one hundred years of light quanta (nobel lecture)

Nobel Lecture: One hundred years of light quanta*

Roy J. Glauber

Harvard University, Cambridge, Massachusetts 02138, USA

�Published 17 November 2006�

DOI: 10.1103/RevModPhys.78.1267

It can’t have escaped you, after so many recent re-minders, that this year marks the one hundredth birth-day of the light quantum. I thought I would tell you thismorning a few things about its century long biography.Of course we have had light quanta on Earth for eons, infact ever since the good Lord said “let there be quantumelectrodynamics”—which is a modern translation, ofcourse, from the biblical Aramaic. So in this talk I’ll tryto tell you what quantum optics is about, but there willhardly be enough time to tell you of the many new di-rections in which it has led us. Several of those are di-rections that we would scarcely have anticipated as all ofthis work started.

My own involvement in this subject began somewherearound the middle of the last century, but I would like todescribe some of the background of the scene I enteredat that point as a student. Let’s begin, for a moment,even before the quantum theory was set in motion byPlanck. It is important to recall some of the remarkablethings that were found in the 19th century, thanks prin-cipally to the work of Thomas Young and AugustinFresnel. They established within the first 20 years of the19th century that light is a wave phenomenon, and thatthese waves, of whatever sort they might be, interpen-etrate one another like waves on the surface of a pond.The wave displacements, in other words, add up alge-braically. That’s called superposition, and it was foundthus that if you have two waves that remain lastingly instep with one another, they can add up constructively,and thereby reinforce one another in some places, orthey can even oscillate oppositely to one another, andthereby cancel one another out locally. That would bewhat we call destructive interference.

Interference phenomena were very well understoodby about 1820. On the other hand, it wasn’t at all under-stood what made up the underlying waves until the fun-damental laws of electricity and magnetism were gath-ered together and augmented in a remarkable way byJames Clerk Maxwell, who developed thereby the elec-trodynamics we know today. Maxwell’s theory showedthat light waves consist of oscillating electric and mag-netic fields. The theory has been so perfect in describingthe dynamics of electricity and magnetism over labora-tory scale distances, that it has remained precisely intact.

It has needed no fundamental additions in the yearssince the 1860s, apart from those concerning the quan-tum theory. It serves still, in fact, as the basis for thediscussion and analysis of virtually all the optical instru-mentation we have ever developed. That overwhelmingand continuing success may eventually have led to a cer-tain complacency. It seemed to imply that the field ofoptics, by the middle of the 20th century, scarcelyneeded to take any notice of the granular nature of light.Studying the behavior of light quanta was then left tothe atomic and elementary particle physicists—whoseinterests were largely directed toward other phenomena.

The story of the quantum theory, of course, really be-gins with Max Planck. Planck in 1900 was confrontedwith many measurements of the spectral distribution ofthe thermal radiation that is given off by a hot object. Itwas known that under ideally defined conditions, that is,for complete �or black� absorbers and correspondinglyperfect emitters this is a unique radiation distribution.The intensities of its color distribution, under such idealconditions, should depend only on temperature and notat all on the character of the materials that are doing theradiating. That defines the so-called blackbody distribu-tion. Planck, following others, tried finding a formulathat expresses the shape of that blackbody color spec-trum. Something of its shape was known at low frequen-cies, and there was a good guess present for its shape athigh frequencies.

The remarkable thing that Planck did first was simplyto devise an empirical formula that interpolates betweenthose two extremes. It was a relatively simple formulaand it involved one constant which he had to adjust inorder to fit the data at a single temperature. Then havingdone that, he found his formula worked at other tem-peratures. He presented the formula to the GermanPhysical Society on October 19, 1900 �Planck, 1900a�and it turned out to be successful in describing stillnewer data. Within a few weeks the formula seemed tobe established as a uniquely correct expression for thespectral distribution of thermal radiation.

The next question obviously was: Did this formulahave a logical derivation of any sort? There Planck, whowas a sophisticated theorist, ran into a bit of trouble.First of all he understood from his thermodynamic back-ground that he could base his discussion on nearly anymodel of matter, however oversimplified it might be, aslong as it absorbed and emitted light efficiently. So hebased his model on the mechanical system he under-stood best, a collection of one-dimensional harmonic os-

*The 2005 Nobel Prize for Physics was shared by Roy J.Glauber, John L. Hall, and Theodor W. Hänsch. This lecture isthe text of Dr. Glauber’s address on the occasion of the award.

REVIEWS OF MODERN PHYSICS, VOLUME 78, OCTOBER–DECEMBER 2006

0034-6861/2006/78�4�/1267�12� ©The Nobel Foundation, 20051267

http://dx.doi.org/10.1103/RevModPhys.78.1267

cillators, each of them oscillating rather like a weight atthe end of a spring. They had to be electrically charged.He knew from Maxwell exactly how these charged oscil-lators interact with the electromagnetic field. They bothradiate and absorb in a way he could calculate. So thenhe ought to be able to find the equilibrium betweenthese oscillators and the radiation field, which acted as akind of thermal reservoir—and which he never madeany claim to discuss in detail.

He found that he could not secure a derivation for hismagic formula for the radiation distribution unless hemade an assumption which, from a philosophical stand-point, he found all but unacceptable. The assumptionwas that the harmonic oscillators he was discussing hadto possess energies that were distributed, not as the con-tinuous variables one expected, but confined instead todiscrete and regularly spaced values. The oscillators offrequency � would have to be restricted to energy valuesthat were integer multiples, i.e., n-fold multiples �withn=0,1 ,2 ,3 , . . .� of something he called the quantum ofenergy, h�.

That number h was, in effect, the single number thathe had to introduce in order to fit his magic formula tothe observed data at a single temperature. So he wassaying, in effect, that these hypothetical harmonic oscil-lators representing a simplified image of matter couldhave only a sequence amounting to a “ladder” of energystates. That assumption permits us to see immediatelywhy the thermal radiation distribution must fall off rap-idly with rising frequency. The energy steps between theoscillator states grow larger, according to his assump-tion, as the frequency rises, but thermal excitation ener-gies, on the other hand, are quite restricted in magnitudeat any fixed temperature. High frequency oscillators atthermal equilibrium would never even reach the firststep of excitation. Hence there tends to be very littlehigh frequency radiation present at thermal equilibrium.Planck presented this revolutionary suggestion �Planck,1900b� to the Physical Society on December 14, 1900,although he could scarcely believe it himself.

The next great innovation came in 1905 from theyoung Albert Einstein, employed still at the Bern PatentOffice. Einstein first observed that Planck’s formula forthe entropy of the radiation distribution, when he exam-ined its high frequency contributions, looked like theentropy of a perfect gas of free particles of energy h�.That was a suggestion that light itself might be discretein nature, but hardly a conclusive one.

