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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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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.

Einstein™s introduction of photons was the ¬rst step toward a true quantum the-

ory of light”just as the Bohr model of the atom was the ¬rst step toward quantum

mechanics”but there is an important di¬erence between these parallel developments.

The transition from classical electromagnetic theory to the photon model is even more

radical than the corresponding transition from classical mechanics to quantum me-

chanics. If one thinks of classical mechanics as a game like chess, the pieces are point

particles and the rules are Newton™s equations of motion. The solution of Newton™s

equations determines a unique trajectory (q (t) , p (t)) for given initial values of the

position q (0) and the momentum p (0) of a point particle. The game of quantum me-

chanics has the same pieces, but di¬erent rules. The initial situation is given by a

wave function ψ (q), and the trajectory is replaced by a time-dependent wave function

ψ (q, t) that satis¬es the Schr¨dinger equation. The situation for classical electrody-

o

namics is very di¬erent. The pieces for this game are the continuous electric and

magnetic ¬elds E (r, t) and B (r, t), and the rules are provided by Maxwell™s equations.

Einstein™s photons are nowhere to be found; consequently, the quantum version of the

game requires new pieces, as well as new rules. A conceptual change of this magnitude

should be approached with caution.

In order to exercise the caution recommended above, we will discuss the experimen-

tal basis for the quantum theory of light in several stages. Section 1.1 contains brief

descriptions of the experiments usually considered in this connection, together with

a demonstration of the complete failure of classical physics to explain any of them.

In Section 1.2 we will introduce Einstein™s photon model and show that it succeeds

brilliantly in explaining the same experimental results.

In other words, the photon model is su¬cient for the explanation of the experi-

ments in Section 1.1, but the question is whether the introduction of the photon is

necessary for this purpose. The only way to address this question is to construct an

alternative model, and the only candidate presently available is semiclassical elec-

trodynamics. In this approach, the charged particles making up atoms are described

by quantum mechanics, but the electromagnetic ¬eld is still treated classically.

In Section 1.3 we will attempt to explain each experiment in semiclassical terms.

In this connection, it is essential to keep in mind that corrections to the lowest-order

approximation”of the semiclassical theory or the photon model”would not have

been detectable in the early experiments. As we will see, these attempts have varying

degrees of success; so one might ask: Why consider the semiclassical approach at all?

The answer is that the existence of a semiclassical explanation for a given experimental

result implies that the experiment is not sensitive to the indivisibility of photons,

which is a fundamental assumption of Einstein™s model (Einstein, 1987a). In Einstein™s

own words:

According to the assumption to be contemplated here, when a light ray is spreading

from a point, the energy is not distributed continuously over ever-increasing spaces,

but consists of a ¬nite number of energy quanta that are localized in points in space,

move without dividing, and can be absorbed or generated only as a whole.

The early experiments

As an operational test of photon indivisibility, imagine that light containing exactly

one photon falls on a transparent dielectric slab (a beam splitter) at a 45—¦ angle

of incidence. According to classical optics, the light is partly re¬‚ected and partly

transmitted, but in the photon model these two outcomes are mutually exclusive. The

photon must go one way or the other. In Section 1.4 we will describe an experiment that

very convincingly demonstrates this all-or-nothing behavior. This single experiment

excludes all variants of semiclassical electrodynamics. Experiments of this kind had to

wait for technologies, such as atomic beams and coincidence counting, which were not

fully developed until the second half of the twentieth century.

1.1 The early experiments

1.1.1 The Planck spectrum

In the last half of the nineteenth century, a considerable experimental e¬ort was made

to obtain precise measurements of the spectrum of radiation emitted by a so-called

blackbody, an idealized object which absorbs all radiation falling on it. In practice,

this idealized body is replaced by a blackbody cavity, i.e. a void surrounded by a

wall, pierced by a small aperture that allows radiation to enter and exit. The interior

area of the cavity is much larger than the area of the hole, so a ray of light entering

the cavity would bounce from the interior walls many times before it could escape

through the entry point. Thus the radiation would almost certainly be absorbed before

it could exit. In this way the cavity closely approximates the perfect absorptivity of

an ideal blackbody. Even when no light is incident from the outside, light is seen

to escape through the small aperture. This shows that the interior of a cavity with

heated walls is ¬lled with radiation. The blackbody cavity, which is a simpli¬cation of

the furnaces used in the ancient art of ceramics, is not only an accurate representation

of the experimental setup used to observe the spectrum of blackbody radiation; it

also captures the essential features of the blackbody problem in a way that allows for

simple theoretical analysis.

Determining the spectral composition (that is, the distribution of radiant energy

into di¬erent wavelengths) of the light emitted by a cavity with walls at temperature

T is an important experimental goal. The wavelength, », is related to the circular

frequency ω by » = c/ν = 2πc/ω, so this information is contained in the spectral

function ρ (ω, T ), where ρ (ω, T ) ∆ω is the radiant energy per unit volume in the

frequency interval ω to ω + ∆ω. The power per unit frequency interval emitted from

the aperture area σ is cρ (ω, T ) σ/4 (see Exercise 1.1). In order to measure this quantity,

the various frequency components must be spectrally separated before detection, for

example, by refracting the light through a prism. If the prism is strongly dispersive

(that is, the index of refraction of the prism material is a strong function of the

wavelength) distinct wavelength components will be refracted at di¬erent angles.

For moderate temperatures, a signi¬cant part of the blackbody radiation lies in

the infrared, so it was necessary to develop new techniques of infrared spectroscopy

in order to achieve the required spectral separation. This e¬ort was aided by the

discovery that prisms cut from single crystals of salt are strongly dispersive in the

infrared part of the spectrum. The concurrent development of infrared detectors in

The quantum nature of light

Fig. 1.1 Distribution of energy in the spec-

trum of a blackbody at various temperatures.

(Reproduced from Richtmyer et al. (1955,

Chap. 4, Sec. 64).)

the form of sensitive bolometers2 allowed an accurate measurement of the blackbody

spectrum. The experimental e¬ort to measure this spectrum was initiated in Berlin

around 1875 by Kirchho¬, and culminated in the painstaking work of Lummer and

Pringsheim in 1899, in which the blackbody spectrum was carefully measured in the

temperature range 998 K to 1646 K. Typical results are shown in Fig. 1.1.

The theoretical interpretation of the experimental measurements also required a

considerable e¬ort. The ¬rst step is a thermodynamic argument which shows that the

blackbody spectrum must be a universal function of temperature; in other words, the

spectrum is entirely independent of the size and shape of the cavity, and of the material

composition of its walls. Consider two separate cavities having small apertures of

identical size and shape, which are butted against each other so that the two apertures

coincide exactly, as indicated in Fig. 1.2. In this way, all the radiation escaping from

each cavity enters the other. The two cavities can have interiors of di¬erent volumes

and arbitrarily irregular shapes (provided that their interior areas are su¬ciently large

compared to the aperture area), and their walls can be composed of entirely di¬erent

materials. We will assume that the two cavities are in thermodynamic equilibrium at

the common temperature T .

