The Concept of the Photon

The concept of the photonIt has its logical foundation in the quantum theoryof radiation. But the "fuzzy-ball" pictureof a photon often leads to unnecessary difficulties.

The idea of the photon has stirred theimaginations of physicists ever since1905 when Einstein originally proposedthe use of light quanta to explain thephotoelectric effect. This concept is for-malized in the quantum theory of radia-tion, which has had unfailing success inexplaining the interaction of electro-magnetic radiation with matter, seem-ingly limited only by the ability ofphysicists to perform the indicatedcalculations. Nevertheless, it has itsconceptual problems-various infinitiesand frequent misinterpretations. Con-sequently an increasing number ofworkers are asking, "to what extent isthe quantized field really necessary anduseful?" In fact the experimental re-sults of the photoelectric effect wereexplained by G. Wentzel in 1927 with-out the quantum theory of radiation.Similarly most electro-optic phenomenasuch as stimulated emission, reaction ofthe emitted field on the emitting,atom,resonance fluorescence, and so on, donot require the quantization of the fieldfor their explanation. As we will see,these processes can all be quantitativelyexplained and physically understood interms of the semiclassical theory of thematter-field interaction in which theelectric field is treated classically whilethe atoms obey the laws of quantum me-chanics. The quantized field is funda-mentally required for accurate descrip-

MarianScully and MurraySargent are bothat the University of Arizona in Tucson.Scully, professor of physics and opticalsciences, is an AlfredP. Sloan Fellowandis supported in part by Professor PeterFranken. Sargent is assistant professorof optical sciences. Together with W. E.Lamb,Jr, they have written a text on quan-tum optics, which will appear shortly; thisbook willgive a more complete account ofthe photon concept than can be presentedhere.

tions of certain processes involving fluc-tuations in the electromagnetic field:for example, spontaneous emission, theLamb shift, the anomalous magneticmoment of the electron, and certain as-pects of blackbody radiation. (TheCompton effect also fits here, but seelater under references 8b and c.) Herewe will outline how the photon conceptoriginated and developed, where it is notrequired and is often misused, and fi-nally where it plays an essential role inthe understanding of physical phenom-ena.1 In our discus(>ionwe will attemptto give a logically consistent definitionof the word "photon"-a statement farmore necessary than one might think,for so many contradictory uses exist ofthis elusive beast. In particular consid-er the original coining of the word by G.N. Lewis:2

"[because it appears to spend] only aminute fraction of its existence as acarrier of radiant energy, while therest of the time it remains an im-portant structural element withinthe atom . . . , I therefore take theliberty of proposing for this hypo-thetical new atom which is not lightbut plays an essential part in everyprocess of radiation, the name pho-ton!"

(our exclamation point). Clearly thepresent usage of the word is very dif-ferent.

From Maxwell to SchrodingerAlthough the nature of light has been

a subject of wonder since day one3 (itwas on a Monday), the conceptualbasis for the understanding of radiativephenomena begins with James ClerkMaxwell and Heinrich Hertz. Whileit is true that Isaac Newton, ChristianHuygens, Thomas Young, and manyothers contributed mightily to ourunderstanding of optics, the Maxwell-Hertz demonstration that light is made

of the same stuff as electric and mag-netic fields must be regarded as thefirst insight into the inner workings ofthe radiation field. According to theirdescription, light is radiated by ac-celerating charges and is an electro-magnetic excitation (of an "aether").

Among other things, it must havebeen the far-ranging success of Maxwellin explaining electromagnetic phe-nomena that led 19th-century physi-cists to state that there were really onlytwo clouds on the horizon of physics atthe beginning of the 20th century.Interestingly enough, both of theseclouds involved electromagnetic radia-tion. The first cloud, namely the nullresult of the Michelson-Morley ex-periment, led to special relativity,which is the epitome of classical me-chanics, and really capped things offin a logical way. The second cloud, theRayleigh-Jeans catastrophe and thenature of blackbody radiation, led tothe beginnings of quantum mechanics,which, of course, was a radical changein physical thought up to that point.Note that while both of these problemsinvolve the radiation field, neither(initially) involved the concept of aphoton. That is, neither Einstein norLorentz in the first instance nor MaxPlanck in the second called upon theparticulate nature of light for the ex-planation of the observed phenomena.Relativity is strictly classical, andPlanck only quantized energies of theoscillators in the walls of his cavity,not the field. Up to this point (before1905) the discreteness of light quantawas never invoked.

