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photomultipliers PMre¬‚ and PMtrans , when gated by the ¬rings of the trigger pho-
tomultiplier PMgate . An individual photon hν2 , upon leaving the beam splitter, can
cause either of the photomultipliers PMre¬‚ or PMtrans to ¬re, but these two possi-
ble outcomes are mutually exclusive. This experiment convincingly demonstrates the
indivisibility of Einstein™s photons.

1.5 Spontaneous down-conversion light source
In more recent times, the cascade emission of correlated pairs of photons used in the
photon indivisibility experiment has been replaced by spontaneous down-conversion. In
this much more convenient and compact light source, atomic beams”which require the
extensive use of inconvenient vacuum technology”are replaced by a single nonlinear
crystal. An ultraviolet laser beam enters the crystal, and excites its atoms coherently
to a virtual excited state. This is followed by a rapid decay into pairs of photons γ1
and γ2 , as shown in Fig. 1.12 and discussed in detail in Section 13.3.2. This process
may seem to violate the indivisibility of photons, so we emphasize that an incident
UV photon is absorbed as a whole unit, and two other photons are emitted, also as
whole units. Each of these photons would pass the indivisibility test of the experiment
discussed in Section 1.4.
Just as in the similar process of radioactive decay of an excited parent nucleus
into two daughter nuclei, energy and momentum are conserved in spontaneous down-
conversion. Due to a combination of dispersion and birefringence of the nonlinear

Nonlinear
crystal
γ1
Fig. 1.12 The process of spontaneous down-
γ0
conversion, γ0 ’ γ1 + γ2 by means of a non-
γ2
linear crystal.
¾
The quantum theory of light

crystal, the result is a highly directional emission of light in the form of a rainbow of
many colors, as seen in the jacket illustration.
The uniquely quantum feature of this rainbow is the fact that pairs of photons
emitted on opposite sides of the ultraviolet laser beam, are strongly correlated with
each other. For example, the detection of a photon γ1 by a Geiger counter placed
behind pinhole 1 in Fig. 1.12 is always accompanied by the detection of a photon γ2
by a Geiger counter placed behind pinhole 2. The high directionality of this kind of
light source makes the collection of correlated photon pairs and the measurement of
their properties much simpler than in the case of atomic-beam light sources.

1.6 Silicon avalanche-photodiode photon counters
In addition to the improved light source discussed in the previous section, solid-state
technology has also led to improved detectors of photons. Photon detectors utilizing
photomultipliers based on vacuum-tube technology have now been replaced by much
simpler solid-state detectors based on the photovoltaic e¬ect in semiconductor crystals.
A photon entering into the crystal produces an electron“hole pair, which is then pulled
apart in the presence of a strong internal electric ¬eld. This ¬eld is su¬ciently large
so that the acceleration of the initial pair of charged particles produced by the photon
leads to an avalanche breakdown inside the crystal, which can be thought of as a chain
reaction consisting of multiple branches of impact ionization events initiated by the
¬rst pair of charged particles. This mode of operation of a semiconductor photodiode
is called the Geiger mode, because of the close analogy to the avalanche ionization
breakdown of a gas due to an initial ionizing particle passing through a Geiger counter.
Each avalanche breakdown event produces a large, standardized electrical pulse
(which we will henceforth call a click of the photon counter), corresponding to the
detection of a single photon. For example, many contemporary quantum optics ex-
periments use silicon avalanche photodiodes, which are single photon counters with
quantum e¬ciencies around 70% in the near infrared. This is much higher than the
quantum e¬ciencies for photomultipliers in the same wavelength region. The solid-
state detectors also have shorter response times”in the nanosecond range”so that
fast coincidence detection of the standardized pulses can be straightforwardly imple-
mented by conventional electronics. Another important practical advantage of solid-
state single-photon detectors is that they require much lower voltage power supplies
than photomultipliers. These devices will be discussed in more detail in Sections 9.1.1
and 9.2.1.