To reach a stronger conclusion he turned to an exami-nation of the photoelectric effect, which had first beenobserved in 1887 by Heinrich Hertz. Shining monochro-matic light on metal surfaces drives electrons out of themetals, but only if the frequency of the light exceeds acertain threshold value characteristic of each metal. Itwould have been most reasonable to expect that as oneshines more intense light on those metals the electronswould come out faster, that is, with higher velocities inresponse to the stronger oscillating electric fields—butthey don’t. They come out always with the same veloci-ties, provided that the incident light is of a frequency

higher than the threshold frequency. If it were belowthat frequency there would be no photoelectrons at all.

The only response that the metals make to increasingthe intensity of light lies in producing more photoelec-trons. Einstein had a naively simple explanation for that�Einstein, 1905�. The light itself, he assumed, consists oflocalized energy packets and each possesses one quan-tum of energy. When light strikes the metal, each packetis absorbed by a single electron. That electron then fliesoff with a unique energy, an energy which is just thepacket energy h� minus whatever energy the electronneeds to expend in order to escape the metal.

It took until about 1914–1916 to secure an adequateverification of Einstein’s law for the energies of the pho-toelectrons. When Millikan succeeded in doing that, itseemed clear that Einstein was right, and that light doesindeed consist of quantized energy packets. It was thusEinstein who fathered the light quantum, in one of theseveral seminal papers he wrote in the year 1905.

To follow the history a bit further, Einstein began torealize in 1909 that his energy packets would have amomentum which, according to Maxwell, should betheir energy divided by the velocity of light. These pre-sumably localized packets would have to be emitted insingle directions if they were to remain localized, or toconstitute “Nadelstrahlung” �needle radiation�, very dif-ferent in behavior from the broadly continuous angulardistribution of radiation that would spread from har-monic oscillators according to the Maxwell theory. Arandom distribution of these needle radiations wouldlook appropriately continuous, but what was disturbingabout that was the randomness with which these needleradiations would have to appear. That was evidently thefirst of the random variables in the quantum theory thatbegan disturbing Einstein and kept nettling him for therest of his life.

In 1916 Einstein found another and very much morecongenial way of deriving Planck’s distribution by dis-cussing the rate at which atoms radiate. Very little wasknown about atoms at that stage save that they must becapable of absorbing and giving off radiations. An atomlodged in a radiation field would surely have its constitu-ent charges shaken by the field oscillations, and thatshaking could lead either to the absorption of radiationor to the emission of still more radiation. Those were theprocesses of absorption or emission induced by the priorpresence of radiation. But Einstein found that thermalequilibrium between matter and radiation could only bereached if, in addition to these induced processes, thereexists also a spontaneous process, one in which an ex-cited atom emits radiation even in the absence of anyprior radiation field. It would be analogous to radioac-tive decays discovered by Rutherford. The rates atwhich these processes take place were governed by Ein-stein’s famous B and A coefficients, respectively. Theexistence of spontaneous radiation turned out to be animportant guide to the construction of quantum electro-dynamics.

Some doubts about the quantized nature of light in-evitably persisted, but many of them were dispelled by

1268 Roy J. Glauber: One hundred years of light quanta

Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006

Compton’s discovery in 1922 that x-ray quanta are scat-tered by electrons according to the same rules as governthe collisions of billiard balls. They both obey the con-servation rules for energy and momentum in much thesame way. It became clear that the particle picture oflight quanta, whatever were the dilemmas that accompa-nied it, was here to stay.

The next dramatic developments of the quantumtheory, of course, took place between the years 1924 and1926. They had the effect of ascribing to material par-ticles such as electrons much of the same wavelike be-havior as had long since been understood to characterizelight. In those developments de Broglie, Heisenberg,Schrödinger, and others accomplished literal miracles inexplaining the structure of atoms. But, however muchthis invention of modern quantum mechanics succeededin laying the groundwork for a more general theory ofthe structure of matter, it seemed at first to have littlenew bearing on the understanding of electromagneticphenomena. The spontaneous emission of light persistedas an outstanding puzzle.

Thus there remained a period of a couple of yearsmore in which we described radiation processes in termsthat have usually been called “semiclassical.” Now theterm “classical” is an interesting one—because, as youknow, every field of study has its classics. Many of theclassics that we are familiar with go back two or threethousand years in history. Some are less old, but allshare an antique, if not ancient, character. In physics weare a great deal more precise, as well as contemporary.Anything that we understood or could have understoodprior to the date of Planck’s paper, December 14, 1900,is to us “classical.” Those understandings are our clas-sics. It is the introduction of Planck’s constant thatmarks the transition from the classical era to our mod-ern one.

The true “semiclassical era,” on the other hand, lastedonly about two years. It ended formally with the discov-ery by Paul Dirac �1927, 1927� that one must treat thevacuum, that is to say, empty space, as a dynamical sys-tem. The energy distributed through space in an electro-magnetic field had been shown by Maxwell to be a qua-dratic expression in the electric and magnetic fieldstrengths. Those quadratic expressions are formallyidentical in their structure to the mathematical expres-sions for the energies of mechanical harmonic oscilla-tors. Dirac observed that even though there may notseem to be any organized fields present in the vacuumthose mathematically defined oscillators that describedthe field energy would make contributions that couldnot be overlooked. The quantum mechanical nature ofthe oscillators would add an important but hitherto ne-glected correction to the argument of Planck.

Planck had said the energies of harmonic oscillatorsare restricted to values n times the quantum energy h�and the fully developed quantum mechanics had shownin fact that those energies are not nh� but �n+ 1

2�h�. All

of the intervals between energy levels remained un-changed, but the quantum mechanical uncertainty prin-ciple required that additional 1

2h� to be present. We can

never have a harmonic oscillator completely empty ofenergy because that would require its position coordi-nate and its momentum simultaneously to have the pre-cise values zero.

So, according to Dirac, the electromagnetic field ismade up of field amplitudes that can oscillate harmoni-cally. But these amplitudes, because of the ever-presenthalf quantum of energy 1

2h�, can never be permanentlyat rest. They must always have their fundamental exci-tations, the so-called “zero-point fluctuations” going on.The vacuum then is an active dynamical system. It is notempty. It is forever buzzing with weak electromagneticfields. They are part of the ground state of emptiness.We can withdraw no energy at all from those fluctuatingelectromagnetic fields. We have to regard them nonethe-less as real and present even though we are denied anyway of perceiving them directly.

An immediate consequence of this picture was theunification of the notions of spontaneous and inducedemission. Spontaneous emission is emission induced bythose zero-point oscillations of the electromagnetic field.Furthermore it furnishes, in a sense, an indirect way ofperceiving the zero-point fluctuations by amplifyingthem. Quantum amplifiers tend to generate backgroundnoise that consists of radiation induced by those vacuumfluctuations.

