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photon indivisibility experiment has been replaced by spontaneous down-conversion.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