Now suppose that the blackbody spectrum were not universal, but depended, for

example, on the material of the walls. If the left cavity were to emit a greater amount

of radiation than the right cavity, then there would be a net ¬‚ow of energy from left

to right. The right cavity would then heat up, while the left cavity would cool down.

The ¬‚ow of heat between the cavities could be used to extract useful work from two

bodies at the same temperature. This would constitute a perpetual motion machine of

the second kind, which is forbidden by the second law of thermodynamics (Zemansky,

1951, Chap. 7.5). The total ¬‚ow of energy out of each cavity is given by the integral of

2 These devices exploit the temperature dependence of the resistivity of certain metals to measure

the deposited energy by the change in an electrical signal.

The early experiments

Fig. 1.2 Cavities ± and β coupled through a

Temperature = 6 common aperture.

its spectral function over all frequencies, so this argument shows that the integrated

spectral functions of the two cavities must be exactly the same.

This still leaves open the possibility that the spectral functions could di¬er in

certain frequency intervals, provided that their integrals are the same. Thus we must

also prove that net ¬‚ows of energy cannot occur in any frequency interval of the

blackbody spectrum. This can be seen from the following argument based on the

principle of detailed balance. Suppose that the spectral functions of the two cavities,

ρ± and ρβ , are di¬erent in the small interval ω to ω + ∆ω; for example, suppose that

ρ± (ω, T ) > ρβ (ω, T ). Then the net power ¬‚owing from ± to β, in this frequency

interval, is

1

c [ρ± (ω, T ) ’ ρβ (ω, T )] σ∆ω > 0 , (1.1)

4

where σ is the common area of the apertures. If we position absorbers in both ± and β

that only absorb at frequency ω, then the absorber in β will heat up compared to that

in ±. The two absorbers then provide the high- and low-temperature reservoirs of a

heat engine (Halliday et al., 1993, Chap. 22“6) that could deliver continuous external

work, with no other change in the system. Again, this would constitute a perpetual

motion machine of the second kind. Therefore the equality

ρ± (ω, T ) = ρβ (ω, T ) (1.2)

must be exact, for all values of the frequency ω and for all values of the temperature T .

We conclude that the blackbody spectral function is universal; it does not depend on

the material composition, size, shape, etc., of the two cavities. This strongly suggests

that the universal spectral function should be regarded as a property of the radiation

¬eld itself, rather than a joint property of the radiation ¬eld and of the matter with

which it is in equilibrium.

The thermodynamic argument given above shows that the spectral function is uni-

versal, but it gives no clues about its form. In classical physics this can be determined

by using the principle of equipartition of energy. For an ideal gas, this states that

the average energy associated with each degree of freedom is kB T /2, where T is the

temperature and kB is Boltzmann™s constant. For a collection of harmonic oscillators,

the kinetic and potential energy each contribute kB T /2, so the thermal energy for

each degree of freedom is kB T .

In order to apply these rules to blackbody radiation, we ¬rst need to identify and

count the number of degrees of freedom in the electromagnetic ¬eld. The thermal

radiation in the cavity can be analyzed in terms of plane waves eks exp (ik · r), where

The quantum nature of light

eks is the unit polarization vector and the propagation vector k satis¬es |k| = ω/c and

k·eks = 0. There are two linearly independent polarization states for each k, so s takes

on two values. The boundary conditions at the walls only allow certain discrete values

for k. In particular, for a cubical cavity with sides L subject to periodic boundary

conditions the spacing of allowed k values in the x-direction is ∆kx = 2π/L, etc.

3

Another way of saying this is that each mode occupies a volume (2π/L) in k-space,

’3

so that the number of modes in the volume element d3 k is 2 (2π/L) d3 k, where the

factor 2 accounts for the two polarizations. The ¬eld is completely determined by the

amplitudes of the independent modes, so it is natural to identity the modes as the

degrees of freedom of the ¬eld. Furthermore, we will see in Section 2.1.1-D that the

contribution of each mode to the total energy is mathematically identical to the energy

of a harmonic oscillator. The identi¬cation of modes with degrees of freedom shows

that the number of degrees of freedom dnω in the frequency interval ω to ω + dω is

k 2 dk L3 k 2

dnω = 2 dθ sin θ dφ 3 = π 2 c dω , (1.3)

(2π/L)

where θ and φ specify the direction of k. The equipartition theorem for harmonic

oscillators shows that the thermal energy per mode is kB T . The spectral function is

the product of dnω and the thermal energy density kB T /L3 , so we ¬nd the classical

Rayleigh“Jeans law:

ω2

ρ (ω, T ) dω = kB T 2 3 dω . (1.4)

πc

This ¬ts the low-frequency data quite well, but it is disastrously wrong at high

frequencies. The ω-integral of this spectral function diverges; consequently, the total

energy density is in¬nite for any temperature T . Since the divergence of the integral

occurs at high frequencies, this is called the ultraviolet catastrophe.

In an e¬ort to ¬nd a replacement for the Rayleigh“Jeans law, Planck (1959) con-

centrated on the atoms in the walls, which he modeled as a family of harmonic oscil-

lators in equilibrium with the radiation ¬eld. In classical mechanics, each oscillator is

described by a pair of numbers (Q, P ), where Q is the coordinate and P is the momen-

tum. These pairs de¬ne the points of the classical oscillator phase space (Chandler,

1987, Chap. 3.1). The average energy per oscillator is given by an integral over the

oscillator phase space, which Planck approximated by a sum over phase space elements

of area h. Usually, the value of the integral would be found by taking the limit h ’ 0,

but Planck discovered that he could ¬t the data over the whole frequency range by

instead assigning the particular nonzero value h ≈ 6.6 — 10’34 J s. He attempted to

explain this amazing fact by assuming that the atoms could only transfer energy to

the ¬eld in units of hν = ω, where ≡ h/2π. This is completely contrary to a clas-

sical description of the atoms, which would allow continuous energy transfers of any

amount.

This achievement marks the birth of quantum theory, and Planck™s constant h

became a new universal constant. In Planck™s model, the quantization of energy is a

property of the atoms”or, more precisely, of the interaction between the atoms and

the ¬eld”and the electromagnetic ¬eld is still treated classically. The derivation of the

The early experiments

spectral function from this model is quite involved, and the fact that the result is in-

dependent of the material properties only appears late in the calculation. Fortunately,

Einstein later showed that the functional form of ρ (ω, T ) can be derived very simply

from his quantum model of radiation, in which the electromagnetic ¬eld itself consists

of discrete quanta. Therefore we will ¬rst consider the other early experiments before

calculating ρ (ω, T ).