It is worth pointing out a small shift in terminologythat took place in the late 1920s. Once material particleswere found to exhibit some of the wavelike behavior oflight quanta, it seemed appropriate to acknowledge thatthe light quanta themselves might be elementary par-ticles, and to call them “photons” as suggested by G. N.Lewis in 1926. They seemed every bit as discrete as ma-terial particles, even if their existence was more transi-tory, and they were at times freely created or annihi-lated.

The countless optical experiments that had been per-formed by the middle of the 20th century were in one oranother way based on detecting only the intensity oflight. It may even have seemed there wasn’t anythingelse worth measuring. Furthermore, those measure-ments were generally made with steady light beams tra-versing passive media. It proved quite easy therefore todescribe those measurements in simple and essentiallyclassical terms. A characteristic first mathematical stepwas to split the expression for the oscillating electricfield E into two complex conjugate terms:

E = E�+� + E�−�, �1�

E�−� = �E�+��*, �2�

with the understanding that E�+� contains only positivefrequency terms, i.e., those varying as e−i�t for all ��0,and E�−� contains only negative frequency terms ei�t.This is a separation familiar to electrical engineers andmotivated entirely by the mathematical convenience ofdealing with exponential functions. It has no physicalmotivation in the context of classical theory, since thetwo complex fields E�±� are physically equivalent. They

1269Roy J. Glauber: One hundred years of light quanta


furnish identical descriptions of classical theory.Each of the fields E�±��rt� depends in general on both

the space coordinate r and time t. The instantaneousfield intensity at r , t would then be

�E�+��r,t��2 = E�−��r,t�E�+��r,t� . �3�

In practice it was always an average intensity that wasmeasured, usually a time average.

The truly ingenious element of many optical experi-ments, going all the way back to Young’s double-pinholeexperiment, was the means their design afforded to su-perpose the fields arriving at different space-time pointsbefore the intensity observations were made. Thus inYoung’s experiment, shown in Fig. 1, light penetrating asingle pinhole in the first screen passes through two pin-holes in the second screen and then is detected as it fallson a third screen. The field E�+��rt� at any point on thelatter screen is the superposition of two waves radiatedfrom the two prior pinholes with a slight difference intheir arrival times at the third screen, due to the slightlydifferent distances they have to travel.

If we wanted to discuss the resulting light intensities indetail, we would find it most convenient to do that interms of a field correlation function which we shall de-fine as

G�1��r1t1,r2t2� = �E�−��r1t1�E�+��r2t2�� . �4�

This is a complex-valued function that depends, in gen-eral, on both space-time points r1t1 and r2t2. The angularbrackets �¯� indicate that an average value is somehowtaken, as we have noted. The average intensity of thefield at the single point rt is then just G�1��rt ,rt�.

If the field E�+��rt� at any point on the third screenis given by the sum of two fields, i.e., proportional toE�+��r1t1�+E�+��r2t2�, then it is easy to see that the aver-age intensity at rt on the screen is given by a sum of fourcorrelation functions,

G�1��r1t1r1t1� + G�1��r2t2r2t2�

+ G�1��r1t1r2t2� + G�1��r2t2r1t1� . �5�

The first two of these terms are the separate contribu-tions of the two pinholes in the second screen, that is,the intensities they would contribute individually if eachalone were present. Those smooth intensity distributionsare supplemented, however, by the latter two terms of

the sum, which represent the characteristic interferenceeffect of the superposed waves. They are the terms thatlead to the intensity fringes observed by Young.

Intensity fringes of that sort assume the greatest pos-sible contrast or visibility when the cross-correlationterms like G�1��r1t1r2t2� are as large in magnitude as pos-sible. But there is a simple limitation imposed on themagnitude of such correlations by a familiar inequality.There is a formal sense in which cross-correlation func-tions like G�1��r1t1r2t2� are analogous to the scalar prod-ucts of two vectors and are thus subject to a Schwarzinequality. The squared absolute value of that correla-tion function can then never exceed the product of thetwo intensities. If we let x abbreviate a coordinate pairr , t, we must have

�G�1��x1x2��2 � G�1��x1x1�G�1��x2x2� . �6�

The upper bound to the cross-correlation is attained ifwe have

�G�1��x1x2��2 = G�1��x1x1�G�1��x2x2� , �7�

and with it we achieve maximum fringe contrast. Weshall then speak of the fields at x1 and x2 as being opti-cally coherent with one another. That is the definition ofrelative coherence that optics has traditionally used�Born and Wolf, 1959�.

There is another way of stating the condition for op-tical coherence that is also quite useful, particularlywhen we are discussing coherence at pairs of points ex-tending over some specified region in space-time. Let usassume that it is possible to find a positive frequencyfield E�rt� satisfying the appropriate Maxwell equationsand such that the correlation function �4� factorizes intothe form

G�1��r1t1,r2t2� = E*�r1t1�E�r2t2� . �8�

While the necessity of this factorization property re-quires a bit of proof �Titulaer and Glauber, 1965, 1966� itis at least clear that it does bring about the optical co-herence that we have defined by means of the upperbound in the inequality �6� since in that case we have

�G�1��r1t1r2t2��2 = �E�r1t1��2�E�r2t2��2. �9�

In the quantum theory, physical variables such asE�±��rt� are associated not with simple complex numbersbut with operators on the Hilbert-space vectors � � thatrepresent the state of the system, which in the presentcase is the electromagnetic field. Multiplication of theoperators E�+��r1t1� and E�−��r2t2� is not in general com-mutative, and the two operators can be demonstrated toact in altogether different ways on the vectors � � thatrepresent the state of the field. The operator E�+�, inparticular, can be shown to be an annihilation operator.It lowers by 1 the number of quanta present in thefield. Applied to an n-photon state, �n�, it reduces it toan n−1 photon state, �n−1�. Further applications ofE�+��rt� keep reducing the number of quanta present still

FIG. 1. Young’s experiment. Light passing through a pinholein the first screen falls on two closely spaced pinholes in asecond screen. The superposition of the waves radiated bythose pinholes at r1 and r2 leads to interference fringes seen atpoints r on the third screen.



further, but the sequence must end with the n=0 orvacuum state, �vac�, in which there are no quanta left. Inthat state we must finally have

E�+��rt��vac� = 0. �10�

The operator adjoint to E�+�, which is E�−�, must havethe property of raising an n-photon state to ann+1-photon state, so we may be sure, for example, thatthe state E�−��rt��vac� is a one-photon state. Since thevacuum state cannot be reached by raising the numberof photons, we must also require the relation

�vac�E�−��rt� = 0, �11�

which is adjoint to Eq. �10�.The results of quantum measurements often depend

on the way in which the measurements are carried out.The most useful and informative ways of discussing suchexperiments are usually those based on the physics ofthe measurement process itself. To discuss measure-ments of the intensity of light then we should under-stand the functioning of the device that detects or countsphotons.

