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Quantum Optics

J. C. Garrison

Department of Physics

University of California at Berkeley

and

R. Y. Chiao

School of Natural Sciences and School of Engineering

University of California at Merced

1

3

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ISBN 978“0“19“850886“1

Printed in Great Britain

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This book is dedicated to our wives: Florence Chiao and Hillegonda Garrison.

Without their unfailing support and almost in¬nite patience, the task would have

been much harder.

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Preface

The idea that light is composed of discrete particles can be traced to Newton™s Opticks

(Newton, 1952), in which he introduced the term ˜corpuscles™ to describe what we now

call ˜particles™. However, the overwhelming evidence in favor of the wave nature of

light led to the abandonment of the corpuscular theory for almost two centuries. It was

resurrected”in a new form”by Einstein™s 1905 explanation of the photoelectric e¬ect,

which reconciled the two views by the assumption that the continuous electromagnetic

¬elds of Maxwell™s theory describe the average behavior of individual particles of light.

At the same time, the early quantum theory and the principle of wave“particle duality

were introduced into optics by the Einstein equation, E = hν, which relates the energy

E of the light corpuscle, the frequency ν of the associated electromagnetic wave, and

Planck™s constant h.

This combination of ideas marks the birth of the ¬eld now called quantum optics.

This subject could be de¬ned as the study of all phenomena involving the particulate

nature of light in an essential way, but a book covering the entire ¬eld in this general

sense would be too heavy to carry and certainly beyond our competence. Our more

modest aim is to explore the current understanding of the interaction of individual

quanta of light”in the range from infrared to ultraviolet wavelengths”with ordinary

matter, e.g. atoms, molecules, conduction electrons, etc. Even in this restricted domain,

it is not practical to cover everything; therefore, we have concentrated on a set of topics

that we believe are likely to provide the basis for future research and applications.

One of the attractive aspects of this ¬eld is that it addresses both fundamental

issues of quantum physics and some very promising applications. The most striking

example is entanglement, which embodies the central mystery of quantum theory and

also serves as a resource for communication and computation. This dual character

makes the subject potentially interesting to a diverse set of readers, with backgrounds

ranging from pure physics to engineering. In our attempt to deal with this situation, we

have followed a maxim frequently attributed to Einstein: ˜Everything should be made

as simple as possible, but not simpler™ (Calaprice, 2000, p. 314). This injunction, which

we will call Einstein™s rule, is a variant of Occam™s razor : ˜it is vain to do with more

what can be done with fewer™ (Russell, 1945, p. 472).

Our own grasp of this subject is largely the result of fruitful interactions with many

colleagues over the years, in particular with our students. While these individuals are

responsible for a great deal of our understanding, they are in no way to blame for the

inevitable shortcomings in our presentation.

With regard to the book itself, we are particularly indebted to Dr Achilles Spe-

liotopoulos, who took on the onerous task of reading a large part of the manuscript,

and made many useful suggestions for improvements. We would also like to express

our thanks to Sonke Adlung, and the other members of the editorial sta¬ at Oxford

Ú Preface

University Press, for their support and patience during the rather protracted time

spent in writing the book.