1.7 The quantum theory of light
In this chapter we have seen that the blackbody spectrum, the photoelectric e¬ect,
Compton scattering and spontaneous emission are correctly described by Einstein™s
photon model of light, but we have also seen that plausible explanations of these phe-
nomena can be constructed using an extended form of semiclassical electrodynamics.
However, no semiclassical explanation can account for the indivisibility of photons
demonstrated in Section 1.4; therefore, a theory that incorporates indivisibility must
be based on new physical principles not found in classical electromagnetism. In other
¿¼ The quantum nature of light

words, the quantum theory of light cannot be derived from the classical theory; in-
stead, it must be based on new conjectures.4 Fortunately, the quantum theory must
also satisfy the correspondence principle; that is, it must agree with the classical
theory for the large class of phenomena that are correctly described by classical elec-
trodynamics. This is an invaluable aid in the construction of the quantum theory. In
the end, the validity of the new principles can only be judged by comparing predictions
of the quantum theory with the results of experiments.
We will approach the quantum theory in stages, beginning with the electromag-
netic ¬eld in an ideal cavity. This choice re¬‚ects the historical importance of cavities
and blackbody radiation, and it is also the simplest problem exhibiting all of the
important physical principles involved. An apparent di¬culty with this approach is
that it depends on the classical cavity mode functions, which are de¬ned by boundary
conditions at the cavity walls. Even in the classical theory, these boundary condi-
tions are a macroscopic idealization of the properties of physical walls composed of
atoms; consequently, the corresponding quantum theory does not appear to be truly
microscopic. We will see, however, that the cavity model yields commutation relations
between ¬eld operators at di¬erent spatial points which suggest a truly microscopic
quantization conjecture that does not depend on macroscopic boundary conditions.

1.8 Exercises
1.1 Power emitted through an aperture of a cavity
Show that the radiative power per unit frequency interval at frequency ω emitted from
the aperture area σ of a cavity at temperature T is given by
1
P (ω, T ) = cρ (ω, T ) σ .
4
1.2 Spectrum of a one-dimensional blackbody
Consider a coaxial cable of length L terminated at either end with resistors of the same
small value R. The entire system comes into thermal equilibrium at a temperature T .
The dielectric constant inside the cable is unity. All you need to know about this
terminated coaxial cable is that the wavelength »m of the mth mode of the classical
electromagnetic modes of this cable is determined by the condition L = m»m /2, where
m = 1, 2, 3, . . ., and therefore that the frequency νm of the mth mode of the cable is
given by νm = m (c/2L).
(1) In the large L limit, derive the classical Rayleigh“Jeans law for this system. Is
there an ultraviolet catastrophe?
(2) Argue that the analysis in Section 1.2.2-B applies to this one-dimensional system,
so that eqn (1.19) is still valid. Combine this with the result from part (1) to
obtain the Planck distribution.
(3) Sketch the frequency dependence of the power spectrum, up to a proportionality
constant, for the radiation emitted by one of the resistors.
4 We
prefer ˜conjecture™ to ˜axiom™, since an axiom cannot be questioned. In physics there are no
unquestionable statements.
¿½
Exercises

(4) For a given temperature, ¬nd the frequency at which the power spectrum is a
maximum. Compare this to the corresponding result for the three-dimensional
blackbody spectrum.

1.3 Slightly anharmonic oscillator
Given the following Hamiltonian for a slightly anharmonic oscillator in 1D:

p2 1 1
+ mω 2 x2 + »m2 x4 ,
H=
2m 2 4
where the perturbation parameter » is very small.
(1) Find all the perturbed energy levels of this oscillator up to terms linear in ».
(2) Find the lowest-order correction to its ground-state wave function. (Hint: Use
raising and lowering operators in your calculation.)