1.1.2 The photoelectric e¬ect

The infrared part of atomic spectra, contributing to the blackbody radiation discussed

in the last section, does not typically display sharp spectral lines. In this and the

following two sections we will consider e¬ects caused by radiation with a sharply

de¬ned frequency. One of the most celebrated of these is the photoelectric e¬ect:

ultraviolet light falling on a properly cleaned metallic surface causes the emission of

electrons. In the early days of spectroscopy, the source of this ultraviolet light was

typically a sharp mercury line”at 253.6 nm”excited in a mercury arc.

In order to simplify the classical analysis of this e¬ect, we will replace the complex-

ities of actual metals by a model in which the electron is trapped in a potential well.

According to Maxwell™s theory, the incident light is an electromagnetic plane wave

with |E| = c |B|, and the electron is exposed to the Lorentz force F = ’e (E + v — B).

Work is done only by the electric ¬eld on the electron. Hence it will take time for

the electron to absorb su¬cient energy from the ¬eld to overcome the binding energy

to the metal, and thus escape from the surface. The time required would necessarily

increase as the ¬eld strength decreases. Since the kinetic energy of the emitted electron

is the di¬erence between the work done and the binding energy, it would also depend

on the intensity of the light. This leads to the following two predictions. (P1) There

will be an intensity-dependent time interval between the onset of the radiation and

the ¬rst emission of an electron. (P2) The energy of the emitted electrons will depend

on the intensity.

Let us now consider an experimental arrangement that can measure the kinetic

energy of the ejected photoelectrons and the time delay between the arrival of the

light and the ¬rst emission of electrons. Both objectives can be realized by positioning

a collector plate at a short distance from the surface. The plate is maintained at a

negative potential ’Vstop , with respect to the surface, and the potential is adjusted to

a value just su¬cient to stop the emitted electrons. The photoelectron™s kinetic energy

can then be determined through the energy-conservation equation

1

mv 2 = (’e) (’Vstop ) . (1.5)

2

The onset of the current induced by the capture of the photoelectrons determines the

time delay between the arrival of the radiation pulse and the start of photoelectron

emission. The amplitude of the current is proportional to the rate at which electrons

are ejected. The experimental results are as follows. (E1) There is no measurable time

delay before the emission of the ¬rst electron. (E2) The ejected photoelectron™s kinetic

energy is independent of the intensity of the light. Instead, the observed values of

½¼ The quantum nature of light

the energy depend on the frequency. They are very accurately ¬tted by the empirical

relation

1

e = eVstop = mv = ω ’ W ,

2

(1.6)

2

where ω is the frequency of the light. The constant W is called the work function; it is

the energy required to free an electron from the metal. The value of W depends on the

metal, but the constant is universal. (E3) The rate at which electrons are emitted”

but not their energies”is proportional to the ¬eld intensity. The stark contrast between

the theoretical predictions (P1) and (P2) and the experimental results (E1)“(E3) posed

another serious challenge to classical physics. The relation (1.6) is called Einstein™s

photoelectric equation, for reasons which will become clear in Section 1.2.

In the early experiments on the photoelectric e¬ect it was di¬cult to determine

whether the photoelectron energy was better ¬t by a linear or a quadratic dependence

on the frequency of the light. This di¬culty was resolved by Millikan™s beautiful ex-

periment (Millikan, 1916), in which he veri¬ed eqn (1.6) by using alkali metals, which

were prepared with clean surfaces inside a vacuum system by means of an in vacuo

metal-shaving technique. These clean alkali metal surfaces had a su¬ciently small work

function W , so that even light towards the red part of the visible spectrum was able

to eject photoelectrons. In this way, he was able to measure the photoelectric e¬ect

from the red to the ultraviolet part of the spectrum”nearly a threefold increase over

the previously observed frequency range. This made it possible to verify the linear

dependence of the increment in the photoelectron™s ejection energy as a function of

the frequency of the incident light. Furthermore, Millikan had already measured very

accurately the value of the electron charge e in his oil drop experiment. Combining

this with the slope h/e of Vstop versus ν from eqn (1.6) he was able to deduce a value

of Planck™s constant h which is within 1% of the best modern measurements.

1.1.3 Compton scattering

As the study of the interaction of light and matter was extended to shorter wavelengths,

another puzzling result occurred in an experiment on the scattering of monochromatic

X-rays (the K± line from a molybdenum X-ray tube) by a graphite target (Compton,

1923). A schematic of the experimental setup is shown in Fig. 1.3 for the special

Scattering

angle Lead box

θ = 135ο

Detector

Graphite

target Crystal

spectrometer

X-ray tube

(source of

Fig. 1.3 Schematic of the setup used to ob-

Mo K± line)

serve Compton scattering.

½½

The early experiments

case when the scattering angle θ is 135—¦ . The wavelength of the scattered radiation

is measured by means of a Bragg crystal spectrometer using the relation 2d sin φ =

m», where φ is the Bragg scattering angle, d is the lattice spacing of the crystal,

and m is an integer corresponding to the di¬raction order (Tipler, 1978, Chap. 3“

6). Compton™s experiment was arranged so that m = 1. The Bragg spectrometer

which Compton constructed for his experiment consisted of a tiltable calcite crystal

(oriented at a Bragg angle φ) placed inside a lead box, which was used as a shield

against unwanted background X-rays. The detector, also placed inside this box, was

an ionization chamber placed behind a series of collimating slits to de¬ne the angles

θ and φ.

A simple classical model of the experiment consists of an electromagnetic ¬eld of

frequency ω falling on an atomic electron. According to classical theory, the incident

¬eld will cause the electron to oscillate with frequency ω, and this will in turn generate

radiation at the same frequency. This process is called Thompson scattering (Jack-

son, 1999, Sec. 14.8). In reality the incident radiation is not perfectly monochromatic,

but the spectrum does have a single well-de¬ned peak. The classical prediction is that

the spectrum of the scattered radiation should also have a single peak at the same

frequency.

The experimental results”shown in Fig. 1.4 for the scattering angles of θ = 45—¦ ,

90—¦ , and 135—¦ ”do exhibit a peak at the incident wavelength, but at each scattering

(a) (c)

Molybdenum Scattered at

K± line 90o

primary

(d)

(b)

135o

Scattered by

graphite at

45o

6o 30' 7o 7o 30' 6o 30' 7o 7o 30'

Angle from calcite Angle from calcite

Fig. 1.4 Data from the Compton scattering experiment sketched in Fig. 1.3. A calcite crystal

was used as the dispersive element in the Bragg spectrometer. (Adapted from Compton

(1923).)

½¾ The quantum nature of light

angle there is an additional peak at longer wavelengths which cannot be explained by

the classical theory.

