum correlations were inspected. Specifically, the correlation is quantum steering which is studied by Reid, and had been generalized to other systems lately. Barnett and Knight (1985) studied the statistical properties of a single mode of the two modes squeezed state and detect a thermal nature whose effective temperature raises with the degree of squeezing. This effect is almost associated with quantum entanglement of the two modes of squeezed form. Strong associations in a two-mode squeezed form were achieved lately in which reports as high as 10 dB squeezing. Quantum correlation had been revealed beyond two modes squeezing to create entanglement in an ultralarge-scale network of quantum states, utilizing beam splitters and SPDC (Mari and Eisert, 2012; Eberle et al., 2013). While energy and momentum could have any values, the polarization of   could have only one of the two values, right/left or horizontal/ vertical circular. Understanding that the polarization of the 2 SPDC photons is interrelated (Wenger et al., 2004; Genoni et al., 2008). Certainly, the polarization degree of freedom had only two values demonstrate that a photonic quit could be realized by the polarization of a single photon. Kwiat and his collaborators showed how to produce an entangled polarization state utilizing Type I and Type II SPDC processes. The polarization-entangled photon pairs had been utilized to show entanglement swapping, quantum key distributions, and quantum teleportation entanglement swapping. As the bipartite entanglement was shown, there had been an effort to raise the size of entangled photonic networks, and it has demonstrated that as many as 8 photons are entangled (Burnham and Weinberg, 1970; Yokoyama et al., 2013). However, an isolated two-level structure like a quantum dot, an ion, an  }  !     

   %  photons deterministically, SPDC had been at the front of the single-photon production, because of its simplicity (Kim et al., 2005; Ourjoumtsev et al., 2006). Numerous of the quantum information%   }   make sure that single photons are the same so that they interfere with each

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Quantum Optics Applications

 —

  ~ =%   %   %  } “ =‚€  ?  ! of coincidence counting rate would indicate how two photons are identical. Quantum interference had been generalized to a huge number of input and output photons. Lately, technology has advanced to produce rare identical photons from quantum dots to interfere with their non-classicality trial (Aspect et al., 1982; Kwiat et al., 1995, 1999).

2.2.3. Quantum State Engineering The squeezed states and coherent states are represented through Gaussian Wigner functions in the phase space. Phase-shifting and beam splitting, which are usual operations in an optical lab, are termed Gaussian operations as they do not alter the Gaussian nature of input states. The heterodyne and homodyne measurements are then termed as Gaussian measurements (Nha and Carmichael, 2004; Ourjoumtsev et al., 2007). However, the Gaussian operations on Gaussian states are beneficial for particular tasks of quantum information processing like as precision measurements and continuousvariable quantum key distribution, it had been noticed that Gaussian states might not be useful for further purposes of quantum information processing like as computations and simulations. Therefore, have been studies on producing and manipulating non-Gaussian states (Pan et al., 1998; Bouwmeester et al., 2013). Ourjoumtsev et al. showed a single-photon subtraction operation to transform a Gaussian state to a non-Gaussian one. Transferring an initial #!   = {!# %} %      }     #{  ?   ”¸?~}   }  ¤ =”    ^^ studied in what way the coherent superposition states decohere however the   # #  %   >  computing (Erven et al., 2008; Yao et al., 2012). However, it is simple, a single photon reducing operation does not bring a classical state into a nonclassical one. On the second hand, the addition of a single photon could transform any classical state into a nonclassical one. Conceptually, the addition of a single photon could be done through

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41

reverting the subtraction operation (Hong et al., 1987; Rigovacca et al., 2016). Rather of nothing impinged on the vacant input port of the beam splitter, a single photon is invaded, then a single photon is added on the early state conditioned on measuring not anything at the further output mode of  # %  {        =%  ¦ {   ® devised a creative scheme to utilize an SPDC process and transformed a coherent state to a non-Gaussian state with extreme negativity in phase space. A pattern of displacement operations and single-photon additions could transform the vacuum state into an arbitrary quantum state written as where cn is complex (Jeong and Kim, 2002; Munro et al., 2002). Another amazingly non-Gaussian class of states of the electromagnetic     * ¹     %%     {!  coherent states with huge negativity of the Wigner function. The decoherence >  #!   {         {!    

 >  #!    {!    •    ~ * ¹           }   !      #! %  } #  %    ˜ % ~  et al., 1999; Kim, 2008).

2.2.4. Towards Integrated Quantum Photonics One of the benefits of photons as a candidate to apply quantum technology is that their operations do not need cryogenic temperatures, and they are hardly affected by environments. Though, precise alignment of an optical structure might be an issue regarding integrability, scalability, and portability. Integrated photonic chips (photonic circuits) had been fabricated to place all the essential optical elements into a small chip to overwhelmed these problems (Garraway and Knight, 1994; Gilles et al., 1994). As it was shown by Reck et al. any distinct qubit operation could be done by a set of beam splitters and an arbitrary reflectivity beam splitter could be devised through an MZI (Mach-Zehnder interferometer). An MZI is a building block for the arbitrary qubit operations. We could also combine photodetectors and beam splitters to realize two-qubit gate operations like the controlled-phase operations suggested by KLM (Knill, Laflamme, and Milburn). There might have good growth in fabricating photonic chips even to assist photodetector units and polarization states of light. The photodetector is also a significant component. At the moment, the most effective detection of photons is accomplished by utilizing the alteration of resistance of superconducting

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Quantum Optics Applications

wires. The effort in creating photonic chips had been stretched to integrate photon sources as they insert the nonlinear materials on the chip for the onchip SPDC. Utilizing the photonic chips, several tasks of photonic quantum information processing had been shown comprising boson sampling (Reck et al., 1994; Sansoni et al., 2012). As seen above, photons  #  for quantum technology for simulation and computation and information mover for long haul communications. The only method to realize longhaul QKD is utilizing photons. However, photons are comparatively robust against environmental effects, they still decohere and lose entanglement and nonclassical effects. Even yet it is not a pre-requisite, quantum memories > %  #   }¯Ÿ _  }

2.3. CAVITY QED The JCM had been one of the significant tools to characterize the quantummechanical interaction of an atom through a single-mode field where the restoration of Rabi oscillations was a significant manifestation of the quantization of the field, initially discussed by Eberly et al. (1980). Knight and Radmore (1982) studied the restoration when the field is primarily in a thermal state (Grosshans and Grangier, 2002; Spring et al., 2013). A key experimental advancement was done by being able to make the atom initially in their superposition state. This opened a probability to exploit quantum coherences in the cavity QED, which had permitted a way to rebuild the cavity field and to generate a huge superposition of coherent states in a cavity. In specific, a complementarity trial in the Einstein-Bohr dialog was experimentally revealed, based on entanglement among the cavity field and the atom (Briegel et al., 1998). •   [={         {! }  %%   state could be produced in the cavity whose nonclassical characteristics were inspected. One of the primary information-theoretical studies of the      #!   ² "  %  concept of entropy to examine quantum interaction. The time dependence

  %!   {        two interacting subsystems. Knight and colleagues suggested to entangle two atoms sitting in their particular cavities and to teleport the atom state from one cavity to the other utilizing the cavity decay, which activated a discussion on the distributed quantum computation among two remote locations utilizing photons as a messenger (Bertet et al., 2001; Reim et al., 2010).

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We could generalize the theory of cavity QED through a spin-boson interaction model as the atom is equal to spin and the cavity field is bosons. The spin-boson model had been realized in numerous experimental setups like as superconductor circuit QED, nanodiamond in a trap, and ionic motion in a harmonic trap. Non-classicalities of the ionic motion were shown experimentally, which had become a base for an ionic quantum computer. Knight and coworkers had taken marginally a different angle to suggest quantum computation utilizing vibrational coherent states (Wodkiciewz et }^\œ¯”º~ }^^

2.4. TRAPPED IONS Trapped ions are amongst the most fascinating structures, as experiments with exclusively trapped ions managed to oppose Schrödinger’s famous prediction that we would never execute experiments with separate quantum objects. Ion traps allow confining charged atoms exactly well that the light released by one individual atom could be recorded and that the properties of released light permit to draw results on the electronic state of the ion (Brune et al., 1996; Bose et al., 1999). The outlook of working with discrete quantum objects rather of an ensemble and to investigate their properties with an unprecedented level of exactness had resulted in intensive efforts to manipulate trapped ions for the search of fundamental physics and, more lately for the development of      %%  } % !} >      processing (Monroe et al., 1995, 1996; Plenio et al., 1999). The movement of ions in real space could be manipulated by electric — %  { »¼#  „ ˜  }  %  ¼ ~  =    However, there is always at least a single spatial direction where the potential is repulsive. Generally, the attractive potential could, although, be realized #!   #            #! %  !       %      %  %%    Penning traps and traps founded on the second principle are named radiofrequency trap Paul trap or Paul trap. Given the distinctive opportunities to probe and manipulate individual quantum particles, trapped ions had been established into an exceptional test bed for quantum information processing and fundamental physics (Schrödinger, 1935; Paternostro et al., 2005). Earlier to coherent operations on individual ions could be performed, the movement of the ions required to be cooled near to its quantum mechanical

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Quantum Optics Applications

ground state. As established through equilibrium thermodynamics, energy moves from cold to hot systems, which poses a fundamental task for   

        %     μK system in a room temperature environment (Neuhauser et al., 1980). ¢  

 {            

 trapped ions near to their quantum mechanical ground state. Since cooling is essentially an incoherent procedure, it is usually not applied throughout coherent gate operations. However, given the huge temperature difference among the trapped ions and their surroundings, the motions of the ions incline to be heated up as soon as there is no active cooling existing. Such

   

%      emotional states (Paul, 1990; Bergquist et al., 1986). However, ion traps of the primary generations were macroscopic in size; there had been a trend of miniaturization like that ions are stuck in proximity to trap electrodes. The  !         !       an extremely relevant obstacle to coherent operations of the trapped ions. There is the right understanding of the microscopic mechanisms liable for heating, however, the best methods to decrease heating is still simply extending the distance among trap electrodes and ions (Itano and Wineland, 1982; Stenholm, 1986). The heating value of the state-of-the-art tests are low enough for numerous proofs of principle tests; however, heating would be a bottleneck for big-scale quantum information processing needing stable phase coherence over the long times. Still, as we will search further below, ion traps form one of the prominent candidates for scalable quantum computing (Horvath et al., 1997; Häffner et al., 2008).

2.5. OPTOMECHANICS: QUANTUM OPTICS (QO) WITH MECHANICAL OSCILLATORS In section III, we had discussed how light could couple to multi-level systems like atoms to give stimulating nonlinear dynamics in the fields of circuit-QED and cavity-QED. By those internal variables, they could also join to the motion of the atoms as conversed in section IV on trapped ions. Yet, light could pair to the motional variables of a material object more straight as well: through the polarizability of the material or its radiation pressure on the object. In both the mentioned cases, the fundamental lightmatter pairing is of a stimulating “tri-linear” form-the number of a photon in the cavity (quadratic in annihilation/creation operators) pairs to a position (linear) variable of the mechanical object entirely these operators are linear

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(Diedrich et al., 1989; Murao and Knight, 1998). Let us suppose that the mechanical object is a quantum harmonic oscillator with the frequency ¾m and annihilation operator represented by b, interrelating with a cavity field mode of frequency ¾0 and annihilation operator represented by a. The related complete optomechanical Hamiltonian is: (4)

where; the light-matter pairing of strength g is termed the single-photon nonlinearity as it measures the impact of a single photon on mechanics. When the mechanical object is one of the glasses of a Fabry-Perot cavity (supposing the glass is movable and a harmonic oscillator) then it could be shown, through considering a slow (adiabatic) movement of the mirror that:

(5)

and m and L are mass of the portable mirror and the length of the cavity, respectively. In the situation that the portable object is a trapped mechanical

#_   ¿# À  {!  " }  % is given by:

(6)

where; Vbead is the volume of the trapped bead; Vcavity is the cavity mode volume and its electric permittivity (Deslauriers et al., 2006; Safavi-Naini et al., 2011). The above Hamiltonian with its tri-linear pairing is very exciting from the point of view of creating nonclassical states of both the material object             %        nonlinearity from Gaussianity after beginning from Gaussian initial states in the evolution depends on the ratio g/¾m, for example, the relative power of the trilinear term with relative to the harmonic term of the mechanical oscillator (Pace et al., 1993; Goodwin et al., 2016). Taking a regime when g Á¾m >> Âc, where Âc      !   {!}  }  coworkers in parallel with, were one of the initial teams to demonstrate how nonclassical non-Gaussian states like as Schrödinger cat states of the light  could be formed as a consequence of the dynamics stemming from the

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Quantum Optics Applications

overhead optomechanical Hamiltonian . Furthermore, Knight and coworkers presented that the same Hamiltonian allowed the production of nonclassical states of the mechanical object. Such states are recognized to be {!  >    }~  !±}"%  a couple of future systems, the above regime had not been extended to date. Most experimental outcomes in optomechanics have, therefore, to date, been       {         %#       #  ? We would discuss this regime next in short earlier to the proceeding of the details of nonlinear optomechanics (Bose et al., 1997; Chang et al., 2010).

2.5.1. Optomechanics: Linearized Regime When the electromagnetic field approach in a cavity is powerfully driven such that there is a non-zero amplitude «of the field in the cavity, one could redraft the Hamiltonian of Eqn. (4) on basis of displaced creation/annihilation operators . Then the third (interrelation) term of the Hamiltonian becomes the active linearized optomechanical interaction: (7)

where; geff = g|«|2 is the novel (effective) optomechanical pairing strength, which is much tougher than the bare coupling strength g. From the above Hamiltonian, and supposing that the laser of frequency ¾L motivating    {!       »      {! >! ¾0, then three discrete quadratic Hamiltonians could be produced in accordance to the quantity of detuning in the interrelation picture by eradicating terms rotating much quicker than others:

(8) One could physically recognize the blue/red detuned Hamiltonians from the reality that a phonon (mechanical oscillator quantum) had to be added/ subtracted to/from the mechanical model in the directive for the cavity mode to be settled through the driving laser (Mancini et al., 1997; Jacobs et al., 2017). The above gives a rich toolbox for the Gaussian operations on light and mechanical modes. For instance, the beam splitter interrelation HBeamcould be utilized to switch the state of a mechanical object through the Split

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displaced light mode in the cavity. This could be utilized (i) for transmitting the thermal quanta from mechanics to the light in a cavity, which consequently leaks out; this is thus a process for cooling the (potentially macroscopic) mechanical oscillator this is termed cavity cooling, (ii) for producing an interesting nonclassical, possibly non-Gaussian state of the mechanical object through injecting an optical mode in such a form into the cavity and then exchanging its form with the mechanical object (Murch et al., 2008; Johansson et al., 2014). Furthermore, as mechanics could generically interrelate with most quantum systems, like electromagnetic    ={ ! {  "{    >} Beam  %%    %     >  ! Split which would not naturally interact. Ent is the well-recognized two modes squeezing Hamiltonian and is recognized to make continuous-variable EPR entangled state of the two interrelating modes-in this case, one of the modes is mechanical, while the other is electromagnetic. Non-demo is particularly #  %   =     %  (b+b†) of the mechanical oscillator by a homodyne measurement on the electromagnetic     #? %       the mechanical oscillator by direct position measurements through a strong pulse of light (Aspelmeyer et al., 2014; Millen et al., 2015).

2.5.2. Nonlinear Regime If one aims to go further the domain of Gaussian states of the electromagnetic field and the mechanics utilizing optomechanics alone, one had to utilize the Hamiltonian of Eqn. (4) with the complete tri-linear optomechanical interaction (Figure 2.3). The obvious time evolution by supposing an initial state at time t = 0 like as:

Figure 2.3. A superposition of Fock states and is formed inside a cavity. The mirror oscillates around diverse equilibrium separations corresponding to

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Quantum Optics Applications

every Fock state, thus producing a superposition whose components comprise discrete coherent states at scaled times (where m = integer). The co       %%   #    probing the coherence among the cavity states and   Source: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.59.3204. (9)

where; are primary coherent states of the mirror and the %{!}{   %#!+

where;

% Š ~    {!  {  and the n, are coherent states of the mechanical oscillator presented by:

(10)

in which t stands for scaled time ¾mt and k = g/¾m. We perceive that after a time t = 2the mechanical oscillator renews to its original state. There is currently an effective Kerr nonlinearity  (a†a)2     {!          {!  { {      seen under a Kerr-like nonlinearity, that is the production of Schrödinger cat states. The physical cause is that the position of the mirror    “      {!   . Certainly, this is the case. For instance, when the mechanics and the cavity mode disentangle at time t = 2, then for we got a four constituent cat state:

(11)

Therefore, by adjusting the ratio of the mirror frequency and the bare optomechanical coupling and thus varying k one could, in principle, attain all these types of cats at time t = 2.

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It is even more prominent to note that at all times among t = 2 = 0, the mechanical oscillator form is entangled through the field state by the entanglement being extreme when t = . This entangled state is obviously of non-Gaussian nature as its superposed constituents have discrete coherent states, and there is a periodic disentanglement-entanglement dynamic approach. It was revealed by Knight and coworkers that this dynamic approach is very appropriate for examining the coherence among superpositions of discrete states of a macroscopic mechanical oscillator wherein merely the electromagnetic field attached to it required to be measured. This scheme is shown in Figure 2.3. This basic methodology of examining the macroscopic limits of the quantum superposition principle had been accepted and incorporated into numerous subsequent schemes (Pirandola et al., 2003; Stannigel et al., 2010).

