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Introductionto Quantum Optics

From the Semi-classical Approach toQuantized Light

GILBERT GRYNBERG

ALAIN ASPECT

CNRS

CLAUDE FABREUniversité Pierre et Marie Curie Paris VI

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C A M B R I D G E U N I V E R S I T Y P R E S S

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,São Paulo, Delhi, Dubai, Tokyo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

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www.cambridge.orgInformation on this title: www.cambridge.org/9780521551120

c© G. Gilbert, A. Aspect, C. Fabre 2010

Original edition: Introduction aux Lasers et á l’Optique, Ellipses 1997

Published with the assistance of the French Ministry of Culture–National Book Centre

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permission of Cambridge University Press.

First published 2010

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Contents

Foreword xxiiiPreface xxvAcknowledgements xxviii

Part I Semi-classical description of matter–light interaction 1

1 The evolution of interacting quantum systems 31.1 Review of some elementary results of quantum mechanics 41.2 Transition between discrete levels induced by a time-dependent perturbation 51.3 Case of a discrete level coupled to a continuum: Fermi’s golden rule 191.4 Conclusion 32

Complement 1A A continuum of variable width 34Complement 1B Transition induced by a random broadband perturbation 38

2 The semi-classical approach: atoms interacting with a classicalelectromagnetic field 452.1 Atom–light interaction processes 462.2 The interaction Hamiltonian 532.3 Transitions between atomic levels driven by an oscillating electromagnetic

field 642.4 Absorption between levels of finite lifetimes 802.5 Laser amplification 922.6 Rate equations 962.7 Conclusion 104

Complement 2A Classical model of the atom–field interaction:the Lorentz model 105

Complement 2B Selection rules for electric dipole transitions.Applications to resonance fluorescence and optical pumping 120

Complement 2C The density matrix and the optical Bloch equations 140Complement 2D Manipulation of atomic coherences 167Complement 2E The photoelectric effect 179

3 Principles of lasers 1913.1 Conditions for oscillation 193

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xx Contents�

3.2 Description of the amplifying media of some lasers 1993.3 Spectral properties of lasers 2153.4 Pulsed lasers 2213.5 Conclusion: lasers versus classical sources 227

Complement 3A The resonant Fabry–Perot cavity 230Complement 3B The transverse modes of a laser: Gaussian beams 239Complement 3C Laser light and incoherent light: energy density and

number of photons per mode 247Complement 3D The spectral width of a laser: the Schawlow–Townes limit 257Complement 3E The laser as energy source 261Complement 3F The laser as source of coherent light 271Complement 3G Nonlinear spectroscopy 283

Part II Quantum description of light and its interaction with matter 299

4 Quantization of free radiation 3014.1 Classical Hamiltonian formalism and canonical quantization 3024.2 Free electromagnetic field and transversality 3054.3 Expansion of the free electromagnetic field in normal modes 3104.4 Hamiltonian for free radiation 3154.5 Quantization of radiation 3174.6 Quantized radiation states and photons 3194.7 Conclusion 324

Complement 4A Example of the classical Hamiltonian formalism: chargedparticle in an electromagnetic field 325

Complement 4B Momentum and angular momentum of radiation 327Complement 4C Photons in modes other than travelling plane waves 334

5 Free quantum radiation 3415.1 Photodetectors and semi-reflecting mirrors. Homodyne detection of the

quadrature components 3425.2 The vacuum: ground state of quantum radiation 3505.3 Single-mode radiation 3535.4 Multimode quantum radiation 3715.5 One-photon interference and wave–particle duality. An application

of the formalism 3775.6 A wave function for the photon? 3835.7 Conclusion 385

Complement 5A Squeezed states of light: the reduction of quantumfluctuations 387

Complement 5B One-photon wave packet 398

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xxi Contents�

Complement 5C Polarization-entangled photons and violationof Bell’s inequalities 413

Complement 5D Entangled two-mode states 434Complement 5E Quantum information 443

6 Interaction of an atom with the quantized electromagnetic field 4576.1 Classical electrodynamics and interacting fields and charges 4586.2 Interacting fields and charges and quantum description in the

Coulomb gauge 4676.3 Interaction processes 4716.4 Spontaneous emission 4776.5 Photon scattering by an atom 4856.6 Conclusion. From the semi-classical to the quantum treatment

of atom–light interaction 495

Complement 6A Hamiltonian formalism for interacting fields and charges 498Complement 6B Cavity quantum electrodynamics 502Complement 6C Polarization-entangled photon pairs emitted in an

atomic radiative cascade 518

Part III Applying both approaches 527

7 Nonlinear optics. From the semi-classical approach to quantum effects 5297.1 Introduction 5297.2 Electromagnetic field in a nonlinear medium. Semi-classical treatment 5307.3 Three-wave mixing. Semi-classical treatment 5357.4 Quantum treatment of parametric fluorescence 5457.5 Conclusion 559

Complement 7A Parametric amplification and oscillation. Semi-classicaland quantum properties 560

Complement 7B Nonlinear optics in optical Kerr media 577

8 Laser manipulation of atoms. From incoherent atom optics to atom lasers 5998.1 Energy and momentum exchanges in the atom–light interaction 6008.2 Radiative forces 6048.3 Laser cooling and trapping of atoms, optical molasses 6188.4 Gaseous Bose–Einstein condensates and atom lasers 633

Complement 8A Cooling to sub-recoil temperatures by velocity-selectivecoherent population trapping 651

Index 661

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Foreword

Atomic, molecular and optical physics is a field which, during the last few decades, hasknown spectacular developments in various directions, like nonlinear optics, laser coolingand trapping, quantum degenerate gases, quantum information. Atom–photon interactionsplay an essential role in these developments. This book presents an introduction to quantumoptics which, I am sure, will provide an invaluable help to the students, researchers andengineers who are beginning to work in these fields and who want to become familiar withthe basic concepts underlying electromagnetic interactions.

Most books dealing with these subjects follow either a semi-classical approach, wherethe field is treated as a classical field interacting with quantum particles, or a full quantumapproach where both systems are quantized. The first approach is often oversimplified andfails to describe correctly new situations that can now be investigated with the developmentof sophisticated experimental techniques. The second approach is often too difficult forbeginners and lacks simple physical pictures, very useful for an initial understanding of aphysical phenomenon. The advantage of this book is that it gives both approaches, startingwith the first, illustrated by several simple examples, and introducing progressively thesecond, clearly showing why it is essential for the understanding of certain phenomena.The authors also clearly demonstrate, in the case of non-linear optics and laser cooling,how advantageous it may be to combine both approaches in the analysis of an experimentalsituation and how one can get from each point of view useful, complementary physicalinsights. I believe that this challenge to present and to illustrate both approaches in a singlebook has been taken up successfully. Whatever their ultimate interests, the readers of thiswork will be exposed to an important example of a broad and vibrant field of researchand they will better understand the intellectual enrichment and the technical developmentswhich result from it.

To write a book on such a broad topic, the authors must obviously possess wide knowl-edge of the field, they must have thought long and hard about the basic concepts and aboutthe different levels of complexity with which one can approach the topics. They mustalso have a deep and concrete knowledge about experimental and technical details and themany problems which daily confront a laboratory researcher. Having worked extensivelywith them, I know the authors of this work fulfil these requirements. I have the highestadmiration for their enthusiasm, their scientific rigour, their ability to give simple and pre-cise physical explanations, and their quest to illuminate clearly the difficult points of thesubject without oversimplification. Each of them has made many original contributions tothe development of this important field of physics, and they and their younger collaboratorsfor this book work at the cutting edge of modern quantum optics. In reading the book, I amtherefore not surprised to find their many fine qualities reflected in the text. The general

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xxiv Foreword�

organisation of the main chapters and complementary sections allows reading on manydifferent levels. When the authors discuss a new physical problem, they begin the analy-sis with the simplest possible model. A large variety of experiments and applications arepresented with clear diagrams and explanations and with constant attention to highlightingthe guiding principles, the orders of magnitude and the problems which remain open.