The next chapter in the history of thephoton concept came when Einsteinapplied Planck's quantization ideas tothe photoelectric effect. The situationhere was very different from that en-visioned by Planck. Einstein invokedthe existence of discrete bundles or

Laser pUlse photographed in flight.This is an ultrashort green pulse obtainedas the second harmonic of 1.06-micronlight from a neodymium-doped glass laser.To make the photograph the pulse waspassed through a water cell; the scale onthe cell wall is in millimeters. The camerashutter was a Kerr cell. triggered by aninfrared pUlse (1.06 micron) from the samelaser; exposure time was about 10picosec, the same period as the durationof these ultrashort pulses. During theexposure the pulse moved about 2.2 mm(right to left), the velocity of light in thecell being approximately 2.2 X 1010 cm/sec. (Photograph by Michel Duguay, BellLaboratories. )

quanta of light energy (photons) toexplain the ejection of photoelectronsfrom solids. This was distinct fromPlanck's idea in which only matter wasquantized, as is illustrated in the fol-lowing passage from George Gamow'sdelightful little book, The Thirty YearsThat Shook Physics4

"Having let the spirit of the quantumout of the bottle, Max Planck washimself scared to death of it andpreferred to believe the packages ofenergy arise not from the propertiesof the light waves themselves butrather from the internal propertiesof the atoms which can emit andabsorb radiation only in certaindiscrete quantities. Radiation is likebutter, which can be bought or re-turned to the grocery store only inquarter-pound packages, althoughthe butter as such can exist in anydesired amount .... Only five yearsafter the original Planck proposal,the light quantum was established asa physical entity existing indepen-dently of the mechanism of its emis-sion or absorption by atoms. Thisstep was taken by Albert Einstein inan article published in 1905, the yearof his first article on the Theory ofRelativity. Einstein indicated thatthe existence of light quanta rushingfreely through space represents anecessary condition for explainingempirical laws of the photoelectriceffect; that is, the emission of elec-trons from the metallic surfaces ir-radiated by violet or ultraviolet rays."

We shall return to the photoelectriceffect later; however, we note thatPlanck's "butter-ball quantum" idea isnot completely absurd, and, in fact, amodern version of it is being recon-sidered by some modern theoretical

physicists.8The next cornerstone is the realization

that matter itself has a wave-like side toits personality. The first to put this ina concrete mathematical form wasErwin Schrodinger. He wrote hisfamous equation for the wave functionof an atom, y;(r, t) in terms of its Hamil-tonian as

ih :t y; (r,t) = 3Cy; (r,t) (1)

and is responsible for demonstratingthat the wave nature of matter is es-sential for its understanding.

Semiclassical theory

Atoms require quantum theory in thedescription of their behavior, becauseamong other things, classical mechanicstells us that orbiting (therefore acceler-ating) electrons in atoms should radiateand spiral into the nucleus in contradic-tion of observed results! A surprisinglysuccessful theory of the atom-field in-teraction can be obtained in which theatoms obey the laws of quantum me-chanics and the electric field is treatedclassically according to Maxwell'sequations-that is to say, without theconcept of the photon. This semi-classical theory is important for ourpresent purposes for two reasons: Firstit is important to understand whichclassical phenomena do not need orlogically imply quantized fields for theirexplanation, and second, the semiclassi-cal theory accounts quantitatively formost radiation-matter interactions.In this part of our article we will supportthis contention by reviewing the semi-classical description of:~ the response of an atom to a reso-nant, monochromatic field~ the self-consistent treatment of theatom-field interaction

Radiation-induced transitions. Thisdiagram shows how the electric field ofequation (2) induces transitions from theenergy levels Ul00 to U210. Thecorresponding wave function is given by asuperposition of states.Figure 1

~ stimulated emission~ resonance fluorescence~ the photoelectric effect

Consider first a hydrogenic atom withenergy eigenstates Unlm, and supposethe atom is initially in its ground (Is)state, UIOO, with energy hWIOO. We ir-radiate the atom by a light beam repre-sented by the linearly polarized, plane-wave electric field

E(Y,t) = z Eo cos (pt - Ky) (2)

where the (circular) frequency p is near-ly resonant with the Is -- 2p (UIOO --

U210) transitions, that is, v "" W = ~lO -

WIOO. Radiation so polarized inducestransitions from the UIOO level to theU210 level, causing the wave functiony;(r,t) to become a linear superpositionof the two eigenfunctions as depicted infigure 1. The time development ofy;(r,t) is determined by the Schrodingerequation (1) whose Hamiltonian in-cludes the electric dipole interactionenergy

3Cl = -er·E (3)

The resulting z dependence of y;(r,t)varies in time as shown in figure 2a.There the probability density y;*y; os-cillates back and forth across the (posi-tively charged) nucleus with frequencyW = W210 - WIOO. Hence an ensemble ofN such systems located in a volume(small compared to a cubic wavelength)about the position Ro produces anaverage oscillating dipole moment,namelypeR') =