1.1.4 Bothe™s coincidence-counting experiment

During the early development of the quantum theory, Bohr, Kramers, and Slater raised

the possibility that energy and momentum are not conserved in each elementary quan-

tum event”such as Compton scattering”but only on the average over many such

events (Bohr et al., 1924). However, by introducing the extremely important method

of coincidence detection”in this case of the scattered X-ray photon and of the

recoiling electron in each scattering event”Bothe performed a decisive experiment

showing that the Bohr“Kramers“Slater hypothesis is incorrect in the case of Compton

scattering; in fact, energy and momentum are both conserved in every single quantum

event (Bothe, 1926). In the experiment sketched in Fig. 1.5, X-rays are Compton-

scattered from a thin, metallic foil, and registered in the upper Geiger counter. The

thin foil allows the recoiling electron to escape, so that it registers in the lower Geiger

counter.

When viewed in the wave picture, the scattered X-rays are emitted in a spherically

expanding wavefront, but a single detection at the upper Geiger counter registers the

absorption of the full energy ω of the X-ray photon, and the displacement vector

linking the scattering point to the Geiger counter de¬nes a unique direction for the

momentum k of the scattered X-ray. This is an example of the famous collapse of

the wave packet.

When viewed in the particle picture, both the photon and the electron are treated

like colliding billiard balls, and the principles of the conservation of energy and mo-

mentum ¬x the momentum p of the recoiling electron. The detection of the scattered

X-ray is therefore always accompanied by the detection of the recoiling electron at the

lower Geiger counter, provided that the second counter is carefully aligned along the

uniquely de¬ned direction of the electron momentum p. Coincidence detection became

possible with the advent, in the 1920s, of fast electronics using vacuum tubes (triodes),

which open a narrow time window de¬ning the approximately simultaneous detection

of a pair of pulses from the upper and lower Geiger counters.

Later we will see the central importance in quantum theory of the concept of an

entangled state, for example, a superposition of products of the plane-wave states

of two free particles. In the case of Compton scattering, the scattered X-ray pho-

ton and the recoiling electron are produced in just such a state. The entanglement

Low-pressure box

Source of Geiger

X-rays counter

(Mo K± line) hν'

hν Foil

S

Fig. 1.5 Schematic of Bothe™s coincidence de-

e

tection of a Compton-scattered X-ray from a

Geiger

thin, metallic foil, and of the recoil electron

counter

from the same scattering event.

½¿

Photons

between the electron and the photon produced by their interaction enforces a tight

correlation”determined by conservation of energy and momentum”upon detection of

each quantum scattering event. It was just such correlations which were ¬rst observed

in the coincidence-counting experiment of Bothe.

1.2 Photons

In one of his three celebrated 1905 papers Einstein (1987a) proposed a new model of

light which explains all of the experimental results discussed in the previous sections.

In this model, light of frequency ω is supposed to consist of a gas of discrete photons

with energy = ω. In common with material particles, photons carry momentum as

well as energy. In the ¬rst paper on relativity, Einstein had already pointed out that

the relativistic transformation laws governing energy and momentum are identical to

those governing the frequency and wavevector of a plane wave (Jackson, 1999, Sec.

11.3D). In other words, the four-component vector (ω, ck) transforms in the same way

as (E, cp) for a material particle. Thus the assumption that the energy of a light

quantum is ω implies that its momentum must be k, where |k| = (ω/c) = (2π/»).

The connection to classical electromagnetic theory is provided by the assumption that

the number density of photons is proportional to the intensity of the light.

This is a far reaching extension of Planck™s idea that energy could only be trans-

ferred between radiation and matter in units of ω. The new proposal ascribes the

quantization entirely to the electromagnetic ¬eld itself, rather than to the mechanism

of energy exchange between light and matter. It is useful to arrange the results of

the model into two groups. The ¬rst group includes the kinematical features of the

model, i.e. those that depend only on the conservation laws for energy and momentum

and other symmetry properties. The second group comprises the dynamical features,

i.e. those that involve explicit assumptions about the fundamental interactions. In

the ¬nal section we will show that even this simple model has interesting practical

applications.

1.2.1 Kinematics

A The photoelectric e¬ect

The ¬rst success of the photon model was its explanation of the puzzling features

of the photoelectric e¬ect. Since absorption of light occurs by transferring discrete

bundles of energy of just the right size, there is no time delay before emission of the

¬rst electron. Absorption of a single photon transfers its entire energy ω to the bound

electron, thereby ejecting it from the metal with energy e given by eqn (1.6), which

now represents the overall conservation of energy. The energy of the ejected electron

therefore depends on the frequency rather than the intensity of the light. Since each

photoelectron emission event is caused by the absorption of a single photon, the number

of electrons emitted per unit time is proportional to the ¬‚ux of photons and thereby

to the intensity of light. The photoelectric equation implied by the photon model is

kinematical in nature, since it only depends on conservation of energy and does not

assume any model for the dynamical interaction between photons and the electrons in

the metal.

½ The quantum nature of light

B Compton scattering

The existence of the second peak in Compton scattering is also predicted by a kine-

matical argument based on conservation of momentum and energy. Consider an X-ray

photon scattering from a weakly bound electron. In this case it is su¬cient to consider

a free electron at rest and impose conservation of energy and momentum to determine

the possible ¬nal states as shown in Fig. 1.6.

For energetic X-rays the electron may recoil at velocities comparable to the velocity

of light, so it is necessary to use relativistic kinematics for this calculation (Jackson,

1999, Sec. 11.5). The relativistic conservation laws for energy and momentum are

mc2 + ω = E + ω , k = k + p, (1.7)

where p and E = m2 c4 + c2 p2 are respectively the ¬nal electron momentum and

energy, |k| = ω/c, and |k | = ω /c. Since the recoil kinetic energy of the scattered

electron (K = E ’ mc2 ) is positive, eqn (1.7) already explains why the scattered

quantum must have a lower frequency (longer wavelength) than the incident quantum.

Combining the two conservation laws yields the Compton shift

∆» ≡ » ’ » = »C (1 ’ cos θ) , (1.8)

in wavelength as a function of the scattering angle θ (the angle between k and k ),

where the electron Compton wavelength is

h

»C = = 0.0048 nm . (1.9)

mc

This simple argument agrees quite accurately with the data in Fig. 1.4, and with

other experiments using a variety of incident wavelengths. The fractional wavelength

shift for Compton scattering is bounded by ∆»/» < 2»C /». This shows that ∆»/» is

negligible for optical wavelengths, » ∼ 103 nm; which explains why X-rays were needed

to observe the Compton shift.

E, p ω', k'

E = mc , p = 0

2

ω, k

Fig. 1.6 Scattering of an incident X-ray quan-

tum from an electron at rest.

½

Photons

The argument leading to eqn (1.8) seems to prove too much, since it leaves no

room for the peak at the incident wavelength, which is also evident in the data. This

is a consequence of the assumption that the electron is weakly bound. In carrying out

the same kinematic analysis for a strongly bound electron, the electron mass m in eqn

(1.9) must be replaced by the mass M of the atom. Since M m, the resulting shift

is negligible even at X-ray wavelengths, and the peak at the incident wavelength is

recovered.