This work will allow a broad audience an easier access to a field of science which con-tinues to see spectacular developments. I believe that science is not simply a matter ofexploring new horizons. One must also make the new knowledge readily available and wehave in this book, a beautiful example of such a pedagogical effort. I would like finallyto evoke the memory of Gilbert Grynberg who participated with Alain Aspect and ClaudeFabre in the writing of a preliminary, much less developed, French version of this book andwho passed away in 2003. Gilbert was an outstanding physicist, a fine person, and had anexceptional talent for explaining in the clearest possible way the most difficult questions.I think that the present book is the best possible tribute to be paid to him.

Claude Cohen-TannoudjiParis, September 2009

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Preface

Since its invention in 1960, the laser has revolutionized both the study of optics and ourunderstanding of the nature of light, prompting the emergence of a new field, quantumoptics. Actually, it took decades until the words quantum optics took their current precisemeaning, referring to phenomena which can be understood only by quantizing the elec-tromagnetic field describing light. Surprisingly enough, such quantum optics phenomenahardly existed at the time that the laser was invented, and almost all optics effects could befully understood by describing light as a classical electromagnetic field; the laser was noexception. As a matter of fact, to understand how a laser works, it suffices to use the semi-classical description of matter–light interaction, where the laser amplifying medium, madeof atoms, molecules, ions or semi-conductors, is given a quantum mechanical treatment,but light itself is described by classical electromagnetic waves.

The first part of our book is devoted to presentation of the semi-classical approach andits use in describing various optical phenomena. It includes an elementary exposition ofthe physics of lasers, and some applications of this ubiquitous device. After recallingin Chapter 1 some basic results of the quantum mechanical description of interactioninduced transitions between the atomic energy levels, we use these results in Chapter 2to show how the interaction of a quantized atom with a classical electromagnetic waveleads to absorption or stimulated emission, and to derive the process of laser amplifi-cation that happens when a wave propagates in an inverted medium. Chapter 3 givesan elementary exposition of the physics of laser sources and of the properties of laserlight.

Although the quantum theory of light existed since its development by Dirac in the early1930s, quantum optics theory in its modern sense started when Roy Glauber showed, in theearly 1960s, how to apply it to classical optics devices such as the Michelson stellar inter-ferometer or the Hanbury Brown and Twiss intensity interferometer. At that time it couldhave appeared to be an academic exercise without consequence, since the only known phe-nomenon that demanded quantization of light was spontaneous emission, and it was notclear whether quantum theory was at all useful for describing light freely propagating farfrom the source. Actually, Glauber developed a clear quantum formalism to describe opticsphenomena, and introduced the important notion of quasi-classical states of light, a theoret-ical tool that allowed physicists to understand why all available sources of light, includinglasers, delivered light whose properties could be totally understood in the framework ofthe semi-classical approach. But in doing so, he paved the way for the discovery of newphenomena which can be understood only if light is considered as a quantum system. Itbecame possible to build sources delivering single photon wave packets, pairs of entangledphotons, squeezed beams of light. . .

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xxvi Preface�

The second part of our book is devoted to the presentation of the quantum theory oflight and its interaction with matter, and its use in describing many phenomena of modernquantum optics. We show in Chapter 4 how it is possible to write the dynamical equa-tions of a classical electromagnetic field, i.e. Maxwell equations, in a form allowing us touse the canonical quantization procedure to quantize the electromagnetic field, and obtainthe notion of a photon. We then use our results, in Chapter 5, to describe some fullyquantum effects observed in experiments with single photons, squeezed light or pairs ofentangled photons. It is remarkable that many of these experiments, whose first goal wasto demonstrate the highly counter-intuitive, non-classical properties of new types of lightstates, turned out to stimulate the emergence of a new field, quantum information, whereone uses such properties to implement new ways of processing and transmitting data. InChapter 6, we show how to use the quantum optics formalism to describe the interactionbetween light and atoms. We will then revisit in this new framework the phenomena ofabsorption and stimulated emission, already studied in Chapter 2. Moreover, we will nowbe able to give a consistent treatment of spontaneous emission.

Having introduced the full quantum optics formalism and reviewed some remarkablephenomena that could not have been discovered without such a formalism, we would notlike to leave the reader with the impression that he/she can now forget the semi-classicalapproach. Both approaches are definitely useful. On the one hand, there is no reason to usethe, usually more involved, fully quantum analysis, when the situation does not demand it.After all, nobody would use quantum mechanics to describe the motion of planets. Simi-larly, no experimentalist studying fusion plasmas with intense lasers would start using thequantum formalism of light. What is important then is to be able to recognize when thefull quantum theory is necessary, and when one can content oneself with the semi-classicalmodel. To help the reader to develop their intuition about this point, we present, in the thirdpart of this book, two topics, non-linear optics in Chapter 7, laser cooling and trappingof atoms in Chapter 8, where it is convenient to ‘juggle’ between the two approaches,each being better adapted to one or the other particular phenomenon. As ‘the cherry on thecake’, we will give in Chapter 8 an elementary presentation of atomic Bose–Einstein con-densates, and emphasize the analogy between such a system, where all atoms are describedby the same matter wave, and a laser beam where all photons are described by the samemode of the electromagnetic field. When we started to write the first French version of thisbook, we had never dreamt of being able to finish it with a presentation on atom lasers.

This book is composed of chapters, in which we present the fundamental concepts andsome applications to important quantum optics phenomena, and of complements, whichpresent supplementary illustrations or applications of the theory presented in the mainchapter. The choice of these examples is, of course, somewhat arbitrary. We present themas a snap-shot of the current state of a field which is rapidly evolving. Complements ofanother type are intended to give some supplementary details about a derivation or aboutconcepts presented in the chapter.

The prerequisite for using this book is to have followed an elementary course on bothelectromagnetism (Maxwell’s equations) and quantum mechanics (Schrödinger formula-tion in the Dirac formalism of bras and kets, with application to the harmonic oscillator).The book is then self-consistent, and can be used for an advanced undergraduate, or for

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xxvii Preface�

a first graduate course on quantum optics. Although we do not make use of the mostadvanced tools studied at graduate school, we make all efforts to provide the reader withsolid derivations of the main results obtained in the chapters. For example, to quantizeelectromagnetic waves, first in free space, and then in interaction with charges, we do notuse the Lagrangian formalism, but we introduce enough elements of the Hamiltonian for-malism to be able to apply the canonical quantization rules. We are thus able to provide thereader with a solid derivation of the basic quantum optics formalism rather than bringingit in abruptly. On the other hand, when we want to present in a Complement a particu-larly important and interesting phenomenon, we do not hesitate to ask the reader to admita result which results from more advanced courses.

We have done our best to merge the French teaching tradition of logical and deductiveexposition with the more pragmatic approach that we use as researchers, and as advisorsto Ph.D. and Masters students. We have taught the content of this book for many years toadvanced undergraduate or beginning graduate students, and this text represents the resultsof our various teaching experiences.

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Acknowledgements

In this book, we refer to a number of textbooks in which general elementary resultsof quantum mechanics are established, in particular the book by Jean-Louis Basdevantand Jean Dalibard,1 which we indicate by the short-hand notation ‘BD’, and the one byClaude Cohen-Tannoudji, Bernard Diu and Franck Laloë,2 which we denote by ‘CDL’.On the more advanced side, we sometimes refer to more rigorous demonstrations, orto more advanced developments, that can be found in the two books written by ClaudeCohen-Tannoudji, Jacques Dupont-Roc and Gilbert Grynberg, to which we refer under theshort-hand notations ‘CDG I’ and ‘CDG II’, respectively.3,4

It is not possible to mention all those who have contributed to or influenced thiswork. We would first like to acknowledge, however, our principal inspiration, ClaudeCohen-Tannoudji, whose lectures at the Collège de France we have had the good fortuneto be able to follow for three decades. At the other end of the spectrum, we also owe alot to our students at Ecole Polytechnique, Ecole Normale Supérieure, Institut d’OptiqueGraduate School, Université Pierre et Marie Curie, as well as the many graduate studentswe have advised towards Masters or Ph.D. work. By their sharp questioning, never con-tent with a vague answer, they have forced us to improve our lectures year upon year.We cannot cite all of the colleagues with whom we have taught, and from whom wehave borrowed many ideas and materials, but we cannot omit to mention the names ofManuel Joffre, Emmanuel Rosencher, Philippe Grangier, Jean-François Roch, FrançoisHache, David Guéry-Odelin, Jean-Louis Oudar, Hubert Flocard, Jean Dalibard, Jean-LouisBasdevant. In addition, Philippe Grangier was kind enough to write Complement 5E onquantum information.