2.6. QUANTUM TECHNOLOGIES 2.6.1. Quantum Simulators It is merely in a previous couple of decades that such simulators had been suggested and realized for precise idealized models of numerous body physics (whose dynamics are yet very difficult to forecast under general conditions). Primarily and initially implemented through ultra-cold atom systems, other avenues like superconducting arrays, trapped ions, integrated photonic chips joined matter-light systems, and, most lately, solid-state platforms are similarly being studied. These simulators, where the interrelation among the modules is hard-wired to understand a precise physical model, are termed “analog” quantum simulators (Braginsky et al., 1980; Vanner et al., 2011). There is, of course, also the exertion to build a universal quantum computer in numerous platforms as described in section VIB. Such machines could be programmed to mimic an extensive variety of diverse (ideally any) complex quantum structures and phenomena therein. Here multiple pulses might be utilized to control the structures as per the program and realize a complicated evolution in terms of a series of numerous shorter evolutions. Every of these smaller unitary evolution steps could be thought of as a quantum gate, with the complete simulation being a quantum circuit. Due to the quantum circuit methodology for execution, in principle, quantum error rectification could be incorporated to make such simulators extremely accurate. Such simulators are named “digital” quantum simulators, and through existing implementations are still without error correction, these have been already

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Quantum Optics Applications

implemented in photonic chips and superconducting qubit architectures (Figure 2.4) (Bose et al., 1999; Armour et al., 2002).

Figure 2.4. Illustration of a quantum simulator. Source: https://www.assignmentpoint.com/science/physics/quantum-simulators.html.

We initiate from the zone of ultra-cold atoms where a Bose-Einstein compressed gas of atoms is exposed to a lattice potential produced by light. The light is extremely detuned from the inner atomic resonances and generates an intensity relying on periodic potential which is mainly conservative. Due to their exactly small masses, atoms (an atom utilized usually is Rubidium) could tunnel over micron separations among nearby %      Š  }    %  site could interrelate with the contact interaction, which could be increased through a Feshbach resonance. The tunneling energy t, the atom-atom on-site interrelation energy U and the chemical potential (the energy of a particular atom placed in a site) μ, respectively in frequency units, together gave rise to the Bose-Hubbard Hamiltonian (Marshall et al., 2003; Scala et al., 2013).

(12)

where;   #    ¯  produces a particle in the itch site. Two stable points of this model are the Mott insulator phase for U >> t   >  "# of atoms in every site (relying on μ  %‡%  t >> U

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     %‡   suppose a non-zero value. In an outstanding experiment, it was exposed that one could observe a quantum phase transition from one to the other phases as the ratio t/U is varied. Likewise, experiments had now been made with fermionic atoms, where the phase move is from a metallic to a Mott phase (Jaksch et al., 1998; Wan et al., 2016). This opens the opportunity to controllably simulate, and thus understand, emergent numerous body phenomena in a condensed matter like high-temperature superconductivity, which are supposed to take place in doped Mott insulators. Furthermore, the Mott insulator phase with precisely one atom per site gave rise to an antiferromagnetic Heisenberg %     % =%          {   atom. This is due to the atoms, being banned to doubly occupy a site, could only swap their positions by a second-order procedure with frequency J  t2/U, which is equal to swapping their pseudo-spins with their positions  "} }  }   %“#“    (here are the Pauli matrices) performing among their pseudo-spins. With internal level relying on hoppings, numerous anisotropic Heisenberg models could therefore be synthesized, thus also initially the door for exotic spin liquid phases to be grasped. A similar philosophy with N degenerate atomic levels like in alkaline earth metals takes to models with SU(N) symmetry and ultimately, simulators for toughly coupled quantum     ~ >     !           { (Greiner et al., 2002; Duan et al., 2003). Note that BH is obtained through adding an on-site nonlinearity (in QO terms a Kerr nonlinearity) to a free (hopping) model (quadratic model). It is fairly natural that the idea of an analogous model for the photons, where they hop among a nonlinearity given by two-level structures (for example, atoms) trapped in every cavity and distinct electromagnetic cavities, would originate from former students and associates of Peter Knight. Similarly, with another liberated group, they had introduced the so-termed JaynesCummings-Hubbard model, whose ultimate realization seems to be linked with microwave cavities, each interrelating with superconducting qubits. This utilizes the basic fact that the JCM had an anharmonic level system that multiple polaritonic excitations (atom + photon combined excitations) at a site charges more energy than distinct excitations. Therefore, these     %   € =%‡=~> %  transition, spin models with atomic interior levels performing as pseudo

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Quantum Optics Applications

spins, and, from thereon, even fractional Quantum Hall States utilizing twodimensional arrays of associated cavities (Mintert and Wunderlich, 2001; Simon et al., 2011). Trap Ions had given fertile ground for the quantum simulation tests. The harmonic degrees of freedom are perfectly suited to mimic the behavior of light, and ideas to put on cavity electrodynamics and optical elements were amongst the main proposals for quantum simulations with trapped ions. Amongst the various ideas for quantum simulation of more complicated systems, are relativistic dynamics and phase transitions in anharmonically interrelating many-body structures (Porras and Cirac, 2004). Proposals focusing on the qubit-like grades of freedom proposed trapped ions as a quantum simulator of spin models of solid-state systems or molecules. The following developments covered the realization of more than pairwise interrelations or localization phenomena induced through disorder. With ideas to put on frustration in interrelating magnetism and quantum%           }       

 for quantum simulations is adequately advanced to utilize trapped ions for simulations that surpass the capacities of existing classical computers. As a consequence, trapped ions are currently a popular platform for proposals  ?>         computational resources. Amongst the more lately developed ideas is the application of two-dimensional spectroscopy to trapped ions (Richerme et al., 2014; Roushan et al., 2017). Though the spin-spin coupling simulated in an ion trap could be uniaxial/Ising, for example.,           another direction (which could be a laser driving) or a stroboscopic (digital simulator) technique whereby the spin bases are altered regularly from to and/or by suitable pulses, could utilize to produce more general Hamiltonians. Likewise, superconducting qubit ranges naturally had either Ising couplings by direct (say, capacitive) interactions among neighboring qubits or joins when the interactions are arbitrated by an adiabatically eradicated microwave bus. Nonetheless, these had been %   # ‡"#          %   utilizing the method of basis rotation through regular pulses and even for digital simulation of itinerant fermionic particles through using the JordanWigner transformation mapping spins to fermions (Crespi et al., 2013; Smith et al., 2016). Diverse simulators had different limitations and advantages. For instances, measurements of extended range entanglement and correlations

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 #             usually fall with distance, however, photonic chips could bring very detached photons to interfere at a usual beam splitter-this had been utilized to verify the dynamics of extended range entanglement development following a slake in an emulated spin model. Now it is worth saying that the quantum simulators are too opening novel vistas: both ion-trap simulators and ultra-cold atomic are very well quarantined from their environments distinct to usual solid-state many-body structures of nature. They, therefore, enable the study of extended time non-equilibrium dynamics (more exactly unitary evolution) of a non-eigenstate of a many-body Hamiltonian. This in return permits exploring important phenomena like dynamics of excitations (which could, in turn, had quantum technology applications, as well as to inquire notions of statistical physics like localization and ergodicity in the quantum regime (Angelakis et al., 2007). It is worthwhile emphasizing that one of the important lessons that quantum simulators teach us is that physical structures that are quite different could be made to mimic each other. On this part, we point out Peter Knight’s effort on the cavity-QED and optical executions of the coined quantum walk, which could both stimulate a similar phenomenon (a natural phenomenon which this mimic, on the second hand, is the spin reliant motion of a physical particle a Dirac like equation in the continuum range).

2.6.2. Quantum Computing Quantum computing is one of the most possibly transformative applications of quantum mechanics to technology. Signifying data quantum mechanically permits one to calculate in a fundamentally novel way. Information could be stored in quantum degrees of freedom, like the polarization state of a photon or the internal state of an ion, where entanglement and coherence permit for information processing in a nonclassical way (Hartmann et al., 2006). The qubit or quantum bit is the basic constructing block of a quantum computer demonstrating logical 1 and 0 in a couple of orthogonal quantum states . In a quantum computer, a catalog of quantum bits stores data in a coherent superposition of these states. This data is manipulated through quantum logic gates, in correspondence to the logic gates in a conventional ¿  À %      %=># operations like the Hadamard gate, which produces superposition states through the basis states respectively. The CNOT or controlled-NOT gate is a two-qubit gate in which the state of the additional

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>#‡%%   % {       ># In particular, CNOT gates could produce entanglement amongst the qubits (Figure 2.5) (Greentree et al., 2006; Cho et al., 2008).

Figure 2.5. Applications of quantum computing. Source: https://foresite.com/what-basics-do-i-need-to-know-about-quantumcomputing/.

Quantum algorithms for issues like factoring numbers, had been displayed to run with a run time which scales polynomial with the scope of the issue (for example, the number to be factored), however, the bestrecognized classical algorithms had exponential run-time scaling. Quantum computers, therefore, offer the potential to resolve issues beyond the range of conventional computing technology. Any quantum structure with two or more distinct states could represent a qubit, and there is thus a wealth of probable physical realizations of a quantum computer. The execution of quantum computing remains an area of vibrant research, and still, it is not obvious which type or types of physical structure would be the most appropriate platforms for developing architectures for big-scale quantum computations (Houck et al., 2012). Atomic physics and QO research had delivered several candidate systems for quantum computing, comprising two of the presently leading candidates, linear optical quantum computing and ion trap quantum computing, which we would describe here. They had also encouraged a third leading execution, quantum computing with superconducting qubits, where superconducting circuits intended to mimic matter and light and its interaction, realize a quickly emerging technology for quantum computing   = *  }

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2.6.2.1. Ion Trap Quantum Computing Ion traps signify one of the prominent candidates for scalable quantum computing. A central feature in the utilization of trapped ions for quantum simulations or sensing and also quantum information processing is the capability to realize coherent gate operations of qubits, as normally encoded in the electronic states of ions. Realizing single-qubit gates is relatively simple, however, attaining entangling gates that need a coherent interaction is much more exciting. Due to their mutual Coulomb repulsion, the distance among two trapped ions is excessively large for direct interrelations (like as dipoledipole interactions) to be large, and the electronic degrees of diverse trapped ions are non-interacting (Hensgens et al., 2017). It was, therefore, necessary to find an extra degree of freedom that could mediate an interaction, and a mode of combined oscillation of a pair of trapped ions, which is constantly referred to as bus mode, is the most usually employed solution. Interactions amongst a motional degree of freedom and ionic qubits could be realized in terms of driving with a coherent light field of frequency near to resonance with the transition frequency of the two qubit states . As this transition is accompanied by the emission or absorption of a photon, and this photon has a limited momentum, the electronic transition also impacts the emotional state of a gathering of ions. Alternating the driving field precisely on resonance with the qubit transition allows us to minimize this effect and to move carrier transitions that left the emotional state of the ion’s invariant. Selecting a detuning that equals the resonance frequency of the bus mode, though, makes the carrier transition vigorously forbidden and consequences in a blue or red sideband transition; for example, a transition among the qubit states and creation of a phonon or additional annihilation (Gray et al., 2016).        ? –Ÿ‚       of carrier transition and sideband transition on two diverse ions. It needs the bus mode to be initially cooled to its ground state so that the state in one qubit could be printed in the bus mode through a red sideband transition. A sideband transition (comprising the third level) on a second ion encourages a phase shift that is formed on the state of the bus mode, and therefore on the original state of the initial qubit. Scripting the state of the bus mode rear to the qubit encoded in the initial ion concludes a controlled phase gate, which with some extra single-qubit gates, realized through carrier transitions consequences in the desired CNOT gate.

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The prospect of quantum logic encouraged vast activities towards the realization of controlled interactions, both among qubit and motional degrees of freedom. While first ideas for the realization of two-qubit gates were depended on a motional mode cool to its quantum mechanical ground state and need substantial time for the execution of a gate, the technical contest of cooling trapped ions near to the ground state and to sustain phase coherence over extended times was a strong inspiration to develop ideas for fast gates with feeble needs to the properties of the bus mode (Berry et al., 2015; Banchi et al., 2016). The main step towards quantum gates that need no ground-state cooling was the hint to realize the mediation of entangling interrelations by Raman transitions in which variations in the emotional state are only simulated. —   £ #=>! #   !   phonon number, an intervention of numerous path amplitudes makes the   %          ? with this driving pattern therefore even works for incoherent thermal early states. A practical constraint to low temperatures, though, is executed by the fact that contributions of sideband transitions comprising variations by more than one motional quantum required to be ignored, which is valid only at adequately low temperatures. The capability to realize entangling gates without the requirement to cool the ion’s motion to its entire ground state-assisted experiments considerably, and is surely the main step from fundamental proof of principle tests to a practical technology. These advances also encouraged substantial additional theoretical work on the    %  }%  *   Ä }] With several practical ideas for the realization of quantum gates that are  #  > }% ! =~   %% ions are appropriate for the proof-of-principle representation of quantum information processing. The next rational step is the query of scalability, i.e., the realization of hardware that could be assembled to a big computational unit with the potential to surpass the capacities of existing classical  % „{   #!  ‡"#}  => !  { sources, regulating of trapped ions without complicated laser systems is a route aggressively pursued towards the decrease of experimental complexity. Ÿ          lack of laser-control, certainly induces dephasing, however encoding of qubits in dressed states that are unaffected to such noise in leading order gives excellence phase coherence (Barends et al., 2015).

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With the capability to implement quantum gates with adequately high fidelity, quantum error modification could be employed to increase the exactness of quantum information processing. Since quantum error modification needs encoding of a qubit in at least three ions, the experimental realization of quantum error modification is rather challenging. Once realized experimentally, though, it got recognized as a much-applied element in the transformation of an ion trap trial to a quantum computer. With today’s experimental capabilities to regulate on the order of 10 ions, also lately developed topological protection had been executed (Pitsios et al., 2017). Theoretical contributions (beyond the progress of the general framework

>       ! #   %  quantum error modifying schemes to the needs of trapped ion experiments.        %     of a qubit in numerous ions rather well, and the requirement to perform complicated measurements is a bottleneck that could be overwhelmed     % !          arrangement of trapped ions or more independent solutions in which dissipation overwhelms the requirement of active measurements and corrections (Jördens et al., 2008). Trapped ions are surely a prime instance of a quantum optical structure that is on the threshold of being converted into a technology. The driving force behind the growths over the previous decades was surely the prospect of quantum computation, however, also quantum simulations had recognized  {   } {•   # {   of decreasing decoherence and noise in experiments, theoretical work had notably backed to advances with ideas to realize preferred coherent dynamics, even in experiments that do not attain perfect suppression of imperfections (Hadley and Bose, 2008). The outlook of quantum information processing positively sparked various activities towards the growth of ion trap technology that is able of implementing proper quantum algorithms. Given the various experimental challenge in the realization of the preferred technologies, theoretical developments are surpassing the experimental progress. That is, despite the numerous theoretical ideas to realize and develop quantum gates, the experimental actuality is yet far away from a quantum computer surpassing the capacities of prevailing classical computers (Gorshkov et al., 2010). Beyond discovering the potential of an ion trap quantum simulator or quantum computer, there are also huge theoretical efforts to assist the

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experimental growth of the essential technology. As any hardware of %        { }    {   # # decoherence and other system inadequacies, optimal control theory is     % !       !    #         experimental hardware (Banerjee et al., 2013). !    %  {! # %%      decrease of coupling to environmental noise, and it could also be utilized to improve the robustness of trapped ion quantum gates. For the degrees of liberty that encode qubits, it looks natural to target decoupling from entire sources of noise. However, for auxiliary degrees of liberty utilized to mediate interactions, this is not essentially the only feasible option. Certainly, coherent quantum gates could also be mediated through dissipative motional  }    ? ‡    

  entangling gates or by energetically exploiting motional dissipation (Gou and Knight, 1996).

2.6.2.2. Linear Optical Quantum Computing Photons are natural applicants to signify quantum bits. For instance, states might be represented as orthogonal polarizations (e.g., vertical vs. horizontal) or through path degrees of freedom (which arm the photon travels below in an interferometer). Linear optical networks comprising of phase shifters and beam splitters, or equivalently polarization rotators and polarizing beam splitters, permit entire single qubit quantum logic gates to be realized. As discussed above, such networks also permit arbitrary unitary mode transformations. It is hence natural to inquire whether universal quantum computing could be  %      %”º~ }^^œ To accomplish a universal set of quantum gates, need the addition of a two-qubit gate like the CNOT gate. Inappropriately, CNOT gates cannot be attained through linear optical networks alone. Photonic ways interfere in linear optical networks; however, photons do not directly interrelate. However, linear optical transformations can take to entanglement among modes (for instance, in Hong-OuMandel interference, debated above), a linear optical network cannot realize an entangling quantum gate on the photon qubits (Bermudez et al., 2007; Retzker et al., 2008). Ÿ         ? %  =%   interaction, and the Kerr effect could be the base of entangling photonic

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quantum gates. Kerr nonlinearities in natural constituents are too weak for convenient quantum gates to be realized. Procedures that exploit atom-light interactions for tougher optical nonlinearities continue an active area of research (Figure 2.6) (Wunderlich et al., 2003).

Figure 2.6. “Fusion gates” realize a parity measurement through a single polarizing beam splitter followed through measurement. They could be utilized to combine entangled multiphoton states into bigger cluster states, resources for the universal quantum computing through single-qubit measurement alone. Source: https://www.sciencedirect.com/science/article/abs/pii/S007967271730 0186.

