output beam does not appreciably a¬ect the Poissonian photocount distribution

function.

(2) For the G-case, the coherence time „coh is determined by the time required for

a speckle feature to cross the pinhole. For a rapidly rotating disc this is shorter

than the integration time of the photon detection system. As explained above,

this results in incoherent light described by a Bose“Einstein distribution peaked

at n = 0.

(3) The measurement process occurs at the photocathode surface of the photomulti-

plier tube, which, for unit quantum e¬ciency, emits n photoelectrons if n photons

impinge on it. However, unity quantum e¬ciency is not an essential requirement for

this experiment, since an analysis for arbitrary quantum e¬ciencies, when folded

½½

Properties of coherent states

in with a Bernoulli distribution function, shows that the Poissonian photoelec-

tron distribution still always results from an initial Poissonian photon distribution

(Loudon, 2000, Sec. 6.10). Similarly, a Bose“Einstein photoelectron distribution

function always results from an initial Bose“Einstein photon distribution.

(4) The condition that the laser be far above threshold is often not satis¬ed by real

continuous-wave lasers. The Scully“Lamb quantum theory of the laser predicts

that there can be appreciable deviations from the exact Poissonian distribution

when the small-signal gain of the laser is comparable to the loss of output mirrors.

Nevertheless, a skewed bell-shape curve that roughly resembles the Poissonian

distribution function is still predicted by the Scully“Lamb theory.

In sum, Arecchi™s experiment gave the ¬rst partial evidence that lasers emit a

coherent state, in that the observed photon count distribution is nearly Poissonian.

However, this photon-counting experiment only gives information concerning the di-

agonal elements n |ρ| n = p(n) of the density matrix. It gives no information about

the o¬-diagonal elements n |ρ| m when n = m. For example, this experiment cannot

distinguish between a pure coherent state |± , with |±| = n, and a mixed state for

which n |ρ| n happens to be a Poissonian distribution and n |ρ| m = 0 for n = m.

We shall see later that quantum state tomography experiments using optical homo-

dyne detection are sensitive to the o¬-diagonal elements of the density operator. These

experiments provide evidence that the state of a laser operating far above threshold

is closely approximated by an ideal coherent state.

In an extension of Arecchi™s experiment, Meltzer and Mandel (1971) measured the

photocount distribution function as a laser passes from below its threshold, through

its threshold, and ends up far above threshold. The change from a monotonically

decreasing photocount distribution below threshold”associated with the thermal state

of light”to a peaked one above threshold”associated with the coherent state”was

observed to agree with the Scully“Lamb theory.

5.4 Properties of coherent states

One of the objectives in studying coherent states is to use them as an alternate set

of basis functions for Fock space, but we must ¬rst learn to deal with the peculiar

mathematical features arising from the fact that the coherent states are eigenfunctions

of the non-hermitian annihilation operator a.

5.4.1 The displacement operator

The relation (3.83) linking the Heisenberg and Schr¨dinger pictures combines with the

o

explicit solution (5.40) of the Heisenberg equation to yield U † (t) aU (t) = ae’iωt +

± (t). For N = a† a, the identity exp (iθN ) a exp (’iθN ) = exp (’iθ) a (see Appendix

C.3, eqn (C.65)) allows this to be rewritten as

U † (t) aU (t) = eiωN t ae’iωN t + ± (t) , (5.43)

which in turn implies

†

U (t) eiωN t a U (t) eiωN t = a + ± (t) . (5.44)

½¾ Coherent states

Thus the physical model for generation of a coherent state in Section 5.2 implies that

there is a unitary operator which acts to displace the annihilation operator by ± (t).