Martine Maguer, Dominique Toustou, and all the team of Véronique Pellouin at theCentre Polymedia of Ecole Polytechnique have done an impressive and professional job inpreparing the manuscript with its figures. We would like also to thank the Centre Nationaldu Livre, of the French Ministry of Culture, for its important financial support in thetranslation of our French text.

1 J.-L. Basdevant and J. Dalibard, Quantum Mechanics, Springer (2002).2 C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, Wiley (1977).3 C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and Atoms – Introduction to quantum

electrodynamics, Wiley (1989).4 C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-photon Interactions: Basic processes and

applications, Wiley (1992).

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xxix Acknowledgements�

Special acknowledgement

This book has three authors, who wrote the original French textbook on which it is based.5

Sadly, as we had just started to prepare the English version, Gilbert Grynberg passedaway, and for several years we were discouraged and not able to carry on working on theEnglish version. Eventually, we realized that the best demonstration of all that we owe toour former friend and colleague was to resume this project. But we realized then that almosta decade after writing the French version, quantum optics had evolved tremendously, andwe had also personally evolved in the ways in which we understood and taught the sub-ject. The original French book, therefore, had not only to be translated but also widelyrevised and updated. In this long-term enterprise, we have been fortunate to have fantastichelp from our younger colleagues (and former students) Fabien Bretenaker and AntoineBrowaeys. For the past three years they have devoted innumerable hours to helping uscomplete the revised version, and without their help this would not have been possible.There is not a single chapter that has not been strongly influenced by their thorough crit-icisms, their strong suggestions, and their contributions to the rewriting of the text, not tospeak of the double checking of equations. Moreover, they bring to this book the point ofview of a new generation of physicists who have been taught quantum optics in its modernsense, in contrast to we who have seen it developing while we were already engaged inresearch. For their priceless contribution, we can only express to Fabien Bretenaker andAntoine Browaeys our immense gratitude. Gilbert would have been happy to have suchwonderful collaborators.

Alain Aspect and Claude Fabre,Palaiseau, Paris, July 2009.

5 Gilbert Grynberg, Alain Aspect, Claude Fabre, Introduction aux lasers et à l’Optique Quantique, cours del’Ecole Polytechnique, Ellipses, Paris (1997).

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377 5.5 One-photon interference and wave–particle duality�

Clearly state (5.132) can now be written

|1〉 = b†m|0〉, (5.141)

that is, |1m〉, the mode m being associated with b†m.

The generalized mode m is associated with a classical electromagnetic field that is notmonochromatic, but which can be written, generalizing (4C.8),

E(+)m (r, t) =

∑�

c�E(1)� ε� ei(k�·r−ω�t). (5.142)

This is a classical wave packet (see Complement 5B). To conclude, the non-classical natureof the multimode one-photon state is not the result of entanglement, and must be consideredas a simple generalization of the non-classical nature of a single-mode number state.

�Comment There is a necessary and sufficient condition for a radiation state |ψ〉 to be expressed in the form ofa single-mode state of a generalized mode m. The state |ψ〉 can be written in the form |1m〉 if andonly if all the vectors a�|ψ〉 are proportional to one another (or zero), with the modes � forming acomplete basis. For (5.132), we do indeed have a�|1〉 = c�|0〉, i.e. a vector proportional to |0〉 ifc� �= 0.

Note further that the multimode quasi-classical state (5.122) can itself be considered as a single-mode state, for a generalized mode constructed from the associated classical field. More precisely, itis a quasi-classical state in the mode whose spatial distribution is proportional to α1 eik1·r+α2 eik2·r.

Such states, which reduce to the form |1m〉, are called intrinsic single-mode states. The radiationthey describe is intrinsically coherent.14

5.5 One-photon interference and wave–particle duality. Anapplication of the formalism

5.5.1 Mach–Zehnder interferometer in quantum optics

The Mach–Zehnder interferometer (Figure 5.8) is the prototype for all amplitude divisiontwo-wave interferometers. The incoming mode at port (1) is divided by the beamsplitterSin. Mode 2 is combined with mode 1 on Sin. The mirrors M3′ and M4′ can then recombinethe two waves 3′ and 4′ on the beamsplitter Sout. Two detectors D3 and D4 are placedat the output ports. The mirrors M3′ and M4′ can be displaced to control the path lengthdifference:

δL′ = L3′ − L4′ = [SinM3′Sout]− [SinM4′Sout]. (5.143)

14 C. Fabre, Quantum Optics, from One Mode to Many Modes, in Cours des Houches (2007), available online athttp://hal-sfo.ccsd.cnrs.fr/docs/00/27/05/37/PDF/CoursLesHouchesFabre2.pdf and N. Treps et al., PhysicalReview A71, 013820 (2005).


378 Free quantum radiation�

(3)D4(4)

(2)(1)

Sin

Sout

(a)

Coincidence

(b)

(4′) M4′

(3′)

M3′

D3

(4′) D4′

(3′)

D3′

Sin

�Figure 5.8 Mach–Zehnder two-wave interferometer. (a) The photodetection signals measured byphotodetectors D3 and D4 depend on the optical path difference [SinM3′Sout] − [SinM4′Sout].(b) Two detectors D3′ and D4′ can be inserted into the arms of the interferometer,monitored by a coincidence circuit to check for the single-photon characteristic.

Square brackets denote the optical paths, taking into account the refractive index of thepaths and any phase-shift at the mirrors. To simplify, we consider a monochromatic incom-ing mode (frequency ω), polarized normally to the figure. All resulting waves (3′, 4′, 3, 4)have the same frequency and polarization. We can then use scalar fields.

To calculate the photodetection signals at D3 and D4, we follow the prescription givenin Section 5.1.2, i.e. we express the outgoing fields E(+)

3 and E(+)4 as a function of the

incoming fields E(+)1 and E(+)

2 , using the same relations as those relating the outgoing

classical fields E(+)3 and E(+)

4 to the incoming classical fields E(+)1 and E(+)

2 . The originof the phases of the outgoing fields is taken on the beamsplitter Sout, while that of theincoming fields is taken at Sin. We then apply (5.8–5.9) at Sin, to relate E(+)

3′ and E(+)4′ to

E(+)1 and E(+)

2 . At Sout, we take into account the fact that the beamsplitter is reversed by

putting a minus sign in the coefficient relating E(+)3 and E(+)

3′ (reflection inside the glass).Multiplying by the propagation factors between Sin and Sout along paths 3′ and 4′, we thushave

E(+)3 = 1√

2

(−E(+)

3′ eikL3′ + E(+)4′ eikL4′

)(5.144)

E(+)4 = 1√

2

(E(+)

3′ eikL3′ + E(+)4′ eikL4′

), (5.145)

where k = ω/c. After a straightforward calculation, we obtain

E(+)3 = eikL′

(− i sin k

δL′

2E(+)

1 − cos kδL′

2E(+)

2

)eikr3 (5.146)

E(+)4 = eikL′

(cos k

δL′

2E(+)

1 + i sin kδL′

2E(+)

2

)eikr4 , (5.147)

where L′ = (L3′ + L4′ )/2, and the factors exp{ikr3} and exp{ikr4} stand for propagationfrom Sout to the detectors.