By joining linear optical networks with photon measurement, KLM   ¢ ‡  €#   ¢€ %       ~ as the CNOT gate could be realized, through a mechanism known as measurement-induced nonlinearity. The entangling gates realized in this way are non-deterministic, they only prosper some of the time, when a % %  = ‚  }  ‡ %} #    quantum state (by measurement) of the qubits in the procedure (Bermudez et al., 2009, 2010). This non-deterministic property knows that one is required to take additional steps to attain convenient near-deterministic computation. KLM proposed a technique for enhancing the success probability of the gate adequately high that error modifying codes could tolerate the residual failure rate. This mechanism though was very complex and need a large number of extra photons to implement each CNOT gate (Bermudez et al., 2011a, b). Many alternate approaches had been suggested since this work, which considerably decreases this overhead. A common approach is to utilize an alternate model of quantum computation, measurement-based quantum computation. Now, rather than realizing quantum computations through quantum gates, equal computations are accomplished through single-qubit measurements upon an entangled resource state recognized as a cluster

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state. Since photon measurement and linear optics permit measurement of single qubits in the entire bases, all that is then needed for universal quantum computation is to generate an entangled multiphoton cluster state. Nielsen " #    = –Ÿ‚    % !   to cluster state construction, decreasing the overhead related to KLM’s approach considerably. Rudolph and Browne presented that HongOu-Mandel type interference joined with photon measurement gives a direct approach to making cluster states through so-termed fusion gates, a form of parity measurement. This decreases the resource overhead further, gives entangled photon Bell sets and active switching of photons were accessible. Numerous challenges persist for the huge scale implementation of linear

%  >   %   ƒ    !     high-quality single photons or (superior) entangled photon pairs or triples, complicated interferometers with certain active switching, and highly  “ {}  %%      had been established, and linear optical quantum computing resides a strong candidate for the realization of huge-scale quantum computing. In the near term, linear optics had become a strong applicant for the demonstration of so-termed quantum supremacy, computational demonstrations of chores which would be too tough for existing classical computers to duplicate. The presentation of a quantum algorithm of appropriate complexity might be one method to achieve this, however, a potentially simpler alternate is offered by sampling issues. A sampling issue is a challenge to generate output bit-strings sampled as per a particular probability distribution. Properties of the distribution could make certain distributions hard to realize on a classical computer, and this hardness could be proven given certain expectations, which are extensively believed by computer scientists, hold to be accurate (Gessner et al., 2014). ”  = %    !  %#! Arkhipov and Aaronson utilizing linear optical circuits. The output of a circuit, with numerous single photons as input, and numerous single-photon detectors at output could generate a distribution for which there is strong proof that it is classically hard. These arguments utilize elegant methods    % }  %  %     %"!¯ {} they could be understood to follow from a link among matrix permanents and linear optics. A matrix permanent is a function like the better-recognized      !} {}     

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signs that characterize the determinant. Not like the determinant, for which     {   }       {    algorithm for strong evidence and permanent computation that no such algorithm would be found. Scheel exhibited that the permanent increases naturally in the calculation of multiphoton states after a linear optical transformation (Schlawin et al., 2014). The look of the permanent in linear optical network transformations is an initiative of the potential for quantum supremacy later exposed by the Boson Sampling issue.

2.6.2.3. Networked Quantum Computing While linear optical quantum computing and ion trap quantum computing each had distinct advantages, a further promising area is a hybrid approach that joins features from each. Ion traps had the benefit of high reliability deterministic for multi-qubit and single-qubit gates. The principle test is in their scalability. Linear optical quantum computing, though, gives simple interferometric tools to produce qubit-qubit entanglement (Lemmer et al., 2015). The main idea of networked quantum computation is to accept a modular architecture, where distinction trap registers of a minor number

># _       % #   = induced nonlinearity to generate long-distance entanglement among separate   %   %  #   %%    modular design should assist scalability since the numerous ion trap registers could share a uniform design (Figure 2.7) (Zippilli et al., 2014).

Figure 2.7. Spatially detached qubits (here ions stuck in separate cavities) could be entangled through interference on a beam splitter and the emission of photons followed through photon detection. The deletion of which-path information on the beam splitter takes to the entanglement. Source: https://www.sciencedirect.com/science/article/abs/pii/S007967271730 0186.

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The idea of manipulating photon interference to tangle spatially detached qubits could be traced back to previous work by the groups of Peter Zoller and Peter Knight, who displayed that if objects release photons and those photons interfere, the removal of which-path information because of the photonic interference could, after a successive photon detection, permit for the production of entanglement among the releasing objects. This is demonstrated in Figure 2.7. Entanglement arises because of the removal of which path information together with the conservation of coherence in the procedure. Alteration of this idea had been suggested which make the procedure   #}%} % %     ~ ” }  %  !   #   %   

% =” disparity experiment (Fukuhara et al., 2013). Realizing such a networked method to quantum computing is now one of the main goals of Networked Quantum Information Technologies (NQIT) Hub of the UK National Quantum Technologies program, a determined and substantial investment from the UK government towards the realization of quantum technologies. The NQIT hub is forming an NQIT 20:20 prototype, where 20 ion trap modules would be linked through a linear optical network. — {       !  ‡         potential and scalability of the networked method to quantum computation (Jurcevic et al., 2014).

2.6.2.4. Superconducting Qubits Superconducting qubits are one of the two prominent approaches (together with spin qubits) for realizing quantum computing in the solid-state. We put attention to superconducting qubits here as their project had integrated several ideas from QO. Circuit QED is a straight analog of cavity QED. In-circuit QED, an       %   ? #!   > !  –

% %    superconducting island of the so-termed Cooper-pair box which interrelates    { %  %   {    Lately, researchers had developed a circuit QED structure less susceptible to charge noise to enhance the coherence time, which is named transmons. The transmons are one of the tough contenders to realize qubits for quantum computing with extensive coherence scalability and time (Bose, 2007). We could regulate the interaction strength of the circuit QED structure. Certainly, the interaction strength could be made very huge in contrast to the

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spin transition frequency. With the informal controllability of the interface, the circuit QED structure had realized some quantum simulation protocols. When the interaction strength is as huge as the transition frequency, the rotating-wave estimation is not effective, and the dynamics of the circuit QED structure are termed by the Rabi model rather than the JCM. Once the counter-rotating terms are no extended ignored, a substantially large number of states become joined and a quasi-equilibrium state seems even though the entire dynamics are described through a closed system (Compagno et al., 2017).   ># %     {   elements of quantum computing, from extended coherence times and the execution of a universal set of gates to the demonstration of the building # ~ >     

2.7. SUMMARY AND OUTLOOK In the 1970s, when neither single photons nor single atoms, had yet been straight observed, a little group of physicists founded QO. For the previous 40 years, we had observed enormous development in understanding quantum mechanics and influencing single photons and single atoms. Some widespread textbooks still explain quantum mechanics as a law of nature at the subatomic scale. Though our capability to regulate nonclassical interactions among photons and atoms is getting better and we are now able to observe quantum behavior in an object comprising billions of atoms. Researchers are implementing quantum mechanics to tackle unrequited questions in numerous branches of science like as high-efficiency energy transport in photosynthesis. We are in a stimulating era where our deepest theories of physics could be tested utilizing quantum optical tabletop experiments and where quantum mechanical ideas had reached the main role in the growth of new technologies (Schreiber et al., 2015; Smith et al., 2016).  %      { % # ? by the UK government with considerable investment, by EPSRC, in the National Quantum Technologies program. Peter Knight played a vital role in convincing the UK government to fund this ambitious program and assist shape its actions on its Strategic Advisory Board and beyond (Meyer, 1996). Research in this program is controlled by 4 Quantum Technology Hubs. A hub centered at Birmingham University is emerging quantum improved Sensors and Metrology, however, the QUANTIC hub lead by Glasgow University is working towards Quantum Enhanced Imaging. The NQIT

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hub with it headquarter at Oxford University is emerging a networked approach to quantum technologies as explained above. Lastly, the Quantum Communications Hub, led by York University, is emerging quantum communication protocols for safe communication for communication with security based on quantum mechanical principles (Knight et al., 2003; Sanders et al., 2003). The European Commission had recently announced an ambition for a ‡  %   { %%          of quantum technologies development and research across Europe, and there are strategies for similar huge scale research investments around the •   # ~   }        ambitious and transformative technologies looks bright.

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Chapter

3

Integration of Quantum Optics and Photonics

CONTENTS 3.1. Introduction ...................................................................................... 80 3.2. Plasmonics and Nanophotonics: Controlling Optical Fields and Propagation on the Nanoscale ....................................... 84 3.3. Coherent Electromagnetic Fields: Attosecond Time Scales and X-Ray Photon Energies ............................................................ 88 3.4. Optomechanical Interactions: From Single-Molecule Mechanics to the Macroscopic Quantum States ............................. 91 3.5. Seeing Beyond the Diffraction Limit and New Imaging Modalities .... 95 3.6. Creating and Controlling Quantum Coherence with Light................. 97 3.7. Controlling Molecules with Light and Light with Molecules ............ 100 3.8. Observing the Universe: Optics and Photonics for Astronomy and Astrophysics .......................................................................... 103 References ............................................................................................. 106

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3.1. INTRODUCTION Photonics and Optics is a unified rational discipline with a history courting back to the earliest growth of mankind’s scientific belief. It is founded on rigorous theories of electromagnetism, comprehensive where suitable to quantum electrodynamics (QED), joined with an understanding of lightmatter interfaces in materials (Kimble et al., 1992; Rokhsari and Vahala, 2004). At the similar time, the field of photonics and optics is near without parallel in terms of the extent of its effect on other fields of science and its effect on a wide range of technologies. Regardless of the long history of photonics and optics, it is also a field that has experienced and is experiencing dramatic growth in its impact. This growth is driven by constant technological and scientific breakthroughs: from exceptional control of single photons to pulses of electromagnetic radiation of constantly enhancing intensity and reducing duration to novel coherent sources of x-ray radiation (Figure 3.1) (Lodahl et al., 2017; Barik et al., 2018).

Figure 3.1. Diagram illustration of the 2D scanning mechanism utilizing quantum optics. Source: https://www.semanticscholar.org/paper/Quantum-Optics-and-Photonics-1.-Experimental-a-and-Ezekiel-Ham/02ede3cde5d80810e3b9cfd19cd6381  œ  ¿   ?À      

    " {   and subcellular structure with heretofore-unattainable precision (Liao and  }    {   {  %    %{ { { £  % progress in ultrahigh-resolution optical traps and the utilization of orbital angular momentum to carry exact amounts of torque individual to molecules, as shown in Figure 3.6. Another vital arena of advance had been the amalgamation of controlled mechanical manipulation of molecules with single-molecule spectroscopy. This permits one to determine the impact of force through exact spectroscopic signatures from the molecules, for instance, through variations in resonant energy transfer among chromophores (Figure 3.7).

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Figure 3.7. – #    ¿  ?À   %%!% es to individual molecules with details of the consequences of the force utilizing single-molecule spectroscopy through FRET. Source: https://science.sciencemag.org/content/318/5848/279.

   %     #%! { direction where quantum mechanics plays a vital role like both the mechanical {#        { >  %          {   }  %   }  }  optical physics (for example., ultracold science), low-temperature physics, nanoscience, condensed-matter physics, QO, and quantum information ƒ%}   { !   ‚    physics for exquisite regulating of light-matter interactions; for knowing the mechanical effects of light and the related routes to laser cooling; for  ?    =    %    >  limit of measurements; and for knowing of decoherence and quantum noise. Nanoscience and technology have given us the tools to make an extensive variety of nanomechanical and ultrasensitive microdevices, system with high-Q resonance and robust light-matter interaction, and to describe these structures with spatial resolution below to the atomic level. These latter progress mirror other breakthroughs in the utilization of mechanical sensing to measure properties on the atomic gauge with extreme exactness, comprising single electron spin detection thru magnetic resonance, yoctogram scale mass sensing, atto meter scale dislocation sensing and zepto newton scale force sensing (Hugel et al., 2002; Yin et al., 2013). The main focus of research in quantum optomechanics had been the study of the interface of light with mechanical vibrations by the utilization of high-Q resonant structures. In addition to attaining a deep understanding of the diverse coupling regimes, researchers had been capable to apply a technique similar to side-band cooling of trapped ions that were made

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within the molecular, optical, and atomic physics community to calm the vibrations. In this approach, improvement by an optical resonance put the rate of anti-Stokes Raman scattering procedures to surpass that of Stokes Raman  ƒ  !}    !  {#  } taking to a laser-cooling procedure. This technique had now reached the main milestone in attaining cooling to the quantum limit of an average mode use below one vibrational quantum (Lombardo and Twamley, 2015; Clarke and Vanner, 2018). While this situation is simply reached in molecular systems with the vibrational frequencies in the middle-of far-infrared, accomplishing  >      %{#    %!  regime, one where quantum mechanics interact more obviously with the macroscopic world. Depending on such fundamental light-matter interactions permits these phenomena to be examined for mechanical resonances over a huge range of frequencies and masses, as shown in Figure 3.8. Certainly, the original light-matter interactions are appropriate for determining basic properties for the sensing of movement in gravitational wave exposure in the LIGO (Laser Interferometer Gravitational-wave Observatory) facility (Arndt et al., 2009).

Figure 3.8. Cavity optomechanics inquiries are presently being pursued on structures from individual atoms to macroscopic masses, covering more than 10 orders of magnitude in frequency and 20 orders of magnitude in mass. * + %+{  exists between an atom that interacts with a quantum particle and a single       %  ¯       {       #        } > ?     Therefore, we can observe the Jaynes-Cummings dynamics with trapped ions. Recently, D. Wineland et al. (1998) have demonstrated this concept in a series of experiments. A Schrödinger cat state was also produced by them, they did it by preparing a trapped ion that was positioned between wave packets that were spatially separated, and they formed these packets by combining different vibrational quantum states (Udem et al., 2002; Venturini, 2005; Stranius et al., 2018). Apart from the microwave region experiments, An et al. (1994) have also realized that an atom in the visible range, emits laser. Moreover, Miller et al. (2005) has studied the effects of cavity QED in the spectral region (Yamamoto and Slusher, 1993; Berman, 1994).

4.9.2. Phenomena of Quantum Interference In classical optics, interference occurs when we add at least two wave amplitudes having different phases. In classical physics, there is no phase of a classical particle; hence only waves possess the capability of raising interference (Schulz-Ruhtenberg et al., 2009; Schmidt and Hawkins, 2010). By contrast, in quantum mechanics, interference is not limited to only waves; it is a general feature but turns up at a point where the measurement outcome can arrive through several indistinguishable paths, the amplitudes of probability must be added for calculating the measurement result. Therefore, both waves and particles present themselves in the description of quantum mechanical interference. The photon’s particle aspect is apparent as at the same time, we cannot detect it at two different positions, for instance, if the photon is detected at a point, the probability of detecting it at a different point vanishes; hence light is treated in terms of the wave only in classical optics, so photon interference effects cannot be explained by classical optics (Phillips, 1998; Nesvizhevsky et al., 2003). To achieve photon interference experimentally, we have to generate %  %   # #!%   %  ‡  or parametric down-conversion. In this process, inside the crystal, a “pumped”

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photon (ultraviolet one) decays into the idler and signal, that are highly correlated, as both are red photons. When both photons arrive at detectors   #  } }        %     #} interferences are observed. L. Mandel and co-workers introduced a setup that is being used for investigating quantum phenomena such as the quantum eraser, Bell inequalities, photon tunneling, and the electron paramagneticresonance paradox (Chiao, Kwiat, and Steinberg, 1994; Chiao and Steinberg, 1998; Shih, 1998). We can also generate the entangled photon pairs by using a parametric down-conversion of type-II. They serve as the base for the experiments being performed that are correlated with quantum information (Sols et al., 1989; Stamper-Kurn et al., 1998; Steane, 1998).

4.9.3. Quantum State Measurement On a quantum system, all the information is accommodated in the wave function. In recent years, the experimental technology and the theoretical knowledge were obtained and gathered for preparing a specific system, i.e., atoms or light in a specific quantum state. One of the examples is the above discussed squeezed state preparation. To test the primary predictions made by quantum mechanics, fascinating possibilities are offered by the capability of engineering the quantum states (Popelka et al., 2005; Polak et al., 2018). Moreover, practical implications are also involved; highprecision interferometry might be allowed by particularly designed laser fields (for instance, squeezed light), aiming to reduce the quantum noise. Successful measurement is also required by the engineering of states. In a single measurement, we cannot determine a quantum state’s wave function (Schleich and Raymer, 1997). Therefore, in a measurement, we have to repeatedly reproduce the initial conditions to sample the data properly. Not only theoretically, but there has also been exceptional progress in experimental work as well for forming strategies for the reconstruction of numerous quasi-probability distributions and for other quantum state representations. In the quantum-mechanical phase, as the uncertainty principle forbids the point like sampling, instead of it, a tomographic method is adopted. This problem is outmaneuvered by investigating circular discs, thin slices, or quasi-probability distribution’s infinitesimal rings in phase space and due to this, quasi-probability distributions (Q function or the Wigner function) are acquired as the quantum state representation (Reynaud et al., 1992; Predehl et al., 2012). £ ! [   •          Š    %      }       ˜

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quantization is performed in comparison to the harmonic oscillator. Light’s single-mode dynamics are correlated with a quantum particle dynamic in a

 %  *   {  }¯* Ä }^*% !}   %            {  #} } conjugate variables that correspond to momentum and position. Therefore, we can immediately transfer the phase-space considerations to light. A

 !   %       !        —  # %} ~  "        coherent. The time when the light’s phase shifts, the signal that appears in the two arms is analyzed, resulting in the Wigner function’s reconstruction. Recently, at NIST in Boulder, the group of D. Wineland et al. (1998) were able to determine a single ion’s vibrational quantum state while stored in a Paul trap. Another fascinating and new area of research is quantum state measurements; it is opening a highly innovative window. Now, complete information of the elementary quantum system can be extracted, i.e., a single light mode, a single trapped particle, or a molecule and for determining the wave function of these systems (Royer, 1989; Roth et al., 2008).