1.2.2 Dynamics

A Emission and absorption of light

The dynamical features of the photon model were added later, in conjunction with the

Bohr model of the atom (Einstein, 1987b, 1987c). The level structure of a real atom

is quite complicated, but for a ¬xed frequency of light only the two levels involved

in a quantum jump describing emission or absorption of light at that frequency are

relevant. This allows us to replace real atoms by idealized two-level atoms which have

a lower state with energy 1 , and a single upper (excited) state with energy 2 . The

combination of conservation of energy with the photoelectric e¬ect makes it reasonable

(following Bohr) to assume that the atoms can absorb and emit radiation of frequency

ω = ( 2 ’ 1 ) / . In this spirit, Einstein assumed the existence of three dynamical

processes, absorption, spontaneous emission, and stimulated emission. The simplest

cases of absorption and emission of a single photon are shown in Fig. 1.7.

Einstein originally introduced the notion of spontaneous emission by analogy with

radioactive decay, but the existence of spontaneous emission is implied by the princi-

ple of time-reversal invariance: i.e. the time-reversed ¬nal state evolves into the time-

reversed initial state. We will encounter this principle later on in connection with

Maxwell™s equations and quantum theory. In fact, time-reversal invariance holds for

all microscopic physical phenomena, with the exception of the weak interactions. These

photon atom atom

BEFORE AFTER

(a) Absorption of a single photon

photon

atom atom

AFTER

BEFORE

(b) Spontaneous emission

Fig. 1.7 (a) An atom in the ground state jumps to the excited state after absorbing a single

photon. (b) An atom in the excited state jumps to the ground state and emits a single photon.

½ The quantum nature of light

very small e¬ects will be ignored for the purposes of this book. For the present, we

will simply illustrate the idea of time reversal by considering the motion of classi-

cal particles (such as perfectly elastic billiard balls). Since Newton™s equations are

second order in time, the evolution of the mechanical system is determined by the

initial positions and velocities of the particles, (r (0) , v (0)). Suppose that at time

t = „ , each velocity is somehow reversed3 while the positions are unchanged so that

(r („ ) , v („ )) ’ (r („ ) , ’v („ )). More details on this operation”which is called time

reversal”are found in Appendix B.3.3. With this new initial state, the particles will

exactly reverse their motions during the interval („, 2„ ) to arrive at (r (2„ ) , v (2„ )) =

(r (0) , ’v (0)), which is the time-reversed form of the initial state. A mathematical

proof of this statement, which also depends on the fact that the Newtonian equations

are second order in time, can be found in standard texts; see, for example, Bransden

and Joachain (1989, Sec. 5.9).

In the photon model, the reversal of velocities is replaced by the reversal of the

propagation directions of the photons. With this in mind, it is clear that Fig. 1.7(b) is

the time-reversed form of Fig. 1.7(a). Absorption of light is a well understood process

in classical electromagnetic theory, and in principle the intensity of the ¬eld can be

made arbitrarily small. This is not the case in Einstein™s model, since the discreteness

of photons means that the weakest nonzero ¬eld is one describing exactly one photon,

as in Fig. 1.7(a). If we extrapolate the classical result to the absorption of a single

incident photon, then time-reversal invariance requires the existence of the process of

spontaneous emission, pictured in Fig. 1.7(b).

This argument can also be applied to the situation illustrated in Fig. 1.8, in which

many photons in the same mode are incident on an atom in the ground state. The

absorption event shown in Fig. 1.8(a) is evidently the time-reversed version of the

process shown in Fig. 1.8(b). Consequently, the principle of time-reversal invariance

implies the necessity of the second process, which is called stimulated emission.

Since the N photons in Fig. 1.8(a) are all in the same mode, this argument also shows

that the stimulated photon must be emitted into the same mode as the N ’ 1 incident

photons. Thus the stimulated photon must have the same wavevector k, frequency ω,

and polarization s as the incident photons. The identical values of these parameters”

which completely specify the state of the photon”for the stimulated and stimulating

photons implies a perfect ampli¬cation of the incident light beam by the process of

stimulated emission (ignoring, for the moment, the process of spontaneous emission).

This is the microscopic origin of the nearly perfect directionality, monochromaticity,

and polarization of a laser beam.

B The Planck distribution

We now consider the rates of these processes. Absorption and stimulated emission

both vanish in the absence of atoms and of light, so for low densities of atoms and

low intensities of radiation it is natural to assume that the absorption rate W1’2 from

the lower level 1 to the upper level 2, and the stimulated emission rate W2’1 ”from

the upper level 2 to the lower level 1”are both jointly proportional to the density of

3 This is hard to do in reality, but easy to simulate. A movie of the particle motions in the interval

(0, „ ) will display the time-reversed behavior in the interval („, 2„ ) when run backwards.

½

Photons

photon

photons atom

atom

BEFORE AFTER

(a) Absorption from a multi-photon state

photon

atom photons atom

BEFORE AFTER

(b) Stimulated emission

Fig. 1.8 (a) An atom in the ground state jumps to the excited state after absorbing one

of the N incident photons. (b) An atom in the excited state illuminated by N ’ 1 incident

photons jumps to the ground state and leaves N photons in the ¬nal state.

atoms and the intensity of the light. We further assume that the two-level atoms are

placed inside a cavity at temperature T , so that the light intensity is proportional to

the spectral function ρ (ω, T ). Therefore we expect that

W1’2 = B1’2 N1 ρ (ω, T ) , (1.10)

W2’1 = B2’1 N2 ρ (ω, T ) , (1.11)

where N1 and N2 are respectively the number of atoms in the lower level 1 and the

upper level 2. The rate S2’1 of spontaneous emission can only depend on N2 :

S2’1 = A2’1 N2 , (1.12)

since spontaneous emission occurs in the absence of any incident photons. The phe-

nomenological Einstein A and B coe¬cients, A2’1 , B2’1 , and B1’2 , are assumed to

be properties of the individual atoms which are independent of N1 , N2 , and ρ (ω, T ).

By studying the situation in which the atoms and the radiation ¬eld are in thermal

equilibrium, it is possible to derive other useful relations between the rate coe¬cients,

and thus to determine the form of ρ (ω, T ). The total rate T2’1 for transitions from

the upper state to the lower state is the sum of the spontaneous and stimulated rates,

T2’1 = A2’1 N2 + B2’1 N2 ρ (ω, T ) , (1.13)

and the condition for steady state”which includes thermal equilibrium as an impor-

tant special case”is T2’1 = W1’2 , so that

[A2’1 + B2’1 ρ (ω, T )] N2 = B1’2 ρ (ω, T ) N1 . (1.14)

Since the atoms and the radiation ¬eld are both in thermal equilibrium with the walls

of the cavity at temperature T , the atomic populations satisfy Boltzmann™s principle,

½ The quantum nature of light

e’β

N1 1

= eβ ω

= ’β , (1.15)

N2 e 2

where β = 1/kB T . Using this relation in eqn (1.14) leads to

A2’1

ρ (ω, T ) = . (1.16)

B1’2 exp (β ω) ’ B2’1

This solution has very striking consequences. In the limit of in¬nite temperature

(β ’ 0), the spectral function approaches a constant value:

A2’1

ρ (ω, T ) ’ . (1.17)

B1’2 ’ B2’1

On the other hand, it seems natural to expect that the energy density in any ¬nite

frequency interval should increase without bound in the limit of high temperatures.