J. C. Garrison and R. Y. Chiao

July 2007

Contents

Introduction 1

1 The quantum nature of light 3

1.1 The early experiments 5

1.2 Photons 13

1.3 Are photons necessary? 20

1.4 Indivisibility of photons 24

1.5 Spontaneous down-conversion light source 28

1.6 Silicon avalanche-photodiode photon counters 29

1.7 The quantum theory of light 29

1.8 Exercises 30

2 Quantization of cavity modes 32

2.1 Quantization of cavity modes 32

2.2 Normal ordering and zero-point energy 47

2.3 States in quantum theory 48

2.4 Mixed states of the electromagnetic ¬eld 55

2.5 Vacuum ¬‚uctuations 60

2.6 The Casimir e¬ect 62

2.7 Exercises 65

3 Field quantization 69

3.1 Field quantization in the vacuum 69

3.2 The Heisenberg picture 83

3.3 Field quantization in passive linear media 87

3.4 Electromagnetic angular momentum— 100

3.5 Wave packet quantization— 103

3.6 Photon localizability— 106

3.7 Exercises 109

4 Interaction of light with matter 111

4.1 Semiclassical electrodynamics 111

4.2 Quantum electrodynamics 113

4.3 Quantum Maxwell™s equations 117

4.4 Parity and time reversal— 118

4.5 Stationary density operators 121

4.6 Positive- and negative-frequency parts for interacting ¬elds 122

4.7 Multi-time correlation functions 123

4.8 The interaction picture 124

4.9 Interaction of light with atoms 130

Ü Contents

4.10 Exercises 145

5 Coherent states 148

5.1 Quasiclassical states for radiation oscillators 148

5.2 Sources of coherent states 153

5.3 Experimental evidence for Poissonian statistics 157

5.4 Properties of coherent states 161

5.5 Multimode coherent states 167

5.6 Phase space description of quantum optics 172

5.7 Gaussian states— 187

5.8 Exercises 190

6 Entangled states 193

6.1 Einstein“Podolsky“Rosen states 193

6.2 Schr¨dinger™s concept of entangled states

o 194

6.3 Extensions of the notion of entanglement 195

6.4 Entanglement for distinguishable particles 200

6.5 Entanglement for identical particles 205

6.6 Entanglement for photons 210

6.7 Exercises 216

7 Paraxial quantum optics 218

7.1 Classical paraxial optics 219

7.2 Paraxial states 219

7.3 The slowly-varying envelope operator 223

7.4 Gaussian beams and pulses 226

7.5 The paraxial expansion— 228

7.6 Paraxial wave packets— 229

7.7 Angular momentum— 230

7.8 Approximate photon localizability— 232

7.9 Exercises 234

8 Linear optical devices 237

8.1 Classical scattering 237

8.2 Quantum scattering 242

8.3 Paraxial optical elements 245

8.4 The beam splitter 247

8.5 Y-junctions 254

8.6 Isolators and circulators 255

8.7 Stops 260

8.8 Exercises 262

9 Photon detection 265

9.1 Primary photon detection 265

9.2 Postdetection signal processing 280

9.3 Heterodyne and homodyne detection 290

9.4 Exercises 305

Ü

Contents

10 Experiments in linear optics 307

10.1 Single-photon interference 307

10.2 Two-photon interference 315

10.3 Single-photon interference revisited— 333

10.4 Tunneling time measurements— 337

10.5 The meaning of causality in quantum optics— 343

10.6 Interaction-free measurements— 345

10.7 Exercises 348

11 Coherent interaction of light with atoms 350

11.1 Resonant wave approximation 350

11.2 Spontaneous emission II 357

11.3 The semiclassical limit 369

11.4 Exercises 379

12 Cavity quantum electrodynamics 381

12.1 The Jaynes“Cummings model 381

12.2 Collapses and revivals 384

12.3 The micromaser 387

12.4 Exercises 390

13 Nonlinear quantum optics 391

13.1 The atomic polarization 391

13.2 Weakly nonlinear media 393

13.3 Three-photon interactions 399

13.4 Four-photon interactions 412

13.5 Exercises 418

14 Quantum noise and dissipation 420

14.1 The world as sample and environment 420

14.2 Photons in a lossy cavity 428

14.3 The input“output method 435

14.4 Noise and dissipation for atoms 442

14.5 Incoherent pumping 447

14.6 The ¬‚uctuation dissipation theorem— 450

14.7 Quantum regression— 454

14.8 Photon bunching— 456

14.9 Resonance ¬‚uorescence— 457

14.10 Exercises 466

15 Nonclassical states of light 470

15.1 Squeezed states 470

15.2 Theory of squeezed-light generation— 485

15.3 Experimental squeezed-light generation 492

15.4 Number states 495

15.5 Exercises 497

16 Linear optical ampli¬ers— 499

Ü Contents

16.1 General properties of linear ampli¬ers 499

16.2 Regenerative ampli¬ers 502

16.3 Traveling-wave ampli¬ers 510

16.4 General description of linear ampli¬ers 516

16.5 Noise limits for linear ampli¬ers 523

16.6 Exercises 527

17 Quantum tomography 529