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38. Hall, J. L., Ye, J., Diddams, S. A., Ma, L. S., Cundiff, S. T., & Jones, D. J., (2001). Ultrasensitive spectroscopy, the ultrastable lasers, the     }     !    #+—       physics and metrology. IEEE Journal of Quantum Electronics, 37(12), 1482–1492. 39. Hänsch, T. W., & Inguscio, M., (1994). Frontiers in Laser Spectroscopy: Varenna on Lake Como (Vol. 120, pp. 1–33). Villa Monastero. NorthHolland. 40. Hänsch, T. W., & Schawlow, A. L., (1975). Cooling of gases by laser radiation. Optics Communications, 13(1), 68, 69. 41. Haroche, S., & Kleppner, D., (1989). Cavity quantum electrodynamics. Phys. Today, 42(1), 24–30. 42. Haroche, S., (1998). Entanglement, decoherence and the quantum/ classical boundary. Physics Today, 51(7), 36–42. 43. Hemmerich, A., & Hänsch, T. W., (1993). Two-dimensional atomic crystal bound by light. Physical Review Letters, 70(4), 410. 44. Höffges, J. T., Baldauf, H. W., Lange, W., & Walther, H., (1997). “ !     ‡     Journal of Modern Optics, 44(10), 1999–2010. 45. Hong, H. G., & An, K., (2012). Quantum theory of frequency pulling in the cavity-QED microlaser. Physical Review A, 85(2), 023836. 46. ƒ{  }€}“ }§}* }•}¡•  }*–}^\[  %      { !  –Ò”   SO2. The Journal of Chemical Physics, 78(11), 6531–6540. 47. Jones, R. J., & Diels, J. C., (2001). Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radiofrequency synthesis. Physical Review Letters, 86(15), 3288. 48. Kasevich, M., & Chu, S., (1992). Laser cooling below a photon recoil with three-level atoms. Physical Review Letters, 69(12), 1741. 49.  #} “ §} ¡ € } ¢} ^œœ £   ‡    "   #   Physical Review A, 15(2), 689. 50. Kinjo, M., & Rigler, R., (1995). Ultrasensitive hybridization analysis ‡    %  %!Nucleic Acids Research, 23(10), 1795–1799. 51. Kneipp, K., Kneipp, H., Itzkan, I., Dasari, R. R., & Feld, M. S., (1999). Ultrasensitive chemical analysis by Raman spectroscopy. Chemical Reviews, 99(10), 2957–2976.

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Chapter

5

Quantum-Mechanical Concepts in Optical Coherence

CONTENTS 5.1. Introduction .................................................................................... 144 5.2. History............................................................................................ 146 5.3. The Quantum Optic’s Present Status................................................ 153 5.4. Optical Frequency Comb Techniques and Laser-Based Precision Spectroscopy ............................................. 154 References ............................................................................................. 164

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5.1. INTRODUCTION The most appropriate example of a quantum object’s dual nature is provided by light; Maxwell’s electromagnetic theory is verified by oscillatory properties of light while the modern quantum theory is endorsed the photons and “lumpiness” of light (Klein, 1961; Planck, 1978; Glauber, 2007). Most of modern technology is founded on electromagnetic phenomena. This phenomenon appears in all electrical motors, similarly, oscillatory behavior is utilized by the communication devices in vital ways. Our mobile phones and radio receivers are based on the capability of radiation through      = %  % %  >! †, 1905, 1917; Dutta et al., 1995). Contrary to it, every radiation detecting device must absorb the energy of radiation into the medium of material. This energy occurs in the form of packets, named photons. Absorbing a photon will result in creating an excitation that can be   % * ^]}     ~  †, we are well aware that absorbing a quantum of radiation will raise only a single photoemission electron from a solid (Nobel Prize, 1921). Hence, photons are not counted by the detector; it only counts photoelectrons, so we always have indirect information about the photon’s behavior. For the observation process, it is mandatory for a photon to get absorbed, and thus the photon is no longer available (Dirac, 1927; Brown and Twiss, 1956). Between these pictures, the dualism may appear contradicting. In actual they do, from an important example in quantum theory that is called complementarity, namely the possibility of displaying either particle or wave properties; in spite of the fact that their emergence is observed in mutually exclusive limits (Hioe and Eberly, 1981; Simon et al., 1987). Talking fundamentally, there exists a need for reconciling these two descriptions. We should be aware of the manifestation of the seemingly smooth oscillations

        %        %! >   “} we need a microscopic theory for accounting the interaction between the material and photons, as well as a macroscopic theory for accounting the phase properties of photons (Figure 5.1) (Purcell, 1956; Mandel et al., 1964; Jibu et al., 1994).

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Figure 5.1. Optical coherence tomography’s schematic illustration. Source: https://sites.google.com/site/obeluwa/research/oct/introduction-to-oct.

In Physics, this year’s Nobel Prize falls in the domain of the following aspects of light: ˆ

ˆ

The initial part belongs to Roy J. Glauber. He showed the formulation of the quantum theory for describing the process of detection (Glauber, 2007). This also helped in distinguishing the # {    

 }>  % and lasers) and thermal light sources. The sophistry of quantum electrodynamics (QED) is used by this theory in describing photon absorption. Higher-order correlations can be obtained by comparing the outcomes of several such detectors, which can display the attributes of quantum radiation (Sudarshan, 1963; Klauder and Sudarshan, 2006). The other part belongs jointly to Theodor W. Hänsch and John L. Hall. They have shown their contributions in developing precision spectroscopy based on laser; they have included the optical frequency comb technique (Helstrom, 1967; Glauber et al., 2005). Their work has made it possible to test fundamental theories, to determine the matter’s quantum structure with complete accuracy, and their work also has some other important applications such as in setting an absolute limit for the set-up’s performance, quantum effects, and in precision measurements. ƒ  !}      is connected with the latter one (Glauber, 1963; Scarcelli et al., 2004).

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5.2. HISTORY 5.2.1. Quantum Theory Emergence Even before the start of the 20th century, spectral observations turned up to be accurate enough that scientists were able to conclude that the black body emission was not in accordance with the prevailing theories. The theoretical model of this type of system was a closed resonant cavity, and its spectral properties were represented under the name “Hohlraumstrahlung.” The issue was that the resulting expressions disagreed with the experimental data (Romer, 2004; Valeur and Berberan-Santos, 2011). The seminal contribution was made by Max Planck at this point (Nobel Prize, 1918); he exerted his profound thermal entropy knowledge to the standard phenomenological approach and made an ad hoc   “          "% %   !  ƒ ‚ # 1900, Planck (1978) obtained the result that was successful; however, it did not brief the physics behind it. Nonetheless, Planck continued considering the energy exchange between the material radiators that constitute the thermal {    —    % }  !>  " % # # %%   ” ?   applied to entropy and “after a few weeks of the most strenuous work of my life;” Planck was able to derive a formula (Klein, 1961; Singh and Vandersloot, 2005). According to the assumption of Planck the energy quantization is linked              %      {!   table was turned by Einstein; he realized that Planck’s theory algebraic form allows an interpretation with respect to radiation lumpiness, for instance, the radiation was taken into consideration as it consists of particles; by 1926, Lewis termed these particles like photons. This hypothesis was applied by Einstein to many physical phenomena (Einstein, 1905; Kiefer, 1994). The photoelectric effect is one of these phenomena, and on it, he won the Nobel Prize of 1921. A common misconception needs to be cleared that at that time when Einstein published this idea, no accurate data on electron photoemission was present. The results were later provided when several investigators continued his work, in this matter, R. A. Millikan showed convincing demonstrations and gained his reward as a Nobel Prize in 1923 (Joos and Zeh, 1985; Ambjørn et al., 2004).  }       {       %%       of energy lumps that are emitted and absorbed in integer units only. In

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theoretical works, this picture was elaborated and extended, and with passing time, it emerged as a basic need for future developments. In 1913, it was used in the hydrogen spectrum theory represented by Bohr (Nobel Prize, 1922). Light quantization by this time was well established. Nonetheless, Maxwell’s theory is serving as a base in electrical engineering. In terms of well-established phases and continuous energy distributions, the radiation is described by this theory. As recognized by Einstein (1917), both these pictures are contradicting each other]: “These properties of the elementary processes required by equation (12) in Einstein (1917) make it seem practically unavoidable that one must construct an essentially quantumtheoretical theory of radiation. The weakness of the theory lies, on the one hand, in the fact that it does not bring any nearer the connection with the wave theory and, on the other hand, in the fact that it leaves moment and direction of the elementary process to “chance;” all the same, I have  %   #!    ÀŠ] (Landsman and Linden, 1991; Faulkner et al., 2011; Durey et al., 2018).

Figure 5.2. Quantum theory applications. Source: https://www.pinterest.com/pin/816840451140429364/.

5.2.2. The Field Quantization’s Emergence Once Schrödinger, Heisenberg, and others developed quantum theory, the need for quantizing the electromagnetic fields became obvious. The mapping of the electromagnetic theory on the harmonic oscillators was known; hence, the procedure appeared to be obvious. This approach was first considered in detail by Dirac (1927). He concludes: “There is thus a complete harmony between the wave and light-quantum description of the interaction (McMullan and Tsutsui, 1995; Steinacker, 2009). We shall

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build up the theory from the light-quantum point of view, and show that the Hamiltonian transforms naturally into a form which resembles that for waves.” Using his theoretical adroitness, Dirac succeeded to obtain the expression for spontaneous emission rate that is considered as a characteristic quantum effect. P. Jordan, along with collaborators R. Peierls, W. Pauli, and L. D. Landau, further developed the quantized electromagnetic field theory (Cetto et al., 2012; Oza et al., 2014). “ {}    } %%   were prevented. From classical physics, a problem was inherited that the        {   %  {            >    !˜     %%     !#!  After World War II, S. Tomonaga, R. P. Feynman, and J. Schwinger solved this issue (Nobel Prize, 1965). Their renormalization program came out to be successful, and now, it is regarded as a base for all modern approaches (Yang and Sivakumar, 2010; Yang and Sivakumar, 2010).

5.2.3. Quantum Considerations Enter Optics When scientists were fully aware of the tools required for handling the QED, they were mainly practiced on high-energy processes. To some extent, it is derived from the collision experiment’s rapid development and in the view of fact that the relativistic invariance requirement played a major part in creating this theory. Natively, it was still believed that the disagreement between Planck’s and Maxwell’s treatments would not significantly affect the optical observations. But this blissful indifference states blissful indifference (Ottaviani et al., 2006; Goulding et al., 2007). An interferometric method was investigated by Twiss and Hanbury (1956) in 1954–1956, for determining the distant stellar object’s angular extension, and laboratory measurements were also made by them. Between photocurrents, the intensity-intensity correlation was observed by them, according to which if recorded in separated detectors, a bump is displayed at a point when in optical path lengths, the difference of zero is present between the signals. In fact, it was found that the correlation function , at x=y takes twice its value in contrast to the value for widely separated arguments (y and x). According to the authors, it is a consequence of quantum theory (Kanis et al., 1994): “The experiment shows beyond question that the photons in the two coherent beams of light are correlated and that this correlation is

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preserved in the process of photoelectric emission.” Purcell (1956) indicated that a classical interpretation of the effect might be present, though it was still assumed by him that it is a vindication of the light’s quantum features. These arguments initiated a great interest in the quantum consideration’s relation to optical observations. It gained more interest in 1960, when the laser was invented, as it  % #! % { {  !   from the thermal ones. As a result, the emergence of two different points of view was noticed.    }   >     are proportional to (n + 1), n   # %   —   %    of Hanbury Brown and Twiss, only a few photons are induced and, on this assumption, the factor of two was accounted; one photon might be inducing another photon. Many photons contribute to laser, and a giant effect might be predicted by one. According to another point of view, an impression lingered that

!%% %%#! >  } !%%!   amplitudes. Hence, a random function theory would be accounting for the observed effects. As an example, we quote Mandel et al. (1964): “In the conditions under   ‡    ! #!%   }  semi-classical treatment applies as readily to the light of non-thermal origin     }   =   !       !À    "%  %  % {   ‡     = classical point of view. In 1963, Roy Glauber published the correct theory, and this one served as a base for all lateral theoretical considerations (Rolo and Vasilevskiy, 2007).

5.2.4. Optical Interference Experiment’s Quantum Theory Glauber (1963) presented the primary characteristics of his optical coherence’s quantum theory. In the same year, he expanded the formal features of his theory in two articles (Scarcelli et al., 2004). This material formed the base on which quantum optics (QO) is developed to the present time (Mandel, 1964; Loudon, 1983). The following points were made by Glauber in the publication of 1963: In photon correlation experiments, the detection must be based on a QED’ consistent application. Thus, the experiments of all multi-photons must form

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its basis on the certainty that once a photon gets absorbed, the present state

     ~  "  # %  {  ~ %         ƒ%}   having n photons only can undergo correlations up to nth order. For the processes of optical detection, this implicit the usage of expectation values that are ordered normally (Eichmann et al., 1993; Risby and Tittel, 2010). As the processes of consecutive absorption are formed on the basis

       }           ?       % # ¯    %          ƒ%} "% “ #!  ^] are explained by a constant calculation of the interference effects of two%     { „  ‡  % %! {+  = + It is explaining the factor of two at a point when x = y. The light phase is important in interference experiments, and in regard to coherent states}   ##%”!    distribution on these, the idea of a quasi-distribution into QO was introduced by Glauber. These represent the state’s quantum descriptions, having simple relations to phase space distributions (classical ones). As shown by Glauber, in certain cases, a diagonal representation can be given to them in coherent states. Though, they do display obvious non-classical features; for instance,  % # #! # ˜ % {!      #!       straightforward quantum states. A classical interpretation can be given to a state if there is a positive distribution. According to what Glauber has shown, a Gaussian distribution  % #!     } }  }‡     !    _ —               ! correlations of the type shown by the Hanbury Brown and Twiss (Figure 5.3).

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Figure 5.3. Three experimental setups for Glauber’s correlation function’s measurement of light that is being emitted by CdSe/CdS DRs and their clusters using an ICCD camera. * + %+   % }   quantum communications. In all discussed situations, it is necessary to develop a proper understanding of the basic theory, as there exists a fundamental limit for the things to be achieved; quantum noise cannot be eliminated, but we can eliminate the technical noise. The possibility for testing the foundational features of quantum theory

     %%    % >  %%  % the fact that many successful applications are developed by using quantum

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theory, but any consensus is still not reached by the interpretation of the theory. Hence, there still exists a need of pushing experiments further to gain insights to achieve a compact formalism (De Natale et al., 2005). Contrary to it, a tool is offered by the coherent state representation for taking this theory into the classical regime. Here the phase and amplitude of    # ={  #• ?   ultra-high precision measurements and communication.

5.4. OPTICAL FREQUENCY COMB TECHNIQUES AND LASER-BASED PRECISION SPECTROSCOPY 5.4.1. The Requirement for Precision Measurements of Optical Frequencies and Atomic Structures The historical background of science shows that unprecedented precision measurements have stimulated many advances revealing new phenomena and structures. In atomic spectroscopy, this is particularly true at a point where increasing spectral resolution results in the observation of atomic hyperfine structure (because of the nuclear spin), volume isotopic shifts (because of the isotopic specie’s different charge distributions of the nuclei), and fine structure (because of the electronic spin). Electric and external magnetic fields give rise to Stark and Zeeman energy level structures. The Lamb shift is raised by the quantum electrodynamic effects that are more subtle. We are probably going to detect new phenomena by pushing to the highest ever resolution and precision. Ultimately, we will be able to achieve a precision that may approach one part in 1018. At extremely high precision, questions regarding the constancy of frequencies of optical transitions can be raised over time. Similarly, possible asymmetries between antimatter and matter might also be revealed. The probability of determining the optical transition frequencies precisely corresponds to obtaining better atomic clocks. In turn, this will allow better space navigation, better GPS systems, and astronomical telescope array’s improved control. This year, the Nobel Prize to T. W. Hänsch and J. L. Hall is based on such developments.

5.4.2. Historical Background In the spectrometers, many technical features are present that limit the achievable resolution. However, to the resolution, more basic limits are present; for instance, a Doppler broadening is raised by atom movement,

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such broadening directly corresponds to the transition frequency. This broadening is nearly 1 GHz for a transition frequency of 1015 Hz in the visible region. However, for transitions, it becomes negligible in the microwave or radiofrequency regions. More generally, the excited state’s finite lifetime raises a natural linewidth. To spectral resolution, this latter limitation is associated with the limited time that can be availed for the measurement. In the 100-nanosecond range, for an exciting lifetime, the natural linewidth appears to be in the range of MHz and exceedingly small for a metastable level. Many different methods have been developed to accomplish the hunt of achieving a highly spectral resolution. We can make the internal level splitting measurements in a durable long atomic state, and by doing so, it becomes possible to attain benefit from the small natural linewidth. It is being practiced in the technique of atomic beam magnetic resonance (Nobel Prize to Rabi, 1944), where between the two structure levels (hyperfine ones) at 9.2 GHz, in 133Cs the transition frequency can be evaluated with the precision of 15 digits, forming the basis for the present definition of the second. We can attain this precision by increasing the time of interaction of the particles by the means of Ramsey fringe techniques (Nobel Prize to Ramsey, 1989). A resolution as low as the Heisenberg limit in optical double or optical pumping resonance experiments is also provided by optical resonance techniques (1966, Nobel Prize to Kastler). In the visible regime, the Doppler broadening is usually very detrimental, we can eliminate it by the means of special techniques, i.e., two-photon spectroscopy or saturation. In 1981, for their contributions in the laser spectroscopy development, A. L. Schawlow and N. Bloembergen were awarded the Nobel Prize. Reducing the velocities provide a solution for the problem raised by the Doppler effect, we can reduce the velocities through laser cooling techniques. Light exerted forces can also be utilized for bringing about the atom’s spatial confinement (Nobel Prize to Phillips and Cohen-Tannoudji, 1997). Electromagnetic field arrangements can trap ions (Nobel Prize to Paul and Dehmelt, 1989). Bose-Einstein condensation (BEC) was made possible by a combination of trapping and cooling with evaporative cooling, producing coherent matter (Nobel Prize to Wieman, Cornell, and Ketterle, 2001).

