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semiclassical model.

1.3.1 The Planck distribution
This seems to be the simplest of the experiments under consideration, but ¬nding a
semiclassical explanation turns out to involve some subtle issues. Suppose we make
the following assumptions.
(a) The electromagnetic ¬eld is described by the classical form of Maxwell™s equations.
(b) The electromagnetic ¬eld is an independent physical system subject to the stan-
dard laws of statistical mechanics.
With both assumptions in force the equipartition argument in Section 1.1.1 inevitably
leads to the Rayleigh“Jeans distribution and the ultraviolet catastrophe. This is phys-
ically unacceptable, so at least one of the assumptions (a) or (b) must be abandoned.
At this point, Planck chose the rather risky alternative of abandoning (b), and Einstein
took the even more radical step of abandoning (a).
¾½
Are photons necessary?

Our task is to ¬nd some way of retaining (a) while replacing Planck™s ad hoc
procedure by an argument based on a quantum mechanical description of the atoms
in the cavity wall. There does not seem to be a completely satisfactory way to do this,
so a rough plausibility argument will have to su¬ce. We begin by observing that the
derivation of the Planck distribution in Section 1.2.2-B does not explicitly involve the
assumption that light is composed of discrete quanta. This suggests that we ¬rst seek
a semiclassical origin for the A and B coe¬cients, and then simply repeat the same
argument.
The Einstein coe¬cients B1’2 (for absorption) and B2’1 (for stimulated emission)
can both be evaluated by applying ¬rst-order, time-dependent perturbation theory”
which is reviewed in Section 4.8.2”to the coupling between the atom and the classical
electromagnetic ¬eld. In both processes the electron remains bound in the atom, which
is small compared to typical optical wavelengths. Thus the interaction of the atom
with the classical ¬eld can be treated in the dipole approximation, and the interaction
Hamiltonian is
Hint = ’d · E , (1.30)
where d is the electric dipole operator, and the ¬eld is evaluated at the center of mass
of the atom. Applying the Fermi-golden-rule result (4.113) to the absorption process
leads to
2
π |d12 |
B1’2 = , (1.31)
2
30
where d12 is the matrix element of the dipole operator. A similar calculation for stimu-
lated emission yields the same value for B2’1 , so the equality of the two B coe¬cients
is independently veri¬ed.
The strictly semiclassical theory used above does not explain spontaneous emis-
sion; instead, it predicts A = 0. The reason is that the interaction Hamiltonian (1.30)
vanishes in the absence of an external ¬eld. If no external ¬eld is present, an atom in
any stationary state”including all excited states”will stay there permanently. On the
other hand, spontaneous emission is not explained in Einstein™s photon model either;
it is built in by assumption at the beginning. Since the present competition is with the
photon model, we are at liberty to augment the strict semiclassical theory by simply
assuming the existence of spontaneous emission. With this assumption in force, Ein-
stein™s rate arguments (eqns (1.10)“(1.21)) can be used to derive the ratio A/B. Note
that these equations refer to transition rates within the two-level atom; they do not
require the concept of the photon. Combining this with the independently calculated
value of B1’2 given in eqn (1.31) yields the correct value for the A coe¬cient. This
line of argument is frequently used to derive the A coe¬cient without bringing in the
full blown quantum theory of light (Loudon, 2000, Sec. 1.5).
The extra assumptions required to carry out this semiclassical derivation of the
Planck spectrum may make it appear almost as ad hoc as Planck™s argument, but it
does show that the photon model is not strictly necessary for this purpose.

1.3.2 The photoelectric e¬ect
By contrast to the derivation of the Planck spectrum, Einstein™s explanation of the
photoelectric e¬ect depends in a very direct way on the photon concept. In this case,
¾¾ The quantum nature of light

however, the alternative description using the semiclassical theory turns out to be much
more straightforward. For this calculation, the electrons in the metal are described by
quantum mechanics, and the light is described as an external classical ¬eld. The total
electron Hamiltonian is therefore H = H0 + Hint , where H0 is the Hamiltonian for an
electron in the absence of any external electromagnetic ¬eld and Hint is the interaction
term. For a single electron in a weak external ¬eld, the standard quantum mechanical
result”reviewed in Appendix C.6”is
e
Hint = ’ A (r, t) · p , (1.32)
m
where r and p are respectively the quantum operators for the position and momentum.
In the usual position-space representation the action of the operators is rψ (r) =
rψ (r) and pψ (r) = ’i ∇ψ (r). The c-number function A (r, t) is the classical vector
potential”which can be chosen to satisfy the radiation-gauge condition ∇ · A = 0”
and it determines the radiation ¬eld by
‚A
E=’ , B = ∇—A. (1.33)
‚t
For a monochromatic ¬eld with frequency ω, the vector potential is
1
A (r, t) = E0 e exp (ik · r ’ ωt) + CC , (1.34)
ω
where e is the unit polarization vector, E0 is the electric ¬eld amplitude, |k| = ω/c,
and e · k = 0. Another application of Fermi™s golden rule (4.113) yields the rate

| 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

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

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