17.1 Classical tomography 529

17.2 Optical homodyne tomography 532

17.3 Experiments in optical homodyne tomography 533

17.4 Exercises 537

18 The master equation 538

18.1 Reduced density operators 538

18.2 The environment picture 538

18.3 Averaging over the environment 539

18.4 Examples of the master equation 542

18.5 Phase space methods 546

The Lindblad form of the master equation—

18.6 556

18.7 Quantum jumps 557

18.8 Exercises 576

19 Bell™s theorem and its optical tests 578

19.1 The Einstein“Podolsky“Rosen paradox 579

19.2 The nature of randomness in the quantum world 581

19.3 Local realism 583

19.4 Bell™s theorem 589

19.5 Quantum theory versus local realism 591

19.6 Comparisons with experiments 596

19.7 Exercises 600

20 Quantum information 601

20.1 Telecommunications 601

20.2 Quantum cloning 606

20.3 Quantum cryptography 616

20.4 Entanglement as a quantum resource 619

20.5 Quantum computing 630

20.6 Exercises 639

Appendix A Mathematics 645

A.1 Vector analysis 645

A.2 General vector spaces 645

A.3 Hilbert spaces 646

A.4 Fourier transforms 651

A.5 Laplace transforms 654

A.6 Functional analysis 655

A.7 Improper functions 656

Ü

Contents

A.8 Probability and random variables 659

Appendix B Classical electrodynamics 661

B.1 Maxwell™s equations 661

B.2 Electrodynamics in the frequency domain 662

B.3 Wave equations 663

B.4 Planar cavity 669

B.5 Macroscopic Maxwell equations 670

Appendix C Quantum theory 680

C.1 Dirac™s bra and ket notation 680

C.2 Physical interpretation 683

C.3 Useful results for operators 685

C.4 Canonical commutation relations 690

C.5 Angular momentum in quantum mechanics 692

C.6 Minimal coupling 693

References 695

Index 708

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Introduction

For the purposes of this book, quantum optics is the study of the interaction of indi-

vidual photons, in the wavelength range from the infrared to the ultraviolet, with ordi-

nary matter”e.g. atoms, molecules, conduction electrons, etc.”described by nonrela-

tivistic quantum mechanics. Our objective is to provide an introduction to this branch

of physics”covering both theoretical and experimental aspects”that will equip the

reader with the tools for working in the ¬eld of quantum optics itself, as well as its

applications. In order to keep the text to a manageable length, we have not attempted

to provide a detailed treatment of the various applications considered. Instead, we try

to connect each application to the underlying physics as clearly as possible; and, in

addition, supply the reader with a guide to the current literature. In a ¬eld evolving

as rapidly as this one, the guide to the literature will soon become obsolete, but the

physical principles and techniques underlying the applications will remain relevant for

the foreseeable future.

Whenever possible, we ¬rst present a simpli¬ed model explaining the basic physical

ideas in a way that does not require a strong background in theoretical physics. This

step also serves to prepare the ground for a more sophisticated theoretical treatment,

which is presented in a later section. On the experimental side, we have made a serious

e¬ort to provide an introduction to the techniques used in the experiments that we

discuss.

The book begins with a survey of the basic experimental observations that have led

to the conclusion that light is composed of indivisible quanta”called photons”that

obey the laws of quantum theory. The next six chapters are concerned with building

up the basic theory required for the subsequent developments. In Chapters 8 and

9, we emphasize the theoretical and experimental techniques that are needed for the

discussion of a collection of important experiments in linear quantum optics, presented

in Chapter 10.

Chapters 11 through 18 contain a mixture of more advanced topics, including cavity

quantum electrodynamics, nonlinear optics, nonclassical states of light, linear optical

ampli¬ers, and quantum tomography.

In Chapter 19, we discuss Bell™s theorem and the optical experiments performed

to test its consequences. The ideas associated with Bell™s theorem play an important

role in applications now under development, as well as in the foundations of quantum

theory. Finally, in Chapter 20 many of these threads are drawn together to treat topics

in quantum information theory, ranging from noise suppression in optical transmission

lines to quantum computing.

