1.3.1 The Planck distribution

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

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

the following assumptions.

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

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

dard laws of statistical mechanics.

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

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

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

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

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

¾½

Are photons necessary?

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

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

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

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

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

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

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

argument.

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

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

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

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

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

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

Hamiltonian is

Hint = ’d · E , (1.30)

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

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

leads to

2

π |d12 |

B1’2 = , (1.31)

2

30

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

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

is independently veri¬ed.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1.3.2 The photoelectric e¬ect

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

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

¾¾ The quantum nature of light

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

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

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

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

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

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

result”reviewed in Appendix C.6”is

e

Hint = ’ A (r, t) · p , (1.32)

m

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

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

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

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

and it determines the radiation ¬eld by

‚A

E=’ , B = ∇—A. (1.33)

‚t

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

1

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

ω

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

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

2π

| f |Hint | i |2 δ ( ’ ’ ω)

Wf i = (1.35)

f i

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

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

result is valid for observation times t

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

¾¿

Are photons necessary?

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

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

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

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

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

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

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

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

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

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

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

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

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

1.3.3 Compton scattering

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

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

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

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

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

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

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

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

o

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

in this calculation.

1.3.4 Conclusions

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

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

¾ The quantum nature of light

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

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

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

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

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

that bear on this question.

1.4 Indivisibility of photons

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

depend on the indivisibility of individual photons.

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

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

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

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

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

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

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

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

5

Dν

2

Fig. 1.10 The lowest three energy levels of

Dν

the calcium atom allow the cascade of two suc-

cessive transitions, in which two photons hν1

and hν2 are emitted in rapid succession. The

5

intermediate level has a short lifetime of 4.7 ns.

¾

Indivisibility of photons

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

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

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

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

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

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

photon pairs.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

the existence of individual quanta of the electromagnetic ¬eld.

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

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

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

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

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

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

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

semiclassical theory of the photoelectric e¬ect.

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

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

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

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

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

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

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

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

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

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

gate width w.

¾ The quantum nature of light

N1 Nrefl

D

PMrefl

Ncoinc

PMgate

ν2

ν1

D

D S

BS

Ntrans

PMtrans

gated

gate pulse,

counters

width w

Fig. 1.11 The photon-indivisibility experiment of Grangier, Roger, and Aspect. The detec-

tion of the ¬rst burst of light, of frequency ν1 , of a calcium-atom cascade produces a gate

pulse of width w during which the outputs of the photomultipliers PMtrans and PMre¬‚ de-

tecting the second burst of light, of frequency ν2 , are recorded by the gated counters. The

™ ™

rate of gate openings is Ngate = N1 . The probabilities of detection during the gate openings

™ ™ ™ ™ ™ ™

are ptrans = Ntrans /N1 , pre¬‚ = Nre¬‚ /N1 for singles, and pcoinc = Ncoinc /N1 for coincidences.

(Adapted from Grangier et al. (1986).)

If a burst of radiation at ν1 has been detected, the burst of radiation of frequency ν2

from the second transition is necessarily directed toward the beam splitter BS, which

partially re¬‚ects and partially transmits the light falling on it. The two beams pro-

duced in this way are directed toward the two photomultipliers PMre¬‚ and PMtrans .

The outputs of PMre¬‚ and PMtrans are used to drive the gated counters Nre¬‚ and

Ntrans , which record every pulse from the two photomultipliers, and also to drive a co-

incidence counter Ncoinc , which responds only when both of these two photomultipliers

produce current pulses simultaneously within the speci¬ed open-gate time interval w.

Therefore, the probabilities for the individual counters to ¬re (singles probabilities) are

™ ™ ™ ™ ™ ™

given by pre¬‚ = Nre¬‚ /Ngate and ptrans = Ntrans /Ngate , where Ngate ≡ N1 is the rate

™ ™

of gate openings”the count rate of photomultiplier PMgate ”and Nre¬‚ and Ntrans are

™

the count rates of PMre¬‚ and PMtrans , respectively. The coincidence rate Ncoinc is the

rate of simultaneous ¬rings of both detectors PMre¬‚ and PMtrans during the open-gate

™ ™

interval w; consequently, the coincidence probability is pcoinc = Ncoinc/Ngate . The ex-

™ ™ ™

periment consists of measuring the singles counting rates Ngate , Nre¬‚, Ntrans , and the

™

coincidence rate Ncoinc .