The only way to avoid this contradiction is to impose

B1’2 = B2’1 = B , (1.18)

i.e. the rate of stimulated emission must exactly equal the rate of absorption for a

physically acceptable spectral function. This is an example of the principle of detailed

balance (Chandler, 1987, Sec. 8.3), which also follows from time-reversal symmetry.

Substituting eqn (1.18) into eqn (1.16) yields the new form

A 1

ρ (ω, T ) = , (1.19)

B exp (β ω) ’ 1

where we have further simpli¬ed the notation by setting A2’1 = A. In the low

temperature”or high energy”limit, ω kB T (β ω 1), the energy density is

A

ρ (ω, T ) = exp (’β ω) . (1.20)

B

This is Wien™s law, and it indeed agrees with experiment in the high energy limit.

By contrast, in the low energy limit, ω kB T ”i.e. the photon energy is small

compared to the average thermal energy”the classical Rayleigh“Jeans law is known

to be correct. This allows us to determine the ratio A/B by comparing eqn (1.19) to

eqn (1.4), with the result

ω3

A

= . (1.21)

π 2 c3

B

Thus the standard form for the Planck distribution,

ω3 1

ρ (ω, T ) = , (1.22)

exp (β ω) ’ 1

π 2 c3

is completely ¬xed by applying the powerful principles of thermodynamics to two-level

atoms in thermal equilibrium with the radiation ¬eld inside a cavity.

½

Photons

Einstein™s argument for the A and B coe¬cients correctly correlates an impressive

range of experimental results. On the other hand, it does not provide an explanation

for the quantum jumps involved in spontaneous emission, stimulated emission, and

absorption, nor does it give any way to relate the A and B coe¬cients to the micro-

scopic properties of atoms. These features will be explained in the full quantum theory

of light which is presented in the following chapters.

1.2.3 Applications

In addition to providing a framework for understanding the experiments discussed in

Section 1.1, the photon model can also be used for more practical applications. For

example, let us model an absorbing medium as a slab of thickness ∆z and area S

containing N = n∆zS two-level atoms, where n is the density of atoms. The energy

density of light in the frequency interval (ω, ω + ∆ω) at the entrance face is u (ω, z) =

ρ (z, ω) ∆ω, where ρ (z, ω) is the spectral function of the incident light. The incident

¬‚ux is then cu (ω, z), so energy enters and leaves the slab at the rates cu (ω, z) S and

cu (ω, z + ∆z) S, respectively, as pictured in Fig. 1.9.

By energy conservation, the di¬erence between these rates is the rate at which

energy is absorbed in the slab. In order to calculate this correctly, we must provide

a slightly more detailed model of the absorption process. So far, we have used an

all-or-nothing picture in which absorption occurs at the sharply de¬ned frequency

( 2 ’ 1 ) / . In reality, the atoms respond in a continuous way to light at frequency ω.

This is described by a line shape function L (ω), where L (ω) ∆ω is the fraction of

atoms for which ( 2 ’ 1 ) / lies in the interval (ω, ω + ∆ω). In succeeding chapters we

will encounter many mechanisms that contribute to the line shape, but in the spirit

of the photon model we simply assume that L (ω) is positive and normalized by

∞

dωL (ω) = 1 . (1.23)

0

We ¬rst consider the case that all of the atoms are in the ground state, then eqn (1.10)

yields

[cu (z + ∆z) ’ cu (z)] S = ’ ( ω) (Bρ (z, ω)) (L (ω) ∆ωn∆zS) . (1.24)

In the limit ∆z ’ 0 this becomes a di¬erential equation:

du (z, ω)

= ’ ωnBL (ω) u (z, ω) ,

c (1.25)

dz

c uz + ∆z

c uz

Fig. 1.9 Light in the frequency interval

(ω, ω + ∆ω) falls on a slab of thickness

∆z and area S. The incident ¬‚ux is

cu (z, ω) = cρ (z, ω) ∆ω, where ρ (z, ω) is the

∆z spectral function.

¾¼ The quantum nature of light

with the solution

nL (ω) B ω

u (z, ω) = u (0, ω) e’±(ω)z , where ±(ω) = . (1.26)

c

This is Beer™s law of absorption, and ±(ω) is the absorption coe¬cient.

In the opposite situation that all atoms are in the upper state, stimulated emission

replaces absorption, and the same kind of calculation leads to

du (z, ω)

c = ωnBL (ω) u (z, ω) , (1.27)

dz

with the solution

nL (ω) B ω

u (z, ω) = u (0, ω) e± (ω)z , ± (ω) = . (1.28)

c

In this case we get negative absorption, that is, the ampli¬cation of light.

If both levels are nondegenerate, the general case is described by densities n1 and

n2 for atoms in the lower and upper states respectively, with n1 + n2 = n. In the

previous results this means replacing n by n1 in the ¬rst case and n by n2 in the

second. In this situation,

(n2 ’ n1 ) L (ω) B ω

du (z, ω)

= g(ω)u (z, ω) , where g(ω) = . (1.29)

dz c

For thermal equilibrium n1 > n2 , so we get an absorbing medium, but with a popu-

lation inversion, n2 > n1 , we ¬nd instead a gain medium with gain g(ω) > 0. This

is the principle behind the laser (Schawlow and Townes, 1958).

1.3 Are photons necessary?

Now that we have established that the photon model is su¬cient for the interpretation

of the experiments described in Section 1.1, we ask if it is necessary. We investigate

this question by attempting to describe each of the principal experiments using a

semiclassical model.

1.3.1 The Planck distribution

This seems to be the simplest of the experiments under consideration, but ¬nding a

semiclassical explanation turns out to involve some subtle issues. Suppose we make

the following assumptions.

(a) The electromagnetic ¬eld is described by the classical form of Maxwell™s equations.

(b) The electromagnetic ¬eld is an independent physical system subject to the stan-

dard laws of statistical mechanics.

With both assumptions in force the equipartition argument in Section 1.1.1 inevitably

leads to the Rayleigh“Jeans distribution and the ultraviolet catastrophe. This is phys-

ically unacceptable, so at least one of the assumptions (a) or (b) must be abandoned.

At this point, Planck chose the rather risky alternative of abandoning (b), and Einstein

took the even more radical step of abandoning (a).

¾½

Are photons necessary?

Our task is to ¬nd some way of retaining (a) while replacing Planck™s ad hoc

procedure by an argument based on a quantum mechanical description of the atoms

in the cavity wall. There does not seem to be a completely satisfactory way to do this,

so a rough plausibility argument will have to su¬ce. We begin by observing that the

derivation of the Planck distribution in Section 1.2.2-B does not explicitly involve the

assumption that light is composed of discrete quanta. This suggests that we ¬rst seek

a semiclassical origin for the A and B coe¬cients, and then simply repeat the same

argument.