These relations can be used to calculate the photodetection signalsw3 andw4 for variousincoming radiation states in mode 1, with incoming mode 2 empty, so that the incomingradiation has the form

|ψin〉 = |ϕ1〉 ⊗ |02〉. (5.148)

The photodetection signal w3(r3, t) is obtained by writing

w3(r3, t) = s‖E(+)3 (r3)|ψin〉‖2 (5.149)

and expressing E(+)3 and |ψin〉 by (5.146) and (5.148), respectively. As the incoming state

at 2 is empty, the term E(+)2 |ψin〉 is zero. What remains is

w3(t) = s sin2(

kδL′

2

)‖E(+)

1 |ϕ1(t)〉‖2 (5.150)

and likewise,

w4(t) = s cos2(

kδL′

2

)‖E(+)

1 |ϕ1(t)〉‖2. (5.151)

The output signal from the interferometer thus depends on neither the position of the detec-tors, nor the average length L′ of the interferometer arms. As in the classical calculation,it depends only on the path difference δL′. However, it may in principle also depend onthe type of field state entering by port (1). We shall now examine various cases for theincoming field.

5.5.2 Quasi-classical incoming radiation

Consider first the case in which the incoming field at port (1) is in a quasi-classical state:

|ϕ1(t)〉 = |α1 e−iωt〉. (5.152)

Recalling that |ϕ1(t)〉 is an eigenstate of E(+)1 , we obtain

w3 = w1 sin2(

kδL′

2

), (5.153)

where

w1 = s[

E(1)1

]2 |α1|2 (5.154)

is the photodetection probability that would be obtained by placing the detector at theincoming port (1). Likewise, for port (4),

w4 = w1 cos2(

kδL′

2

). (5.155)

The result is strictly identical to what would be obtained from a calculation in classicaloptics, agreeing with the general claim made in Section 5.4.2 that quantum optics gives thesame results as classical optics for quasi-classical states.



The result can be generalized without difficulty (see Complement 5B) to the case of amultimode quasi-classical state of the form (5.122). If the coefficients α� only differ fromzero over a frequency interval �ω� about ω0, and if �ω · δL′/c is small compared with2π, the results (5.153) and (5.155) remain valid with k replaced by k0. If δL′ is greater than1/�ω, a calculation exactly like the classical interference calculation for polychromaticlight shows that the fringe contrast is less than unity.

5.5.3 Particle-like incoming state

Consider now the case in which the incoming state at port (1) is a one-photon state ofmode (1), i.e.

|ϕ1〉 = |11〉, (5.156)

with port (2) remaining empty. The general expressions (5.150) and (5.151) show that theresults (5.153) and (5.155) do in fact remain valid, provided that we take the photodetectionprobability at the incoming port to be the one-photon photodetection probability

w1 = s[

E(1)1

]2. (5.157)

Note that the result has exactly the same form as for a monochromatic classical wave,i.e. the photodetection probability depends sinusoidally on the path difference, andinterference is observed.

If we consider a multimode one-photon state of the form (5.132) at port (1) and nothingat port (2), we obtain

E(+)3 (r3)|ψin(t)〉 =

∑�

E(1)� c�e

i[k�(L′+r3)−ω�t] sin k�δL′

2|0〉. (5.158)

The photodetection signal will thus be

w3(r3, t) = s∣∣∣∑�

E(1)� c�e

i[k�(L′+r3)−ω�t] sin k�δL′

2

∣∣∣2. (5.159)

This expression can be simplified if the distribution of the frequencies ω� has finite width�ω and the path difference L′ is small compared with c/�ω. Denoting the central fre-

quency of the wave packet by ω0 and writing k0 = ω0/c, cos(

k�δL′2

)can be replaced by

cos(

k0δL′2

)in (5.159). The photodetection probability then assumes the form

w3(r3, t) = F(t, r3) sin2(

k0δL′

2

), (5.160)

where the function F(t, r3) is a slowly varying envelope describing the wave packet

(see Complement 5B). The term cos2(

k0δL′2

)is the interference term corresponding to



frequency ω0. Integrating over time t for the whole wave packet, we then obtain the totalprobability of photodetection at 3, keeping the path difference δL′ fixed:

P3 = P1 sin2(

k0δL′

2

). (5.161)

Here P1 is the total photodetection probability for a one-photon wave packet at the interfer-ometer input. The interference signal is clear. An analogous calculation for port (4) wouldyield

P4 = P1 cos2(

k0δL′

2

). (5.162)

As a consequence, even for a one-photon incoming state, which we called a quasi-particle-like state, interference fringes will be observed, as is found with a quasi-classicalstate or classical optical waves. This prediction of quantum optics has been checked exper-imentally (see Figure 5.9). It illustrates the wavelike behaviour of a single photon, or theclaim that a ‘photon can interfere with itself’.

5.5.4 Wave–particle duality for a particle-like state

Consider once again a one-photon state of the form (5.132) arriving at the semi-reflectingmirror Sin (see Figure 5.8), but this time introduce photon-counting photodetectors at

Port 3 Port 4

δL′ δL′

�Figure 5.9 Experimental results giving the counts at ports (3) and (4) of a Mach–Zehnderinterferometer as a function of the path length difference δL′ in the case where theincoming field is a one-photon state. The successive plots correspond to increasingobservation times for each value of the path difference (fixed mirrors). These observationtimes vary from 0.1 to 10 seconds, corresponding to an average number of detections of 1 to100 at each mirror position. Note the complementarity of the two interference patterns ateach port. (Figure, P. Grangier and A. Aspect.)



ports (3′) and (4′) (see Figure 5.8(b)). A coincidence circuit is used to measure the prob-ability of joint detections at (3′) and (4′). We write each field E(+)

3′ and E(+)4′ as a function

of the incoming fields, using (5.8–5.9). As in (5.136), the probability of joint detections iszero, because it only contains terms of the form (a�)2|1�〉 (for a detailed calculation, seeComplement 5B, Section 5B.2). There is thus a distinctly particle-like behaviour here, inthe sense that the photon is detected on one side or the other of Sin, but not both sidesat once. It is tempting to conclude that the photon goes on either one side or the other.But as we saw in Section 5.5.3, the same one-photon wave packet in the interferometerof Figure 5.8(a) gives rise to interference, and this is naturally interpreted by imagining anincoming wave that divides on Sin and propagates in both 3′ and 4′. What we are faced withhere is an example of wave–particle duality: the one-photon wave packet behaves some-times like a wave and sometimes like a particle, depending on whether it is analysed withthe setup of Figure 5.8(a) or that of Figure 5.8(b).

The Bohr complementarity principle is often cited to make the existence of these appar-ently contradictory types of behaviour less disconcerting. According to this principle,contradictory classical behaviour actually arises in incompatible experimental setups. Itis quite true that the setups of Figures 5.8(a) and 5.8(b) cannot be implemented at the sametime. When we analyse the situation, we find that the complementary quantities resultin fact on the one hand from observation of interference – associated with simultaneouspassage through the two ports of the interferometer – and on the other from a precise deter-mination of the path followed – unambiguously revealed by the setup of Figure 5.8(b),since only one of the two detectors D3′ or D4′ gives a signal.

More refined versions of the experiment can be devised,15 in which some limited infor-mation I can be obtained concerning the path followed, without preventing the appearanceof interference with limited contrast C. For these complementary quantities, an analysis ofsuch situations then leads to an inequality,

C2 + I2 ≤ 1. (5.163)

The two situations discussed above correspond to the limiting cases (C = 0, I = 1) or(C = 1, I = 0), respectively.

�Comment Determination of the path requires the use of a particle-like state, like a one-particle wave packet.Indeed, note that, if a quasi-classical radiation state is used, the experimental arrangement ofFigure 5.8(b) gives a non-zero probability for the coincidence count, and the path followed cannotbe unambiguously determined.

5.5.5 Wheeler’s delayed-choice experiment

In conformity with the idea of complementarity, contradictory behaviour is associatedwith incompatible experimental arrangements, and one does indeed find that the observed

15 V. Jacques et al., Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons, PhysicalReview Letters 100, 220402 (2008) and references therein.


383 5.6 A wave function for the photon?�

behaviour depends on the chosen arrangement. One might thus be led to the follow-ing interpretation of complementarity: the quantum system ‘adopts a different behaviour’depending on the apparatus it interacts with.