Such devices generally work by absorbing quanta andregistering each such absorption process, for example,by the detection of an emitted photoelectron. We neednot go into any of the details of the photoabsorptionprocess to see the general nature of the expression forthe photon counting probability. All we need to assumeis that our idealized detector at the point r has negligiblysmall size and has a photoabsorption probability that isindependent of frequency so that it can be regarded asprobing the field at a well defined time t. Then if thefield makes a transition from an initial state �i� to a finalstate �f� in which there is one photon fewer, the probabil-ity amplitude for that particular transition is given by thescalar product—or matrix element

�f�E�+��rt��i� . �12�

To find the total transition probability we must find thesquared modulus of this amplitude and sum it over thecomplete set of possible final states �f� for the field. Theexpression for the completeness of the set of states �f� is

�f

�f��f� = 1,

so that we then have a total transition probability pro-portional to

�f

��f�E�+��rt��i��2 = �f

�i�E�−��rt��f��f�E�+��rt��i�

= �i�E�−��rt�E�+��rt��i� . �13�

It is worth repeating here that in the quantum theorythe fields E�±� are noncommuting operators rather thansimple numbers. Thus one could not reverse their order-ing in the expression �13� while preserving its meaning.In the classical theory we discussed earlier E�+� and E�−�

are simple numbers that convey equivalent information.There is no physical distinction between photoabsorp-tion and emission since there are no classical photons.

The fact that the quantum energy h� vanishes forh→0 removes any distinction between positive andnegative values of the frequency variable.

The initial state of the field in our photon countingexperiment depends, of course, on the output of what-ever light source we use, and very few sources producepure quantum states of any sort. We must thus regardthe state �i� as depending in general on some set of ran-dom and uncontrollable parameters characteristic of thesource. The counting statistics we actually measure thenmay vary from one repetition of the experiment to an-other. The figure we would quote must be regarded asthe average taken over these repetitions. The neatestway of specifying the random properties of the state �i� isto define what von Neumann called the density operator

� = ��i��i�av, �14�

which is the statistical average of the outer product ofthe vector �i� with itself. Expression �14� permits us towrite the average of the counting probability as

��i�E�−��rt�E�+��rt��i�av = Tr��E�−��rt�E�+��rt� . �15�

Interference experiments like those of Young andMichelson, as we have noted earlier, often proceed bymeasuring the intensities of linear combinations of thefields characteristic of two different space-time points.To find the counting probability in a field like E�+��r1t1�+E�+��r2t2�, for example, we will need to know expres-sions like that of Eq. �15� but with two different space-time arguments r1t1 and r2t2. It is convenient then todefine the quantum theoretical form of the correlationfunctions �4� as

G�1��r1t1r2t2� = Tr��E�−��r1t1�E�+��r2t2� . �16�

This function has the same scalar product structure asthe classical function �4� and can be shown likewise toobey the inequality �6�. Once again we can take the up-per bound of the modulus of this cross-correlation func-tion or equivalently the factorization condition �8� to de-fine optical coherence.

It is worth noting at this point that optical experi-ments aimed at achieving a high degree of coherencehave almost always accomplished it by using the mostnearly monochromatic light attainable. The reason forthat is made clear by the factorization condition �8�.These experiments were always based on steady or sta-tistically stationary light sources. What we mean by asteady state is that the function G�1� with two differenttime arguments, t1 and t2, can in fact only depend ontheir difference t1− t2. Optical coherence then requires

G�1��t1 − t2� = E*�t1�E�t2� . �17�

The only possible solution of such a functional equationfor the function E�t� is one that oscillates with a singlepositive frequency. The requirement of monochromatic-ity thus follows from the limitation to steady sources.The factorization condition �8�, on the other hand, de-fines optical coherence more generally for nonsteadysources as well.



Although the energies of visible light quanta are quitesmall on the everyday scale, techniques for detectingthem individually have existed for many decades. Thesimplest methods are based on the photoelectric effectand the use of electron photomultipliers to produce welldefined current pulses. The possibility of detecting indi-vidual quanta raises interesting questions concerningtheir statistical distributions, distributions that should inprinciple be quite accessible to measurement. We mightimagine, for example, putting a quantum counter in agiven light beam and asking for the distribution of timeintervals between successive counts. Statistical problemsof that sort were never, to my knowledge, addressed un-til the importance of quantum correlations began to be-come clear in the 1950s. Until that time virtually all op-tical experiments measured only average intensities orquantum counting rates, and the correlation functionG�1� was all we needed to describe them. It was in thatdecade, however, that several new sorts of experimentsrequiring a more general approach were begun. Thatperiod seemed to mark the beginning of quantum opticsas a relatively new or rejuvenated field.

In the experiment I found most interesting, R. Han-bury Brown and R. Q. Twiss developed a new form ofinterferometry �Hanbury Brown, 1954�. They were inter-ested at first in measuring the angular sizes of radiowave sources in the sky and found they could accom-plish that by using two antennas, as shown in Fig. 2, witha detector attached to each of them to remove the high-frequency oscillations of the field. The noisy low-frequency signals that were left were then sent to a cen-tral device that multiplied them together and recordedtheir time-averaged values. Each of the two detectorsthen was producing an output proportional to the squareof the incident field, and the central device was record-ing a quantity that was quartic in the field strengths.

It is easy to show, by using classical expressions for thefield strengths, that the quartic expression contains a

measurable interference term, and by exploiting it Han-bury Brown and Twiss did measure the angular sizes ofmany radio sources. They then asked themselveswhether they couldn’t perform the same sort of “inten-sity interferometry” with visible light, and thereby mea-sure the angular diameters of visible stars. Although itseemed altogether logical that they could do that, theinterference effect would have to involve the detectionof pairs of photons and they were evidently inhibited inimagining the required interference effect by a state-ment Dirac makes in the first chapter of his famous text-book on quantum mechanics �Dirac, 1958�. In it he isdiscussing why one sees intensity fringes in the Michel-son interferometer, and says in ringingly clear terms,“Each photon then interferes only with itself. Interfer-ence between two different photons never occurs.”

It is worth recalling at this point that interference sim-ply means that the probability amplitudes for alternativeand indistinguishable histories must be added togetheralgebraically. It is not the photons that interfere physi-cally, it is their probability amplitudes that interfere—and probability amplitudes can be defined equally wellfor arbitrary numbers of photons.

