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

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

Fi (r) = Fj (r ) .

d3 r (Eks (r))i (Eks (r ))j (2.31)

V ks

On the other hand, eqn (2.24) leads to

Gj (r ) = 0

d3 r (Eks (r))i (Eks (r ))j (2.32)

V ks

for any longitudinal ¬eld G (r). Thus the integral operator de¬ned by the expression

in curly brackets annihilates longitudinal ¬elds and reproduces transverse ¬elds. Two

operators that have the same action on the entire space of vector ¬elds are identical;

therefore,

(Eks (r))i (Eks (r ))j = ∆⊥ (r ’ r ) . (2.33)

ij

ks

A similar argument applied to the magnetic mode functions leads to the corresponding

result:

(Bks (r))i (Bks (r ))j = ∆⊥ (r ’ r ) . (2.34)

ij

ks

¿

Quantization of cavity modes

C The general cavity

Now that we have mastered the simple rectangular cavity, we proceed to a general

metallic cavity with a bounding surface S of arbitrary shape.3 As we have already

remarked, the di¬erence between this general cavity and the rectangular cavity lies

entirely in the boundary conditions. The solution of the Helmholtz equations (2.11)

and (2.12), together with the general boundary conditions (2.5) and (2.6), has been

extensively studied in connection with the theory of microwave cavities (Slater, 1950).

Separation of variables is not possible for general boundary shapes, so there is no

way to obtain the explicit solutions shown in Section 2.1.1-A. Fortunately, we only

need certain properties of the solutions, which can be obtained without knowing the

explicit forms. General results from the theory of partial di¬erential equations (Za-

uderer, 1983, Sec. 8.1) guarantee that the Helmholtz equation in any ¬nite cavity

has a complete, orthonormal set of eigenfunctions labeled by a discrete multi-index

κ = (κ1 , κ2 , κ3 , κ4 ) that replaces the combination (k, s) used for the rectangular cav-

ity. These normal mode functions E κ (r) and Bκ (r) are real, transverse vector

¬elds satisfying the boundary conditions (2.5) and (2.6) respectively, together with

the Helmholtz equation:

∇2 + kκ E κ = 0 ,

2

(2.35)

∇2 + kκ Bκ = 0 ,

2

(2.36)

where kκ = ωκ /c and ωκ is the cavity resonance frequency of mode κ. The allowed val-

ues of the discrete indices κ1 , . . . , κ4 and the resonance frequencies ωκ are determined

by the geometrical properties of the cavity.

By combining the orthonormality conditions

d3 rE κ · E » = δκ» ,

V

(2.37)

d rBκ · B» = δκ»

3

V

with the completeness of the modes, we can repeat the argument in Section 2.1.1-B

to obtain the general completeness identities

Eκi (r) Eκj (r ) = ∆⊥ (r ’ r ) , (2.38)

ij

κ

Bκi (r) Bκj (r ) = ∆⊥ (r ’ r ) . (2.39)

ij

κ

D The classical electromagnetic energy

Since the cavity mode functions are a complete orthonormal set, general electric and

magnetic ¬elds”and the associated vector potential”can be written as

3 Theterm ˜arbitrary™ should be understood to exclude topologically foolish choices, such as re-

placing the rectangular cavity by a Klein bottle.

¿ Quantization of cavity modes

1

E (r, t) = ’ √ Pκ (t) E κ (r) , (2.40)

0 κ

√

B (r, t) = µ0 ωκ Qκ (t) Bκ (r) , (2.41)

κ

1

A (r, t) = √ Qκ (t) E κ (r) . (2.42)

0 κ

Substituting the expansions (2.40) and (2.41) into the vacuum Maxwell equations

(2.1)“(2.4) leads to the in¬nite set of ordinary di¬erential equations

™ ™

Qκ = Pκ and Pκ = ’ωκ Qκ .

2

(2.43)

For each mode, this pair of equations is mathematically identical to the equations of

motion of a simple harmonic oscillator, where the expansion coe¬cients Qκ and Pκ

respectively play the roles of the oscillator coordinate and momentum. On the basis

of this mechanical analogy, the mode κ is called a radiation oscillator, and the set

of points

{(Qκ , Pκ ) for ’∞ < Qκ < ∞ and ’∞ < Pκ < ∞} (2.44)

is said to be the classical oscillator phase space for the κth mode.

For the transition to quantum theory, it is useful to introduce the dimensionless

complex amplitudes

ωκ Qκ (t) + iPκ (t)

√

±κ (t) = , (2.45)

2 ωκ

which allow the pair of real equations (2.43) to be rewritten as a single complex

equation,

±κ (t) = ’iωκ ±κ (t) ,

™ (2.46)

with the general solution ±κ (t) = ±κ e’iωκ t , ±κ = ±κ (0). The expansions for the ¬elds

can all be written in terms of ±κ and ±— ; for example eqn (2.40) becomes

κ

ωκ

±κ e’iωκ t E κ (r) + CC .