The form of this operator can be derived from the explicit solution of the model

problem, but it is more useful to seek a unitary displacement operator D (±) that

satis¬es

D† (±) aD (±) = a + ± (5.45)

for all complex ±. Since D (±) is unitary, it can be written as D (±) = exp [’iK (±)],

where the hermitian operator K (±) is the generator of displacements. A similar situ-

ation arises in elementary quantum mechanics, where the representation p = ’i d/dq

for the momentum operator implies that the transformation

T∆q ψ (q) = ψ (q ’ ∆q) (5.46)

of spatial translation is represented by the unitary operator exp (’i∆qp/ ) (Brans-

den and Joachain, 1989, Sec. 5.9). This transformation rule for the wave function is

equivalent to the operator relation

e’i∆qp/ qei∆qp/ = q + ∆q . (5.47)

The similarity between eqns (5.45) and (5.47) and the associated fact that a, a† (like

[q, p]) is a c-number together suggest assuming that K (±) is a linear combination of

a and a† :

K (±) = g (±) a† + g — (±) a , (5.48)

where g (±) is a c-number yet to be determined.

One way to work out the consequences of this assumption is to de¬ne the inter-

polating operator a („ ) by

a („ ) = ei„ K(±) ae’i„ K(±) . (5.49)

This new operator is constructed so that it has the initial value a (0) = a and the ¬nal

value a (1) = D† (±) aD (±). In the „ -interval (0, 1), a („ ) satis¬es the Heisenberg-like

equation of motion

da („ )

= [a („ ) , K (±)] = ei„ K(±) [a, K (±)] e’i„ K(±) .

i (5.50)

d„

In the present case, the ansatz (5.48) shows that [a, K (±)] = g (±), so the equation of

motion simpli¬es to

da („ )

= g (±) , (5.51)

i

d„

with the solution a („ ) = a ’ ig (±) „ . Thus eqn (5.45) is satis¬ed by the choice g (±) =

i±, and the displacement operator is

†

’±— a)

D (±) = e(±a . (5.52)

The displacement operator generates the coherent state from the vacuum by

½¿

Properties of coherent states

†

’±— a)

|± = D (±) |0 = e(±a |0 . (5.53)

The simplest way to prove that D (±) |0 is a coherent state is to rewrite eqn (5.45) as

aD (±) = D (±) [a + ±] , (5.54)

and apply both sides to the vacuum state.

The displacement operators represent the translation group in the ±-plane, so they

must satisfy certain group properties. For example, a direct application of the de¬nition

(5.45) yields the inverse transformation as

D ’1 (±) = D† (±) = D (’±) . (5.55)

From eqn (5.45) one can see that applying D (β) followed by D (±) has the same e¬ect

as applying D (± + β); therefore, the product D (±) D (β) must be proportional to

D (± + β):

D (±) D (β) = D (± + β) ei¦(±,β) , (5.56)

where ¦ (±, β) is a real function of ± and β. The phase ¦ (±, β) can be determined by

using the Campbell“Baker“Hausdor¬ formula, eqn (C.66), or”as in Exercise 5.6”by

another application of the interpolating operator method. By either method, the result

is

—

D (±) D (β) = D (± + β) ei Im(±β ) . (5.57)

5.4.2 Overcompleteness

Distinct eigenstates of hermitian operators, e.g. number states, are exactly orthogonal;

therefore, distinct outcomes of measurements of the number operator”or any other

observable”are mutually exclusive events. This is the basis for interpreting |cn |2 =

2

| n| ψ | as the probability that the value n will be found in a measurement of the

number operator. By contrast, no two coherent states are ever orthogonal. This is

shown by using eqn (5.23) to calculate the value

1

± |β = exp ’ |± ’ β| exp (i Im [±— β])

2

(5.58)

2

of the inner product. On the other hand, states with large values of |± ’ β| are ap-

proximately orthogonal, i.e. | ± |β | 1, for quite moderate values of |± ’ β|. The

2

lack of orthogonality between distinct coherent states means that | ±| ψ | cannot be

interpreted as the probability for ¬nding the ¬eld in the state |± , given that it is

prepared in the state |ψ .