N [f d3ry;(r,t)ery;(r,t) 1o(R' - Ro) (4)

which depends on the detun~ng (w - v),the strength of the atom-field inter-actions, and so on. We treat the expec-tation-value expression (4) as an ordi-

nary dipole density radiating, for ex-ample, the far-field electric field

E(R,t) = [w2j(41Ttoc2)l (n X p) X

_ exp[iKIR - Roll .n ---~---+ complex conjugate

IR -Rol(5)

We see that the field induces a dipolemoment in an ensemble of atoms andthat this moment, in turn, contributesa field. So far we have neglected theeffect of the back reaction of lightemitted by the dipole back on itself.This reaction is included by requiringthat the field be self consistent, that is,that the field which the atoms see beconsistent with the field radiated, asoutlined in figure 3. Solving the self-consistent set of equations in figure 3simultaneously we account for the backaction. This effect is one phenomenonsometimes said to require the quantumtheory of radiation. We now apply thesemiclassical method to several otherproblems.

Stimulated emission is the first ofthese. We wish to study a sheet ofatoms in the x-z plane (figure 2b) sub-

ject to the incident electric field of equa-tion (2). We suppose again that theatol!ls have two relevant levels, thistime having atomic decay phenomenaassociated with, for example, colli-sions, and we carry out the appropriatetime integrations to find the dipoledensity as in equation (4). We find thedipole moment density at the point (x,O,z)

P(x, O,z, t) ex

zEoli' sin (vt) + (w - v) cos (vt)l (6)

in which the constant of proportion-ality depends on the number of atomsinvolved, the strength of the atom-fieldinteraction, a Lorentzian involving de-tuning, the atomic decay rate 'Y, and soforth. The things to note from equa-tion (6) are:~ the dipole oscillates at the drivingfrequency v and not the atomic line cen-terw~ the magnitude of the dipole is pro-portional to the field amplitude Eo~ there are components "in phase"(cos vt term) and "in quadrature" (sinvt term) with the inducing field of

Radiating dipoles. The z-dependence ofthe wave function tf;(r,t) in part (a) are fort = 0 and t = 1rj w; the probability densitytf;*tf; oscillates back and forth across thenucleus at frequency w, so yielding anoscillating dipole. Part (b) of the figureshows how a sheet of dipoles radiates anelectric field in phase with the incidentfield (equation 2). The individual fieldsbecome increasingly retarded the furtheroff the axis one goes.Figure 2

equation (2). The former modifies theindex of refraction in the sheet; the lat-ter acts as a source for gain due to stim-ulated emission.

Considering the second of these moreclosely, we note that on resonance thepolarization, equation (6), is propor-tional to sin vt. This is 90 deg out ofphase with the applied field of equation(2), and in view of equation (5), onenotes the radiated field (on axis) is not inphase either. In order to get back intophase with the incident field (indeed inphase with the textbooks!), we add upthe contributions from a sheet of di-poles (integrate over x and z) to find aradiated field proportional to cos vt asis equation (2). This second radiatedfield has the same phase, frequency anddirection as the incident field.

Resonance fluorescence5 is our sec-ond application of semiclassical theoryand is defined to be the emission ofradiation by a ground-state atomic en-semble excited by an optical field.As depicted in figure 4, an incident field(spectral width f) and central freq~encyv is absorbed by the ensemble (spectralwidth 'Y) which, in turn, emits intosome new direction with the same spec-trum as the incident field if r is smallcompared to 'Y. This follows fromequation (7), which shows that the in-duced dipole has the same frequency asthe inducing (driving) field. Alter-natively, for a field whose spectral widthis due to its finite duration Ijr, we un-derstand the atomic response as that of adriven oscillator with the frequency ofthe driving field. The atomic oscil-lator scatters for as long as it is driven,that is, for a time 1jr. The spectralwidth of the emitted radiation there-fore corresponds to the reciprocal of thelifetime, namely, r.

The photoelectric effect5,6,8a is our

The uncertaintyprinciple violated

As an aside, we can note for examplethat the commutation relations for anatom damped by a quantized field aretime independent, because of thepresence of quantized Langevin noisesources associated with the atomicdamping. These sources are quan-tized because the field is. Even if wecould damp the atom with a classicalfield, as Michael Crisp and Ed Jaynessuggest, the associated Langevinsources would commute and allowthe atomic commutation relations todecay to zero in time. (For furtherdiscussion see the paper by MelvinLax, reference 10.)