E (r, t) = i (2.47)

20

κ

One of the chief virtues of the expansions (2.40) and (2.41) is that the orthogonality

relations (2.37) allow the classical electromagnetic energy in the cavity,

1

+ µ’1 B2 ,

Uem = 0E

2

d3 r (2.48)

0

2 V

to be expressed as a sum of independent terms: one for each normal mode,

1

Uem = Pκ + ωκ Q2 .

2 2

(2.49)

κ

2

κ

Each term in the sum is mathematically identical to the energy of a simple harmonic

oscillator with unit mass, oscillator frequency ωκ , coordinate Qκ , and momentum Pκ .

For each κ, eqn (2.43) is obtained from

¿

Quantization of cavity modes

‚Uem ‚Uem

™ ™

and Pκ = ’

Qκ = ; (2.50)

‚Pκ ‚Qκ

consequently, Uem serves as the classical Hamiltonian for the radiation oscillators,

and Qκ and Pκ are said to be canonically conjugate classical variables (Marion

and Thornton, 1995). An even more suggestive form comes from using the complex

amplitudes ±κ to write the energy as

ωκ ±— ±κ .

Uem = (2.51)

κ

κ

Interpreting ±— ±κ as the number of light-quanta with energy ωκ makes this a real-

κ

ization of Einstein™s original model.

2.1.2 The quantization conjecture

The simple harmonic oscillator is one of the very few examples of a mechanical sys-

tem for which the Schr¨dinger equation can be solved exactly. For a classical me-

o

chanical oscillator, Q (t) represents the instantaneous displacement of the oscillating

mass from its equilibrium position, and P (t) represents its instantaneous momentum.

The trajectory {(Q (t) , P (t)) for t 0} is uniquely determined by the initial values

(Q, P ) = (Q (0) , P (0)).

The quantum theory of the mechanical oscillator is usually presented in the coor-

dinate representation, i.e. the state of the oscillator is described by a wave function

ψ (Q, t), where the argument Q ranges over the values allowed for the classical co-

ordinate. Thus the wave functions belong to the Hilbert space of square-integrable

2

functions on the interval (’∞, ∞). In the Born interpretation, |ψ (Q, t)| represents

the probability density for ¬nding the oscillator with a displacement Q from equilib-

rium at time t; consequently, the wave function satis¬es the normalization condition

∞

2

dQ |ψ (Q, t)| = 1 . (2.52)

’∞

In this representation the classical oscillator variables (Q, P )”representing the pos-

sible initial values of classical trajectories”are replaced by the quantum operators q

and p de¬ned by

‚

qψ (Q, t) = Qψ (Q, t) and pψ (Q, t) = ψ (Q, t) . (2.53)

i ‚Q

By using the explicit de¬nitions of q and p it is easy to show that the operators satisfy

the canonical commutation relation

[q, p] = i . (2.54)

For a system consisting of N noninteracting mechanical oscillators”with coordi-

nates Q1 , Q2 , . . . , QN ”the coordinate representation is de¬ned by the N -body wave

function

ψ (Q1 , Q2 , . . . , QN , t) , (2.55)

¼ Quantization of cavity modes

and the action of the operators is

qm ψ (Q1 , Q2 , . . . , QN , t) = Qm ψ (Q1 , Q2 , . . . , QN , t) ,

(2.56)

‚

pm ψ (Q1 , Q2 , . . . , QN , t) = ψ (Q1 , Q2 , . . . , QN , t) ,

i ‚Qm

where m = 1, . . . , N . This explicit de¬nition, together with the fact that the Qm s

are independent variables, leads to the general form of the canonical commutation

relations,

[qm , pm ] = i δmm , (2.57)

[qm , qm ] = [pm , pm ] = 0 , (2.58)

for m, m = 1, . . . , N . This mechanical system is said to have N degrees of freedom.

The results of the previous section show that the pairs of coe¬cients (Qκ , Pκ )

in the expansions (2.40) and (2.41) are canonically conjugate and that they satisfy

the same equations of motion as a mechanical harmonic oscillator. Since the classical

descriptions of the radiation and mechanical oscillators have the same mathematical

form, it seems reasonable to conjecture that their quantum theories will also have the

same form. For the κth cavity mode this simply means that the state of the radiation

oscillator is described by a wave function ψ (Qκ , t). In order to distinguish between

the radiation and mechanical oscillators, we will call the quantum operators for the

radiation oscillator qκ and pκ . The mathematical de¬nitions of these operators are still

given by eqn (2.56), with qκ and pκ replaced by qκ and pκ .