Evidently Hanbury Brown and Twiss remained uncer-tain on this point and undertook an experiment �Han-bury Brown and Twiss, 1956, 1957a, 1957b� to determinewhether pairs of photons can indeed interfere. Their ex-perimental arrangement is shown in Fig. 3. The lightsource is an extremely monochromatic discharge tube.The light from that source is collimated and sent to ahalf-silvered mirror which sends the separated beams totwo separate photodetectors. The more or less randomoutput signals of those two detectors are multiplied to-gether, as they were in the radio-frequency experiments,and then averaged. The resulting averages showed aslight tendency for both of the photodetectors to registerphotons simultaneously �Fig. 4�. The effect could be re-moved by displacing one of the counters and thus intro-ducing an effective time delay between them. The coin-cidence effect thus observed was greatly weakened bythe poor time resolution of the detectors, but it raisedconsiderable surprise nonetheless. The observation of

FIG. 2. The intensity interferometry scheme of HanburyBrown and Twiss. Radio-frequency waves are received and de-tected at two antennas. The filtered low-frequency signals thatresult are sent to a device that furnishes an output propor-tional to their product.

FIG. 3. The Hanbury Brown–Twiss photon correlation experi-ment. Light from an extremely monochromatic discharge tubefalls on a half-silvered mirror which sends the split beam totwo separate photodetectors. The random photocurrents fromthe two detectors are multiplied together and then averaged.The variable time delay indicated is actually achieved by vary-ing the distance of the detector D2 from the mirror.



temporal correlations between photons in a steady beamwas something altogether new. The experiment has beenrepeated several times, with better resolution, and thecorrelation effect has emerged in each case more clearly�Rebka and Pound, 1957; Scarl, 1968�.

The correlation effect was enough of a surprise to callfor a clear explanation. The closest it came to that was aclever argument �Purcell, 1956� by Purcell who used thesemiclassical form of the radiation theory in conjunctionwith a formula for the relaxation time of radio-frequency noise developed in wartime radar studies. Itseemed to indicate that the photon correlation timewould be increased by just using a more monochromaticlight source.

The late 1950s were, of course, the time in which thelaser was being developed, but it was not until 1960 thatthe helium-neon laser �Javan et al., 1961� was on thescene with its extremely monochromatic and stablebeams. The question then arose: What are the correla-tions of the photons in a laser beam? Would they ex-tend, as one might guess, over much longer time inter-vals as the beam became more monochromatic? Ipuzzled over the question for some time, I must admit,since it seemed to me, even without any detailed theoryof the laser mechanism, that there would not be anysuch extended correlation.

The oscillating electric current that radiates light in alaser tube is not a current of free charges. It is a polar-ization current of bound charges oscillating in a direc-tion perpendicular to the axis of the tube �Fig. 5�. If it issufficiently strong it can be regarded as a predeterminedclassical current, one that suffers negligible recoil whenindividual photons are emitted. Such currents, I knew

�Glauber, 1951�, emitted Poisson distributions of pho-tons, which indicated clearly that the photons were sta-tistically independent of one another. It seemed thenthat a laser beam would show no Hanbury Brown–Twissphoton correlations at all.

How then would one describe the delayed-coincidence counting measurement of Hanbury Brownand Twiss? If the two photon counters are sensitive atthe space-time points r1t1 and r2t2, we will need tomake use of the annihilation operators E�+��r1t1� andE�+��r2t2� �which do, in fact commute�. The amplitude forthe field to go from the state �i� to a state �f� with twoquanta fewer is

�f�E�+��r2t2�E�+��r1t1��i� . �18�

When this expression is squared, summed over finalstates �f�, and averaged over the initial states �i� we havea new kind of correlation function that we can write as

G�2��r1t1r2t2r2t2r1t1� = Tr��E�−��r1t1�E�−��r2t2�

�E�+��r2t2�E�+��r1t1� . �19�

This is a special case of a somewhat more generalsecond-order correlation function that we can write�with the abbreviation xj=rjtj� as

G�2��x1x2x3x4� = Tr��E�−��x1�E�−��x2�E�+��x3�E�+��x4� .

�20�

Now Hanbury Brown and Twiss had seen to it that thebeams falling on their two detectors were as coherent aspossible in the usual optical sense. The function G�1�

should thus have satisfied the factorization condition �8�,but that statement doesn’t at all imply any correspond-ing factorization property of the functions G�2� given byEq. �19� or �20�.

We are free to define a kind of second-order coher-ence by requiring a parallel factorization of G�2�,

G�2��x1x2x3x4� = E*�x1�E*�x2�E�x3�E�x4� , �21�

and the definition can be a useful one even though theHanbury Brown–Twiss correlation assures us that nosuch factorization is present in their experiment. If itwere present, the coincidence rate according to Eq. �21�would be proportional to

G�2��x1x2x2x1� = G�1��x1x1�G�1��x2x2� , �22�

that is, to the product of the two average intensitiesmeasured separately—and that is what was not found.Ordinary light beams, that is, light from ordinarysources, even extremely monochromatic ones as in theHanbury Brown–Twiss experiment, do not have anysuch second-order coherence.

We can go on defining still higher-order forms of co-herence by defining nth-order correlation functions G�n�

that depend on 2n space-time coordinates. The useful-ness of such functions may not be clear since carryingout the n-fold delayed coincidence counting experimentsthat measure them would be quite difficult in practice. Itis nonetheless useful to discuss such functions since they

FIG. 4. The photon coincidence rate measured rises slightlyabove the constant background of accidental coincidences forsufficiently small time delays. The observed rise was actuallyweakened in magnitude and extended over longer time delaysby the relatively slow response of the photodetectors. Withideal detectors it would take the more sharply peaked formshown.

FIG. 5. Schematic picture of a gas laser. The standing lightwave in the discharge tube generates an intense transverse po-larization current in the gas. Its oscillation sustains the stand-ing wave and generates the radiated beam.



do turn out to play an essential role in most calculationsof the statistical distributions of photons. If we turn on aphoton counter for any given length of time, for ex-ample, the number of photons it records will be a ran-dom integer. Repeating the experiment many times willlead us to a distribution function for that number. Topredict those distributions �Glauber, 1965� we need, ingeneral, to know the correlation functions G�n� of arbi-trary orders.

Once we are defining higher-order forms of coher-ence, it is worth asking whether we can find fields thatlead to factorization of the complete set of correlationfunctions G�n�. If so, we could speak of those as possess-ing full coherence. Now, are there any such states of thefield? In fact there are lots of them, and some can de-scribe precisely the fields generated by predeterminedclassical current distributions. These fields have the re-markable property that annihilating a single quantum inthem by means of the operator E�+� leaves the field es-sentially unchanged. It just multiplies the state vector byan ordinary number. That is a relation we can write as

E�+��rt�� = E�rt�� , �23�

where E�rt� is a positive frequency function of the space-time point rt. It is immediately clear that such statesmust have indefinite numbers of quanta present. Only inthat way can they remain unchanged when one quantumis removed. This remarkable relation does in fact holdfor all of the quantum states radiated by a classical cur-rent distribution, and in that case the function E�rt� hap-pens to be the classical solution for the electric field.