This time dependence of the com-mutation relations then implies a viola-tion of the uncertainty principle. Thispoint was noted first (in a somewhatdifferent way) by Niels Bohr and LeonRosenfeld.

final example of semiclassical theory.Some readers may find this surprising,because the photoelectric effect pro-vided the original impetus for ascribinga particle character to light. The threemain facts of life, photoelectron-wise,are:~ When light shines on a photoemis-sive surface, electrons are ejected with akinetic energy equal to Planck's con-stant times the frequency v of the inci-dent light less some work function cJ>,usually written as

mv2hv = -2-+ cJ> (7)

~ The rate of electron ejection isproportional to the square of the elec-tric field of the incident light (ejection

rate ex Eo2),~ There is not necessarily a time de-lay between the instant the field isturned on and the ejection of photo-electrons.

To explain these three characteristics,we suppose the medium consists ofground-state (Ig» electrons which canmake transitions under the influence ofan applied field, equation (2), to a quasi-continuum consisting of momentumstates Ik) as depicted in figure 5.With the electric-dipole energy, equa-tion (3), and the philosophy of Fermi'sGolden Rule, we find the probabilityfor a transition from the ground stateto the kth excited state within a timet to bePk = 27r(elrkglh]2Eo2to(v-

(fk - fg)/h] (8)

Writing energy fk - fg as mv2/2 + cJ>asin figure 5, we find that the 0 function inequation (8) implies equation (7). Thisresult conflicts with what is often taught,as the following quote from a well knowntext7 illustrates:

"Einstein's photoelectric equationplayed an enormous part in the de-velopment of the modern quantumtheory. But in spite of its generalityand of the many successful applica-tions that have been made of it inphysical theories, the equation hv =mv2/2 + cJ> is, as we shall seepresentiy, based on a concept ofradiation-the concept of 'lightquanta'-completely at variance withthe most fundamental concepts of theclassical electromagnetic theory ofradia tion. "The second fact is also clearly con-

tained in equation (8), since Pk isdirectly proportional to Eo2. Finallythe third point is accounted for, because

Quantumtechanics

i1i~ 1'1')= [Ho - er·E'(R,t)]1'1">at

Statisticalsummation

P(R,t)- eI<'I"lrl'l">a(Rj - R),

Electrodynamics

1,t2 azV' X (V' X E(R,t))+ 2" -2 E(R,t)= - fio -2 P(R,t)

cat at

Self-consistent field:E' = E

equation (8) is nonzero even for smalltimes, a fact underlined by Peter A.Franken.6 "As for the time delays (inthe photoelectric effect], quantummechanics teaches us that the rate isestablished when the perturbation isturned on [after several optical cycles]."

In fact, for the majority of quantumoptical calculations the semiclassicaltheory proves most adequate. We notethat in addition to those examplesabove, nonlinear optics,7a much of lasertheory,7b pulse-propagation phenom-ena7e and even "photon" ech07d are allbest explained without photons. Thatthe list of successes of semiclassicaltheory is impressive is further illustratedby the following quote from the recentpaper of Michael D. Crisp and Ed T.Jaynes,8

"Even though it is generally be-lieved that a full quantum-electro-dynamic treatment is necessary inorder to obtain all radiative effectscorrectly, many calculations involv-ing the interaction of radiation andmatter were first done without quan-tizing the electromagnetic field. Thusis the case of the photoelectric ef-fect,8a the scattering of radiationfrom a free electron (Klein-Nishinaformula)8b stimulated emission andabsorption of radiation by an atom,8eand vacuum polarization,8d the cor-rect predictions were first obtainedby semiclassical methods."

They continue with the assertion that,while spontaneous emission and theLamb shift are generally conceded to re-quire the quantized field, the self-consistent semiclassical theory does sur-prisingly well even here. In fact theyderive a "Lamb shift" that is order-of-magnitude correct from their semi-classical calculation! However we

Self-consistent equations demonstratingthat an assumed field E' (R,t) perturbs theith atom according to the laws of quantummechanics and induces an electric dipoleexpectation value. Values for atomslocalized at R are added to yieldmacroscopic polarization, P(R,t). Thispolarization acts as a source in Maxwell'sequations for a field E(R,t). The loop iscompleted by the self-consistencyrequirement that the field assumed, E',is equal to the fieldproduced, E.Figure 3

\NVVV

should note that their semiclassical ex-planation of spontaneous emission runsinto conceptual difficulties for the caseof atoms excited to a single eigenstate,because the initial atomic dipole p ofequation (5) vanishes, resulting in in-finite lifetimes. One can argue that thisexcited state is metastable much like apencil standing on its point-that is,only a small fluctuation is required toget things started.9 As we shall see,the quantized field readily providessuch fluctuations. Furthermore, in thesemiclassical theory, the electroniccommutation relations1o are not neces-sarily preserved in time, and hence theuncertainty principle for the matter canbe violated. (See box on opposite page.)

Finally, and most importantly, thequantitative successesll of quantumelectrodynamics are so impressive thatwe are virtually compelled to quantizethe field as well as the atoms. In viewof these facts, we now turn to thephoton concept as it is embodied in thequantum theory of radiation.