The Einstein coe¬cients B1’2 (for absorption) and B2’1 (for stimulated emission)

can both be evaluated by applying ¬rst-order, time-dependent perturbation theory”

which is reviewed in Section 4.8.2”to the coupling between the atom and the classical

electromagnetic ¬eld. In both processes the electron remains bound in the atom, which

is small compared to typical optical wavelengths. Thus the interaction of the atom

with the classical ¬eld can be treated in the dipole approximation, and the interaction

Hamiltonian is

Hint = ’d · E , (1.30)

where d is the electric dipole operator, and the ¬eld is evaluated at the center of mass

of the atom. Applying the Fermi-golden-rule result (4.113) to the absorption process

leads to

2

π |d12 |

B1’2 = , (1.31)

2

30

where d12 is the matrix element of the dipole operator. A similar calculation for stimu-

lated emission yields the same value for B2’1 , so the equality of the two B coe¬cients

is independently veri¬ed.

The strictly semiclassical theory used above does not explain spontaneous emis-

sion; instead, it predicts A = 0. The reason is that the interaction Hamiltonian (1.30)

vanishes in the absence of an external ¬eld. If no external ¬eld is present, an atom in

any stationary state”including all excited states”will stay there permanently. On the

other hand, spontaneous emission is not explained in Einstein™s photon model either;

it is built in by assumption at the beginning. Since the present competition is with the

photon model, we are at liberty to augment the strict semiclassical theory by simply

assuming the existence of spontaneous emission. With this assumption in force, Ein-

stein™s rate arguments (eqns (1.10)“(1.21)) can be used to derive the ratio A/B. Note

that these equations refer to transition rates within the two-level atom; they do not

require the concept of the photon. Combining this with the independently calculated

value of B1’2 given in eqn (1.31) yields the correct value for the A coe¬cient. This

line of argument is frequently used to derive the A coe¬cient without bringing in the

full blown quantum theory of light (Loudon, 2000, Sec. 1.5).

The extra assumptions required to carry out this semiclassical derivation of the

Planck spectrum may make it appear almost as ad hoc as Planck™s argument, but it

does show that the photon model is not strictly necessary for this purpose.

1.3.2 The photoelectric e¬ect

By contrast to the derivation of the Planck spectrum, Einstein™s explanation of the

photoelectric e¬ect depends in a very direct way on the photon concept. In this case,

¾¾ The quantum nature of light

however, the alternative description using the semiclassical theory turns out to be much

more straightforward. For this calculation, the electrons in the metal are described by

quantum mechanics, and the light is described as an external classical ¬eld. The total

electron Hamiltonian is therefore H = H0 + Hint , where H0 is the Hamiltonian for an

electron in the absence of any external electromagnetic ¬eld and Hint is the interaction

term. For a single electron in a weak external ¬eld, the standard quantum mechanical

result”reviewed in Appendix C.6”is

e

Hint = ’ A (r, t) · p , (1.32)

m

where r and p are respectively the quantum operators for the position and momentum.

In the usual position-space representation the action of the operators is rψ (r) =

rψ (r) and pψ (r) = ’i ∇ψ (r). The c-number function A (r, t) is the classical vector

potential”which can be chosen to satisfy the radiation-gauge condition ∇ · A = 0”

and it determines the radiation ¬eld by

‚A

E=’ , B = ∇—A. (1.33)

‚t

For a monochromatic ¬eld with frequency ω, the vector potential is

1

A (r, t) = E0 e exp (ik · r ’ ωt) + CC , (1.34)

ω

where e is the unit polarization vector, E0 is the electric ¬eld amplitude, |k| = ω/c,

and e · k = 0. Another application of Fermi™s golden rule (4.113) yields the rate

2π

| f |Hint | i |2 δ ( ’ ’ ω)

Wf i = (1.35)

f i

for the transition from the initial bound energy level i into a free level f . This

1/ω. For optical ¬elds ω ∼ 1015 s’1 , so

result is valid for observation times t

eqn (1.35) predicts the emission of electrons with no appreciable delay. Furthermore,

the delta function guarantees that the energy of the ejected electron satis¬es the

photoelectric equation. Finally the matrix element f |Hint | i is proportional to E0 , so

the rate of electron emission is proportional to the ¬eld intensity. Therefore, this simple

semiclassical theory explains all of the puzzling aspects of the photoelectric e¬ect,

without ever introducing the concept of the photon. This point is already implicit in

the very early papers of Wentzel (1926) and Beck (1927), and it has also been noted

in much more recent work (Mandel et al., 1964; Lamb and Scully, 1969). The energy

conserving delta function in eqn (1.35) reproduces the kinematical relation (1.6), but

it only appears at the end of a detailed dynamical calculation.

Most techniques for detecting photons employ the photoelectric e¬ect, so an expla-

nation of the photoelectric e¬ect that does not require the existence of photons is a bit

upsetting. Furthermore, the response of other kinds of detectors (such as photographic

emulsions, solid-state photomultipliers, etc.) is ultimately also based on the photoelec-

tric e¬ect. Therefore, they can also be entirely described by the semiclassical theory.

This raises serious questions about the interpretation of some experiments claiming to

¾¿

Are photons necessary?

demonstrate the existence of photons. An early example is a repetition of Young™s two

slit experiment (Taylor, 1909), which used light of such low intensity that the average

energy present in the apparatus at any given time was at most ω. The result was a

slow accumulation of spots on a photographic plate. After a su¬ciently long exposure

time, the spots displayed the expected two slit interference pattern. This was taken as

evidence for the existence of photons, and apparently was the basis for Dirac™s (1958)

assertion that each photon interferes only with itself. This interpretation clearly de-

pends on the assumption that each individual spot on the plate represents absorption

of a single photon. The semiclassical explanation of the photoelectric e¬ect shows that

the results could equally well be interpreted as the interference of classical electromag-

netic waves from the two slits, combined with the semiclassical quantum theory for

excitation of electrons in the photographic plate. In this view, there is no necessity for

the concept of the photon, and thus for the quantization of the electromagnetic ¬eld.