Any state vector that obeys the relation �23� will alsoobey the adjoint relation

� �E�−��rt� = E*�rt�� . �24�

Hence the nth-order correlation function will indeedfactorize into the form

G�n��x1, . . . ,x2n� = E*�x1� ¯ E*�xn�E�xn+1� ¯ E�x2n�

�25�

that we require for nth-order coherence. Such states rep-resent fully coherent fields, and delayed coincidencecounting measurements carried out in them will revealno photon correlations at all. To explain, for example,the Hanbury Brown–Twiss correlations we must use notpure coherent states but mixtures of them, for which thefactorization conditions like Eq. �25� no longer hold. Tosee how these mixtures arise, it helps to discuss themodes of oscillation of the field individually.

The electromagnetic field in free space has a con-tinuum of possible frequencies of oscillation, and a con-tinuum of available modes of spatial oscillation at anygiven frequency. It is often simpler, instead of discussingall these modes at once, to isolate a single mode anddiscuss the behavior of that one alone. The field as awhole is then a sum of the contributions of the indi-vidual modes. In fact when experiments are carried out

within reflecting enclosures the field modes form a dis-crete set, and their contributions are often physicallyseparable.

The oscillations of a single mode of the field, as wehave noted earlier, are essentially the same as those of aharmonic oscillator. The nth excitation state of the oscil-lator represents the presence of exactly n light quanta inthat mode. The operator that decreases the quantumnumber of the oscillator is usually written as a, and theadjoint operator—which raises the quantum number byone unit as a†. These operators then obey the relation

aa† − a†a = 1, �26�

which shows that their multiplication is not commuta-tive. We can take the field operator E�+��rt� for the modewe are studying to be proportional to the operator a.Then any state vector for the mode that obeys the rela-tion �23� will have the property

a� � = �� , �27�

where � is some complex number. It is not difficult tosolve for the state vectors that satisfy the relation �27�for any given value of �. They can be expressed as a sumtaken over all possible quantum-number states �n�,n=0,1 ,2 , . . . that takes the form

�� = e−�1/2��2�n=0

��n

n!�n� , �28�

in which we have chosen to label the state with the ar-bitrary complex parameter �. The states �� are fullycoherent states of the field mode.

The squared moduli of the coefficients of the state �n�in Eq. �28� tell us the probability for the presence of nquanta in the mode, and those numbers do indeed forma Poisson distribution, one with the mean value of nequal to ��2. The coherent states form a complete set ofstates in the sense that any state of the mode can beexpressed as a suitable sum taken over them. As wehave defined them they are equivalent to certain oscilla-tor states defined by Schrödinger �1926� in his earliestdiscussions of wave functions. Known thus from the verybeginning of wave mechanics, they seemed not to havefound any important role in the earlier development ofthe theory.

Coherent excitations of fields have a particularlysimple way of combining. Let us suppose that one exci-tation mechanism brings a field mode from its emptystate �0� to the coherent state ��1�. A second mechanismcould bring it, for example, from the state �0� to the state��2�. If the two mechanisms act simultaneously the re-sulting state can be written as ei��1+�2� where ei is aphase factor that depends on �1 and �2, but we don’tneed to know it since it cancels out of the expression forthe density operator

� = ��1 + �2��1 + �2� . �29�

This relation embodies the superposition principle forfield excitations and tells us all about the resulting quan-



tum statistics. It is easily generalized to treat the super-position of many excitations. If, say, j coherent excita-tions were present, we should have a density operator

� = ��1 + ¯ + �j��1 + ¯ + �j� . �30�

Let us suppose now that the individual excitation am-plitudes �j are in one or another sense random complexnumbers. Then the sum �1+ ¯ +�j will describe a suit-ably defined random walk in the complex plane. In thelimit j→� the probability distribution for the sum�=�1+ ¯ +�j will be given by a Gaussian distributionwhich we can write as

P�� =1

�n�e−��2/�n�, �31�

in which the mean value of ��2, which has been writtenas �n�, is the mean number of quanta in the mode.

The density operator that describes this sort of ran-dom excitation is a probabilistic mixture of coherentstates,

� =1

�n� � e−��2/�n��d2� , �32�

where d2� is an element of area in the complex plane.When we express � in terms of m-quantum states byusing the expansion �28�, we find

� =1

1 + �n� �m=0

� � �n�1 + �n�

m

�m��m� . �33�

This kind of random excitation mechanism is thus al-ways associated with a geometrical or fixed-ratio distri-bution of quantum numbers �Fig. 6�. In the best knownexample of the latter, the Planck distribution, we have�n�= �eh�/kT−1�−1, and the density operator �33� thencontains the familiar thermal weights e−mhv/kT.

There is something remarkably universal about thegeometrical sequence of n-quantum probabilities. Theimage of chaotic excitation we have derived it from, onthe other hand, excitation in effect by a random collec-tion of lasers, may well seem rather specialized. It maybe useful therefore to have a more general way of char-acterizing the same distribution. If a quantum state isspecified by the density operator �, we may associatewith it an entropy S given by

S = − Tr�� ln �� , �34�

which is a measure, roughly speaking, of the disordercharacteristic of the state. The most disordered, or cha-otic, state is reached by maximizing S, but in finding themaximum we must observe two constraints. The first is

Tr � = 1, �35�

which says simply that all probabilities add up to 1. Thesecond is

Tr��a†a� = �n� , �36�

which fixes the average occupation number of the mode.When S is maximized, subject to these two con-

straints, we find indeed that the density operator � takesthe form given by Eq. �33�. The geometrical distributionis thus uniquely representative of chaotic excitation.Most ordinary light sources consist of vast numbers ofatoms radiating as nearly independently of one anotheras the field equations will permit. It should be no sur-prise then that these are largely maximum entropy orchaotic sources. When many modes are excited, the lightthey radiate is, in effect, colored noise and indistinguish-able from appropriately filtered blackbody radiation.

For chaotic sources, the density operator �32� permitsus to evaluate all of the higher-order correlation func-tions G�n��x1 , . . . ,x2n�. In fact they can all be reduced�Glauber, 1965� to sums of products of first-order corre-lation functions G�1��xixj�. In particular, for example, theHanbury Brown–Twiss coincidence rate correspondingto the two space-time points x1 and x2 can be written as

G�2��x1x2x2x1� = G�1��x1x1�G�1��x2x2�

+ G�1��x1x2�G�1��x2x1� . �37�

The first of the two terms on the right side of this equa-tion is simply the product of the two counting rates thatwould be measured at x1 and x2 independently. The sec-ond term is the additional delayed coincidence rate de-tected first by Hanbury Brown and Twiss, and it is in-deed contributed by a two-photon interference effect. Ifwe let x1=x2, which corresponds to zero time delay intheir experiment, we see that

G�2��x1x1x1x1� = 2�G�1��x1x1�2, �38�

or the coincidence rate for vanishing time delay shouldbe double the background or accidental rate.