5.4.3. High-Precision Laser-Based Spectroscopy The laser and maser development for which in 1964, the Nobel Prize was rewarded to A. M. Prokhorov, N. G. Basov, and Ch. H. Townes, has provided oscillators of high-frequency having a narrow bandwidth when single-mode operation, continuous-wave is achieved. Owing to the laser resonator action,

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we can achieve a line-width that is much narrower as compared to the width of the transition. Acoustic noise, mechanical vibrations (along with many others) greatly limit the stability of the output and the line-width. We can strongly reduce  #! ~ >!      =% {} especially designed interferometer’s sharp interference fringes by making the use of electronic feedback. We can even achieve the stability of less than the one Hz level. We can further lock such lasers to sharp molecular and atomic transitions. J. L. Hall, along with collaborators (at JILA, Boulder; operated by NIST, National Institute of Standards and Technology, and the University of Colorado), developed the strong frequency stabilization schemes that allow making fundamental measurements (Figure 5.5).

Figure 5.5. Laser induced spectroscopy’s illustration. Source: https://physicstoday.scitation.org/doi/10.1063/1.2709559.

Atomic hydrogen is regarded as the fundamental atomic system that can be studied. It allows the most appropriate and correct theoretical calculations for confronting the precise experimental data. In 1972, Schawlow and “Ñ   %   %  %!    a kind on hydrogen. In this measurement, a tuneable (narrow-band) dye laser could be used for solving the issue of the Lamb shift. Then, the hydrogen spectroscopy was pushed to its limit by Hänsch along with his students, they did this in a sequence of papers that measured the optical frequency of 1s2s, Lamb shifts, and the Rydberg constant, initially at Stanford University,

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later on (after 1986) at MPQ in Garching and the Ludwig-Maximilians Universität, München. Extremely narrow lines are allowed by the upper state’s long lifetime; hence this makes 1s-2s transition to be of particular interest (Figure 5.6).

Figure 5.6. This set-up was formed for determining the precision in atomic hydrogen’s 1s-2s transition. At 486 nm, a stabilized narrow-band laser is frequency-doubled at 243 nm for inducing a two-photon doppler-free transition in hydrogen that is cryogenically cooled. We can achieve longer interaction times by using delayed irradiation to restrict the detection to the slow tail and also through pulsing the hydrogen beam. Through an optical frequency comb generator, the blue laser’s frequency is measured (in the cesium atomic clock terms). Source: https://www.nobelprize.org/uploads/2018/06/advanced-physicsprize2005.pdf.

In 1970, Chebotayev, along with coworkers, suggested a method of twophoton Doppler-free absorption spectroscopy. At about 243 nm, it uses two photons. Figure 5.6 shows the setup in which it is employed. This precise transition can avail the advantage only at a time when we stabilize the output frequency of the laser in accordance with the principles suggested by Hall (1984, 2001). Sometimes suspended at conditions that are highly temperature-stabilized and in a vacuum, ultrastable resonance cavities are utilized. This methodology also involves active devices using electro-optic phase modulators and acousto-optic frequency shifters, as shown in Figure 5.6. In 1984, a major step was taken that Hall and Hänsch have described in a joint paper (Hall and Hänsch, 1984). Now, the 1s-2s interval is calculated to be 2,466,061,413,187,103 (46) Hz, and a value of 109,737.31568525 (73) cm–1 is evaluated for the Rydberg constant. Further

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groups, like the ones in Oxford and Paris, have made important contributions in the hydrogen’s precision spectroscopy, though Hänsch´s group is still regarded as the leading one. Highly stabilized lasers have been used by Hall for primarily essential measurements, this includes the experiments of the Michelson-Morley, and also of the Kennedy-Thorndyke (Brillet and Hall, 1979; Riis et al., 1988). Through Hall and Hänsch’s (1984) work, now the optical laser spectroscopy’s precision is similar to the microwave atomic clock’s precision at the 10–15 level. Though lateral improvements might be limited for the latter ones, it is expected that optical clocks will soon outperform the microwave techniques. Figure 5.7 shows these trends (Hils and Hall, 1990).

Figure 5.7. In precision spectroscopy, the development in relative accuracy is shown. In both, microwave and optical-based systems, 10–15 level has been achieved. Source: https://www.nobelprize.org/uploads/2018/06/advanced-physicsprize2005.pdf.

         {#   continuous development phase. Since 1889, the authority that is managing the matters of the physical unit is the General Conference on Weights and Measures  ^  %#  { 

    %  \  ¢   ^œ}     #    ˜^^[}œœ    [[–}  !% transition. We can now determine the velocity of light with better precision

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by multiplying wavelength and frequency for a stable source of radiation. ƒ       #    ! spectral line, which becomes a constraint. A process in which there was a  { { ”    #} } it was coupled with the second and it was stated in 1983 that in the vacuum, the velocity of light is 299.792,458 m/s, having an uncertainty of zero. It can be concluded that light travels a distance of 1 meter in 1/299.792,458 s (Evenson et al., 1972; Jennings et al., 1983). Š   {        #?>! expressed in the unit of meter, the frequency of it should be measured, and  { !  ˜{  #{      measurement. Determining an optical frequency of 1015 Hz by contrasting   >!  !% –  “?} in complications. Long chains of phase-locked and highly stabilized lasers whose frequency was multiplied and later combined with stable sources of the microwave were developed in the highly specialized laboratories. Still, scientists were able to determine a really small number of optical transitions. Š       %   %% }   ˜     #  unusable. Hence, for accurately measuring the optical frequencies, a new way was urgently required.

5.4.4. The Advancements in Optical Frequency Comb Techniques In high precision metrology, a crucial development in the form of optical frequency comb techniques insightfully solved many major problems. New measurement methods are formed on the basis of the fundamental relations that exist between cavity modes in a laser that is continuously operating and its interference, which results in short pulse’s repetitive strains. However, according to a prerequisite, the cavity modes, separated by: »Ø!  # techniques was practically affected and became limited. Telle et al. (1990) realized a frequency chain that is comprised of nearinfrared or visible laser oscillators only. The measuring optical frequencies task was primarily facilitated with this step. In the late 1990s, the utilization of the frequency comb structure was started on the titanium sapphire mode-locked lasers for bridging large frequency intervals to relate optical frequencies with the cesium clock in a frequency chain that is simpler (Udem et al., 1999a, b). A particular mode frequency fn can be represented as the mode separation’s integer multiple (i.e., the pulse repetition rate frep = 1/T) that adds a carrier-envelope offset frequency fCEO: fn = nfrep + fCEO where; fCEO»!{   ! contributed to precision metrology with L. Hollberg, J. C. Bergquist, D. J. Wineland, and other researchers. In comparisons of the optical transition’s frequency, precisely measured at a time gap of 1–4 years, several groups investigated the drifts that were possible in the fundamental constants. These groups include Hänsch´s group and the group at NIST. So far, no drifts could be established having an uncertainty of some parts within 1015 (Tearney et al., 1997; Niering et al., 2000; Fercher et al., 2003). Hänsch and Hall´s group have a major contribution in developing optical frequency comb techniques. They elaborated on the details of optical frequency measurement schemes that were previously used, and they worked for only a few selected frequencies. Through their developments, they replace them by the 1 × 1 m2 size set-up, which is better for precision measurements and can be used by any frequency and are available at the commercial level. In optical frequency measurements, a true revolution has occurred that paves the way for creating optical clocks having a precision of about 1: 1018. Reviews can be seen in Hall et al. (2001); and Holzwarth et al. (2001).

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5.4.5. The Utility of Optical Comb Techniques and Precision Laser Spectroscopy, and Further Developments As discussed earlier, Hall and Hänsch’s (1984) developed techniques, allowing highly precise optical frequency standards and spectroscopy, have numerous applications of fundamental, as well as of practical nature. Fundamental theories through them can be tested at the highest level of precision, regarding the isotropy of space, possible drifts in the constant, possible asymmetries present between anti-matter and matter (antihydrogen and hydrogen), and relativistic effects. Apart from these, we can refine the global positioning systems, can accurately synchronize the large astronomical telescope arrays, and can facilitate the deep space navigation. Recently, the Hänsch group extended the comb technique to the region of VUV and XUV. JILA group also has a contribution in this cause. For extending them, they used harmonic generation in xenon gas that was encased in an external cavity reliant on intense circulating femtosecond pulses. Usually, femtosecond pulse generated high harmonics requires an extremely high intensity such as the one obtained by pulses when they use –²—  %=% %  }    %    ~“? is employed. Afterward, a sharp frequency comb is not generated, it can be a case only when the full rate of repetition of 100 MHz is retained. Highresolution laser spectroscopy is seemingly within the reach, for instance, in He+, for the interesting 1s-2s transition (in contrast to hydrogen multiplied with a factor of 4 to attain higher frequencies). Ultimately, in the X-ray region, atomic clocks may emerge (Youngquist et al., 1987; Hee et al., 1995). If the high harmonics (that are equidistantly spaced) are phase-locked together, we can form attosecond pulses, but they need to be locked in an analogous way with the mode-locked laser’s case, present in the visible region. In a single attosecond pulse generation process, also for studying  %     =  !  }#   have a stabilized optical phase. In experiments where primary femtosecond pulses are being used, phase stabilization is essentially important (Huang et al., 1991; Rauch and Werner, 2015). In time-domain experiments, there exists great importance of the frequency comb technique. Hänsch is closely in contact with F. Krausz (along with his collaborators, now at the MPQ, formerly were at the Technical University of Vienna, now at the MPQ)      %    %%  ƒ?  } ^^®¯²   }^^]

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37. Huang, D., Swanson, E. A., Lin, C. P., Schuman, J. S., Stinson, W. G., – }•}¡²  }–—}^^‚% 

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Chapter

6

Mode Converter Linear Optical Devices

CONTENTS 6.1. Introduction .................................................................................... 172 6.2. The Mode Converter Basis Set ......................................................... 172 6.3. Derivation of Example Results......................................................... 177 References ............................................................................................. 181

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6.1. INTRODUCTION It is easy to form devices that convert specific types of inputs to specific types of outputs which have thriving attention in optics. Some of the latest examples are optical isolators based on time-varying dielectrics, mode converters, specifically those that convert more than one distinct input to more than one distinct output, and devices that decode or unscramble the outputs of multimode optical fibers (Jiao et al., 2005; Berkhout et al., 2010; Zhu et al., 2011). A wide range of upgraded and advanced devices may be available due to the thriving capabilities of nanophotonic fabrication technologies. We need fine and uncomplicated access to the mathematics of such devices for design and for comprehending limits to what devices can be created (Gabrielli and Lipson, 2011; Wu et al., 2011). Normally, orthogonal output beams are not led by the shining distinct orthogonal beams on the input device with some scatterer; orthogonal inputs do not ensure to always give orthogonal outputs for linear operators (Ye et al., 2015). Nonetheless, it is shown that all linear optical devices are # ! #      %        %   %        % %#!% “}   % { # ! #    {¯  !}     % #! %     %¯ these orthogonal sets can be called the “mode converter” basis sets. In addition, these modes can specify all that this linear device can do (Liu et al., 2012; Miller, 2012, 2013). Š  }  % # % #    % # and drawbacks in optical devices by practicing mathematics. This chapter is concerned with the illustration of the use of this approach after verifying the main result: Firstly, we formulate an expression for the misalignment tolerance of an effective mode coupler which gives an uncomplicated and general result. Secondly, we verify it is not possible to combine the loss-less beam of two orthogonal modes into one (Thaniyavarn, 1986; Tang et al., 2017).

6.2. THE MODE CONVERTER BASIS SET 6.2.1. Device Operator A device, linear optical component, produces output light beams or/and pulses linearly when light beams or/and pulses are shined. Some linear

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operator D always describes this kind of linear device that is known as a device operator. Thus, the mathematical function of D takes an input function ÚI %    % ÚO, i.e.: ÚOIÚI

(1)

 !     %      %   % } }    } %!   ?} {!  in the input space, what function in the output space is to be created. The % # #% ÚI will be in one mathematical function (Hilbert) space, i.e., the input space HI  % # # % ÚO will, in general, be in the other, which we can assume of as the output space HO (Carolan et al., 2015). We are usually allowed to pick these spaces (and therefore the outputs and inputs in which we are involved) to be whatsoever we need for the device of our concern (Markov et al., 2012; Matthès et al., 2019). The operator D is very much alike to the scattering matrix S confronted with circuits or waves and can be known as a general version of S. A key  ?                  #    mathematics about input or output functions. More generally, with the S  "  { }  %%  ‡{  of the input modes that are a set of monochromatic modes (Zou et al., 2002; Tabia, 2012). Still, we need to be able to look at the circumstances where the output waves are, e.g., in a somewhat different one, and the input waves are in one physical surface or volume. D must not be a square matrix (or at least requires not to begin that way) (Figure 6.1).

Figure 6.1.     ®% { % {   

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The input wave could be described by using a large basis of functions, like (i) a plane wave basis, and (ii) a restricted amount of spatial modes to represent the output wave, like a finite amount of modes in a waveguide, e.g., D demands not be unitary, due to probable gain or loss in elements in the device, and also modes might be physically scattered into modes that are not involved in our mathematical output space-for instance, the output volume might actually be missed by a physically scattered wave (Whitney, 1971; Kok et al., 2007). It should be noted that we are clearly permitting the possibility that our device could consist of elements that change with time and/or space. The mapping it provides between input and output functions is a static one,   }   ! {   {   %   ‡  clearly in time. We shall generally consider the device as varying only in space below for the assurance of the debate above (Rao et al., 2008; Heo et al., 2016). On the other hand, we know that changes in time are also studied by mathematics. Distinctly, the word “mode” here actually covering the concept of pulses in time as being modes; we can have modes with combinations of temporal and spatial variations, we can also have pulse modes as well which are orthogonal in time just like we can have spatial modes orthogonal in space (Yu and Fan, 2009; Dai et al., 2012). In the device’s characteristics, there is a predetermination of any similar variations in time to retain the linearity, e.g., explicit time dependence of         "     }       on the output or inputs. (Likewise, in the real space of the materials, any variations are determined in advance and are also not reliant on the outputs or inputs of the device).

6.2.2. Example Optical Systems A wide variation of physical systems can be elaborated by the simple mathematical approach. Optical systems with output and input spaces are depicted in Figure 6.1. A conventional optical system drawing light from one surface to another is demonstrated in Figure 6.1(a). The output space is functions of position on the output surface and the input space is actually waves that are functions of the position in the input aperture. An input coupler for a waveguide is illustrated in Figure 6.1(b). Light is centered on the front surface of the device and is coupled into a waveguide supporting a spatial mode (Orta et al., 1995; Armstrong, 2009). Figure 6.1(c) could be a filter or an optical fiber dispersion compensator. In a single-mode optical

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fiber, the input wave is in a single spatial mode, whereas the mathematical input space is waves as a function of time overpassing the input surface in some window of time, and likewise for the output space (probably with a different window in time). Various center wavelengths of pulses could be the input functions, for instance, with various lags in every case in the output pulses overpassing the output surface, as in some pulse dispersioncompensating or dispersing device, or they could be monochromatic waves, with the device functioning as a filter that either absorbs or carries waves of various incident frequencies. An ordinary configuration seeing coupling between volumes rather than surfaces is depicted in Figure 6.1(d). The input physical volume is picked to coincide with the device volume itself, having a wave for the input field in the truancy of the device itself that would have been in the device volume. The output space has not only a surface but also has a physical volume (Casperson and Tovar, 1998). ƒ {      !         # ¢~}     Š #"      ‡{ %}    %  #    the output surface, but with variant mathematical spaces; the output space would be backward propagating waves passing over the same surface, while the input space would be forward propagating waves passing over that surface (Lugiato and Lefever, 1987).

6.2.3. Singular Value Decomposition of the Device Operator The singular value decomposition of the operator D depends on the focus of our mathematics. That decomposition is one that grants us to write:

(2)

Likewise, this means we can write: (3)

¯   ! %       "      {  ÚDIm ÚDOm) is U (V) is as its column vectors and a diagonal matrix having diagonal complex number elements sDm is Ddiag. “}     ÚDOm  ÚDIm each organize orthogonal sets in their corresponding spaces HO and HI, and sDm are the singular values. The solutions to the two eigenvalue problems are these functions.

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(4)

(5)

It is to be noted that Hermitian (self-adjoint) operators or matrices are both D†D and DD† even if the matrix D is not. These two equations have the equal eigenvalues |sDm |2. This decomposition can be applied for a very wide range of device operators that can be logically taken to essentially incorporate all those of practical concern. The clear-cut explanation of the class of operators for which this is achievable is kind of technical.   >   = ÚDOm ÚDIm for which the singular value sDm is non-zero-are absolute in the following distinguishing way: ˆ

In the output space, any function that can be produced by the device can be created by some function in the input space that is   ÚDIm respective to non-zero singular values; ˆ In the output space, from some function in the input space, any function that can be created by the device can be written as a          ÚDOm relevant to nonzero singular values. Š    %   { }    ÚDOm  ÚDIm will be called the absolute orthogonal sets, and it would be understood that only those combined with non-zero singular values are included; all the functions can #!"%  % %%      for the device operation. Commonly, these functions will also be taken to be normalized, offering orthonormal basis sets for the spaces of interest (Yura and Hanson, 1987). The common result that we have: Since we can write every linear optical device in view of linear operator D between inputs and outputs, and there   %     % ÚDIm that will boost, one at a time,    %  %      % ÚDOm, with non?  % sDm, so we can fundamentally always operate the singular value decomposition of D.