Extending this procedure to describe the general state of the cavity ¬eld introduces

a new complication. The classical state of the electromagnetic ¬eld is represented by

functions E (r, t) and B (r, t) that, in general, cannot be described by a ¬nite number

of modes. This means that the classical description of the cavity ¬eld requires in¬-

nitely many degrees of freedom. A naive interpretation of the quantization conjecture

would therefore lead to wave functions ψ (Q1 , Q2 , . . .) that depend on in¬nitely many

variables. Mathematical techniques to deal with such awkward objects do exist, but it

is much better to start with abstract algebraic operator relations like eqns (2.57) and

(2.58), and then to choose an explicit representation that is well suited to the problem

at hand.

The formulation of quantum mechanics used above is called the Schr¨dinger o

picture; it is characterized by time-dependent wave functions and time independent

operators. The Schr¨dinger-picture formulation of the quantization conjecture for the

o

electromagnetic ¬eld therefore consists of the following two parts.

(1) The time-dependent states of the electromagnetic ¬eld satisfy the superposition

principle: if |Ψ (t) and |¦ (t) are two physically possible states, then the super-

position

± |Ψ (t) + β |¦ (t) (2.59)

is also a physically possible state. (See Appendix C.1 for the bra and ket notation.)

½

Quantization of cavity modes

(2) The classical variables Qκ = Qκ (t = 0) and Pκ = Pκ (t = 0) are replaced by time-

independent hermitian operators qκ and pκ :

Qκ ’ qκ and Pκ ’ pκ , (2.60)

that satisfy the canonical commutation relations

[qκ , pκ ] = i δκκ , [qκ , qκ ] = 0 , and [pκ , pκ ] = 0 , (2.61)

where κ, κ range over all cavity modes.

The statements (1) and (2) are equally important parts of this conjecture.

Another useful form of the commutation relations (2.61) is provided by de¬ning

the dimensionless, non-hermitian operators

ωκ qκ ’ ipκ

ωκ qκ + ipκ

and a† = √

√

aκ = (2.62)

κ

2 ωκ 2 ωκ

for the κth mode of the radiation ¬eld. A simple calculation using eqn (2.61) yields

the equivalent commutation relations

aκ , a† = δκκ , [aκ , aκ ] = 0 . (2.63)

κ

To sum up: by examining the problem of the ideal resonant cavity, we have been

led to the conjecture that the radiation ¬eld can be viewed as a collection of quantized

simple harmonic oscillators. The quantization conjecture embodied in eqns (2.59)“

(2.61) may appear to be rather formal and abstract, but it is actually the fundamental

physical assumption required for constructing the quantum theory of the electromag-

netic ¬eld. New principles of this kind cannot be deduced from the pre-existing theory;

instead, they represent a genuine leap of scienti¬c induction that must be judged by

its success in explaining experimental results.

In the following section, we will combine the canonical commutation relations with

some basic physical principles to construct the Hilbert space of state vectors |Ψ , and

thus obtain a concrete representation of the operators qκ and pκ or aκ and a† for aκ

single cavity mode. In Section 2.1.2-C this representation will be generalized to include

the in¬nite set of normal cavity modes.

A The single-mode Fock space

In this section we will deal with a single mode, so the mode index can be omitted.

Instead of starting with the coordinate representation of the wave function, as in eqn

(2.53), we will deduce the structure of the Hilbert space of states by the following

argument. According to eqn (2.49) the classical energy for a single mode is

1

Uem = P 2 + ω 2 Q2 , (2.64)

2

where the arbitrary zero of energy has been chosen to correspond to the classical

solution Q = P = 0, representing the oscillator at rest at the minimum of the potential.

¾ Quantization of cavity modes

In quantum mechanics the standard procedure is to apply eqn (2.60) to this expression

and to interpret the resulting operator as the (single-mode) Hamiltonian

12

p + ω2 q2 .

Hem = (2.65)

2

It is instructive to rewrite this in terms of the operators a and a† by solving eqn (2.62)

to get

ω

a + a† and p = ’i a ’ a† .

q= (2.66)

2ω 2

Substituting these expressions into eqn (2.65)”while remembering that the operators

a and a† do not commute”leads to

1 ω ω

2 2

a ’ a† a + a†

’

Hem = +

2 2 2

ω

aa† + a† a .

= (2.67)

2

By using the commutation relation (2.63), this can be written in the equivalent form

1

Hem = ω a† a + . (2.68)

2

The superposition principle (2.59) is enforced by the assumption that the states of

the radiation operator belong to a Hilbert space. The structure of this Hilbert space is

essentially determined by the fact that Hem is a positive operator, i.e. Ψ |Hem | Ψ 0

for any |Ψ . To see this, set |¦ = a |Ψ and use the general rule ¦ |¦ 0 to conclude

that

ω

Ψ |Hem | Ψ = ω Ψ a† a Ψ +

2

ω

= ω ¦ |¦ + 0. (2.69)

2

In particular, this means that all eigenvalues of Hem are nonnegative. Let |φ be an

eigenstate of Hem with eigenvalue W ; then a |φ satis¬es

Hem a |φ = {[Hem , a] + aHem } |φ

= W a |φ + [Hem , a] |φ . (2.70)

The commutator is given by

[Hem , a] = ω a† a, a

= ω a† [a, a] + a† , a a

= ’ ωa , (2.71)

so that

Hem a |φ = (W ’ ω) a |φ . (2.72)

Thus a |φ is also an eigenstate of Hem , but with the reduced eigenvalue (W ’ ω).