Such a claim should not be interpreted too naively, e.g. by thinking that the incomingphoton at the beamsplitter Sin will adopt wavelike or particle-like behaviour, depending onwhether the detectors D3′ and D4′ are absent or present. The point is that one can imagine,as suggested by J. A. Wheeler, that the decision to introduce these detectors might be madeonly after the arrival of the photon at the incoming beamsplitter Sin. But the setup proposedby Wheeler is even more radical. His idea was, rather than introducing further detectorsD3′ and D4′ , to use a removable outgoing beamsplitter Sout in the setup of Figure 5.8(a).When it is in place, we have an interferometer, but when it is withdrawn, a photodetec-tor at D3 or D4 indicates unambiguously which path has been followed. If the light traveltime L′/c in the interferometer is longer than the duration of the one-photon wave packet,there is in principle no difficulty in deciding to withdraw or introduce Sout after the pas-sage of the photon at Sin. More precisely, it is enough for the two events to be separatedby a spacelike interval, in the relativistic sense, to be sure that the decision to withdrawor introduce the beamsplitter cannot influence the behaviour of the photon entering theinterferometer, unless one accepts the possibility of some influence travelling faster thanlight.

The experiment has been carried out, and the result is unambiguous: if the beamsplitterSout is in place when the one-photon wave packet arrives there, interference fringes areobserved, whatever the situation when it entered the interferometer.16 And conversely, ifSout is absent at the time of passage of the wave packet, no fringe is observed and one cansay without ambiguity whether the photon has followed path 3′ or path 4′. If one insistedon using a classical picture of the wave – associated with the idea of travelling in botharms at the same time – or the classical picture of a particle – choosing one path or theother, but not both – then one must conclude, to paraphrase Wheeler’s own words, that thebehaviour adopted in the interferometer depends on the choice made at the moment of leav-ing it. This spectacular manifestation of wave–particle duality is what Feynman called thegreat mystery of quantum physics. But it should be stressed that the formalism of quantumoptics provides a consistent account of these different kinds of behaviour, whatever diffi-culty we may find in accepting them in the light of our experience of classical waves andparticles.

5.6 A wave function for the photon?

For a one-photon state of the form (5.132) in free space, the probability of photodetectionat r (5.134) takes the form

w(r, t) = s|U(r, t)|2, (5.164)

16 V. Jacques et al., Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment, Science 315,966 (2007).

9780521551120com5b CUP/GYA June 2, 2010 7:31 Page-398

5B Complement 5B One-photon wave packet

One-photon sources are important elements in quantum optics. The archetypal example isan atom raised to an excited state at time t = 0, then de-exciting with emission of a singlephoton. The development of this kind of source depends on progress with experimentaltechniques, e.g. the possibility of isolating a single atom, molecule or quantum well. Inthis complement, we present the formalism for describing the corresponding radiation, anduse it to discuss some spectacular experiments which bring out properties quite incompat-ible with a classical description of the electromagnetic field. We begin in Section 5B.2 bydescribing the anti-correlation between detections on either side of a semi-reflecting mir-ror, establishing the quantitative difference with a classical field. Section 5B.3 discusses aquantum optical effect that was only demonstrated at the beginning of the twenty-first cen-tury, namely the quantum coalescence of two one-photon wave packets on a semi-reflectingmirror, which occurs even when the two photons were emitted by independent atoms. Ananalogous effect, the Hong–Hou–Mandel effect, is discussed in Chapter 7. These effectsexemplify quantum interference involving two photons. Finally, Section 5B.4 is concernedwith quantum calculations involving quasi-classical states. As we now know, this leads toresults that are identical to the predictions of semi-classical theory.

5B.1 One-photon wave packet

5B.1.1 Definition and single photodetection probability

Consider a one-photon state of the form (5.132), i.e.

|1〉 =∑�

c�|0, . . . , n� = 1, 0, . . .〉 =∑�

c�|1�〉, (5B.1)

with∑�

|c�|2 = 1. It is an eigenstate of N = ∑�

N� with eigenvalue 1, but it is not an

eigenstate of N�, and nor is it an eigenstate of the Hamiltonian HR = ∑�

hω�(N� + 1/2),

whenever there are two non-zero coefficients c� corresponding to two modes of differentfrequencies. It changes in time and the state is given at time t by

|1(t)〉 =∑�

c� e−iω�t|1�〉. (5B.2)


399 Complement 5B One-photon wave packet�

Using (5.4), the photodetection signal at time t and point r is given by

w(r, t) = s∥∥∥E(+)(r)|1(t)〉

∥∥∥2 = s

∣∣∣∣∣∑�

c�E(1)� ε�e

i(k�.r−ω�t)∣∣∣∣∣2

= s|E(+)(r, t)|2, (5B.3)

with

E(+)(r, t) =∑�

c�E(1)� ε�e

i(k�.r−ω�t). (5B.4)

Note that this single photodetection probability is exactly the same as would befound for the classical field (5B.4), using the semi-classical model of photodetection(Equation 5.2).

5B.1.2 One-dimensional wave packet

In classical electromagnetism, we know how to construct wave packets occupying abounded region of spacetime. We consider a set of coefficients c� different from zero forvalues of k� distributed over a volume of k-space of extent δkx, δky, δkz about a value k0.We thus obtain, at time t = 0, a wave packet localized in a volume of real space withdimensions of the order of (δkx)−1, (δky)−1, (δkz)−1. When the same set of coefficients c�is substituted into (5B.1), we thus obtain a photodetection probability (5B.3) that differsfrom zero only within some bounded region.

The volume of this region generally increases without limit as time goes by, and this ineach space dimension, but there are specific forms for which the spreading effect does notoccur. An example is the one-dimensional wave packet we are about to discuss.

Consider the case in which the wavevectors k� associated with the non-zero coefficientsc� are all parallel to the same unit vector u, i.e.

k� = ω�

cu = �2π

Lu, (5B.5)

where L is an arbitrary quantization length. The function (5B.4) then takes the form

E(+)(r, t) =∑�

c�E(1)� ε� eiω�(r·u/c−t), (5B.6)

and the photodetection probability (5B.3) depends on space and time only through thequantity

τ = t − r·u/c. (5B.7)

The photodetection signal thus propagates without distortion at speed c in the directionspecified by u.

This kind of wave packet is not very realistic, in the sense that it extends infinitely in theplane perpendicular to u. One can imagine a cylindrical beam of cross-section S⊥, propa-gating in the direction u. If the transverse dimensions are much larger than the wavelength,diffraction will be negligible, and the cylindrical beam is an approximate solution for the



D

A

�Figure 5B.1 A single atom A, placed at the focus of a parabolic mirror, emits a collimated one-photonwave packet, well modelled by the one-dimensional wave packet of Section 5B.1.2. Thedetector D is wide enough to detect the whole wave packet.

equations of electromagnetism. The beam obtained by placing a very small source, e.g. anexcited atom at the focus of a parabolic mirror, is adequately described by this model (seeFigure 5B.1).

Let us consider the case where all the modes have the same polarization ε. The coef-ficients c� then depend only on the frequency ω�, and a wave packet can be formed byconsidering a distribution peaking at some ω0, described by

c(ω�) = f (ω� − ω0), (5B.8)

where f (�) is a function centred on 0 and having a typical half-width δω that is smallcompared with ω0. The function (5B.6) will then be proportional to the Fourier transformf (τ ) of f (�), yielding a wave packet with width of the order of 1/δω. To carry out thecalculation explicitly, the sum

∑� in (5B.6) is replaced by an integral, introducing the

one-dimensional mode density deduced from (5B.5):

d�

dω�= L

2πc. (5B.9)

The quantization volume is S⊥L, and the constant E(1)� is thus,

E(1)� =

√hω�

2ε0LS⊥. (5B.10)

As it varies little over the interval δω, it can be replaced by its value at ω0 and broughtoutside the integral. The final result is

E(+)(r, t) = ε

√hω0

2ε0LS⊥L

2πc

∫ +∞−∞

dω� c(ω�)e−iω�τ

= ε

√hω0L

4πε0c2S⊥e−iω0(t− u·r

c ) f(

t − u · rc

),

(5B.11)

with

f (τ ) = 1√2π

∫ +∞−∞

d� f (�) e−i�τ . (5B.12)