The quantum theory of radiationIn 1927, P. A. M. Dirac12 quantized

the radiation field in addition to theatom, and the photon concept wasfor the first time placed on a logicalfoundation. We outline here the quan-tum theory of radiation in a form suit-able for our purposes. (The present

Upper-stateatomicpopulation

treatment, which uses E and B insteadof the vector potential, follows that ofreference 2.) For simplicity, we considera one-dimensional cavity of length Lthat has perfectly reflecting mirrors.We take the electric and magnetic fieldsto be polarized in the z and x directionsrespectively and to be single modes ofthe cavity, as shown in figure 6. Therewe see that the electric and magneticfields act as position and momentum co-ordinates. The corresponding energy inthe cavity is given by the volume inte-gral of the electric and magnetic fielddensities:

s [foE2 + /-lOH2]:JC = 2 d(volume)

p2 + fl2q2

2

which is just the energy of a simpleharmonic oscillator for a particle oscil-lating with frequency fl, mass M andspring constant Mfl2. A more generalmultimode field is represented by a col-lection of such oscillators, one for eachmode. To quantize the field (that is,to introduce the photon or particle na-ture of the radiation) we treat the elec-tric-field "position" coordinate q andthe magnetic-field "momentum" paccording to the laws of quantum me-chanics. We require the commutationrelations [q,p] = ih; [q,q] = [P,p] = o.

1Iscatt ;::::-

-.---- r

\NVVV

Resonance fluorescence. Inpart (a)incidentlightis scattered byan atom witha lifetime1/'Y. Inpart (b) the incidentlighthas a spectrum centered at frequencyv and corresponds to a wavetrainofduration 1If. Scattered lighthas width1If, is centered at v and lasts for time1If; 'Yis much greater than f.Figure4.

Our single-mode field is then describedby the quantum-mechanical wave func-tion

y;(q,t) = L Cn(t) cPn(q) (10)n=O

where ICnl2 is the probability that theradiation oscillator is excited to thenth-energy eigenstate characterized bythe eigenfunction cPn(q) (the usual Her-mite polynomial multiplied by a Gaus-sian) and having energy hfl(n + 1/2).This n-quantum state is said to be the"n-photon" state; that is, cPo(q) hasno photons (the vacuum), cPl(q) has onephoton, and so on. We note that the in-troduction of the wave function y;(q, t)(Schrodinger picture) or equivalentlythe noncommutativity of the operatorsp and q (Heisenberg picture) has the

L,g>

Electric and magnetic fields ina one-dimensional cavity, polarized in the z and xdirections respectively. Part (a) showssingle-mode standing waves proportionalto coordinates q and P. withconstants ofproportionality (Y and {3respectively. Inpart (b) we see the simple harmonicoscillator energy-level diagram resultingfrom quantization of the field. The nthlevel of the quantized oscillator, rPn(q).corresponds to the state having n "photons,"while the vacuum is associated with <Po(q)·Figure 6

Photoelectric effect. Anincidentelectromagnetic field interacts with asystem in its ground state Ig), causingtransitions to occur to excited states Ik).that is. ejecting an electron.Figure 5.

effect of bringing out the wave side of"particles" (say, electrons) and theparticle side of "waves" (say, electricwaves).

The q and p operators serve to showthat the single-mode electromagneticfield is dynamically equivalent to asimple harmonic oscillator. A moreconvenient and physically revealing setof operators is the annihilation oper-ator a ex Hq + ip and its adjoint at, thecreation operator. As their names sug-gest. these operators annihilate and cre-ate photons when acting on photonnumber states; in other words, a(at)lowers (raises) rPn to <Pn-l (<Pn+1)'They are not Hermitian and hence donot themselves represent observables.However, the electric field is given bythe Hermitian combination

where {; is the electric field "perphoton" and the Hamiltonian is

X = hHla+a + [a,a+]/21 (12)= hH(a+a + 1/2)

We emphasize that it is the introduc-tion of the commutation relations [q,p]= ih, or equivalently [a,a+] = 1, thatleads to the photon concept.

The first thing to note about thequantized field is that it has fluctua-tions, even in the absence of "photons."In fact, denoting the vacuum state (0photons) by 10), we find the Hamil-tonian, equation (12), has the "zero-point" expectation value (OIX 10) =h!2/2, the electric field of equation (11)has vanishing expectation value, butthat the vacuum average of the fieldsquared is

Thus the field has fluctuations about avanishing mean in the vacuum. Thezero-point energy hH/2 is given by avolume integral of (0IE210) and istherefore called the "energy" of thevacuum fluctuations. We shall out-line the success of these considerations in

Time evolution of the expectation value(E) of the electric field operator, andvariance !:;.E indicated by error barsassociated with the minimum uncertaintywave packet, are shown in part (a).Part (b) shows the time evolution of awave packet with minimum uncertainties!:;.Eand !:;.H.Figure 7

accounting for the statistical fluctua-tions of light quanta, spontaneous emis-sion, the Lamb shift, and so forth, in afew paragraphs below. However, let usfirst find out how we regain the classicalfield, equation (2), from the quantized(photon) field corresponding to the ap-propriate state vector in equation (10).