We have written this book for readers who are already familiar with elementary

quantum mechanics; in particular, with the quantum theory of the simple harmonic

oscillator. A corresponding level of familiarity with Maxwell™s equations for the clas-

¾ Introduction

sical electromagnetic ¬eld and with elementary optics is also a prerequisite. On the

mathematical side, some pro¬ciency in classical analysis, including the use of partial

di¬erential equations and Fourier transforms, will be a great help.

Since the number of applications of quantum optics is growing at a rapid pace,

this subject is potentially interesting to people from a wide range of scienti¬c and

engineering backgrounds. We have, therefore, organized the material in the book into

two tracks. Sections marked by an asterisk are intended for graduate-level students

who already have a ¬rm understanding of quantum theory and Maxwell™s equations.

The unmarked sections will, we hope, be useful for senior level undergraduates who

have had good introductory courses in quantum mechanics and electrodynamics. The

exercises”which form an integral part of the text”are marked in the same way.

The terminology and notation used in the book are”for the most part”standard.

We employ SI units for electromagnetic quantities, and impose the Einstein summa-

tion convention for three-dimensional vector indices. Landau™s ˜hat™ notation is used for

quantum operators associated with material particles, e.g. q, and p, but not for similar

operators associated with the electromagnetic ¬eld. The expression ˜c-number™”also

due to Landau” is employed to distinguish ordinary numbers, either real or com-

plex, from operators. The abbreviations CC and HC respectively stand for complex

conjugate and hermitian conjugate. Throughout the book, we use Dirac™s bra and

ket notation for quantum states. Our somewhat unconventional notation for Fourier

transforms is explained in Appendix A.4.

1

The quantum nature of light

Classical physics began with Newton™s laws of mechanics in the seventeenth century,

and it was completed by Maxwell™s synthesis of electricity, magnetism, and optics in the

nineteenth century. During these two centuries, Newtonian mechanics was extremely

successful in explaining a wide range of terrestrial experiments and astronomical ob-

servations. Key predictions of Maxwell™s electrodynamics were also con¬rmed by the

experiments of Hertz and others, and novel applications have continued to emerge up

to the present. When combined with the general statistical principles codi¬ed in the

laws of thermodynamics, classical physics seemed to provide a permanent foundation

for all future understanding of the physical world.

At the turn of the twentieth century, this optimistic view was shattered by new ex-

perimental discoveries, and the ensuing crisis for classical physics was only resolved by

the creation of the quantum theory. The necessity of explaining the stability of atoms,

the existence of discrete lines in atomic spectra, the di¬raction of electrons, and many

other experimental observations, decisively favored the new quantum mechanics over

Newtonian mechanics for material particles (Bransden and Joachain, 1989, Chap. 4).

Thermodynamics provided a very useful bridge between the old and the new theories.

In the words of Einstein (Schilpp, 1949, Autobiographical Notes, p. 33),

A theory is the more impressive the greater the simplicity of its premises is, the more

di¬erent kinds of things it relates, and the more extended is its area of applicability.

Therefore the deep impression which classical thermodynamics made upon me. It

is the only physical theory of universal content concerning which I am convinced

that, within the framework of the applicability of its basic concepts, it will never be

overthrown (for the special attention of those who are skeptics on principle).

Unexpected features of the behavior of light formed an equally important part

of the crisis for classical physics. The blackbody spectrum, the photoelectric e¬ect,

and atomic spectra proved to be inconsistent with classical electrodynamics. In his

characteristically bold fashion, Einstein (1987a) proposed a solution to these di¬culties

by o¬ering a radically new model in which light of frequency ν is supposed to consist

of a gas of discrete light quanta with energy = hν, where h is Planck™s constant.

The connection to classical electromagnetic theory is provided by the assumption

that the number density of light quanta is proportional to the intensity of the light.

We will follow the current usage in which light quanta are called photons, but this

terminology must be used with some care.1 Conceptual di¬culties can arise because

1 According to Willis Lamb, no amount of care is su¬cient; and the term ˜photon™ should be banned

from physics (Lamb, 1995).

The quantum nature of light

this name suggests that photons are particles in the same sense as electrons, protons,

neutrons, etc. In the following chapters, we will see that the physical meaning of the

word ˜photon™ evolves along with our understanding of experiment and theory.