1.4 Photoionization
A simple model for photoionization is de¬ned by the vector potential A and the
interaction Hamiltonian Hint given respectively by eqns (1.34) and (1.32).
Assume that the initial electron is in a bound state with a spherically symmetric
wave function r |i = φi (r) and energy i = ’ b (where b > 0 is the binding energy)
and that the ¬nal electron state is the plane wave r |f = L’3/2 eikf ·r (this is the
Born approximation).
(1) Evaluate the matrix element f |Hint | i in terms of the initial wave function φi (r).
(2) Carry out the integration over the ¬nal electron state, and impose the dipole
|k|”in eqn (1.35) to get the total transition rate in the
approximation”kf
limit ω b.
(3) Divide the transition rate by the ¬‚ux of photons (F = I0 / ω, where I0 is the
intensity of the incident ¬eld) to obtain the cross-section for photoemission.

1.5 Time-reversal symmetry applied to the time-dependent Schr¨dinger
o
equation
(1) Show that the time-reversal operation t ’ ’t, when applied to the time-dependent
Schr¨dinger equation for a spinless particle, results in the rule
o

ψ ’ ψ—

for the wave function.
(2) Rewrite the wave function in Dirac bra-ket notation explained in Appendix C.1,
and restate the above rule using this notation.
(3) In general, how does the scalar product for the transition probability amplitude
between an initial and a ¬nal state ¬nal| initial behave under time reversal?
2
Quantization of cavity modes

In Section 1.3 we remarked that both classical mechanics and quantum mechanics deal
with discrete sets of mechanical degrees of freedom, while classical electromagnetic
theory is based on continuous functions of space and time. This conceptual gap can be
partially bridged by studying situations in which the electromagnetic ¬eld is con¬ned
by material walls, such as those of a hollow metallic cavity. In such cases the classical
¬eld is described by a discrete set of mode functions. The formal resemblance between
the discrete cavity modes and the discrete mechanical degrees of freedom facilitates
the use of the correspondence-principle arguments that provide the surest route to the
quantum theory.
In order to introduce the basic ideas in the simplest possible way, we will begin by
quantizing the modes of a three-dimensional cavity. We will then combine the 3D cavity
model with general features of quantum theory to explain the Planck distribution and
the Casimir e¬ect.

2.1 Quantization of cavity modes
We begin with a review of the classical electromagnetic ¬eld (E, B) con¬ned to an
ideal cavity, i.e. a void completely enclosed by perfectly conducting walls.

2.1.1 Cavity modes
In the interior of a cavity, the electromagnetic ¬eld obeys the vacuum form of Maxwell™s
equations:
∇· E = 0, (2.1)

∇· B = 0, (2.2)

‚E
∇ — B = µ0 (Amp`re™s law) ,
e (2.3)
0
‚t
‚B
∇—E =’ (Faraday™s law) . (2.4)
‚t
The divergence equations (2.1) and (2.2) respectively represent the absence of free
charges and magnetic monopoles inside the cavity.1 The tangential component of the

1 As of this writing, no magnetic monopoles have been found anywhere, but if they are discovered
in the future, eqn (2.2) will remain an excellent approximation.
¿¿
Quantization of cavity modes

electric ¬eld and the normal component of the magnetic induction must vanish on the
interior wall, S, of a perfectly conducting cavity:

n (r) — E (r) = 0 for each r on S , (2.5)
n (r) · B (r) = 0 for each r on S , (2.6)

where n (r) is the normal vector to S at r.
Since the boundary conditions are independent of time, it is possible to force a
separation of variables between r and t by setting E (r, t) = E (r) F (t) and B (r, t) =
B (r) G (t), where F (t) and G (t) are chosen to be dimensionless. Substituting these
forms into Faraday™s law and Amp`re™s law shows that F (t) and G (t) must obey
e

dG (t) dF (t)
= ω1 F (t) , = ω2 G (t) , (2.7)
dt dt
where ω1 and ω2 are separation constants with dimensions of frequency. Eliminating
G (t) between the two ¬rst-order equations yields the second-order equation