We often hear that large quantumnumbers correspond to the classicallimit. This is a misleading point of theview here, for the expectation value ofthe field in an n-photon state In) van-ishes, and this fact is true even for n ~00. The actual classical limit consists ofa superposition of photon states, and thisfact naturally leads us to a discussion ofphoton statistics. Essentially we de-sire a state of the field 11f( t» thatyields the classical field of equation (2)for the expectation value (E), thesquare of equation (2) for (E2), andso on-that is, a field with precise ampli-tude and phase. But we must recallthat the electric and magnetic fields cor-respond to position and momentum,which obey the uncertainty principle,so that

The best we can do is to take the mini-mum uncertainty case (for all time)for which equality in equation (14)holds. This is described by the co-herent (particle) packet13

la) = exp (-a*a/2) L (an/vnrTln) (15)n=O

This state is the eigenstate of the an-nihilation operator a with eigenvaluea. (If statements such as this turnthe reader off, we invite him to anni-hilate them from his copy.) We see infigure 7 that the probability density forthis state, l(qla)i2, oscillates back andforth in the harmonic oscillator wellwithout change in shape; that is, itcoheres. The amplitude of the classi-cal field, equation (2), is related to thecomplex constant a and the electric

A

field "per photon" S by Eo/2 = Sial.We see that this "most classical"

state is not a single-photon numberstate, but rather a superposition withthe Poisson probability of having nphotons given by

Pn = exp (-aa*)(aa*)n/n! (16)

The average photon number (n) isthus lal2, from which we see that the i~-tensity ( 8jal>2o: (n)M!. Equation (16)

<E(l»

<E(l» ± AE

defines the photon statistical distribu-tion for the coherent state. It is inter-esting to compare it with that for ther-mal radiation and that for a laser14 asshown in figure 8.

It is perhaps worthwile to note that1he distinction between the thermal andcoherent distributions is by no meansmerely academic, for the thermal im-plies a Hanbury-Brown-Twiss correla-tion, while the coherent does not. This

ca.>-I-::Jm 0.02«CDoa:Cl.

0.00o

correlation is the measure of the excessprobability of finding double photoelec-tron emission over that given by a purelyrandom sequence of events such as rain-drops on a roof. In fact, the probabilityof double photoelectron excitation istwice as large for the "single-frequency"blackbody distribution as for the purelycoherent, Poisson, distribution. It isfor this reason that photoelectrons pro-duced by a purely coherent light beamare said to be completely uncorrelated-there is no bunching. In general a lightbeam is completely characterized notjust by its spectral density, but by allhigher order correlations as well.

For many problems of interest, it isconvenient to expand the radiation state(or density operator) in terms of thecoherent states la) instead of the num-ber states. This is accomplished13 interms of the P(a) "distribution" de-fined by the density-operator expansion

p=fd2aP(a)la}(al (17)

where integration is carried out over thecomplex a plane. For the coherent statelao), P(a) = (J2(a - ao). For "single-mode" thermal radiation

P(a) = exp (-jaI2 j(n» (18)

P(a) gives one a measure of deviationfrom the classical state.

We now have at our disposal a simplemethod to show the validity of Ein-stein's15 correct (but not immediatelyaccepted 16) interpretation of the fluc-tuations in the blackbody spectrum.

He claimed that, although the Plancklaw could be accounted for with a clas-sical field (as Planck originally derivedit), the energy fluctuations (~:JC)2con-tained a part due to the wave nature oflight plus a part due to its particlecharacter. His formula is (in our no-tation)

(~:JC)2= Xhn + :JC2jg(n) (19)

where the average energyX = hn(n)g(n) (20)

and g(n) is the density-of-states fac-tor. He identified the first term inequation (19) with the particle characterand the second with the wave. Noteveryone agreed. In fact, using the in-formation the3 (1913) available, Wil-helm Wien16 argued that there was nospecial reason to attribute the fluctua-tions to two separate causes (particleand wave).