Although they are not mutually orthogonal, the coherent states are complete. A

necessary and su¬cient condition for completeness of the family {|± } is that a vector

|ψ satisfying

ψ |± = 0 for all ± (5.59)

is necessarily the null vector, i.e. |ψ = 0. A second use of eqn (5.23) allows this

equation to be expressed as

½ Coherent states

∞

±n

√ c— = 0 ,

F (±) = (5.60)

n

n!

n=0

where c— = ψ |n . This relation is an identity in ±, so all derivatives of F (±) must

n

also vanish. In particular,

√—

n

‚

F (±) = n!cn = 0 , (5.61)

‚± ±=0

so that cn = 0 for all n 0. The completeness of the number states then requires

|ψ = 0, and this establishes the completeness of the coherent states.

The coherent states form a complete set, but they are not linearly independent

vectors. This peculiar state of a¬airs is called overcompleteness. It is straightforward

to show that any ¬nite collection of distinct coherent states is linearly independent,

so to prove overcompleteness we must show that the null vector can be expressed as a

continuous superposition of coherent states. Let u1 = Re ± and u2 = Im ±, then any

linear combination of the coherent states can be written as

∞ ∞

du2 z (u1 , u2 ) |u1 + iu2 ,

du1 (5.62)

’∞ ’∞

where z (u1 , u2 ) is a complex function of the two real variables u1 and u2 . It is custom-

ary to regard z (u1 , u2 ) as a function of ±— and ±, which are treated as independent

variables, and in the same spirit to write

du1 du2 = d2 ± . (5.63)

For brevity we will sometimes write z (±) instead of z (±— , ±) or z (u1 , u2 ), and the

same convention will be used for other functions as they arise. Any confusion caused

by these various usages can always be resolved by returning to the real variables u1

and u2 .

In this new notation the condition that a continuous superposition of coherent

states gives the null vector is

d2 ±z (±— , ±) |± = 0 ,

|z = (5.64)

where the integral is over the entire complex ±-plane and z (±— , ±) is nonzero on

some open subset of the ±-plane. The number states are both complete and linearly

independent, so this condition can be expressed in a more concrete way as

1 2

d2 ±e’|±| z (±— , ±) ±n = 0 for all n

n |z = √ /2

0. (5.65)

n!

By using polar coordinates (± = ρ exp iφ) for the integration these conditions become

∞ 2π

1 n+1 ’ρ2 /2

√ dφz (ρ, φ) einφ = 0 for all n

dρρ e 0. (5.66)

n! 0 0

½

Properties of coherent states

In this form, one can see that the desired outcome is guaranteed if the φ-dependence

of z (ρ, φ) causes the φ-integral to vanish for all n 0. This is easily done by choosing

z (ρ, φ) = g (ρ) ρm exp (imφ) for some m > 0; that is,

z (±— , ±) = g (|±|) ±m , with m > 0 . (5.67)

The linear dependence of the coherent states means that the coe¬cients in the generic

expansion

d2 ±F (±— , ±) |±

|ψ = (5.68)

are not unique, since replacing F (±— , ±) by F (±— , ±) + z (±— , ±) yields the same vector

|ψ .

In spite of these unfamiliar properties, the coherent states satisfy a completeness

relation, or resolution of the identity,

d2 ±

|± ±| = I , (5.69)

π

analogous to eqn (2.84) for the number states. To prove this, we denote the left side

of eqn (5.69) by I and evaluate the matrix elements

d2 ±

n |I| m = n |± ± |m

π

∞ 2π

ρn+m ’ρ2 dφ i(n’m)φ

√

= dρ ρ e e

π

n!m!

0 0

= δnm . (5.70)

Thus I has the same matrix elements as the identity operator, and eqn (5.69) is

established.

Applying this representation of the identity to a state |ψ gives the natural”but

not unique”expansion

d2 ±

|ψ = |± ± |ψ . (5.71)

π

The completeness relation also gives a useful formula for the trace of any operator:

∞

d2 ± d2 ±

|± ±| X n |± ± |X| n

Tr X = Tr =

π π n=0

d2 ±

± |X| ± .

= (5.72)

π

5.4.3 Coherent state representations of operators

The completeness relation (5.69) is the basis for deriving useful representations of

operators in terms of coherent states. For any Fock space operator X, we easily ¬nd

the general result

½ Coherent states

d2 ± d2 β

|± ±| X |β β|

X=

π π

d2 ± d2 β

|± ± |X| β β| .