dF (t)
= ω1 ω2 F (t) , (2.8)
dt
which has exponentially growing solutions for ω1 ω2 > 0 and oscillatory solutions for
ω1 ω2 < 0. The exponentially growing solutions are not physically acceptable; therefore,
we set ω1 ω2 = ’ω 2 < 0. With the choice ω1 = ’ω and ω2 = ω for the separation
constants, the general solutions for F and G can written as F (t) = cos (ωt + φ) and
G (t) = sin (ωt + φ).
One can then show that the rescaled ¬elds2 E ω (r) = 0 / ωE (r) and B ω (r) =

B (r) / µ0 ω satisfy
∇ — E ω (r) = kBω (r) , (2.9)

∇ — Bω (r) = kE ω (r) , (2.10)

where k = ω/c. Alternately eliminating E ω (r) and Bω (r) between these equations
produces the Helmholtz equations for E ω (r) and Bω (r):

∇2 + k 2 E ω (r) = 0 , (2.11)

∇2 + k 2 Bω (r) = 0 . (2.12)

A The rectangular cavity
The equations given above are valid for any cavity shape, but explicit mode functions
can only be obtained when the shape is speci¬ed. We therefore consider a cavity in the
form of a rectangular parallelepiped with sides lx , ly , and lz . The bounding surfaces

2 Dimensional convenience is the o¬cial explanation for the appearance of in these classical
normalization factors.
¿ Quantization of cavity modes

are planes parallel to the Cartesian coordinate planes, and the boundary conditions
are
n — Eω = 0
on each face of the parallelepiped ; (2.13)
n · Bω = 0
therefore, the method of separation of variables can be used again to solve the eigen-
value problem (2.11). The calculations are straightforward but lengthy, so we leave
the details to Exercise 2.2, and merely quote the results. The boundary conditions can
only be satis¬ed for a discrete set of k-values labeled by the multi-index
πnx πny πnz
κ ≡ (k, s) = (kx , ky , kz , s) = , , ,s , (2.14)
lx ly lz
where nx , ny , and nz are non-negative integers and s labels the polarization. The
allowed frequencies
2 1/2
2 2
πnx πny πnz
ωks = c |k| = c + + (2.15)
lx ly lz

are independent of s. The explicit expressions for the electric mode functions are

E ks (r) = Ekx (r) esx (k) ux + Eky (r) esy (k) uy + Ekz (r) esz (k) uz , (2.16)

Ekx (r) = Nk cos (kx x) sin (ky y) sin (kz z) ,
Eky (r) = Nk sin (kx x) cos (ky y) sin (kz z) , (2.17)
Ekz (r) = Nk sin (kx x) sin (ky y) cos (kz z) ,

where the Nk s are normalization factors. The polarization unit vector,

es (k) = esx (k) ux + esy (k) uy + esz (k) uz , (2.18)

must be transverse (i.e. k · es (k) = 0) in order to guarantee that eqn (2.1) is satis¬ed.
The magnetic mode functions are readily obtained by using eqn (2.9).
Every plane wave in free space has two possible polarizations, but the number of
independent polarizations for a cavity mode depends on k. Inspection of eqn (2.17)
shows that a mode with exactly one vanishing k-component has only one polariza-
tion. For example, if k = (0, ky , kz ), then E ks (r) = Ekx (r) esx (k) ux . There are no
modes with two vanishing k-components, since the corresponding function would van-
ish identically. If no components of k are zero, then es can be any vector in the plane
perpendicular to k. Just as for plane waves in free space, there is then a polariza-
tion basis set with two real, mutually orthogonal unit vectors e1 and e2 (s = 1, 2).
If no components vanish, Nk = 8/V , but when exactly one k-component vanishes,
Nk = 4/V , where V = lx ly lz is the volume of the cavity. The spacing between the
discrete k-values is ∆kj = π/lj (j = x, y, z); therefore, in the limit of large cavities
(lj ’ ∞), the k-values become essentially continuous. Thus the interior of a su¬-
ciently large rectangular parallelepiped cavity is e¬ectively indistinguishable from free
space.
¿
Quantization of cavity modes