We see that Einstein was indeedcorrect, by using the p(a) distribu-tion of equation (18), to calculate therequired averages as the following two-line derivation indicates. We note that(:JC) hn(n) (drol?1~inl{the hnj2which ultimately cancels equation (19»and find(:JC2)= (hn)2 f d2a exp (-laI2j(n» x

(alat aat ala)

= (hn)2 f d2a exp (-jaI2 j(n» x(alat a[a,atl + at at aala)

= (n) (hH)2 + 2(hn)2 (n)2 (21)

in which the first term resulted from the

Photon statistical distributions comparedfor filtered blackbody (part a), laser(part b) and purely coherent (part c)light beams.Figure 8

commutation relation [a,atl = 1, thatis, from the quantum character of thefield, and the second from the wavecharacter of a classical average over in-tensity. Calculating the mean-square-deviation density, (~:JC)2 = [(:JC2)-(:JC)21g(n), we find Einstein's formula,equation (19). We see that he cor-rectly identified the particle and wavecontributions, a noteworthy feat and atribute to his insight inasmuch as thequantum theory of radiation was notdeveloped until twenty years later!

As mentioned in our semiclassicaldiscussion, the quantum theory ofradiation accounts neatly for sponta-neous emission.17 To see this, we use theelectric-field operator, equation (11),in the electric-dipole perturbation ener-gy of equation (4). Using the FermiGolden Rule we find the spontaneoustransition rate (inverse of atomic life-time)

/' = (erab)2n3 j(hrrc3) (22)

The emitted radiation is not perfectlymonochromatic, for the exponential de-cay implied by equation (22) yields aLorentzian frequency profile with width2/'. We note that the sum over finalstates of the absolute value squared ofequation (21), which enters the GoldenRule, is, in fact, proportional to thevacuum expectation value of E2; that is,the vacuum fluctuations "stimulate"the atom to emit spontaneously.

Perhaps the greatest triumph of thephoton concept is the explanation of theLamb shift18 between, for example, the2S1/2and 2p1/2 levels in a hydrogenicatom. According to the relativistic Dir-ac theory these levels have the sameenergy, in contradiction of the experi-mentally observed frequency splitting of1057.8MHz. We can understand theshift intuitively19 by picturing theelectron forced to fluctuate about its"Dirac" position because of the fluctu-ating vacuum field. Its average dis-placement (~r), is zero, but thesquared displacement has a small posi-tive value from this mean position,«~r)2). This deviation may changethe potential energy the electron ex-periences in the Coulomb field of thenucleus. To determine how much, weexpand the energy in a second-orderTaylor series. Noting that the first-order term vanishes in the average over

fluctuations, that the fluctuations areisotropic, and that "'12(I/r) = o(r), wefind the energy shift(LlV)f1uet = (V(r + Llr) - V(r)f1uet

= (1/2)"'12( V)(1/3) «Llr)2)f1uet= (l /6)e2o(r)( (Llr)2)f1uel (23)

We may now calculate the shift of theenergy eigenvalues hWnlm by calculat-ing the matrix element S d3runlm *(Ll V)rluetUnlm. Because only s states·have nonzero probabilities for being at r= 0, only these states are shifted (inthis approximation). Computation of«.:lr)2) requires more discussion, andwe refer the reader again to the texts fora complete discussion. However weemphasize that «Llr)2) is nonzero onlybecause of the (l/2)[a,atlhH vacuumfluctuations and is a direct consequenceof the quantized field; that is [a,at]r" O. The changes in potential energyaccount for 1040 MHz of the 1057.8-MHz shift observed between the 2S1/2and 2pl12 states in atomic hydrogen.When various relativistic correctionsand infinities are taken care of, thetheory agrees beautifully with the ex-perimental results and provides an im-pressive confirmation of the quantumtheory of radiation.

The anomalous magnetic momentof the electron20 is more difficult tointerpret in simple physical terms be-cause the origin of the spin is buried inthe relativistic theory of the electron.Nevertheless, the origin is due to themodification of circulating electroniccurrents (and hence the magneticmoment), as they are affected by the

References

1. M. Sargent III, M. O. Scully, W. E.Lamb Jr, Quantum Electronics, to bepublished.

2. G. N. Lewis, Nature 1I8, 874 (1926).3. Genesis, 1. (For the purists only; it was

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5. W. E. Lamb Jr, M. O. Scully, Polar-ization Matter and Radiation (Jubileevolume in honor of Alfred Kastler),Presses Univ. de France, Paris (1969).

6. P. A. Franken, "Collisions of Light withAtoms," in Atomic Physics (B. Beder-son, V. W. Cohen, F. M. J. Pichanick,eds.) Plenum, New York (1969), page377.

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7a. N. Bloembergen, Nonlinear Optics,Benjamin, New York (1965).

7b. W. E. Lamb Jr, Phys. Rev. 134, A1429(1964). For more recent developmentssee M. Sargent III, M. O. Scully, "Phys-ics of Laser Operation," Chapter 2 inLaser Handbook, (F. T. Arecchi, E. O.Schultz-DuBois, eds.), North-Holland,

fluctuating electromagnetic vacuumfields and vacuum polarization.