Since a lowers the energy by ω, repeating this process would eventually generate

¿

Quantization of cavity modes

states of negative energy. This is inconsistent with the inequality (2.69); therefore, the

Hilbert space of a consistent quantum theory for an oscillator must include a lowest

energy eigenstate |0 satisfying

0| a† = 0 ,

a |0 = 0 , (2.73)

and

ω

Hem |0 = |0 . (2.74)

2

In the case of a mechanical oscillator |0 is the ground state, and a is a lowering

operator. A calculation similar to eqns (2.70) and (2.71) leads to

Hem a† |φ = (W + ω) a† |φ , (2.75)

which shows that a† raises the energy by ω, so a† is a raising operator. The idea

behind this language is that the mechanical oscillator itself is the object of interest.

The energy levels are merely properties of the oscillator, like the energy levels of an

atom.

The equations describing the radiation and mechanical oscillators have the same

form, but there is an important di¬erence in physical interpretation. For the electro-

magnetic ¬eld, it is the quanta of excitation”rather than the radiation oscillators

themselves”that are the main objects of interest. This shift in emphasis incorpo-

rates Einstein™s original proposal that the electromagnetic ¬eld is composed of discrete

quanta. In keeping with this view, it is customary to replace the cumbersome phrase

˜quantum of excitation of the electromagnetic ¬eld™ by the term photon. The in-

tended implication is that photons are physical objects on the same footing as massive

particles. The subtleties associated with treating photons as particles are addressed

in Section 3.6. Since a removes one photon, it is natural to call it the annihilation

operator, and a† , which adds a photon, is naturally called a creation operator. In

this language the ground state of the radiation oscillator is referred to as the vacuum

state, since it contains no photons.

The number operator N = a† a satis¬es the commutation relations

N, a† = a† ,

[N, a] = ’a , (2.76)

so that the a and a† respectively decrease and increase the eigenvalues of N by one.

Since N |0 = 0, this implies that the eigenvalues of N are the the integers 0, 1, 2, . . ..

The eigenvectors of N are called number states, and it is easy to see that N |n =

n |n implies

n

|n = Zn a† |0 , (2.77)

where Zn is a normalization constant. The Hamiltonian can be written as Hem =

(N + 1/2) ω, so the number states are also energy eigenstates: Hem |n = (n + 1/2)

ω |n . The commutation relations (2.76) can be used to derive the results

√

√

1

n |n ’ 1 , and a† |n = n + 1 |n + 1 .

Zn = √ , n |n = δnn , a |n =

n!

(2.78)

Quantization of cavity modes

(1)

The Hilbert space HF for a single mode consists of all linear combinations of

number states, i.e. a typical vector is given by

∞

|Ψ = Cn |n . (2.79)

n=0

(1)

The space HF is called the (single-mode) Fock space. In mathematical jargon”see

(1) (1)

Appendix A.2”HF is said to be spanned by the number states, or HF is said to be

the span of the number states. Since the number states are orthonormal, the expansion

(2.79) can be expressed as

∞

|Ψ = |n n |Ψ . (2.80)

n=0

For any state |φ the expression |φ φ| stands for an operator”see Appendix C.1.2”

that is de¬ned by its action on an arbitrary state |χ :

(|φ φ|) |χ ≡ |φ φ |χ . (2.81)

This shows that |φ φ| is the projection operator onto |φ , and it allows the expansion

(2.80) to be expressed as

∞

|ψ = |n n| |ψ . (2.82)

n=0

The general de¬nition (2.81) leads to

n |) = |n n |n n | = δnn (|n n|) ;

(|n n|) (|n (2.83)

therefore, the (|n n|)s are a family of orthogonal projection operators. According to

eqn (2.82) the projection operators onto the number states satisfy the completeness

relation

∞

|n n| = 1 . (2.84)

n=0

B Vacuum ¬‚uctuations of a single radiation oscillator

A standard argument from quantum mechanics (Bransden and Joachain, 1989, Sec.