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The values sDm}  ¿  {À#  ÚDOm ÚDIm are unique, and can always be figured out given D (at least within phase factors and normalization and the typical arbitrariness of orthogonal linear merger of degenerate eigenfunctions). When expressed in these basis sets, there is a diagonal resulting operator or device matrix. Thus, from particular orthogonal input modes to particular orthogonal output modes, any linear optical device can be written as a mode converter (Wendelin, 2003).

6.3. DERIVATION OF EXAMPLE RESULTS Now, to solve two specific difficulties, we will apply the properties of    = { #  ÚDIm  ÚDOm, the singular values sDm and corresponding device operator D: (1) Firstly, the alignment tolerance of an efficacious mode coupler will be formulated; (2) secondly, we will confirm that it is not possible to the loss-less combination of the power from two orthogonal modes. We are unaware of formerly issued derivations either of this common expression for alignment tolerance or of a proper clue in view of modes of the failure of these loss-less combinations (Newman and Sardeshmukh, 1995; Mabuchi and Zoller, 1996).

6.3.1. Mathematical Preliminaries The following reasoning does not mean about a peculiarity about the type of device considering-the given outcomes are general-but it may be straightforward for the purpose of preciseness to imagine, e.g., a focused location to a small waveguide of some optoelectronic device or an optical fiber are coupled (Kolner and Nazarathy, 1989; Mahalati et al., 2012). This coupling will be made by some optical coupler device, which can always be expressed by a device operator D of the nature being a linear optical component that  # {  {ÚI that are shined onto the input face of the  %   % {ÚO merely within the waveguide, as shown in Figure 6.1(b) could be the input functions of this coupler (Kawano et al., 1985; Reid et al., 2009). Here, we can achieve the singular value decomposition of the operator D in assumption. With corresponding singular values sDm, we get the particular

    %  % ÚDIm ÚDOm— earlier, in Appendix A, we can select the functions to be standardized so that ÚDOmÚDOm = 1 and ÚDImÚDIm = 1 resemble energy (in the event of pulses) or unit power (in the event of stable beams) or in every case (Solgaard, 2009).

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There will be at least single with the largest possible magnitude, |smax | among the set of values of sDm. There may be various different couples of functions (and therefore values of the index m) that all own the same magnitude of singular value, where all of those magnitudes are equal to |smax |; in such situation, any set of orthogonal linear consolidation of this subset of input functions can be formed freely (and the identical linear consolidation of  #  %    ?  ÚDOm ÚDIm, as commonly used in managing the numerous eigenfunctions of degenerate eigenvalues (Lee and Kim, 2006; Romero-Garcia et al., 2013). From both of the above results of examples, it can be presumed that we are handling a loss-less coupler with 100% adaptability that is to couple from an input mode (or modes) of interest to the output mode of interest (for example, in the waveguide). This set of functions must be one of the achievable single value decompositions sets with the highest magnitude, smax, of single value if we %%     %    %%   { ƒ          =  %  % } is clear that we cannot have a large single value than this as that would have to approach to more than 100% adaptability, which is not possible. |smax| = |smax|2 = 1 for the coupler with 100% adaptability as we have selected energy or power normalization for the support functions in each space (Zheng et al., 2002; Tong et al., 2015). Either (i) as deliberated above, we are not restricted to compose linear combinations in such a way that our function pair of interest for 100% adaptability mode relates with the coupling of one of the function sets having magnitude |smax | of a single value (ii) the particular mode of interest in the guide and interrelating the input mode are so far unusually the only function pair that has magnitude |smax | of single value; with the increasing m relates to gradually smaller magnitudes of single values, we can number the single value functions in such a way that in either case (i) or      % ÚDO1 ÚDI1.

           Mode Couplers —    %   ÚImis, respective to some misaligned %        %   ÚDI1. Because relative efficiencies of coupling have a great concern, this input function is also taken to have normalized power or energy, so ÚImisÚImis = 1. Now, this misaligned input function, i.e., ÚImisÚImis = 1 can be decomposed into a linear combination of

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  {  %  }}ÚDIm surplus probably any other  ÚN        ÚDIm (i.e., ÚN ÚDIm = 0 for all m), delivering:

(6)

where (7)

We are considering the mode coupler device which is itself completely aligned having 100% adaptability with the output waveguide in our misalignment; if these were manufactured together, as in an inverse taper coupler or simply taper, for example (Almeida et al., 2003; Broaddus et al., 2010), then then this could be the case. Conversely, just like in a lensed #}  %%    %{      with the input optics. In our case, where we have seen misalignment with the output waveguide, mathematics can fairly work the opposite way (Kreyszig, 1978; Shao and Xin, 2019).

6.3.3. Proof of Impossibility of Loss-Less Beam Combination of Multiple Modes Utilizing an optical device having no amplification technique within it, the perfectly loss-free coupling of beams is taken over here from probably some amount of input modes into single output mode. As discussed, one pair of %!  %  } ÚDO1  ÚDI1, are considered. Undoubtedly, for the device operator D, these two modes are a combination of the mode  { #    —      %   ÚDextra,

    ÚDI1—#! ÚDextra     ÚDI1, then there is   ÚDI1 "  ÚDextraÚDextra is a linear combination

  ÚDIm for mÁ <  %         ÚDIm  }  %   %  ÚDextra into   % ÚDO1¯     % ÚDIm pairing only into

% ÚDOm because of the one-to-one correspondence of the modeconverter basis functions, any such power is either paired into the other

    }ÚDOm for mÁ}  %   !  * } for any linear optical element, loss-free pairing from two orthogonal modes into one is not feasible (without an amplification procedure).

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According to the second law of thermodynamics, if we could operate this kind of loss-less coupling from two modes into one, then we could generate a device that from the outputs of two cooler black bodies would heat up a warmer black body, which is against the Second Law (Kawano, 1986; Lee et al., 2013). Clearly, at low temperatures, the device could comprise of two black body radiators, each of it coupled with those two distinct single modes coupled lossless into a single-mode filter through various singlemode output filters at the input to the other black body. Hence, in this way, we could add more power to the third black body. After that, it is radiated by one of the two cooler black bodies. The third black body gets warmer (at high temperature) than either of the two cool black bodies. No doubt, such theories are generally examined in the language of imaging optics schemes and are not surprisingly fixed in terms of modes, and are studied in the background of the Constant Radiance (or Brightness) Theorems (Boyd, 1983). Hence, the proof that we have discussed in this chapter is not dependent on the Constant Radiance (or Brightness) Theorems or of the Second Law. However, why such kind of scheme is impossible for any linear optics? This question has raised a minor argument (Hanson and Yakovlev, 2002; Miller, 2008).

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26. Miller, D. A., (2008). Quantum Mechanics for Scientists and Engineers (Vol. 1, pp. 1–33). Cambridge University Press. 27. Miller, D. A., (2012). All linear optical devices are mode converters. Optics Express, 20(21), 23985–23993. 28. €}  —} [ *=  {    %  component. Photonics Research, 1(1), 1–15. 29. Newman, M., & Sardeshmukh, P. D., (1995). A caveat concerning singular value decomposition. Journal of Climate, 8(2), 352–360. 30. Orta, R., Savi, P., Tascone, R., & Trinchero, D., (1995). Synthesis of %==       %  ! IEEE Photonics Technology Letters, 7(12), 1447–1449. 31. Rao, A. G., Devi, A. U., & Rajagopal, A. K., (2008). Positive-operatorvalued-measure view of the ensemble approach to polarization optics. JOSA A, 25(4), 874–880. 32. Reid, M. D., Drummond, P. D., Bowen, W. P., Cavalcanti, E. G., Lam, P. K., Bachor, H. A., & Leuchs, G., (2009). Colloquium: The EinsteinPodolsky-Rosen paradox: From concepts to applications. Reviews of Modern Physics, 81(4), 1727. 33. Romero-Garcia, S., Marzban, B., Merget, F., Shen, B., & Witzens, J., (2013). Edge couplers with relaxed alignment tolerance for pick-andplace hybrid integration of III–V lasers with SOI waveguides. IEEE Journal of Selected Topics in Quantum Electronics, 20(4), 369–379. 34. Shao, X., & Xin, Y., (2019). A scheduling algorithm based on the singular value decomposition heuristic method in a distributed manufacturing system. Expert Systems, 36(4), e12433. 35. Solgaard, O., (2009). Photonic Microsystems: Micro and Nanotechnology Applied to Optical Devices and Systems (Vol. 1, pp. 1–33). Springer Science & Business Media. 36. Tabia, G. N. M., (2012). Experimental scheme for qubit and qutrit symmetric informationally complete positive operator-valued measurements using multiport devices. Physical Review A, 86(6), 062107. 37.  }£}  }}¡Ÿ ~  }±}œƒ   # unitary optical mode converter using MMI couplers. IEEE Photonics Technology Letters, 29(12), 971–974. 38. Thaniyavarn, S., (1986). Wavelength-independent, optical-damageimmune LiNbO3 TE-TM mode converter. Optics Letters, 11(1), 39–41.

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39. Tong, X., Ye, Z., Xu, Y., Liu, S., Li, L., Xie, H., & Li, T., (2015). A novel subpixel phase correlation method using singular value decomposition       %   IEEE Transactions on Geoscience and Remote Sensing, 53(8), 4143–4156. 40. Wendelin, T., (2003). SolTRACE: A new optical modeling tool for concentrating solar optics. In: International Solar Energy Conference (Vol. 36762, pp. 253–260). 41. Whitney, C., (1971). Pauli-algebraic operators in polarization optics. JOSA, 61(9), 1207–1213. 42. Wu, M. C., Hsiao, F. C., & Tseng, S. Y., (2011). Adiabatic mode conversion in multimode waveguides using chirped computergenerated planar holograms. IEEE Photonics Technology Letters, 23(12), 807–809. 43. Ye, H., Wang, D., Yu, Z., Zhang, J., & Chen, Z., (2015). Ultra-compact broadband mode converter and optical diode based on linear rod-type photonic crystal waveguide. Optics Express, 23(8), 9673–9680. 44. Yu, Z., & Fan, S., (2009). Integrated nonmagnetic optical isolators based on photonic transitions. IEEE Journal of Selected Topics in Quantum Electronics, 16(2), 459–466. 45. Yura, H. T., & Hanson, S. G., (1987). Optical beam wave propagation through complex optical systems. JOSA A, 4(10), 1931–1948. 46. Zheng, D., Hoo, K. A., & Piovoso, M. J., (2002). Low-order model     # %   ! #!   #    singular value decomposition and the Karhunen-Loève expansion. Industrial and Engineering Chemistry Research, 41(6), 1545–1556. 47. Zhu, B., Taunay, T. F., Fishteyn, M., Liu, X., Chandrasekhar, S., Yan, M. F., & Dimarcello, F. V., (2011). 112-Tb/s space-division multiplexed •€    ®=#œ#!{   %  {

  >¾4}¾3}¾2} ¾1, and not curved polarized along the same x-axis: (7) Considering that all the four waves are propagating in the z-direction.; kj = nj¾j/c, nj being the refractive index. Substituting Eqn. (7) in Eqn. (6), we have:

(8) Pj (j= 1, 2, 3, 4) comprises of a huge number of terms which includes the %    Š  }P4 can be elaborated as:

(9) where;

(10)

Cross phase modulation and self-phase modulation are caused by the term proportional to E4 in Eqn. (9). Four-wave mixing is caused by other terms. Only when both, the wave vectors, and frequencies matches, that is, the condition of phase matching, these effects become noticeable. On the other hand, the phase matching is not required by the third-order effects like stimulated inelastic scattering phenomena (Alfano and Shapiro, 1970; Haglund, 1998; Lu et al., 2017). Eqn. (9) shows the two kinds of four-wave mixing terms. At frequency ¾4  ¾1 Ý ¾2 Ý ¾3, three photons transmission of their energy to a one%   !      £“*—¾1 +¾3 ¾1 ¾2 Þ ¾3}           %    ~  =     } ¾1¾2¾3, or frequency conversion to the wave. Eqn. ^ %       %   >¾2 ¾1 are eradicated at the same time of formation of two photons at frequencies

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¾4 ¾3 ¾4ݾ3¾2ݾ1• ¾1¾2, it is comparatively simple to achieve the phase matching condition. Substantially, the demonstration {!   £   — >}¾4 and ¾3} % %% { ¾1 generates two sidebands placed uniformly. Since just three distinct frequencies are assumed, the moderately degenerate four-wave mixing was formerly termed as three-wave mixing (Stolen et al., 1974; Agrawal, 1989; Bosshard et al., 1995).

7.4. PARAMETRIC PROCESSES Since all frequencies are in the same transparency region of the medium, all the fields set the same degrees of freedom of the medium, therefore, the interacting fields frequencies were softly pretended to be similar in the above discussion of NLO effects. Since the response of valence electrons can be considered practically instantaneous and they provide only the optical nonlinearities, the transparency region expands above the vibrational frequencies and below the beginning of electronic changes (Levenson et al., 1985; DeSalvo et al., 1992; Osgood Iii et al., 1998). Nonetheless, if one of the frequencies is reduced into a dissimilar  % !     %  # %    { distinct degrees of freedom. Let us say, the coupling of ionic motion with the electrons build up further methods for nonlinear polarization as also achieve the molecular reorientation, for frequencies under the vibrational frequencies: these further methods may alter a few of their symmetry characteristics, for example, the introduction of contrasting time constants and also breaking down of the Kleinmann symmetry relations which can be fairly critical when we examine very short interactions of the light pulse, at            {  magnitude (Shelby et al., 1986; Asobe et al., 1995; Verbiest et al., 1997). The electrons especially react as free and the dipole almost dies and the symmetry characteristics as well subject to approximation, the nearly clear #µ(2n) à     { !!  ! the X-ray region when any of the frequencies are greater than the electronic transitions. As a matter of fact, without getting familiar with the time and space average we must get going from microscopic Maxwell-Lorentz equations and the multipolar expansion should be discarded as this leads to the equations of macroscopic: hence, the microscopic current distribution and electron density is the relevant quantity (Bar-Joseph et al., 1986; Khurgin, 1987; Bai and Han, 2007).

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7.5. ELECTRO-OPTICS This involves a dc of the low-frequency electric field that alters the refractive index of a material. One has the static Kerr effect or quadratic electro-optic effect, if the difference is quadratic in the static field, however, one has a Pockel’s effect or linear electrooptic in case of linear change in the static field (McKinstrie et al., 2005; Dell’Anno et al., 2006; â , 2009). The Pocket’s effect is connected with a second-order nonlinear polarization: (11) The latter corresponds to a polarization of third order and can appear for materials without inversion symmetry only although: (12) Any material can experience this.

7.6. MAGNETO-OPTICS As compared to the electric field, the oscillating magnetic field combined with a beam of light is highly weak to activate any considerable effects. However, as a result of Zeeman splitting (the degenerate magnetic levels persist and the strengths of oscillator and the concomitant alterations of the energy spacing), an extreme static magnetic field H0 is observed which activates substantial alterations in the optical characteristics (Dechoum et al., 2000; Takesue and Shimizu, 2010). The lowest order magneto-optic effect is in analogous with the polarization: (13) When the direction of the light beam propagation and H0 coincide, it gives us the Faraday effect. Since we have discussed Pockel’s effect before,   ## !    }  }    {"    "    ƒ to be noted that in this case, these two components inside the medium sense distinct phase shifts after traveling an L distance and also that for left and right circular polarization, the refractive indices become distinct: the angle by which the polarization state goes through is given (Wu and Zhang, 1997; Takesue and Shimizu, 2010; Lin et al., 2015).

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(14)

7.7. PHOTOREFRACTIVITY The photorefractive effect was observed as a result of an inhomogeneous change in the refractive index, the light field is the cause of their induction when the crystal experience extreme laser radiation, defocusing, and scattering of the light. It was observed that strong, uniform illumination erase these changes, that was maintained even after the light had been switched off. The Kerr effect induces the light refractive index alteration in conventional NLO materials; however, this index change is basically different from that (Koopmans et al., 2000; Ge et al., 2009). The Kerr effect is an instantaneous response refractive index change and is quadratically reliant on the lightfield amplitude. It is popular for providing the necessary intensity-dependent phase shift for an optical pulse in optical soliton generation in fibers. It needs comparatively high intensities because it is a second-order effect. Opposite to the photorefractive effect, the Kerr effect results in the refractive index changes that are entirely local, such as, when the light hits the material, the change appears at the spatial coordinate (Zak et al., 1991; Wojs et al., 1996). Distinctly, by interfering with laser beams, the leaning on carrier transport through inhomogeneous excitation is the photorefractive effect. In the refractive index, the refractive index is being damaged due to the diffused carriers, which leads to a non-local change. This allows the refractive index to alter at mW laser radiation levels. In comparison with the other NLO effects (the Kerr effect), the photorefractive effect comprises of two of the dominant #      {!*      a slow process that arises as one of the disadvantages. Therefore, materials  #  %  !                 { index. Consequently, by interfering beams, we can see the formation of the grating (Partovi et al., 1990; Zhang et al., 1992). As it is not fundamental to develop any emulsion in order to preserve information through interference between an object beam and reference beam, hence, this photorefractive effect is often also termed as real-time holography. Holographic storage was       ã %   ˜ =   mass storage devices anticipated erasable holographic memories. Due to the versatility of phase-conjugation through degenerate four-wave mixing, it is highly applicable in the analysis of material parameters which generally use two-beam coupling, which is an exceptional case of usual photorefractive beam-coupling and image-deblurring applications (Winiarz et al., 2002). For

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any material to be photorefractive, it is essential that the given material must a semi-insulator or insulator. The second essential demand for the material to be photorefractive is the clarity in the function of wavelength region. It is important for devices for optical signal processing and holography. The third important demand is the liability of the material to generate carriers to move. Photorefractive research is a versatile topic that holds solid-state physics, crystal growth, optics, electromagnetic wave coupling, and electronics. For the fabrication of optical devices, the unity with other materials utilized and problems faced to develop large-sized optical quality materials with spectral sensitivities and feedback speeds convenient for feasible laser wavelengths have been the major complications experienced. KnbO3, LiNbO3, BaTiO3, Ba2NaNb5O15, BSO, Ba2SrNb5O15, KTN, and so on are the common photorefractive materials. For the NLO and electro-optic properties, many of the above materials are famous and common. Due to the sensitivity in the infrared region and immense mobilities of the carrier, semiconductors have #    }   }  %    } holds a commercial potential due to the arrival of the infrared semiconductor lasers. InP, GaAs, GaP, CdTe, and CdS are a few of the semiconductor materials.     >     ~ }       quantum-like GaAs/AlGaAs can intensify the photorefractive effect over  {    — #  = {  # %      ~ –*†      = %>  # ~   is very close to the bandgap as a result of the network between the refractive index and the absorption via Kramers-Kronig relations. The use of graded-gap > { %  %         #! %         the spatial variation of the band energy levels gives one more concern in the semiconductor photorefractive effect. For the everyday photorefractive effect, very few intensities of illumination are mandatory than those required. As the charge transport is not caused by charge concentration gradients and rather   } } =    ! (Ohashi et al., 1982; Margulis and Osterberg, 1987).