The photodetection probability per unit time and per unit detector area is, therefore,

w(r, t) = shω0L

4πε0c2S⊥

∣∣∣f (t − u · rc

)∣∣∣2 , (5B.13)

and the detection probability per unit time over the whole detector is

dPdt= s

hω0L

4πε0c2

∣∣∣f (t − u · rc

)∣∣∣2 . (5B.14)

The total probability of photodetection, integrated over a time interval covering the wholewave packet, is obtained by integrating (5B.14). With the form (5B.12) for the Fouriertransform, the Parseval–Plancherel relation gives∫

dτ |f (τ )|2 =∫

d�|f (�)|2. (5B.15)

The right-hand side is easily evaluated by writing down the normalization condition for the|c�|2, given the density of modes (5B.9):

∑�

|c�|2 = L

2πc

∫d�|f (�)|2 = 1. (5B.16)

The final result is

P = shω0

2ε0c. (5B.17)

As one would hope, the arbitrary length L disappears from the final expression.For a single photon, a perfect detector must give P = 1, and the sensitivity s is, therefore,

sperfect = 2ε0c

hω0. (5B.18)

A real detector generally has lower sensitivity than this, by a factor (less than unity) calledthe quantum efficiency or quantum yield of the detector. For certain spectral ranges, thereare detectors with quantum efficiencies very close to unity.

5B.1.3 Spontaneous emission photon

Spontaneous emission by a single atom in an excited state gives a one-photon wave packet.Indeed, consider a two-level atom in the excited state |b〉 in the radiation vacuum at timet = 0 (situation studied in Section 6.4 of Chapter 6). The initial state |b; 0〉 is coupled by thequantum interaction Hamiltonian with all states of the form |a; 1�〉, where |a〉 is the groundstate of the atom, and |1�〉 represents a one-photon state of mode �. From the detailed inves-tigation of the coupling between a discrete state and a continuum, discussed in Chapter 1,we can calculate the final state of the system after a long lapse of time compared with thelifetime of the excited state. It is found that

|ψ〉 =∑�

γ�(t→∞)|a; 1�〉

= |a〉 ⊗∑�

γ�(t→∞)|1�〉.(5B.19)



The radiation part of this state does indeed have the form (5B.1).Suppose now that the atom is placed at the focus of an optical system, e.g. a parabolic

mirror, which transforms all emitted spherical waves into plane waves propagating alongthe same direction. We then obtain a one-dimensional one-photon wave packet.

Generalizing the result (1.88) to the case of spontaneous emission (see Section 6.4), weobtain the coefficients

c� = K

ω� − ω0 + isp/2, (5B.20)

where

K =(sp

c

L

) 12

, (5B.21)

to ensure normalization of c�, with the mode density (5B.9). Note that the emitted lightspectrum is described by a Lorentzian line centred at ω0, with width sp at half-maximum:

|c(ω�)|2 = K2

(ω� − ω0)2 + 2sp4

. (5B.22)

We now write E(+)(r, t) in the form (5B.11), taking the Fourier transform of

f (�) = K

�+ isp2

. (5B.23)

It can be shown that this yields

f (τ ) = K√

2π H(τ ) e−sp

2 τ , (5B.24)

where H(τ ) is the Heaviside step function, equal to 0 for τ < 0 and 1 for τ ≥ 0. We thusobtain a wave packet that begins suddenly at τ = 0 and dies off exponentially. Taking aperfect photodetector that covers the whole cross-section S⊥ of the beam (see Figure 5B.1),the detection probability per unit time (5B.14) is

dPdt= sp H

(t − u·r

c

)e−sp(t− u·r

c ). (5B.25)

Figure 5B.2 shows this probability. The curve can be obtained experimentally by car-rying out many repeats of the following experiment: the atom is excited at time texc andwe then measure the time tdet at which a photon is detected. The histogram giving the dis-tribution of intervals tdet − texc looks like the distribution (5B.25), apart from statisticalfluctuations.

�Comment We have considered the case of a polarized beam to simplify the treatment. In the general case, eachpolarization is handled separately.



ddt

t

Γ −1sp

r.u/c0

�Figure 5B.2 Probability of photodetection at point r and time t for a one-photon wave packet emitted atr = 0 and t = 0. The signal begins at time t = r . u/c, then dies down exponentially. Notethat, for each individual measurement, only one photodetection can be observed. The abovesignal can be obtained by repeating the experiment many times and measuring the timeinterval between excitation of the atom and detection of a photon by the detector.

5B.2 Absence of double detection anddifference with a classical field

In Section 5B.1.1, as in Section 5.4.3 of Chapter 5, we have seen that for a one-photonwave packet the detection probability w(r, t) at a point has the same form as would beobtained for a classical field. Moreover, we showed in Section 5.4.3 that, for this kind ofone-photon state, the probability of double detection at two points r1 and r2 is strictly zero:

w(2)(r1, r2, t) = 0. (5B.26)

This corresponds perfectly with the intuitive picture one would have of a single photon.We also indicated in Section 5.5.1 that the probability of a double detection in the outputports of a beamsplitter is zero in the case of a one-photon state. We give here the detailedcalculation leading to this result, as well as the calculation in the case of a classical wavepacket.

5B.2.1 Semi-reflecting mirror

The one-photon wave packet is now sent through the input port (1) of a semi-reflectingmirror and two detectors are placed at the output ports (3) and (4) (see Figure 5B.3).

We express the field E(+)3 (D3) in terms of the incoming field at ports (1) and (2). At

port (1), the origin 01 is chosen symmetrically to the focus with respect to the apex of theparaboloid. Using (5.8) and including the propagation in the input and output space, weobtain

E(+)3 (D3) = i√

2

∑�1

E(1)�1

eik�1 (r1+r3)a�1 +i√2

∑�2

E(1)�2

eik�2 (r2+r3)a�2 , (5B.27)



A1 u1

u2

r4

u4D4

Sin

u3

D3

r3

r1

01

02

r2

�Figure 5B.3 Photodetection on either side of the semi-reflecting mirror for a one-photon wave packetemitted by an atom whose image is at distance r1 from the mirror. An electronic systemcounts detection events at D3 and D4, as well as the number of joint detections at D3and D4.

where �1 and �2 characterize the incoming modes at ports (1) and (2), respectively.Likewise, for the fields at detector D4,

E(+)4 (D4) = i√

2

∑�1

E(1)�1

eik�1 (r1+r4)a�1 −i√2

∑�2

E(1)�2

eik�2 (r2+r4)a�2 . (5B.28)

The single detection probabilities per unit time are obtained as in Section 5.5, by takingan input state of the form |11〉 ⊗ |02〉. Only the terms of (5B.28) related to modes �1

contribute, and the rest of the calculation is as in Section 5B.1.3. For perfect detectors, wefind

dP3

dt= 1

2spH(τ3)e−spτ3 , (5B.29)

with

τ3 = t − r1 + r3

c, (5B.30)

and

dP4

dt= 1

2spH(τ4)e−spτ4 , (5B.31)

with

τ4 = t − r1 + r4

c. (5B.32)

The joint detection probability is

w(2)(r3, r4, t) = s2 1

4〈1(t)|E(−)

3 (r3)E(−)4 (r4)E(+)

4 (r4)E(+)3 (r3)|1(t)〉. (5B.33)



When E(+)3 and E(+)

4 are replaced by their expressions (5B.27–5B.28) in terms of theincoming field, one obtains, once again, terms∑

�′1

∑�′′1

a�′1 a�′′1 |1�1〉 = 0. (5B.34)

The joint detection probability is zero, even if r3 = r4, i.e. even if the single detectionprobabilities are simultaneously non-zero. We recover here the result discussed above inthe absence of the mirror (see also Section 5.4.3).

�Comment In the above calculations, and remembering that port (2) is empty, the second terms in expres-sions (5B.27) and (5B.28) contribute nothing. However, it is nevertheless important to include thembecause these terms play a role in certain fluctuation calculations, even when port (2) is empty.