5.4) shows that the canonical commutation relations (2.61) for the operators q and p

lead to the uncertainty relation

∆q∆p, (2.85)

2

where the rms deviations ∆q and ∆p are de¬ned by

2 2

Ψ |q 2 | Ψ ’ Ψ |q| Ψ Ψ |p2 | Ψ ’ Ψ |p| Ψ

∆q = , ∆p = , (2.86)

(1)

and |Ψ is any normalized vector in HF . For the vacuum state the relations (2.66)

and (2.73) yield 0 |q| 0 = 0 and 0 |p| 0 = 0, so the uncertainty relation implies that

Quantization of cavity modes

neither 0 q 2 0 nor 0 p2 0 can vanish. For mechanical oscillators this is attributed

to zero-point motion; that is, even in the ground state, random excursions around the

classical equilibrium at Q = P = 0 are required by the uncertainty principle. The

ground state for light is the vacuum state, so the random excursions of the radiation

oscillators are called vacuum ¬‚uctuations. Combining eqn (2.66) with eqn (2.73)

yields the explicit values

ω

0 q2 0 = 0 p2 0 =

, . (2.87)

2ω 2

We note for future reference that the vacuum deviations are ∆q0 = /2ω and ∆p0 =

ω/2, and that these values saturate the inequality (2.85), i.e. ∆q0 ∆p0 = /2. States

with this property are called minimum-uncertainty states, or sometimes minimum-

uncertainty-product states.

The vacuum ¬‚uctuations of the radiation oscillator also explain the fact that the

energy eigenvalue for the vacuum is ω/2 while the classical energy minimum is Uem =

0. Inserting eqn (2.87) into the original expression eqn (2.65) for the Hamiltonian yields

0 |Hem | 0 = ω/2. The discrepancy between the quantum and classical minimum

energies is called the zero-point energy; it is required by the uncertainty principle for

the radiation oscillator. Since energy is only de¬ned up to an additive constant, it would

be permissible”although apparently unnatural”to replace the classical expression

(2.64) by

12 ω

U= p + ω2 q2 ’ . (2.88)

2 2

Carrying out the substitution (2.66) on this expression yields the Hamiltonian

Hem = ωa† a . (2.89)

With this convention the vacuum energy vanishes for the quantum theory, but the

discrepancy between the quantum and classical minimum energies is unchanged. The

same thing can be accomplished directly in the quantum theory by simply subtracting

the zero-point energy from eqn (2.68). Changes of this kind are always permitted, since

only di¬erences of energy eigenvalues are physically meaningful.

C The multi-mode Fock space

Since the classical radiation oscillators in the cavity are mutually independent, the

quantization rule is given by eqns (2.60)“(2.63), and the only real di¬culties stem from

the fact that there are in¬nitely many modes. For each mode, the number operator

Nκ = a† aκ is evidently positive and satis¬es

κ

[Nκ , a» ] = ’δκ» aκ , (2.90)

Nκ , a† = δκ» a† . (2.91)

κ

»

Combining eqn (2.90) with the positivity of Nκ and applying the argument used for

the single-mode Hamiltonian in Section 2.1.2-A leads to the conclusion that there must

be a (multimode) vacuum state |0 satisfying

Quantization of cavity modes

aκ |0 = 0 for every mode-index κ . (2.92)

Since number operators for di¬erent modes commute, it is possible to ¬nd vectors |n

that are simultaneous eigenstates of all the mode number operators:

Nκ |n = nκ |n for all κ ,

(2.93)

n = {nκ for all κ} .

According to the single-mode results (2.77) and (2.78) the many-mode number states

are given by

nκ

a†

κ

√

|n = |0 . (2.94)

nκ !

κ

The total number operator is

a† aκ ,

N= (2.95)

κ

κ

and

N |n = |n .

nκ (2.96)

κ

The Hilbert space HF spanned by the number states |n is called the (multimode)

Fock space.

It is instructive to consider the simplest number states, i.e. those containing exactly

one photon. If κ and » are the labels for two distinct modes, then eqn (2.96) tells us

that |1κ = a† |0 and |1» = a† |0 are both one-photon states. The same equation

κ »

tells us that the superposition

1 1 1

|ψ = √ |1κ + √ |1» = √ a† + a† |0 (2.97)

κ »

2 2 2

is also a one-photon state; in fact, every state of the form

ξκ a† |0

|ξ = (2.98)

κ

κ

is a one-photon state. There is a physical lesson to be drawn from this algebraic

exercise: it is a mistake to assume that photons are necessarily associated with a single

classical mode. Generalizing this to a superposition of modes which form a classical

wave packet, we see that a single-photon wave packet state (that is, a wave packet

that contains exactly one photon) is perfectly permissible.

According to eqn (2.94) any number of photons can occupy a single mode. Further-

more the commutation relations (2.63) guarantee that the generic state a† 1 · · · a† n |0

κ κ

is symmetric under any permutation of the mode labels κ1 , . . . , κn . These are de¬n-

ing properties of objects satisfying Bose statistics (Bransden and Joachain, 1989, Sec.

10.2), so eqns (2.63) are called Bose commutation relations and photons are said to

be bosons.