The Gaussian representation of the density operatorin terms of coherent states is an example of a broaderclass of so-called “diagonal representations” that arequite convenient to use—when they are available. If thedensity operator for a single mode, for example, can bewritten in the form

� =� P��d2� , �39�

then the expectation values of operator products likea†nam can be evaluated as simple integrals over the func-tion P such as

FIG. 6. Geometrical or fixed-ratio sequence of probabilitiesfor the presence of n quanta in a mode that is excited chaoti-cally.



�a†namav =� P��*n�md2� . �40�

The function P�� then takes on some of the role of aprobability density, but that can be a bit misleading sincethe condition that the probabilities derived from � all bepositive or zero does not require P�� to be positivedefinite. It can and sometimes does take on negativevalues over limited areas of the �-plane in certain physi-cal examples, and it may also be singular. It is a member,as we shall see, of a broader class of quasiprobabilitydensities. The representation �39�, and the P-representation, unfortunately is not always available�Sudarshan, 1963; Cahill and Glauber, 1969a, 1969b�. Itcannot be defined, for example, for the familiar“squeezed” states of the field in which one or the otherof the complimentary uncertainties is smaller than thatof the coherent states.

The difference between a monochromatic laser beamand a chaotic beam is most easily expressed in terms ofthe function P��. For a stationary laser beam the func-tion P depends only on the magnitude of � and vanishesunless �� assumes some fixed value. A graph of thefunction P is shown in Fig. 7, where it can be comparedwith the Gaussian function for the same mean occupa-tion number �n� given by Eq. �31�.

How do we measure the statistical properties of pho-ton distributions? A relatively simple way is to place aphoton counter in a light beam behind either a mechani-cal or an electrical shutter. If we open the shutter for agiven length of time t, the counter will register somerandom number n of photons. By repeating that mea-surement sufficiently many times we can establish a sta-tistical distribution for those random integers n. Theanalysis necessary to derive this distribution mathemati-

cally can be a bit complicated since it requires, in gen-eral, a knowledge of all higher-order correlation func-tions. Experimental measurements of the distribution,conversely, can tell us about those correlation functions.

For the two cases in which we already know all corre-lation functions, it is particularly easy to find the photo-count distributions. If the average rate at which photonsare recorded is w, then the mean number recorded intime t is

�n� = wt .

In a coherent beam the result for the probability of nphotocounts is just the Poisson distribution

pn�t� =�wt�n

n!e−wt. �41�

In a chaotic beam, on the other hand, the probability ofcounting n quanta is given by the rather more spread-out distribution

pn�t� =1

1 + wt� wt

1 + wt n

. �42�

These results, which are fairly obvious from the occupa-tion number probabilities implicit in Eqs. �28� and �33�,are illustrated in Fig. 8.

Here is a closely related question that can also beinvestigated experimentally without much difficulty: Ifwe open the shutter in front of the counter at an arbi-trary moment, some random interval of time will passbefore the first photon is counted. What is the distribu-tion of those random times? In a steady coherent beam,in fact, it is just an exponential distribution

Wcoh = we−wt, �43�

while in a chaotic beam it assumes the less obvious form

Wch�t� =w

�1 + wt�2 . �44�

There is an alternative way of finding a distribution oftime intervals. Instead of simply opening a shutter at anarbitrary moment, we can begin the measurement withthe registration of a given photocount at time zero andthen ask what is the distribution, of time intervals untilthe next photocount. This distribution, which we may

FIG. 8. The two P�� distributions of Fig. 7 lead to differentphoton occupation number distributions p�n�: for chaotic exci-tation a geometric distribution, for coherent excitation a Pois-son distribution.

FIG. 9. Time interval distributions for counting experiments ina chaotically excited mode: Wch�t� is the distribution of inter-vals from an arbitrary moment until the first photocount.Wch�0�t� is the distribution of intervals between two successivephotocounts.

FIG. 7. The quasiprobability function P�� for a chaotic ex-citation is Gaussian in form, while for a stable laser beam ittakes on nonzero values only near a fixed value of ��.



write as W�0�t�, takes the same form for a coherentbeam as it does for the measurement described earlier,which starts at arbitrary moments,

Wcoh�0�t� = we−wt = Wcoh�t� . �45�

This identity is simply a restatement of the statisticallyindependent or uncorrelated quality of all photons in acoherent beam.

For a chaotic beam, on the other hand, the distribu-tion Wch�0�t� takes a form quite different from Wch�t�. Itis

Wch�0�t� =2w

�1 + wt�3 , �46�

an expression which exceeds Wch�t� for times for whichwt�1, and is in fact twice as large as Wch�t� for t=0 �Fig.9�. The reason for that lies in the Gaussian distributionof amplitudes implicit in Eqs. �31� and �32�. The veryfact that we have counted a photon at t=0 makes it moreprobable that the field amplitude � has fluctuated to alarge value at that moment, and hence the probabilityfor counting a second photon remains larger than aver-age for some time later. The functions Wch�t� andWch�0�t� are compared in Fig. 8.

All of the experiments we have discussed thus far arebased on the procedure of photon counting, whetherwith individual counters or with several of them ar-ranged to be sensitive in delayed coincidence. The func-tions they measure, the correlation functions G�n�, are allexpectation values of products of field operators writtenin a particular order. If one reads from right to left, theannihilation operator always precedes the creation op-erators in our correlation functions, as they do, for ex-ample, in Eq. �19� for G�2�. It is that so-called “normalordering” that gives the coherent states, and the qua-siprobability density P�� the special roles they occupyin discussing this class of experiments.

But there are other kinds of expectation values thatone sometimes needs in order to discuss other classes ofexperiments. These could, for example, involve sym-

metrically ordered sums of operator products, or evenantinormally ordered products which are opposite to thenormally ordered ones. The commutation relations forthe multiplication of field operators will ultimately relateall these expectation values to one another, but it is of-ten possible to find much simpler ways of evaluatingthem. There exists a quasiprobability density that playsmuch the same role for symmetrized products as thefunction P does for the normally ordered ones. It is, infact, the function Wigner �1932� devised in 1932 as akind of quantum mechanical replacement for the classi-cal phase space density. For antinormally ordered opera-tor products, the role of the quasiprobability density istaken over by the expectation value which for a singlemode is �1/��. The three quasiprobability densi-ties associated with the three operator orderings andwhatever experiments they describe are all members ofa larger family that can be shown to have many proper-ties in common �Cahill and Glauber, 1969a, 1969b�.

