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