Normal ordering and zero-point energy

D Field operators

In the Schr¨dinger picture, the operators for the electric and magnetic ¬elds are”by

o

de¬nition”time-independent. They can be expressed in terms of the time-independent

operators pκ and qκ by ¬rst using the classical expansions (2.40) and (2.41) to write the

initial classical ¬elds E (r, 0) and B (r, 0) in terms of the initial displacements Qκ (0)

and momenta Pκ (0) of the radiation oscillators. Setting (Qκ , Pκ ) = (Qκ (0) , Pκ (0)),

and applying the quantization conjecture (2.60) to these results leads to

1

E (r) = ’ √ pκ E κ (r) , (2.99)

0 κ

1

kκ qκ Bκ (r) .

B (r) = √ (2.100)

0 κ

For most applications it is better to express the ¬elds in terms of the operators aκ and

a† by using eqn (2.66) for each mode:

κ

ωκ

aκ ’ a† E κ (r) ,

E (r) = i (2.101)

κ

20

κ

µ0 ωκ

aκ + a† Bκ (r) .

B (r) = (2.102)

κ

2

κ

The corresponding expansions for the vector potential in the radiation gauge are

1

qκ E κ (r)

A (r) =

0

κ

aκ + a† E κ (r) .

= (2.103)

κ

2 0 ωκ

κ

2.2 Normal ordering and zero-point energy

In the absence of interactions between the independent modes, the energy is additive;

therefore, the Hamiltonian is the sum of the Hamiltonians for the individual modes.

If we use eqn (2.68) for the single-mode Hamiltonians, the result is

ωκ

ω κ a† aκ +

Hem = . (2.104)

κ

2

κ

The previously innocuous zero-point energies for each mode have now become a serious

annoyance, since the sum over all modes is in¬nite. Fortunately there is an easy way

out of this di¬culty. We can simply use the alternate form (2.89) which gives

ω κ a† aκ .

Hem = (2.105)

κ

κ

With this choice for the single-mode Hamiltonians the vacuum energy is reduced from

in¬nity to zero.

Quantization of cavity modes

It is instructive to look at this problem in a di¬erent way by using the equivalent

form eqn (2.67), instead of eqn (2.68), to get

ωκ †

aκ a κ + aκ a † .

Hem = (2.106)

κ

2

κ

Now the zero-point energy can be eliminated by the simple expedient of reversing the

order of the operators in the second term. This replaces eqn (2.106) by eqn (2.105). In

other words, subtracting the vacuum expectation value of the energy is equivalent to

reordering the operator products so that in each term the annihilation operator is to

the right of the creation operator. This is called normal ordering, while the original

order in eqn (2.106) is called symmetrical ordering.

We are allowed to consider such a step because there is a fundamental ambiguity

involved in replacing products of commuting classical variables by products of non-

commuting operators. This problem does not appear in quantizing the classical energy

expression in eqn (2.64), since products of qκ with pκ do not occur. This happy cir-

cumstance is a fortuitous result of the choice of classical variables. If we had instead

chosen to use the variables ±κ de¬ned by eqn (2.45), the quantization conjecture would

be ±κ ’ aκ and ±— ’ a† . This does produce an ordering ambiguity in quantizing eqn

κ κ

(2.51), since ±κ ±— , (±— ±κ + ±κ ±— ) /2, and ±— ±κ are identical in the classical theory,

κ κ κ κ

but di¬erent after quantization. The last two forms lead respectively to the expressions

(2.106) and (2.105) for the Hamiltonian. Thus the presence or absence of the zero-point

energy is determined by the choice of ordering of the noncommuting operators.

It is useful to extend the idea of normal ordering to any product of operators

X1 · · · Xn , where each Xi is either a creation or an annihilation operator. The normal-

ordered product is de¬ned by

: X1 · · · Xn : = X 1 · · · Xn , (2.107)

where (1 , . . . , n ) is any ordering (permutation) of (1, . . . , n) that arranges all of the

annihilation operators to the right of all the creation operators. The commutation

relations are ignored when carrying out the reordering. More generally, let Z be

a linear combination of distinct products X1 · · · Xn ; then : Z : is the same linear

combination of the normal-ordered products : X1 · · · Xn : . The vacuum expectation

value of a normal-ordered product evidently vanishes, but it is not generally true that

Z = : Z : + 0 |Z| 0 .

2.3 States in quantum theory

In classical mechanics, the coordinate q and momentum p of a particle can be precisely

speci¬ed. Therefore, in classical physics the state of maximum information for a system

of N particles is a point q, p = (q1 , p1 , . . . , qN , pN ) in the mechanical phase space.

For large values of N , specifying a point in the phase space is a practical impossibility,

so it is necessary to use classical statistical mechanics”which describes the N -body

system by a probability distribution f q, p ”instead. The point to bear in mind here

is that this probability distribution is an admission of ignorance. No experimentalist

can possibly acquire enough information to determine a particular value of q, p .