We conclude with a couple of re-marks concerning the so-called wave-particle duality and its effect on inter-ference phenomena. From Newton toHuygens to the present, this point hasfascinated and often confounded scien-tists. Even such an outstanding opticstext as Arnold Sommerfeld's21 con-tains opaque remarks in this regardsuch as ". . . the photon theory, atleats in its present state of develop-ment, is unable to account precisely forpolarization and interference phenom-ena." Indeed, the "photon theory," asembodied in the quantum theory ofradiation, does very well even in thesecases. For it is the normal-modefunctions Uk(r) that describe inter-ference phenomena in terms of nodal(dark) and antinodal (bright) regions ofspace. These functions are the samefor both classicial and quantum fields.Hence there is no need to switch fromquantum to classical descriptions or tointroduce a mysterious wave-particledualism in order to explain interfer-ence and diffraction. This point ismade clear in Fermi's article22 on thequantum theory of radiation. He doesa Lippman-fringe calculation in whichlight is emitted from one atom, strikesa mirror perpendicular to its directionof propagation, and is absorbed by asecond atom. The calculation showsthat the probability of excitation of thesecond atom varies periodically with itsdistance from the mirror because ofinterference between the incoming andoutgoing light. Fermi comments that

Amsterdam (1972).7c. S. L. McCall, E. L. Hahn, Phys. Rev.

183, 457 (1969). See also G. L. LambJr, Rev. Mod. Phys. 43, 99 (1971).

7d. 1. D. Abella, N. A. Kurnit, S. R. Hart-man, Phys Rev. 141, 391 (1966). Seealso H. Pendleton, in Proceedings of theInternational Conference on the Phys-ics of Quantum Electronics, (P. L.Kelly, B. Lax, P. E. Tannenwald, eds.),McGraw-Hill, New York (1965), page822.

8. M. D. Crisp, E. T. Jaynes, Phys. Rev.179, 1253(1969).

8a. G. Wentzel, Z. Physik 41,828 (1927).8b. O. Klein, Y. Nishina, A. Physik 52,

853 (1929).8c. O. Klein, Z. Physik 41, 407 (1927).8d. E. A. Uehling, Phys. Rev. 48, 55 (1935).9. M. Crisp, private communication.

10. M. Lax, Phys. Rev. 145, 1I0 (1966).11. R. P. Feynman, Phys. Rev. 76, 769

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"We may conclude that the results ofthe quantum theory of radiation de-scribe this phenomenon in exactly thesame way as the classical theory ofinterference. "

Recent interference experiments23,24involving independent light beams havebeen made possible by the availabilityof coherent laser sources. These mea-surements were largely stimulated byDirac's comment,211"Each photon theninterferes only with itself. Interferencebetween two different photons neveroccurs." The fact that interference be-tween independent lasers is observed isnot puzzling if we recall that the fringesare described by the normal modes ofthe system. Dirac's comment is con-sistent with this experiment in view ofthe fact that the photon is a quantizedexcitation of the normal modes of theentire system.

In conclusion: The photon conceptas contained in the quantum theory ofradiation provides the basis for ex-plaining all known electromagneticphenomena. However, the "fuzzy-ball" picture of a photon often leads tounnecessary confusion. Finally, mostquantum and electro-optical physicsis well understood and quantitativelyexplained semiclassically.

The authors wish to thank Willis E. Lamb Jrfor a careful reading of the manuscript andvaluable criticisms.

This work was supported by the US AirF'orce (Office of Scientific Research) and bythe US Air Force (Kirtland).

C. Cohen, Tannoudji, eds.), Gordon andBreach, New York (1965).

14. M. O. Scully, W. E. Lamb Jr, Phys.Rev. 159, 208 (1967).

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17. V. Weisskopf, E. P. Wigner, Z. Physik63,54 (1930).

18. W. E. Lamb Jr, R. C. Retherford, Phys.Rev. 72, 241 (1947).

19. T. A. Welton, Phys. Rev. 74, 1157(1948); see also V. Weisskopf, Rev.Mod. Phys. 21, 305 (1949).

20. S. Koenig, A. G. Prodell, P. Kusch,Phys. Rev. 88, 191 (1952).

21. A. Sommerfeld, Optics, Academic,New York (1964).

22. E. Fermi, Rev. Mod. Phys. 4, 87 (1932).23. L. Mandel, Nature 198, 255 (1963).24. A most useful collection of papers con-

cerning these topics is found in Co-herence and Fluctuations of Light,Vol. I and II (L. Mandel, E. Wolf, eds.),Dover, New York (1970).

25. P. A. M. Dirac, The Principles ofQuantum Mechanics, Oxford U.P.,London (1958), page 9. 0