The mode functions are eigenfunctions of the hermitian operator ’∇2 , so they are
guaranteed to form a complete, orthonormal set. The orthonormality conditions

d3 rE ks (r) · E k s (r) = δkk δss , (2.19)
V


d3 rB ks (r) · Bk s (r) = δkk δss (2.20)
V
can be readily veri¬ed by a direct calculation, but the completeness conditions are
complicated by the fact that the eigenfunctions are vectors ¬elds satisfying the di-
vergence equations (2.1) or (2.2). We therefore consider the completeness issue in the
following section.

B The transverse delta function
In order to deal with the completeness identities for vector modes of the cavity, it is
useful to study general vector ¬elds in a little more detail. This is most easily done by
expressing a vector ¬eld F (r) by a spatial Fourier transform:

d3 k
F (r) = 3F (k) eik·r , (2.21)
(2π)
so that the divergence and curl are given by

d3 k
∇ · F (r) = i · F (k) eik·r
3k (2.22)
(2π)
and
d3 k
∇ — F (r) = i — F (k) eik·r .
3k (2.23)
(2π)
In k-space, the ¬eld F (k) is transverse if k · F (k) = 0 and longitudinal if k —
F (k) = 0; consequently, in r-space the ¬eld F (r) is said to be transverse if ∇·F (r) =
0 and longitudinal if ∇ — F (r) = 0. In this language the E- and B-¬elds in the cavity
are both transverse vector ¬elds.
Now suppose that F (r) is transverse and G (r) is longitudinal, then an application
of Parseval™s theorem (A.54) for Fourier transforms yields

d3 k
— —
d rF (r) · G (r) = 3F (k) · G (k) = 0 .
3
(2.24)
(2π)
In other words, the transverse and longitudinal ¬elds in r-space are orthogonal in
the sense of wave functions. Furthermore, a general vector ¬eld F (k) can be decom-
posed as F (k) = F (k) + F ⊥ (k), where the longitudinal and transverse parts are
respectively given by
k · F (k)
F (k) = k (2.25)
k2
and
¿ Quantization of cavity modes


F ⊥ (k) = F (k) ’ F (k) . (2.26)
For later use it is convenient to write out the transverse part in Cartesian components:

Fi⊥ (k) = ∆⊥ (k) Fj (k) , (2.27)
ij

where
ki kj
∆⊥ (k) ≡ δij ’, (2.28)
ij
k2
and the Einstein summation convention over repeated vector indices is understood.
The 3 — 3-matrix ∆⊥ (k) is symmetric and k is an eigenvector corresponding to the
eigenvalue zero. This matrix also satis¬es the de¬ning condition for a projection op-
2
erator: ∆⊥ (k) = ∆⊥ (k). Thus ∆⊥ (k) is a projection operator onto the space of
transverse vector ¬elds.
The inverse Fourier transform of eqn (2.27) gives the r-space form

Fi⊥ (r) = d3 r∆⊥ (r ’ r ) Fj (r ) , (2.29)
ij
V

where
d3 k
∆⊥ ⊥
(r ’ r ) ≡ (k) eik·(r’r ) .
3 ∆ij (2.30)
ij
(2π)
The integral operator ∆⊥ (r ’ r ) reproduces any transverse vector ¬eld and annihi-
ij
lates any longitudinal vector ¬eld, so it is called the transverse delta function.
We are now ready to consider the completeness of the mode functions. For any
transverse vector ¬eld F , satisfying the ¬rst boundary condition in eqn (2.13), the
combination of the completeness of the electric mode functions and the orthonormality
conditions (2.19) results in the identity

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