States in quantum theory

In quantum theory, the uncertainty principle prohibits simultaneous determination

of the coordinates and momenta of a particle, but the notions of states of maximum

and less-than-maximum information can still be de¬ned.

2.3.1 Pure states

In the standard interpretation of quantum theory, the vectors in the Hilbert space de-

scribing a physical system”e.g. general linear combinations of number states in Fock

space”provide the most detailed description of the state of the system that is consis-

tent with the uncertainty principle. These quantum states of maximum information

are called pure states (Bransden and Joachain, 1989, Chap. 14). From this point of

view the random ¬‚uctuations imposed by the uncertainty principle are intrinsic; they

are not the result of ignorance of the values of some underlying variables.

For any quantum system the average of many measurements of an observable X

on a collection of identical physical systems, all described by the same vector |Ψ , is

given by the expectation value Ψ |X| Ψ . The evolution of a pure state is described

by the Schr¨dinger equation

o

‚

|Ψ (t) = H |Ψ (t) ,

i (2.108)

‚t

where H is the Hamiltonian.

2.3.2 Mixed states

In the absence of maximum information, the system is said to be in a mixed state.

In this situation there is insu¬cient information to decide which pure state describes

the system. Just as for classical statistical mechanics, it is then necessary to assign a

probability to each possible pure state. These probabilities, which represent ignorance

of which pure state should be used, are consequently classical in character.

As a simple example, suppose that there is only su¬cient information to say that

each member of a collection of identically prepared systems is described by one or the

other of two pure states, |Ψ1 or |Ψ2 . For a system described by |Ψe (e = 1, 2), the

average value for measurements of X is the quantum expectation value Ψe |X| Ψe .

The overall average of measurements of X is therefore

X = P1 Ψ1 |X| Ψ1 + P2 Ψ2 |X| Ψ2 , (2.109)

where Pe is the fraction of the systems described by |Ψe , and P1 + P2 = 1.

The average in eqn (2.109) is quite di¬erent from the average of many measure-

ments on systems all described by the superposition state |Ψ = C1 |Ψ1 + C2 |Ψ2 . In

that case the average is

—

2 2

Ψ |X| Ψ = |C1 | Ψ1 |X| Ψ1 + |C2 | Ψ2 |X| Ψ2 + 2 Re [C1 C2 Ψ1 |X| Ψ2 ] ,

(2.110)

which contains an interference term missing from eqn (2.109). The two results (2.109)

—

and (2.110) only agree when |Ce |2 = Pe and Re [C1 C2 Ψ1 |X| Ψ2 ] = 0. The latter

—

condition can be satis¬ed if C1 C2 Ψ1 |X| Ψ2 is pure imaginary or if Ψ1 |X| Ψ2 = 0.

Since it is always possible to choose another observable X for which neither of these

¼ Quantization of cavity modes

conditions is satis¬ed, it is clear that the mixed state and the superposition state

describe very di¬erent physical situations.

A The density operator

In general, a mixed state is de¬ned by a collection, usually called an ensemble, of

normalized pure states {|Ψe }, where the label e may be discrete or continuous. For

simplicity we only consider the discrete case here: the continuum case merely involves

replacing sums by integrals with suitable weighting functions. For the discrete case, a

probability distribution on the ensemble is a set of real numbers {Pe } that satisfy

the conditions

0 Pe 1 , (2.111)

Pe = 1 . (2.112)

e

The ensemble may be ¬nite or in¬nite, and the vectors need not be mutually orthog-

onal.

The average of repeated measurements of an observable X is represented by the

ensemble average of the quantum expectation values,

Pe Ψe (t) |X| Ψe (t) ,

X (t) = (2.113)

e

where |Ψe (t) is the solution of the Schr¨dinger equation with initial value |Ψe (0) =

o

|Ψe . It is instructive to rewrite this result by using the number-state basis {|n } for

Fock space to get

Ψe (t) |X| Ψe (t) = Ψe (t) |n n |X| m m |Ψe (t) , (2.114)

n m

and

n |X| m Pe m |Ψe (t) Ψe (t) |n

X (t) = . (2.115)

n m e

By applying the general de¬nition (2.81) to the operator |Ψe (t) Ψe (t)|, it is easy

to see that the quantity in square brackets is the matrix element m |ρ (t)| n of the

density operator:

Pe |Ψe (t) Ψe (t)| .

ρ (t) = (2.116)

e

With this result in hand, eqn (2.115) becomes

m |ρ (t)| n n |X| m

X (t) =

m n

m |ρ (t) X| m

=

m

= Tr [ρ (t) X] , (2.117)

½

States in quantum theory

where the trace operation is de¬ned by eqn (C.22). Each of the ket vectors |Ψe in

the ensemble evolves according to the Schr¨dinger equation, and the bra vectors Ψe |

o

obey the conjugate equation

‚

’i Ψe (t)| = Ψe (t)| H , (2.118)

‚t

so the evolution equation for the density operator is

‚

i ρ (t) = [H, ρ (t)] . (2.119)

‚t

By analogy with the Liouville equation for the classical distribution function (Huang,

1963, Sec. 4.3), this is called the quantum Liouville equation. The condition

(2.112), together with the normalization of the ensemble state vectors, means that

the density operator has unit trace,

Tr (ρ (t)) = 1 , (2.120)

and eqn (2.119) guarantees that this condition is valid at all times.