7.8. CURRENT RESEARCH IN NONLINEAR QUANTUM OPTICS (QO) The first step towards the drastic change in communication technologies and data processing is based on the optics instead of electronics because of the extensive use of fiber optics in current communications transmission

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systems. Modern systems are anticipated to soon be greatly improved to include speedily optical and hybrid optical/electrical elements. There is a great need for fundamental research into the characteristics of the materials and also the applied research into the optimal use of these materials (Arakawa and Takahashi, 1989; Perina et al., 2012). It is an essential part of the actions for photon-induced processes. For turning up the photonic data processing technologies, it has been identified that liquid crystalline and NLO polymeric materials are the strong candidates. In general, these materials include highly colored chromophores, covalently attached, or dissolved in a polymeric host material, or molecular fragments showing NLO activity. NLO polymers/liquid crystals retain the very large potential for use in a range of photonic systems, comprising of ultra-fast optical switches, high-density optical data storage media, and high-speed optical modulators. These devices are crucial for persistent progress in the attempt to change information storage and transmission from the electrical to the optical system. The strong consolidation of low cost, remarkable optical qualities, and ease of assembling into device structures is lied in the promise of NLO polymers. For commercial applications, significant research into the growth of NLO polymers has been led by these technologically encouraging properties. However, appropriate material for extensive industrial use has still to be synthesized (Busche et al., 2017; Krasnok et al., 2018). Light-induced modulation of the index of refraction of the material via photorefractive effect is one of the most amusing and technologically promising phenomena suited to some of these materials. The electro-optic effect and photoconductivity are combined with photorefractive materials. The most efficient strategy by far is the photorefractive nonlinear QO for generating beams of light to connect. Holograms can be erased and written in photorefractive media with low power lasers. The energy can be exchanged between the writing beams because of the intrinsic phase shift in the holographic grating. With the extensive use of optical communications technologies and fiber optics, the prime candidates for all-optical data processing applications are the photorefractive materials. They are proposed to be used for associative image processing techniques including image amplification and dynamic holography, high-density optical data storage, spatial light modulation, programmable interconnections in simulations of neural networks, and integrated optics, and associative memories with parallel signal processing. In recent years, the liquid crystals and polymers are considered to be the exceptional candidates for photo-refraction, but enough work in material characterization and development remains to be accomplished.

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15. DeSalvo, R., Hagan, D. J., Sheik-Bahae, M., Stegeman, G., Van, S. E. W., & Vanherzeele, H., (1992). Self-focusing and self-defocusing by cascaded second-order effects in KTP. Optics Letters, 17(1), 28–30. 16. Drummond, P. D., & Gardiner, C. W., (1980). Generalized P-representations in quantum optics. Journal of Physics A: Mathematical and General, 13(7), 2353. 17. Franken, P. A., Hill, A. E., Peters, C. W., & Weinreich, G., (1961). Generation of optical harmonics, Phys. Rev. Lett., 7, 118, 119. 18. Ge, Z., Gauza, S., Jiao, M., Xianyu, H., & Wu, S. T., (2009). Electrooptics of polymer-stabilized blue phase liquid crystal displays. Applied Physics Letters, 94(10), 101104. 19. Giordmaine, J. A., & Miller, R. C., (1965). Tunable coherent parametric oscillation in LiNbO3 at optical frequencies, Phys. Rev. Lett., 14, 973–976. 20. Haglund, Jr. R. F., (1998). Ion implantation as a tool in the synthesis of practical third-order nonlinear optical materials. Materials Science and Engineering: A, 253(1, 2), 275–283. 21. Horowski, M., Odzijewicz, A., & Tereszkiewicz, A., (2003). Some integrable systems in nonlinear quantum optics. Journal of Mathematical Physics, 44(2), 480–506. 22. Khurgin, J., (1987). Second-order susceptibility of asymmetric coupled quantum well structures. Applied Physics Letters, 51(25), 2100–2102. 23. Klauder, J. R., & Sudarshan, E. C., (2006). Fundamentals of Quantum Optics (Vol. 1, pp. 1–33). Courier Corporation. 24. Koopmans, B., Van, K. M., Kohlhepp, J. T., & De Jonge, W. J. M., (2000). Ultrafast magneto-optics in nickel: Magnetism or optics? Physical Review Letters, 85(4), 844. 25. Krasnok, A., Tymchenko, M., & Alù, A., (2018). Nonlinear metasurfaces: A paradigm shift in nonlinear optics. Materials Today, 21(1), 8–21. 26. Kurtz, S. K., & Perry, T. T., (1968). A powder technique for the evaluation of nonlinear optical materials. Journal of Applied Physics, 39(8), 3798–3813. 27. Levenson, M. D., Shelby, R. M., Aspect, A., Reid, M., & Walls, D. F., (1985). Generation and detection of squeezed states of light by    = {"  % #Phys. Rev. A, 32, 1550–1562.

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28. Lin, Q., Armin, A., Nagiri, R. C. R., Burn, P. L., & Meredith, P., (2015). Electro-optics of perovskite solar cells. Nature Photonics, 9(2), 106– 112. 29. Lu, L., Tang, X., Cao, R., Wu, L., Li, Z., Jing, G., & Li, J., (2017). Broadband nonlinear optical response in few-layer antimonene and antimonene quantum dots: A promising optical Kerr media with enhanced stability. Advanced Optical Materials, 5(17), 1700301. 30. € }•} ¡ ‚#} Ê} ^\œ Š =%   #   Opt. Lett., 12, 519–521. 31. McKinstrie, C. J., Yu, M., Raymer, M. G., & Radic, S., (2005). Quantum noise properties of parametric processes. Optics Express, 13(13), 4986–5012. 32. Muhammad, S., Xu, H. L., Zhong, R. L., Su, Z. M., Al-Sehemi, A. G., & Irfan, A., (2013). Quantum chemical design of nonlinear optical materials by sp 2-hybridized carbon nanomaterials: Issues and opportunities. Journal of Materials Chemistry C, 1(35), 5439–5449. 33. Ohashi, M., Kitayama, K., Ishida, Y., & Uchida, N., (1982). Phase     %   #!  = { " %    ## " ! %%Appl. Phys. Lett., 41, 1111–1113. 34. Osgood, I. R. M., Bader, S. D., Clemens, B. M., White, R. L., & Matsuyama, H., (1998). Second-order magneto-optic effects in   % Journal of Magnetism and Magnetic Materials, 182(3), 297–323. 35. Partovi, A., Millerd, J., Garmire, E. M., Ziari, M., Steier, W. H., Trivedi, S. B., & Klein, M. B., (1990). Photorefractive at 1.5 æ in CdTe: V. Applied Physics Letters, 57(9), 846–848. 36. Perina, J., Hradil, Z., & Jurco, B., (2012). Quantum Optics and Fundamentals of Physics (Vol. 63, pp. 1–33). Springer Science & Business Media. 37.   } Ÿ} ¡ ” Ü?!~} — €} \ „      optimal for waveguided nonlinear-quantum-optical processes. Physical Review A, 98(4), 043813. 38. â , M., (2009). Third-order nonlinear optical properties of a oneand two-electron spherical quantum dot with and without a hydrogenic impurity. Journal of Applied Physics, 106(6), 063710.

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39. Sánchez-Burillo, E., Garcia-Ripoll, J., Martín-Moreno, L., & Zueco, D., (2015). Nonlinear quantum optics in the (ultra) strong light-matter coupling. Faraday Discussions, 178, 335–356. 40. Shelby, R. M., Levenson, M. D., Perlmutter, S. H., DeVoe, R. G., & • }Š}^\” # %   %   >     % #Phys. Rev. Lett., 57, 691–694. 41. Stolen, R. H., Bjorkholm, J. E., & Ashkin, A., (1974). Phase-matched  = {" # %  {Appl. Phys. Lett., 24, 308–310. 42. Takesue, H., & Shimizu, K., (2010). Effects of multiple pairs on visibility measurements of entangled photons generated by spontaneous parametric processes. Optics Communications, 283(2), 276–287. 43. Verbiest, T., Houbrechts, S., Kauranen, M., Clays, K., & Persoons, A., (1997). Second-order nonlinear optical materials: Recent advances in chromophore design. Journal of Materials Chemistry, 7(11), 2175– 2189. 44. Wilson, J., & Hawkes, J. F. B., (1999). Optoelectronics: An Introduction (Vol. 1, pp. 1–22). 45. Winiarz, J. G., Zhang, L., Park, J., & Prasad, P. N., (2002). Inorganic: Organic hybrid nanocomposites for photorefractive at communication wavelengths. The Journal of Physical Chemistry B, 106(5), 967–970. 46. Wojs, A., Hawrylak, P., Fafard, S., & Jacak, L., (1996). Electronic structure and magneto-optics of self-assembled quantum dots. Physical Review B, 54(8), 5604. 47. Wu, Q., & Zhang, X. C., (1997). Free-space electro-optics sampling of mid-infrared pulses. Applied Physics Letters, 71(10), 1285–1286. 48. Yariv, A., (1989). Quantum Electronics (Vol. 1, pp. 1–22). 49. Zak, J., Moog, E. R., Liu, C., & Bader, S. D., (1991). Magneto-optics of multilayers with arbitrary magnetization directions. Physical Review B, 43(8), 6423. 50. Zhang, Y., Cui, Y., & Prasad, P. N., (1992). Observation of photo refractivity in a fullerene-doped polymer composite. Physical Review B, 46(15), 9900.

Chapter

8

Recent Advances in Quantum Optics

CONTENTS 8.1. Introduction .................................................................................... 202 8.2. Quantum Measurements................................................................. 202 8.3. Two-Photon Interferometry.............................................................. 205 8.4. Nonlocal Cancellation of Dispersion .............................................. 209 8.5. Dynamic Phase of Electromagnetic Field ........................................ 210 8.6. Quantum Cryptography .................................................................. 210 8.7. Quantum Computing ...................................................................... 214 References ............................................................................................. 216

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8.1. INTRODUCTION Since the assumption of the significance of quantum mechanical effects only when handling weak electromagnetic fields having one or two photons, generally for most engineering applications, Maxwell’s equations are enough. Numerous new phenomena have been discovered because of the latest work in quantum optics (QO) that may have functional applications, few of which will be discussed here. The quantum mechanical properties of distinctive photons are relied on by these effects and cannot be known with the support of Maxwell’s equations (Wen et al., 2013; Chen et al., 2017, 2018). Nonetheless, it is obvious that there is a noteworthy role of quantum    =!      # ! abundant number of photons. For instance, a few of the nonclassical outcomes   #% %%!       the electric motor or in a transformer. A greater reliance on a “quantumengineering” technique might eventually be required for the design of    !  ‚         {  (Dell’Anno et al., 2006; Hines and Kamat, 2014). An overview of various latest advancements in QO at the Applied Physics Laboratory will be provided in this chapter. A thorough theoretical discussion will not be attempted to provide for the source of these effects (Polavarapu, 2002). However, in general, various examples for the illustration of a few

                  light will be given. These examples cover a new sort of stage related to the   }  !%  !, and the elimination of the dispersion encountered by two distant optical pulses. There will also be a description of a totally operational system for reliable communications on the basis of the quantum mechanical uncertainty principle (Glushkov et al., 2008; Lee et al., 2018).

8.2. QUANTUM MEASUREMENTS Often nonclassical outcomes are given by quantum mechanics when two systems or particles are at first permitted to corporate with each other in order to become correlated but then are detached by a large gap. There will be an instant change in the state of the other distant particle by a successive measurement of the characteristics of one of the particles. This procedure is known as the reduction or collapse of the quantum mechanical state of a system (Deutsch, 1983; Peres, 1990).

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The reduction of the state of a system can be demonstrated by assuming two photons diffused from a common origin such that their polarizations are denotes the quantum mechanical state of this identical but not known. system and has the form: (1) Here; y or x polarization for photon 1 is denoted by |y1> and |x1>, and similar notation is applied for the polarization of photon 2. Each terminology given in Eqn. (1) corresponds to a probability amplitude whose square provides the probability of getting either y or x polarizations. In contrast to classical probabilities, all of the probability amplitudes relevant to a {#    >  {      probability of an occurrence (Gisin, 1984; Renes et al., 2004). It can be illustrated that the polarization of a couple of the photons is also completely corresponded in any other coordinate frame, e.g., the yç} xç coordinate frame: (2)

Assume that the polarization of photon 1 was measured and was found that either polarized along the x-axis. Now, as it is known that photon 1 was not polarized in the y-direction, there is a reduction of the second term in Eqn. (1) to zero and the quantum mechanical state of the system rapidly becomes: (3)

The explanation of photon 2 is changed as well, and it is observed that in this way, a measurement taken on one particle can instantly transform the state of a distant particle. It might be suspected that the principles of special relativity would be violated by such a procedure. This aspect of quantum theory is criticized by Podolsky, Einstein, and Rosen in their remarkable paper written in 1935. Nevertheless, if some knowledge is acquired about the state of photon 1, a similar readjustment of the probabilities related to photon 2 would also be needed by the classical probability theory as their characteristics are solidly associated. Consequently, for many years Einstein’s arguments to the quantum theory were typically assumed to be only a concern of philosophy or interpretation (Lindblad, 1973).

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In 1964, that case is changed when John Bell presented that the instantaneous transmission of information from one position to another would be expected for any classical interpretation of such a practice. As illustrated in Figure 8.1, the polarization of the two photons should be calculated together with two randomly selected axes, xç} x, differing by an angle $. With respect to quantum mechanics, it is found that the xç x polarizations are proportional to cos2 $ for the possibility of both of the photons (Sensarn et al., 2009). It is shown by Bell and illustrated in Figure 8.2 that the best linear dependence adjacent to $ = 0 with a sharp angle can be given by any classical theory in which particles cannot transfer any data faster than the speed of light. The probability of simply emission of photons with those specific polarizations is ruled out by the random selection of measurement axes. Excellent compliance has been given with the quantum theory predictions by an enormous number of this kind of experiments (Franson, 1992, 2009).

Figure 8.1. Evaluation of the polarization of a couple of obscure photons to      ! "#!  $ Source: https://www.jhuapl.edu/Content/techdigest/pdf/V16-N04/16-04-Franson.pdf.

Figure 8.2. Probability regarding the measured polarization of a couple of photons will be lied together with two randomly selected axes differing by an angle

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$ %  >  %#! }   maximum correlation in any classical theory is corresponded by the black line where the information cannot be disseminated swiftly than the speed of light. Source: https://www.jhuapl.edu/Content/techdigest/pdf/V16-N04/16-04-Franson.pdf.

The cross products in the square of the sum of corresponding probability amplitudes help to show the vital distinction between the quantum and classical mechanical predictions. The nonclassical form of interference is produced by these cross products that rely on the relative state between numerous probability amplitudes (Strekalov and Dowling, 2002). Although an instantaneous transmission of information would be required by a classical interpretation of these correlations, which is not the quantum mechanical interpretation. As there is no direction to modulate or control the selection of polarization of photon 1 that is selected randomly at the time of measurement, therefore it is not possible to transfer meaningful information faster than the speed of light. Roughly saying, as there can be no external authority over the process, photon 2 anyhow receives this information instantaneously (from a classical point of view), but information cannot be transferred (Ekert et al., 1992; Nagasako et al., 2001).

8.3. TWO-PHOTON INTERFEROMETRY The outcomes of Bell’s experiment were applicable to systems having two degrees of freedom, like the spins of two electrons or the polarizations of two photons. These ideas have been extended to other systems while showing that two faraway interferometers can display very identical correlations (Franson, 1993, 1995a, b). Assume a light source as illustrated in Figure 8.3 that produces two photons simultaneously to analyze the occurrence of such a situation. The two photons move in contrary ways over an arbitrarily spacious distance, later they come across two similar interferometers. The photons are allowed to move by a beam splitter in each interferometer on a shorter or a longer way over the interferometers, whereas the photons are allowed to move by a second beam splitter toward one of two sets of detectors (Einstein et al., 1935; Munro and Reid, 1993). A much larger difference is selected in the length of the two ways than the coherence length of two photons, therefore there would be no classical expectation of interference at all. Consequently, there is a 50/50 chance for the detection of each photon, either the unprimed

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%    }^^[Assume that photon 1 selected to propagate toward detector D1, as pointed by an output pulse coming out of that detector. The state of photon 2 is rapidly reduced by this information for reasons that will be described shortly, in such a way that photon 2 will # {  2 rather than in D2ç  #{   of the total correlation between the selected detectors (unprimed vs primed) provides that the sum of the interferometer state settings is zero ($1 = – $2), while a correlation proportional to cos2[($1 + $2)/2] is given by the measurements made with other state settings. Same as in Bell’s authentic proof, since $2 and $1 can be randomly selected after the photons have been diffused, the afterward consequence is not consistent with any classical theory where the information cannot be carried at velocities larger than the speed of light. The latest issue of    published a more detailed discussion of this experiment and associated topics (Shih and Alley, 1988; Franson, 1989, 1991a, b).