= (5.73)

π π

Since the coherent states are complete, this result guarantees that X is uniquely de¬ned

by the matrix elements ± |X| β . On the other hand, the overcompleteness of the

coherent states suggests that the same information may be carried by a smaller set of

matrix elements.

A An operator X is uniquely determined by ± |X| ±

The diagonal matrix elements n |X| n in the number-state basis”or in any other

orthonormal basis”do not uniquely specify the operator X, but the overcompleteness

of the coherent states guarantees that the diagonal elements ± |X| ± do determine

X uniquely. The ¬rst step in the proof is to use eqn (5.23) one more time to write

± |X| ± in terms of the matrix elements in the number-state basis,

∞ ∞

m |X| n —m n

’|±|2

√

± |X| ± = e ± ±. (5.74)

m!n!

m=0 n=0

Now suppose that two operators Y and Z have the same diagonal elements, i.e.

± |Y | ± = ± |Z| ± ; then X = Y ’ Z must satisfy

∞ ∞

m |X| n —m n

√ ± ± = 0. (5.75)

m!n!

m=0 n=0

This is an identity in the independent variables ± and ±— , so the argument leading to

eqn (5.61) can be applied again to conclude that m |X| n = 0 for all m and n. The

completeness of the number states then implies that X = 0, and we have proved that

if ± |Y | ± = ± |Z| ± for all ± , then Y = Z . (5.76)

B Coherent state diagonal representation

The result (5.76) will turn out to be very useful, but it does not immediately supply us

with a representation for the operator. On the other hand, the general representation

(5.73) involves the o¬-diagonal matrix elements ± |X| β which we now see are appar-

ently super¬‚uous. This suggests that it may be possible to get a representation that

only involves the projection operators |± ±|, rather than the o¬-diagonal operators

|± β| appearing in eqn (5.73). The key to this construction is the identity

an |± ±| a†m = ±n ±—m |± ±| , (5.77)

which holds for any non-negative integers n and m. Let us now suppose that X has

a power series expansion in the operators a and a† , then by using the commutation

½

Multimode coherent states

relation a, a† = 1 each term in the series can be rearranged into a sum of terms in

which the creation operators stand to the right of the annihilation operators, i.e.

∞ ∞

Xnm an a†m ,

A

X= (5.78)

m=0 n=0

where Xnm is a c-number coe¬cient. Since this exactly reverses the rule for normal

A

ordering, it is called antinormal ordering, and the superscript A serves as a reminder

of this ordering rule. By combining the identities (5.69) and (5.77) one ¬nds

∞ ∞

d2 ±

|± ±| a†m

Xnm an

A

X=

π

m=0 n=0

d2 ±X A (±) |± ±| ,

= (5.79)

where

∞ ∞

1

Xnm ±n ±—m

X (±) =

A A

(5.80)

π m=0 n=0

is a c-number function of the two real variables Re ± and Im ±. This construction gives

us the promised representation in terms of the projection operators |± ±|.

5.5 Multimode coherent states

Up to this point we have only considered coherent states of a single radiation oscil-

lator. In the following sections we will consider several generalizations that allow the

description of multimode squeezed states.

5.5.1 An elementary approach to multimode coherent states

A straightforward generalization is to replace the de¬nition (5.18) of the one-mode

coherent state by the family of eigenvalue problems

aκ |± = ±κ |± for all κ , (5.81)

where ± = (±1 , ±2 , . . . , ±κ , . . .) is the set of eigenvalues for the annihilation operators

aκ . The single-mode case is recovered by setting ±κ = 0 for κ = κ. The multimode

coherent state |± ”de¬ned as the solution of the family of equations (5.81)”can

be constructed from the vacuum state by using eqn (5.53) for each mode to get

|± = D (±κ ) |0 , (5.82)

κ

where

D (±κ ) = exp ±κ a† ’ ±— aκ (5.83)

κ κ