A pure state is described by an ensemble consisting of exactly one vector, so that

eqn (2.116) reduces to

ρ (t) = |Ψ (t) Ψ (t)| . (2.121)

This explicit statement can be replaced by the condition that ρ (t) is a projection

operator, i.e.

ρ2 (t) = ρ (t) . (2.122)

Thus for pure states

Tr ρ2 (t) = Tr (ρ (t)) = 1 , (2.123)

while for mixed states

Tr ρ2 (t) < 1 . (2.124)

For any observable X and any state ρ, either pure or mixed, an important statistical

property is given by the variance

2

V (X) = X 2 ’ X , (2.125)

2

where X = Tr (ρX). The easily veri¬ed identity V (X) = (X ’ X ) shows that

V (X) 0, and it also follows that V (X) = 0 when ρ is an eigenstate of X, i.e.

Xρ = ρX = »ρ. Conversely, every eigenstate of X satis¬es V (X) = 0. Since V (X)

is non-negative, the variance is often described in terms of the root mean square

(rms) deviation

2

X2 ’ X

∆X = V (X) = . (2.126)

¾ Quantization of cavity modes

B Mixed states arising from measurements

In quantum theory the act of measurement can produce a mixed state, even if the state

before the measurement is pure. For simplicity, we consider an observable X with a

discrete, nondegenerate spectrum. This means that the eigenvectors |xn , satisfying

X |xn = xn |xn , are unique (up to a phase). Suppose that we have complete infor-

mation about the initial state of the system, so that we can describe it by a pure state

|ψ . When a measurement of X is carried out, the Born interpretation tells us that

the eigenvalue xn will be found with probability pn = | xn |ψ |2 . The von Neumann

projection postulate further tells us that the system will be described by the pure state

|xn , if the measurement yields xn . This is the reduction of the wave packet. Now con-

sider the following situation. We know that a measurement of X has been performed,

but we do not know which eigenvalue of X was actually observed. In this case there is

no way to pick out one eigenstate from the rest. Thus we have an ensemble consisting

of all the eigenstates of X, and the density operator for this ensemble is

pn |xn xn | .

ρmeas = (2.127)

n

Thus a measurement will change the original pure state into a mixed state, if the

knowledge of which eigenvalue was obtained is not available.

2.3.3 General properties of the density operator

So far we have only considered observables with nondegenerate eigenvalues, but in

general some of the eigenvalues xξ of X are degenerate, i.e. there are several linearly

independent solutions of the eigenvalue problem X |Ψ = xξ |Ψ . The number of solu-

tions is the degree of degeneracy, denoted by dξ (X). A familiar example is X = J 2 ,

where J is the angular momentum operator. The eigenvalue j (j + 1) 2 of J 2 has the

degeneracy 2j + 1 and the degenerate eigenstates can be labeled by the eigenvalues

m of Jz , with ’j m j. An example appropriate to the present context is the

operator

a† aks ,

Nk = (2.128)

ks

s

that counts the number of photons with wavevector k. If k has no vanishing com-

ponents, the eigenvalue problem Nk |Ψ = |Ψ has two independent solutions corre-

sponding to the two possible polarizations, so d1 (Nk ) = 2. In general, the common

eigenvectors for a given eigenvalue span a dξ (X)-dimensional subspace, called the

eigenspace Hξ (X). Let

|Ψξ1 , . . . , Ψξdξ (X) (2.129)

be a basis for Hξ (X), then

|Ψξm Ψξm |

Pξ = (2.130)

m

is the projection operator onto Hξ (X).

¿

States in quantum theory

According to the standard rules of quantum theory (see eqns (C.26)“(C.28)) the

conditional probability that xξ is the result of a measurement of X, given that the

system is described by the pure state |Ψe , is

| Ψξm |Ψe |2 = Ψe |Pξ | Ψe .

p (xξ |Ψe ) = (2.131)

m

For the mixed state the overall probability of the result xξ is, therefore,

| Ψξm |Ψe |2 = Tr (ρPξ ) .

Pe

p (xξ ) = (2.132)

e m

Thus the general rule is that the probability for ¬nding a given value xξ is given by the

expectation value of the projection operator Pξ onto the corresponding eigenspace.

Other important mathematical properties of the density operator follow directly

from the de¬nition (2.116). For any state |Ψ , the expectation value of ρ is positive,

Pe | Ψe |Ψ |2

Ψ |ρ| Ψ = 0, (2.133)

e