Figure 8.3. As marked by single-photon detectors D1,’ D1, D2,’ D2, a two-photon interferometer showing nonclassical correlations within output ports selected #!%  &1 &2. Phase shifts $1 and $2 are proposed into the way through each interferometer. Source: https://www.jhuapl.edu/Content/techdigest/pdf/V16-N04/16-04-Franson.pdf.

      %            =%   interferometer, it is essential to look at the formation of photon pairs, that are created with the aid of a nonlinear crystal which is adequate of dividing individual photons into two secondary photons from an ultraviolet laser beam, as demonstrated in Figure 8.4. In quantum theory, the momentum of a photon is proportional to its wave vector k, and its energy is proportional to its angular frequency ¾, hence the conservation of momentum and energy needs that k1 + k2 = k0 and ¾1 + ¾2 = ¾0. Here, k0 and ¾0 are the wave vector and angular frequency of the initial laser photons. The fundamental factor

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of this process is that the two photons are initiated strictly simultaneously but the time is completely ambiguous. Thus, this situation is corresponding to Eqn. (1), where two photons have identical but unfamiliar polarization.

Figure 8.4. From an ultraviolet laser, individual photons being break up into two secondary photons meanwhile conserving momentum and energy in the process. Source: https://www.jhuapl.edu/Content/techdigest/pdf/V16-N04/16-04-Franson.pdf.

The unknown time period of the emission of the pair of photons indicates that more than one approach can be used for the occurrence of a   {}   % # #! % should be added and then squared. There could have three approaches for a pair of photons to be reached the detectors: (i) shorter path may have taken by both, (ii) longer path may have taken by both, or (iii) shorter path may have taken by one whereas longer path by other. Thus, for such a process, the total probability amplitude At can be written in the following mentioned form:

(4)

where; Ass indicates the probability amplitude for the shorter path that was taken by both photons; All represents the probability amplitude for the longer paths that were taken by both; and Als and Asl are the probability amplitudes for the longer path that was taken by one photon and the shorter path that was taken by other respectively. All vary from Ass by a phase determinant that involves the insertion of phase shifts $1 and $2 into the longer paths, also the terms ƅ2êT and ƅ1êT along the shorter and longer paths because of the % %   êT (Meystre and Sargent, 2007).

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If there is a use of high-speed electronics to choose those events only in which the detection of both of the photons is simultaneously, then the longer path must hav0e been taken by both photons via interferometers or the shorter paths must have been taken by both, because the two photons were emitted concurrently. In this case, there cannot be a contribution of the probability amplitudes Asl and Als to the eventual outcome and must be eradicated from Eqn. (4). For coincident events, the total probability amplitude then scales down to: (5)

From the conservation of energy, the phase determinant (ƅ1 + ƅ2êT is merely ƅ0êT and causes a constant phase offset; if that were not the situation, there would be the destruction of the interference pattern by the large dissemination in the frequency of two photons as it happens classically. The total probability of a coincident event Pc is proportional to the square of the probability amplitude given in Eqn. (5), which can be lowered to:

(6)

¯ë    %       detectors, and the exclusion of a constant phase factor of ƅ0êT has been made. Once again, the interference between the probability amplitudes for the long-long and short-short processes lead to this nonclassical outcome,      #          }^^ This kind of two photons experiments was performed by various laboratories quickly after the publication of the author’s theoretical predictions. ² #! ~!}  # "%#! "%     %      £ ! –  ˜  % #     experimental outcomes at the University of California at Berkeley (CohenTannoudji et al., 1997). Figure 8.5 displays some of their data that has been reproduced from.

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Figure 8.5. Outcomes are illustrated for an experiment executed by Ray Chiao’s group, a two-photon interferometer, at the University of California at Berkeley. Interference in the two-photon coincidence counting rate and not in the singlephoton rates is demonstrated. The units conformed to the number of events of every type gained in a 10-s interval. Source: https://link.springer.com/book/10.1007/978-1-4615-2936-1.

    !     %%      %          the ratio at which one of the detectors counted a single photon as a purpose of the phase difference within the two interferometers; this result does not show any interference because of the intensely short coherence length of the photons. After all, a pronounced interference pattern steady with the theory was exhibited by the proportion at which pairs of coincident photons were observed.

8.4. NONLOCAL CANCELLATION OF DISPERSION In a beam of light, assuming the Fourier transform of the electric field can help to understand these nonclassical effects in more detail (Zhong and Wong, 2013). Initially begin with the Fourier transform of a classical electric field at location r1, that can be shown as:

(7)

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where; t indicates time and ck1 are the Fourier coefficients. (For clarity, here a single polarization and a plane wave are being considered.) From Eqn. (7), the product of electric fields at both locations r1 and r2 is given by:

(8)

where; c.c. means the complex conjugate terms. It seems obvious that  %                % Š  {!}  #!ck1ck2.

8.5. DYNAMIC PHASE OF ELECTROMAGNETIC FIELD In applications such as heterodyne and homodyne detection, interferometry, and communication through phase modulation, the phase of the electromagnetic field plays a significant act. Recently, it was shown by the author that there is the latest kind of phase linked with the electromagnetic field, which is completely different from the classical phase usually measured (Franson, 1995a; Noh, 2003). It is directed to as the dynamic phase of the field as it fades in the limit of gradually changing currents (Figure 8.6) (Simon and Kumar, 1988; Rosenblum et al., 2010).

Figure 8.6. Scattering of two classical optical pulses diffused by a source and propagating in two scattered media. The diffusion encountered by one photon can be abolished by that encountered by a faraway photon in quantum optics. Source: https://link.springer.com/book/10.1007/978-1-4615-2936-1.

8.6. QUANTUM CRYPTOGRAPHY In a tale entitled “The Gold Bug,” an adventure associated with a secret code and buried treasure by Edgar Allan Poe. The hero stated while breaking the

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code and finding the treasure: “It may be doubted whether human ingenuity can construct an enigma of the kind which human ingenuity may not, by proper application, resolve” (Figure 8.7) (Bennett et al., 1992; Steinberg et al., 1992).

Figure 8.7. Two coordinate frames pivoted by 45° and utilized for the calculation of the linear polarization of single photons. Source: https://www.jhuapl.edu/Content/techdigest/pdf/V16-N04/16-04-Franson.pdf.

If the x, y coordinate frame measures the polarization of a single photon, then it will be constituted to be polarized along either x or y-axis; either one or other of the two output ports of a birefringent polarization analyzer must be emerged by a single photon. This is an intrinsically quantum mechanical effect from when a continuous range of values can be given by an evaluation of the polarization of a classical beam of light (Rarity et al., 1994; Zhong and Wong, 2013). The following ways are used to generate a secret key in their prototype system: The random and independent selection of the unprimed or primed coordinate frames of Figure 8.7 by two computers. Then a single photon is transmitted by computer 1 with a randomly selected polarization in its coordinate frame (i.e., x, x’, y, or y’ polarization.) Later the polarization of the photon has been measured by computer 2 in its coordinate frame, the selection of coordinate frames is compared by two computers openly but the polarizations are not disclosed either received or transmitted (Ekert, 1991; Mayers, 2001). All those events in which the computers selected various coordinate frames are just ignored, therefore polarizations transferred and received will be completely correlated in the leftover events. If a y’ or y

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polarization is taken to indicate a bit 1, and an x’ or x polarization is taken to indicate a bit 0, then a common series of random bits will be established by a sequence of this kind of operations that can be used as earlier to decode and encode messages transferred via an open communications line (Franson and Ilves, 1994a, b). The surveillance of this technique is because the correct coordinate frame is unknown to an observer and there will be a 50% chance for the selection to be wrong. If the photon is simply absorbed by the spy, then that event will be neglected by the two computers. With the identical polarization, a “substitute” photon is emitted toward computer 2 as got from the measurement on the original photon is the best that a capable eavesdropper can do (Bennett, 1992; Bernnett et al., 1992). For example, if an x polarization had been for the original photon, and the measurement is made in x’, y’ coordinate frame by a spy, then the state of polarization of the photon will inevitably be changed. Consequently into 25% of the polarizations, an unavoidable error is introduced by a listener as measured by computer 2, which can conveniently be used to disclose the existence of any attempted spying. A simple example of the uncertainty principle of quantum mechanics is the introduction of uncertainty into the polarization of a photon using a measurement taken in another coordinate frame (Shor, 1994; Franson and Jacobs, 1995). In Figure 8.8 is shown a fully functional system for quantum cryptography. The fundamental challenge in this kind of a practical system is the fact that a time-dependent change will be produced by the transmission of a single %  {   % #  %  ?  #  # and other aspects. This change in polarization is compensated by a feedback loop by differing the voltages applied to various Pockels cells, which are birefringent crystals whose delay is dependent on the applied voltage. From computer 1 to computer 2, the required voltages are automatically determined by the two computers to transmit an x, x’, y, or y’ photon. The rotation of the plane of polarization by 45° is possible by a third Pockels cell, which permits computer 2 to measure the polarization in either x, y or x’, y’ coordinate frame utilizing a steady polarization analyzer.

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Figure 8.8. Block diagram of a functional system for quantum cryptography, in addition with a feedback loop, to reimburse for time-dependent variations in the %  %  ?   %     % # Source: https://www.jhuapl.edu/Content/techdigest/pdf/V16-N04/16-04-Franson.pdf.

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8.7. QUANTUM COMPUTING The distribution of a secret key is not required by some of the latest traditional cryptography systems. Recognized as public-key systems, these methods depend on the expected burden in factoring large integers. However, recently it was showed by Peter Shor (1994) that with the help of a quantum computer, large numbers can be factored very smoothly, which may ultimately give in the security of public-key systems (Figure 8.9) (Steane, 1998; Walther et al., 2005).

Figure 8.9. A capture from APL, the quantum cryptography laboratory. On the left-hand side of the optical table, there is a prototype system based on a singlephoton polarization. On the right-hand side, laser beams and other equipment shown are part of a two-photon interferometer experiment. Source: https://www.jhuapl.edu/Content/techdigest/pdf/V16-N04/16-04-Franson.pdf.

Quantum computing is established on a system of quantum logic aspects that have no classical counterpart. Maybe the NOT function is the simplest example, which simply changes a TRUE input into a FALSE output, and vice versa. It is potential to build up a logic element called the square root of NOT because quantum computing incorporates probability amplitudes instead of probabilities. The same outcome as the classical NOT is given by this logic element when applied twice. But if it is enforced only once, a nonclassical operation is produced by the square root of NOT that can be consolidated with other nonclassical logic elements to carry out computations in a way that would be impractical classically (Figure 8.10) (Lanyon et al., 2008; Preskill, 2018).

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Figure 8.10. Communication of secure information from computer 1 (i.e., top) to computer 2 (i.e., bottom). The encrypted message was transferred through an open transmission line with the decoded message. Source: https://link.springer.com/book/10.1007/978-1-4615-2936-1.

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44. Shor, P. W., (1994). Algorithms for quantum computation: Discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science (Vol. 1, pp. 124–134). IEEE. 45. Simon, R., & Kumar, N., (1988). A note on the Berry phase for systems having one degree of freedom. Journal of Physics A: Mathematical and General, 21(7), 1725. 46. Steane, A., (1998). Quantum computing. Reports on Progress in Physics, 61(2), 117. 47. Steinberg, A. M., Kwiat, P. G., & Chiao, R. Y., (1992). Dispersion cancellation in a measurement of the single-photon propagation velocity in glass. Physical Review Letters, 68(16), 2421. 48. Strekalov, D., & Dowling, J., (2002). Two-photon interferometry for high-resolution imaging. Journal of Modern Optics, 49(3, 4), 519–527. 49. Walther, P., Resch, K. J., Rudolph, T., Schenck, E., Weinfurter, H., Vedral, V., & Zeilinger, A., (2005). Experimental one-way quantum computing. Nature, 434(7030), 169–176. 50. Wen, J., Zhang, Y., & Xiao, M., (2013). The Talbot effect: Recent advances in classical optics, nonlinear optics, and quantum optics. Advances in Optics and Photonics, 5(1), 83–130. 51. Zhong, T., & Wong, F. N., (2013). Nonlocal cancellation of dispersion in Franson interferometry. Physical Review A, 88(2), 020103.

INDEX

A acousto-optics 187 atoms 32, 35, 36, 37, 42, 43, 44, 50, 51, 63, 68, 70, 71, 75, 76 B bistable operation 187 blue spectral region 117 body emission 146 Bose-Einstein condensation (BEC) 124 C classical algorithm 33, 61 classical mechanical predictions 205 classical mechanics 2 Coherence 97, 110 coherent light pulses 187 coherent monochromatic light 32 communication devices 144 complementarity 144

D data storage 116 detectors 205, 206, 207, 208, 209 diagonal matrix elements 38 diode laser development 116 diverse materials 2 diverse media 2 DNA 118 Doppler broadened line 120 E Einstein 144, 146, 147, 155, 165 electrical motors 144 ^ electricity 4   ®} electromagnetic phenomena 144 electromagnetic radiation 80, 81, 88, 97, 186 electromagnetic theory 2, 4 electromagnetic waves 3 electromagnetism 80 electro-optic effects 187

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emission 204, 207 energy quantization 146 entangled quantum network 34 F # œ Š   Fourier transform 209 G Gaussian distribution 150 gravitational wave detection 32 H harmonic oscillators 147, 152 hidden message 33, 34 Higher-order correlations 145 huge quantum structure 33 human ingenuity 33 I individual photons 3 information 32, 33, 34, 39, 40, 42, 43, 44, 53, 55, 56, 57, 61, 62, 74 interaction 2, 3, 21 interferometry 202, 210, 216, 217, 218, 219 ions 35, 43, 44, 49, 52, 55, 56, 57, 61, 66, 67, 70, 71, 75, 76, 78 L Laser light 116 laser printing 116 laser spectroscopy 116, 118, 123, 124, 125, 126, 135, 136 laser systems 81 light 2, 3, 4, 5, 6, 7, 8, 9, 23, 26, 27, 28

 [}[œ}[^}®]}]] light-matter interfaces 80, 100 light propagation 85, 95, 107 lightwave technology 81 linear electro-optic effect (LEO) 188 linear optical devices 172, 183 linear polarization 187 M macroscopic theory 144  ® magneto-optic 187, 192, 198 materials processing 116 matter 116, 117, 121, 123, 124, 125, 128, 138, 140 Maxwell’s equations 202 mechanical oscillators 35 modern physics 5 modern quantum theory 144 modern technology 144 molecular physics 117, 141 monochromatic waves 175   % #œ N nanophotonic fabrication technologies 172 nanoscience 84, 93, 100, 101, 102 Newton’s laws 2, 4 Newton’s laws of motion 2 nonclassical behavior 36, 38 nonclassical outcomes 202 nonlinear polarization 186, 187, 191, 192 Nonlinear quantum optical (NLO) materials 186 nonlinear spectroscopy 116

Index

223

O

Q

%  %\

% #   ‚Š– 186 optical hydrogen 120 optical isolators 172, 184 optical metamaterials 86

%   \\}^ optical solitons 186 optical spectroscopy 116, 120 optical switching 186 Optics 1, 2, 3, 14, 25, 26, 27, 28 optoelectronics 81, 82 optoelectronic tools 116 orthogonal output beams 172 oscillatory behavior 144

quanta 2, 4 Quantum 2, 3, 7, 8, 9, 10, 11, 12, 13, 14, 22, 23, 24, 25, 26, 27, 28 quantum bit 32, 53 quantum computer 33, 43, 49, 53, 54, 57, 68, 76 quantum electrodynamics (QED) 80, 119 quantum energy 4, 5 quantum-engineering 202 Quantum entanglement 34 quantum hypothesis 4 quantum mechanical effects 202 quantum mechanical expansion 32 quantum mechanical interpretation 205 quantum mechanical state 202, 203 quantum mechanics 2, 5, 9, 12, 13, 14, 16, 18, 21, 22, 23, 24 quantum optics (QO) 2 quantum radiation 145 quantum transitions 149

P %   %] parametric oscillation 188, 189, 197 phase conjugation 187 photon correlation 149, 152 Photonic crystals 85 photons 35, 36, 38, 39, 41, 42, 51, 53, 58, 59, 60, 61, 62, 63, 68, 71, 74, 75 photon science 81 physical process 2 physics 2, 3, 4, 5, 7, 8, 9, 12, 13, 18, 20, 21 Poissonian photon statistics 37 polarizations 203, 204, 205, 211, 212 probability amplitudes 203, 205, 207, 208, 214 propagation 186, 189, 192

R Rabi oscillations 32, 42 radiation 3, 4, 5, 7 recognition 145 resolution 116, 118, 119, 123, 124 rigorous theories 80 S science 116, 117, 125 science and technology (S&T) 81 second harmonic generation (SHG) 187 seminal contribution 146 sensitivity 116, 118

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silent communications 32 solid-state lasers 117 soliton formation 187 spectrometers 154 square matrix 173 STED (stimulated emission depletion) 95 STED (stimulated emission depletion) microscopy 95 T telecommunications 116 thriving attention 172 time-varying dielectrics 172 transformer 202

U ultraviolet 117, 132    %\ V valence electrons 191 vigor power 116 W Wigner function 38, 41 X x-ray radiation 80, 111

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