The developments I have described to you were allrelatively early ones in the development of the field wenow call quantum optics. The further developments thathave come in rapid succession in recent years are toonumerous to recount here. Let me just mention a few. Agreat variety of careful measurements of photon count-ing distributions and correlations of the type we havediscussed have been carried out �Arecchi, 1969; Jake-man and Pike, 1969� and furnish clear agreement withthe theory. They have furthermore shown in detail howthe properties of laser beams change as they rise inpower from below threshold to above it.

The fully quantum mechanical theory of the laser wasdifficult to develop �Scully and Lamb, 1967; Lax, 1968;Haken, 1970� since the laser is an intrinsically nonlineardevice, but only through such a theory can its quantumnoise properties be understood. The theories of a con-siderable assortment of other kinds of oscillators andamplifiers have now been worked out.

Nonlinear optics has furnished us with new classesof quantum phenomena such as parametric down-conversion in which a single photon is split into a pair ofhighly correlated or entangled photons. Entanglementhas been a rich source of the quantum phenomena thatare perhaps most interesting—and baffling—in everydayterms.

It is worth emphasizing that the mathematical toolswe have developed for dealing with light quanta can beapplied equally well to the much broader class of par-ticles obeying Bose-Einstein statistics. These include at-oms of 4He, 23Na, 87Rb, and all of the others which haverecently been Bose condensed by optical means. Whenproper account is taken of the atomic interactions andnonvanishing atomic masses, the coherent state formal-ism is found to furnish useful descriptions of the behav-ior of these bosonic gases.

The formalism seems likewise to apply to subatomicparticles, to bosons that are only short-lived. Pions thatemerge by hundreds or even thousands from the high-energy collisions of heavy ions are also bosons. Pions ofsimilar charge have a clearly noticeable tendency to be

FIG. 10. �Color� Professor Einstein, encountered in the springof 1951 in Princeton, NJ.



emitted with closely correlated momenta, an effectwhich is evidently analogous to the Hanbury Brown–Twiss correlation of photons, and invites the same sortof analysis �Glauber, 2006�.

Particles obeying Fermi-Dirac statistics, of course, be-have quite differently from photons or pions. No morethan a single one of them ever occupies any given quan-tum state. This kind of reckoning associated with fer-mion fields is radically different therefore from the sortwe have associated with bosons, like photons. It hasproved possible, nonetheless, to develop an algebraicscheme �Cahill and Glauber, 1999� for calculating expec-tation values of products of fermion fields that is re-markably parallel to the one we have described for pho-ton fields. There is a one-to-one correspondencebetween the mathematical operations and expressionsfor boson fields, on the one hand, and fermion fields, onthe other hand. That correspondence has promise ofproving useful in describing the dynamics of degeneratefermion gases.

I’d like, as a final note, to share with you an experi-ence I had in 1951, while I was a postdoc at the Institutefor Advanced Study in Princeton. Possessed by the habitof working late at night—in fact on photon statistics�Glauber, 1951� at the time—I didn’t often appear at mydesk early in the day. Occasionally I walked out to theInstitute around noon, and that was closer to the end ofthe work day for Professor Einstein. Our paths thuscrossed quite a few times, and on one of those occasionsI had ventured to bring my camera. He seemed morethan willing to let me take his picture as if acknowledg-ing his role as a local landmark, and he stood for me justas rigidly still. Here, in Fig. 10, is the hitherto unpub-lished result. I shall always treasure that image, and har-bor the enduring wish I had been able to ask him just afew questions about that remarkable year, 1905.

REFERENCES

Arecchi, F. T., 1969, in Quantum Optics, Course XLII, EnricoFermi School, Varenna, 1969, edited by R. J. Glauber �Aca-demic Press, New York�.

Born, M., and E. Wolf, 1959, Principles of Optics �PergamonPress, London�, Chap. X.

Cahill, K. E., and R. J. Glauber, 1969a, Phys. Rev. 177, 1857.Cahill, K. E., and R. J. Glauber, 1969b, Phys. Rev. 177, 1882.Cahill, K. E., and R. J. Glauber, 1999, Phys. Rev. A 59, 1538.Dirac, P. A. M., 1927a, Proc. R. Soc. London 114, 243.Dirac, P. A. M., 1927b, Proc. R. Soc. London 114, 710.Dirac, P. A. M., 1958, The Principles of Quantum Mechanics,

4th ed. �Oxford University Press�, p. 9.Einstein, A., 1905, Ann. Phys. 17, 132.Glauber, R. J., 2006, Quantum Optics and Heavy Ion Physics,

e-print org/nucl-th/0604021.Glauber, R. J., 1951, Phys. Rev. 84, 395.Glauber, R. J., 1965, in Quantum Optics and Electronics �Les

Houches 1964�, edited by C. de Witt, A. Blandin, and C.Cohen-Tannoudji �Gordon and Breach, New York�, p. 63.

Haken, H., 1970, in Encyclopedia of Physics XXV/2c, editedby S. Flügge �Springer-Verlag, Heidelberg�.

Hanbury Brown, R., and R. Q. Twiss, 1954, Philos. Mag. 45,663.

Hanbury Brown, R., and R. Q. Twiss, 1956, Nature �London�177, 27.

Hanbury Brown, R., and R. Q. Twiss, 1957a, Proc. R. Soc.London, Ser. A 242, 300.

Hanbury Brown, R., and R. Q. Twiss, 1957b, Proc. R. Soc.London, Ser. A 243, 29.

Jakeman, E., and E. R. Pike, 1969, J. Phys. A 2, 411.Javan, A., W. R. Bennett, and D. R. Herriot, 1961, Phys. Rev.

Lett. 6, 106.Lax, M., 1968, in Brandeis University Summer Institute Lec-

tures �1966�, Vol. II, edited by M. Cretien et al. �Gordon andBreach, New York�.

Planck, M., 1900a, Ann. Phys. 306, 69.Planck, M., 1900b, Ann. Phys. 306, 719.Purcell, E. M., 1956, Nature �London� 178, 1449.Rebka, G. A., and R. V. Pound, 1957, Nature �London� 180,

1035.Scarl, D. B., 1968, Phys. Rev. 175, 1661.Schrödinger, E., 1926, Naturwiss. 14, 664.Scully, M. O., and W. E. Lamb, Jr., 1967, Phys. Rev. 159, 208.Sudarshan, E. C. G., 1963, Phys. Rev. Lett. 10, 277.Titulaer, U. M., and R. J. Glauber, 1965, Phys. Rev. 140, B676.Titulaer, U. M., and R. J. Glauber, 1966, Phys. Rev. 145, 1041.Wigner, E., 1932, Phys. Rev. 40, 749.



one hundred years of light quanta (nobel lecture)

Documents