According to Einstein™s photon model of light, each atomic transition produces

a single quantum of light which cannot be subdivided. An indivisible quantum with

energy hν2 which has scattered from the beam splitter can only be detected once.

Therefore it must go either to PMre¬‚ or to PMtrans ; it cannot go to both. In the

absence of complicating factors, the photon model would predict that the coincidence

probability pcoinc is exactly zero. Since this is a real experiment, complicating factors

are not absent. It is possible for two di¬erent atoms inside the source region S to emit

two quanta hν2 during the open-gate interval, and thereby produce a false coincidence

count. This di¬culty can be minimized by choosing the gate interval w „ , where

„ is the lifetime of the intermediate level in the cascade, but it cannot be completely

¾

Indivisibility of photons

removed from this experimental arrangement.

Only three general features of semiclassical theories are needed for the analysis of

this experiment: (1) the atom is described by quantum mechanics; (2) each atomic

transition produces a burst of radiation described by classical ¬elds; (3) the photomul-

tiplier current is proportional to the intensity of the incident radiation. The ¬rst two

features are part of the de¬nition of a semiclassical theory, and the third is implied

by the semiclassical analysis of the photoelectric e¬ect. The beam splitter will convert

the classical radiation from the atom into two beams, one directed toward PMre¬‚ and

the other directed toward PMtrans . Therefore, according to the semiclassical theory,

the coincidence probability cannot be zero”even in the absence of the false counts

discussed above”since the classical electromagnetic wave must smoothly divide at the

beam splitter. The semiclassical theory predicts a minimum coincidence rate, which

is proportional to the product of the re¬‚ected and transmitted intensities. The in-

stantaneous intensities falling on PMre¬‚ and PMtrans are proportional to the original

intensity falling on the beam splitter, and the gated measurement e¬ectively averages

over the open-gate interval. Thus the photocurrents produced in the nth gate interval

are proportional to the time averaged intensity at the beam splitter:

tn +w

1

In = dtI (t) , (1.36)

w tn

where the gate is open in the interval (tn , tn + w). The atomic transitions are described

by quantum mechanics, so they occur at random times within the gate interval. This

means that the intensities In exhibit random variations from one gate interval to

another. In order to minimize the e¬ect of these ¬‚uctuations, the counting data from a

sequence of gate openings are averaged. Thus the singles probabilities are determined

from the average intensity

Mgate

1

I= In , (1.37)

Mgate n=1

where Mgate is the total number of gate openings. The singles probabilities are given

by

pre¬‚ = ·re¬‚ w I , ptrans = ·trans w I , (1.38)

where ·re¬‚ is the product of the detector e¬ciency and the fraction of the original

intensity directed to PMre¬‚ and ·trans is the same quantity for PMtrans . Since the

coincidence rate in a single gate is proportional to the product of the instantaneous

photocurrents from PMre¬‚ and PMtrans , the coincidence probability is proportional to

the average of the square of the intensity:

pcoinc = ·re¬‚ ·trans w2 I 2 , (1.39)

with

Mgate

1

2 2

I = In . (1.40)

Mgate n=1

2

By using the identity (I ’ I ) 0 it is easy to show that

¾ The quantum nature of light

2

I2 I , (1.41)

which combines with eqns (1.38) and (1.39) to yield

pcoinc pre¬‚ ptrans . (1.42)

This semiclassical prediction is conveniently expressed by de¬ning the parameter

™ ™

pcoinc Ncoinc Ngate

±≡ = 1, (1.43)

™ ™

pre¬‚ ptrans Nre¬‚ Ntrans

where the latter inequality follows from eqn (1.42). With the gate interval set at

™

w = 9 ns, and the atomic beam current adjusted to yield a gate rate Ngate = 8800

counts per second, the measured value of ± was found to be ± = 0.18 ± 0.06. This

violates the semiclassical inequality (1.43) by 13 standard deviations; therefore, the

experiment decisively rejects any theory based on the semiclassical treatment of emis-

sion. These data show that there are strong anti-correlations between the ¬rings of