5B.2.2 Double detection with a classical wave packet

In classical electromagnetism, one can imagine a polarized one-dimensional wave packetdescribed by the analytic signal,

E(+)cl (r, t) =

∑�

E(1)� c�e

−iω�(t− rc ), (5B.35)

with the same coefficients c� as in Section 5B.1.3, and the form (5B.10) for E(1)� . In

the sense of classical electromagnetism, the energy contained in this wave packet ishω0. Calculations to find the single photodetection probabilities are strictly analogousto those presented above, and expressions (5B.29) and (5B.31) are obtained behind thesemi-reflecting mirror.

Let us now find the double photodetection probability. In the semi-classical model, thereis a wave packet at each detector, and the photodetections at the two detectors are inde-pendent random events. The joint detection probability is thus the product of the singledetection probabilities, i.e.

w(2)(r1, t1; r2, t2) = w(r1, t1) w(r2, t2) (5B.36)

and

d2Pdt3dt4

= dP3

dt· dP4

dt= 1

42

spH(τ3)H(τ4)e−sp(τ3+τ4). (5B.37)

There are clearly some values of τ3 and τ4 for which this joint detection probability differsfrom zero. To compare with the calculation in Section 5B.2.1, consider the case t3 = t4.The simultaneous photodetection probability differs from zero if the two wave packetsoverlap, i.e. if |r3 − r4| is not large compared with c/sp.

In contrast with the one-photon wave packet of quantum optics, a classical wave packetgives rise to a certain probability of joint detection on either side of the semi-reflectingmirror. A lower bound for the simultaneous detection probability can in fact be found forthe semi-classical model of photodetection.



To this end, consider the single detection probabilities for detectors D3 and D4, assumedto be at the same distance from the emitter (r3 = r4). Measurements thus refer to theclassical field taken at the same instant of time, which we shall write as E(+)(t), omittingthe retardation to simplify the notation. The single detection probabilities are

w(D3, t) = s√2|E(+)(t)|2 (5B.38)

w(D4, t) = s√2|E(+)(t)|2, (5B.39)

and the simultaneous detection probability is

w(2)(D3, D4, t) = w(D3, t)w(D4, t) = s2

2|E(+)(t)|4. (5B.40)

In a real experiment, the measurement is repeated many times, which amounts to aver-aging over possibly different values of E(t). The average photodetection probabilities are,therefore,

w(D3) = s√2|E(+)(t)|2 (5B.41)

w(D4) = s√2|E(+)(t)|2 (5B.42)

w(2)(D3, D4) = s2

2|E(+)(t)|4. (5B.43)

Now any real quantity f (t) satisfies the Cauchy–Schwartz inequality:

[f (t)]2 ≥ [f (t)]2. (5B.44)

In the experiment, measurements of the single and simultaneous detection probabilitiesrefer to the same time intervals, and the averages are the same. It follows that

w(2)(D3, D4) ≥ w(D3) · w(D4). (5B.45)

In the case where the classical wave packets all have the same amplitude, equality holds in(5B.45). In any case, for a classical field and the semi-classical photodetection model, thesimultaneous detection probability cannot be less than the product of the single detectionprobabilities.

If we find w(2)(D3, D4) < w(D3) · w(D4) in an experiment, we can be sure that theradiation under investigation cannot be correctly described by classical electromagnetism.The violation of inequality (5B.45) is a criterion used to test single-photon light sources,which play an important role in quantum optics.

The above photodetection probabilities are instantaneous detection probabilities forpointlike detectors, and one may wonder whether there is an analogous inequality whenone considers photodetection probabilities integrated over a finite observation time win-dow θ and over detectors of surface area S⊥. Consider first the case of point detectors,



but measurements integrated over a time lapse θ . We define the single photodetectionprobabilities over θ by

πθ (D3, t) =∫ t+θ

tw(D3, t)dt (5B.46)

πθ (D4, t) =∫ t+θ

tw(D4, t)dt. (5B.47)

The probability of a coincidence detection during θ is obtained by double integration ofw(2)(D3, D4; t3, t4), which is the double detection probability at D3 at time t3 and at D4 attime t4 :

w(2)(D3, D4; t3, t4) = w(D3, t3)w(D4, t4) = s2

2|E(+)(t3)|2 |E(+)(t4)|2. (5B.48)

Given (5B.47–5B.48), this yields

π(2)θ =

∫ t+θ

tdt3

∫ t+θ

tdt4 w

(2)(D3, D4; t3, t4)

= πθ (D3, t)πθ (D4, t).

(5B.49)

When the experiment is repeated many times, the average is taken over a large number ofdifferent samples of E(t) taken during θ . Consider the sample i, between ti and ti + θ , anddefine

λi =∫ ti+θ

tidt|E(+)(t)|2. (5B.50)

The probabilities for this sample are

{πθ (D3)}i = s√2λi (5B.51)

{πθ (D4)}i = s√2λi (5B.52)

and

{π(2)θ (D3, D4)}i = s2

2λ2

i . (5B.53)

Averaging over all the samples i, we obtain the average probabilities Pθ (D3), Pθ (D4) andP(2)θ (D3, D4), which are the quantities obtained at the end of the experiment. Now there is

once again a Cauchy–Schwartz inequality for the numbers λi:

λ2i ≥ (λi)

2, (5B.54)

from which it follows that

P(2)θ (D3, D4) ≥ Pθ (D3) · Pθ (D4). (5B.55)



An analogous argument can be made when considering integration over the whole surfaceof the detectors, and the inequality (5B.55) applies to the probabilities obtained by carry-ing out measurements with finite detectors and finite time windows. In practice it is thisinequality that is used to check whether a source does in fact emit one-photon pulses. Forexample, for the one-photon interference illustrated in Figure 5.9, it was found that

P(2)θ (D3, D4)

Pθ (D3) Pθ (D4)= 0.18, (5B.56)

well below the critical value of one corresponding to the limits of (5B.55). The fact thatthis number is not zero is easy to explain. For one thing, the detectors give a residualsignal even in the absence of any light (thermal noise), and for another, there is a non-zero probability of exciting two individual emitters during the same time window θ . Butthe result (5B.56) nevertheless violates the inequality (5B.55), whatever corrections mightbe made to it. There can be no doubt that this radiation is non-classical, and has a clearsingle-photon character.

5B.3 Two one-photon wave packets ona semi-reflecting mirror

5B.3.1 Single detections

A one-photon wave packet now enters each of the input ports (see Figure 5B.4). The twowave packets are identical, i.e. the incoming radiation has the form

|ψ〉 = |11〉 ⊗ |12〉, (5B.57)

01

r1

02

r2

D4

r4

D3

r3

�Figure 5B.4 Two one-photon wave packets on a semi-reflecting mirror. The two emitters are excited atthe same time, and the two wave packets arrive at the mirror at the same time if r1 = r2.Electronic circuits record single and joint detections at D3 and D4.



where the coefficients c�1 and c�2 (Equation 5B.1) are the same. To be specific, we shalluse the form (5B.20).

The expression (5B.27) for E(+)3 (D3) yields,

E(+)3 (D3)|ψ(t)〉 = i√

2

∑�1

E(1)�1

c�1 e−iω�1

(t− r1+r3

c

)|01〉 ⊗ |12〉

+ i√2

∑�2

E(1)�2

c�2 e−iω�2

(t− r2+r3

c

)|11〉 ⊗ |02〉,

(5B.58)

where |01〉 and |02〉 denote the vacuum at ports (1) and (2). Taking the square of the norm,a calculation as in Section 5B.1.3 yields the single detection probability over the wholedetector as

dP3

dt(t) = 1

2H

(t − r1 + r3

c

)spe

−sp

(t− r1+r3

c

)

+ 1

2H

(t − r2 + r3

c

)spe

−sp

(t− r2+r3

c

).

(5B.59)

An analogous calculation for detector D4 yields

E(+)4 (D4)|ψ(t)〉 = i√

2

∑�1

E(1)�1

c�1 e−iω�1

(t− r1+r4

c

)|01〉 ⊗ |12〉

− i√2

∑�2

E(1)�2

c�2 e−iω�2

(t− r2+r4

c

)|11〉 ⊗ |02〉

(5B.60)

and

dP4

dt(t) = 1

2H

(t − r1 + r4

c

)spe

−sp

(t− r1+r4

c

)

+ 1

2H

(t − r2 + r4

c

)spe

−sp

(t− r2+r4

c

).