1.3.3 Compton scattering

The kinematical explanation for the Compton shift given in Section 1.1.3 is often

o¬ered as conclusive evidence for the existence of photons, but the very ¬rst derivation

(Klein and Nishina, 1929) of the celebrated Klein“Nishina formula (Bjorken and Drell,

1964, Sec. 7.7) for the di¬erential cross-section of Compton scattering was carried

out in a slightly extended form of the semiclassical approximation. The analysis is

more complicated than the semiclassical treatment of the photoelectric e¬ect for two

reasons. The ¬rst is that the electron motion may become relativistic, so that the

nonrelativistic Schr¨dinger equation must be replaced by the relativistic Dirac equation

o

(Bjorken and Drell, 1964, Chap. 1). The second complication is that the radiation

emitted by the excited electron cannot be ignored, since observing this radiation is the

point of the experiment. Thus Compton scattering is a two step process in which the

electron is ¬rst excited by the incident radiation, and the resulting current subsequently

generates the scattered radiation. In the original paper of Klein and Nishina, the

Dirac equation for an electron exposed to an incident plane wave is solved by using

¬rst-order time-dependent perturbation theory. The expectation value of the current-

density operator in the perturbed state is then used as the source term in the classical

Maxwell equations. The radiation ¬eld generated in this way automatically satis¬es

the kinematical relations (1.7), so it again yields the Compton shift given in eqn (1.8).

Furthermore, the Compton cross-section calculated by using the semiclassical Klein“

Nishina model precisely agrees with the result obtained in quantum electrodynamics,

in which the electromagnetic ¬eld is treated by quantum theory. Once again we see that

Einstein™s quantum model provides a beautifully simple explanation of the kinematical

aspects of the experiment, but that the more complicated semiclassical treatment

achieves the same end, while also providing a correct dynamical calculation of the cross-

section. There is again no necessity to introduce the concept of the photon anywhere

in this calculation.

1.3.4 Conclusions

The experiments discussed in Section 1.1 are usually presented as evidence for the

existence of photons. The reasoning behind this claim is that classical physics is in-

¾ The quantum nature of light

consistent with the experimental results, while Einstein™s photon model describes all

the experimental results in a very simple way. What we have just seen, however, is

that an augmented version of semiclassical electrodynamics can explain the same set

of experiments without recourse to the idea of photons. Where, then, is the empirical

evidence for the existence of photons? In the next section we will describe experiments

that bear on this question.

1.4 Indivisibility of photons

The semiclassical explanations of the experimental results in Section 1.1 imply that

these experiments are not sensitive to the indivisibility of photons. Classical electro-

magnetic theory describes light in terms of electric and magnetic ¬elds with contin-

uously variable ¬eld amplitudes, but the photon model of light asserts that electro-

magnetic energy is concentrated into discrete quanta which cannot be further subdi-

vided. In particular, a classical electromagnetic wave must be continuously divisible

at a beam splitter, whereas an indivisible photon must be either entirely transmitted,

or entirely re¬‚ected, as a whole unit. The continuous division of the classical waves

and the discontinuous re¬‚ection-or-transmission choice of the photon are mutually ex-

clusive; therefore, the quantum and classical theories of light give entirely di¬erent

predictions for experiments involving individual quanta of light incident on a beam

splitter. The indivisibility of the photon is a postulate of Einstein™s original model,

and it is a consequence of the fully developed quantum theory of the electromagnetic

¬eld. Since even the most sophisticated versions of the semiclassical theory describe

light in terms of continuously variable classical ¬elds, the decisive experiments must

depend on the indivisibility of individual photons.

Two important advances in this direction were made by Clauser in the context of

a discussion of the experimental limits of validity of semiclassical theories, in particu-

lar the neoclassical theory of Jaynes (Crisp and Jaynes, 1969). For this purpose, the

two-level atom used in previous discussions is inadequate; we now need atoms with at

least three active levels. The ¬rst advance was Clauser™s reanalysis (Clauser, 1972) of

the data from an experiment by Kocher and Commins (1967), which used a three-level

cascade emission in a calcium atom, as shown in Fig. 1.10. A beam of calcium atoms

is crossed by a light beam which excites the atoms to the highest energy level. This

5

Dν

2

Fig. 1.10 The lowest three energy levels of

Dν

the calcium atom allow the cascade of two suc-

cessive transitions, in which two photons hν1

and hν2 are emitted in rapid succession. The

5

intermediate level has a short lifetime of 4.7 ns.

¾

Indivisibility of photons

excitation is followed by a rapid cascade decay, with the correlated emission of two

photons. The ¬rst (hν1 ) is emitted in a transition from the highest energy level to the

short-lived intermediate level, and the second (hν2 ) is emitted in a transition from the

intermediate level to the ground level. These two photons, which are emitted almost

back-to-back with respect to each other, are then detected using fast coincidence elec-

tronics. In this way, a beam of calcium atoms provides a source of strongly correlated

photon pairs.

The light emitted in each transition is randomly polarized”i.e. all polarizations

are detected with equal probability”but the experiment shows that the probabilities

of observing given polarizations at the two detectors are correlated. The correlation

coe¬cient obtained from a semiclassical calculation has a lower bound which is violated

by the experimental data, while the correlation predicted by the quantum theory of

radiation agrees with the data. The second advance was an experiment performed by

Clauser himself (Clauser, 1974), in which the two bursts of light from a three-level

cascade emission in the mercury atom are each passed through beam splitters to four

photodetectors. The object in this case is to observe the coincidence rate between

various pairs of detectors, in other words, the rates at which a pair of detectors both

¬re during the same small time interval. The semiclassical rates are again inconsistent

with experiment, whereas the quantum theory prediction agrees with the data. The

¬rst experiment provides convincing evidence which supports the quantum theory and

rejects the semiclassical theory, but the role of the indivisibility of photons is not easily

seen. The second experiment does depend directly on this property, but the analysis

is rather involved. We therefore refer the reader to the original papers for descriptions

of this seminal work, and brie¬‚y describe instead a third experiment that yields the

clearest and most direct evidence for the indivisibility of single photons, and thus for

the existence of individual quanta of the electromagnetic ¬eld.

The experiment in question”which we will call the photon-indivisibility experi-

ment”was performed by Grangier et al. (1986). The experimental arrangement (shown

in Fig. 1.11) employs a three-level cascade (see Fig. 1.10) in a calcium atom located at

S. Two successive, correlated bursts of light”centered at frequencies ν1 and ν2 ”are

emitted in opposite directions from the source. At this point in the argument, we leave

open the possibility that the light is described by classical electromagnetic waves as

opposed to photons, and assume that detection events are perfectly describable by the

semiclassical theory of the photoelectric e¬ect.

The atoms, which are delivered by an atomic beam, are excited to the highest

energy level shortly before reaching the source region S. The photomultiplier PMgate

is equipped with a ¬lter that screens out radiation at the frequency ν2 of the second

transition, while passing radiation at ν1 , the frequency of the ¬rst transition. The out-

put from PMgate , which monitors bursts of radiation at frequency ν1 , is registered by

the counter Ngate , and is also used to activate (trigger) a device called a gate gener-

ator which produces a standardized, rectangularly-shaped gate pulse for a speci¬ed

time interval, Tgate = w, called the gate width. The outputs of the photomultipliers

PMre¬‚ and PMtrans , which monitor bursts of radiation at frequency ν2 , are registered

by the gated counters Nre¬‚ and Ntrans only during the time interval speci¬ed by the

gate width w.

¾ The quantum nature of light

N1 Nrefl