(5B.61)

At each detector, we find the sum of the probabilities associated with each wave packet,with no interference term. The effects are additive in terms of probabilities, as for inde-pendent classical particles distributed randomly by the beamsplitter between the outputports.

5B.3.2 Joint detections

To calculate the joint detection probability w(2)(D3, D4) on either side of the mirror, wemust take the square of the norm of E(+)

3 (D3) E(+)4 (D4)|ψ(t)〉. Referring to the incoming

space and using (5B.60), this yields



E(+)3 (D3) E(+)

4 (D4)|ψ(t)〉 =

1

2

⎧⎨⎩⎡⎣∑

�1

c�1 E(1)�1

e−iω�1

(t− r1+r4

c

)⎤⎦⎡⎣∑

�2

c�2 E(1)�2

c�2 e−iω�2

(t− r2+r3

c

)⎤⎦

−⎡⎣∑

�1

c�1 E(1)�1

e−iω�1

(t− r1+r3

c

)⎤⎦⎡⎣∑

�2

c�2 E(1)�2

c�2 e−iω�2

(t− r2+r4

c

)⎤⎦⎫⎬⎭ |01〉 ⊗ |02〉.

(5B.62)

Since c�1 and c�2 are the same, the subscripts 1 and 2 can be interchanged in the aboveexpression and it turns out that, when r1 = r2, the expressions in square brackets are thesame and exactly cancel. The probability of joint detection on either side of the mirror iszero. This result contradicts the simple picture developed in Section 5B.3.1, according towhich the single photons are distributed randomly between the output ports of the semi-reflecting mirror, like independent classical particles. The calculation just carried out showsthat, when they perfectly overlap on the beamsplitter (r1 = r2), the two photons never leaveby different ports, but leave together either by port (3) or by port (4). This is referred to as‘quantum coalescence’.

This phenomenon is the result of a quantum interference effect between the quantumamplitudes associated with the two symbolic diagrams shown in Figure 5B.5. The negativesign responsible for the cancellation results from the opposite signs for the reflection termsin the input–output relations for a semi-reflecting mirror.

This purely quantum effect should not be confused with an interference effect betweenclassical fields. What we have here is interference between quantum amplitudes associatedwith the diagrams of Figure 5B.5, each one involving both photons. For interference tooccur, the two processes must be indistinguishable, and this requires the two wave packetsto be identical, i.e. they must have the same distribution of c� and the same emission time,and r1 must be equal to r2. But we should stress that there is no coherence, in the clas-sical sense, between the wave packets, since they are produced by independent emitters.

A2

D4

D3

A1

(a)

A1

A2

D4

D3

(b)

�Figure 5B.5 Schematic diagrams representing the two two-photon processes whose amplitudes interferedestructively. (a) The photon emitted by A1 is detected by D3 and the photon emitted by A2is detected by D4. (b) The photon emitted by A1 is detected by D4 and the photon emittedby A2 is detected by D3.



The experiment has indeed been carried out in this form, using two distinct atoms or ionsexcited independently and re-emitting spontaneous photons.1

�Comments (i) Insofar as two strictly indistinguishable photons are picked up at detector D3 or detector D4, theeffect is related to the bosonic nature of photons. Two fermions with the same characteristicswould never be detected simultaneously at the same detector.

(ii) Quantitative significance can be given to the indistinguishability of the two photons by calculat-ing the joint detection probability when the distances r1 and r2 are not the same. The calculationis straightforward but tedious, starting from (5B.62). After integration, it is found that the totaljoint probability of a detection at D3 and a detection at D4 is given by

P(2)(D3, D4) = 1

2

(1− e−sp

|r2−r1|c

), (5B.63)

whatever the detection times (within the wave packet). If r2 = r1, P(2) = 0. But if |r2 − r1| >c/sp, which means that one can tell whether the photon came from A1 or from A2 by thedetection time (the excitation times are known), the result is P(2)(D3, D4) = 1/2. This is justwhat would be obtained by considering two classical particles, each of which has probability1/2 of being reflected and probability 1/2 of being transmitted. Indeed, in this case, there arefour possible final situations, two of which correspond to joint detection on either side of thesemi-reflecting mirror.

5B.4 Quasi-classical wave packet

Consider a multimode quasi-classical state of the form (5.122):

|ψqc(t)〉 = ��|α�e−iω�t〉, (5B.64)

where all modes have the same direction of propagation k�/k� and the same polarizationε. Each state |α�e−iω�t〉 is an eigenstate of a�, and the α� are taken in the form (5B.20), butwithout imposing the normalization condition (5B.21). The expected number of photonsin this state is equal to

N =∑�

〈ψqc|a†�a�|ψqc〉 =

∑�

|α�|2. (5B.65)

We saw in Section 5.4.2 that |ψqc〉 is an eigenstate of E(+)(r), with eigenvalue

E(+)cl (r, t) = iε�

∑�

E(1)� c�e

−iω�(t− rc )

= ε�E(+)cl (r, t),

(5B.66)

where E(+)cl (r, t) is precisely the analytic signal introduced in (5B.35). It can be calculated

explicitly using (5B.11), (5B.20), (5B.24) and (5B.65), and we obtain

1 J. Beugnon et al., Quantum Interference Between Two Single Photons Emitted by Independently TrappedAtoms, Nature 440, 779 (2006).



E(+)cl (r, t) = √N

√hω0sp

2ε0cS⊥H(

t − r

c

)e−(sp

2 +iω�)(t− r

c ). (5B.67)

Such a classical wave packet has cross-section S⊥. Its amplitude is zero for t <r

c, and

becomes negligible again after a time of the order of −1sp . The classical electromagnetic

energy contained in this packet is Nhω0.The single photodetection probability for the quasi-classical wave packet is

w(r, t) = s‖E(+)(r)|ψ1qc(t)〉‖2,

= s|E(+)cl (r, t)|2,

(5B.68)

where E(+)cl (r, t) is the classical wave packet (5B.67). If we take N = 1, this result is the

same as the one obtained in the semi-classical photodetection model with the classical field(5B.67), which is identical to the probability calculated for the quantum one-photon wavepacket of Section 5B.1.3.

The double photodetection probability is

w(2)(r1, r2, t) = s2‖E(+)(r2)E(+)(r1)|ψ1qc(t)〉‖2. (5B.69)

As |ψ1qc〉 is an eigenstate of E(+)(r1) and E(+)(r2), we have immediately,

w(2)(r1, r2, t) = w(r1, t) · w(r2, t), (5B.70)

which is precisely the result of the semi-classical photodetection model for the classicalfield (5B.67).

Note that the above result remains true even if the expected number of photons N iswell below unity. Now it can be shown that a light pulse that has been highly attenuatedby passing through an absorbent material gives precisely a multimode quasi-classical state,where N can assume a value much less than unity. Such a state is radically different froma one-photon state, since the inequalities (5B.45) or (5B.55) are not violated. It has beenchecked that the inequality (5B.55) remains true even for an expected number of photonsper pulse equal to N = 10−2.2 It is therefore incorrect to claim that a highly attenuatedbeam is made up of single photons.

The properties of a quasi-classical wave packet can thus be interpreted using a classicalmodel of the electromagnetic field. The above properties can also be interpreted in terms ofphotons by recalling that, for a quasi-classical state, the probabilities of having N photonsare not zero when N > 1, but are in fact equal to [P(N = 1)]N /N !. In a quasi-classicalpulse, one thus has a certain probability of having 2 photons, 3 photons, N photons. It isthese groups of photons that give rise to a sufficient number of coincidences on either sideof the semi-reflecting mirror to avoid violation of the inequality (5B.45).

2 The experiment was carried out with the same detection system as the one used to violate the inequality (5B.55)in the case of a one-photon wave packet (see Section 5B.2.2 and in particular 5B.55). The same experimentalsetup was used to observe the interference patterns in Figure 5.9 with one-photon wave packets. P. Grangier,G. Roger and A. Aspect, Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: a NewLight on Single-Photon Interferences, Europhysics Letters 1, 173 (1986).

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