modes, the de¬nition (5.82) raises various mathematical issues, such as the convergence

½ Coherent states

of the in¬nite product. In the following sections, we show how these issues can be dealt

with, but for most applications it is safe to proceed by using the formal in¬nite product.

For later use, it is convenient to specialize the general de¬nition (5.82) of the

multimode state to the case of box-quantized plane waves, i.e.

|± = D (±) |0 , (5.84)

exp ±ks a† ’ ±— aks .

D (±) = D (±ks ) = (5.85)

ks

ks

ks ks

By combining the eigenvalue condition aks |± = ±ks |± with the expression (3.69) for

E(+) , one can see that

E(+) (r) |± = E (r) |± , (5.86)

where

ωk

E (r) = i ±ks eks eik·r (5.87)

2 0V

ks

is the classical electric ¬eld de¬ned by |± .

Coherent states for wave packets—

5.5.2

The incident ¬eld in a typical experiment is a traveling-wave packet, i.e. a superposition

of plane-wave modes. A coherent state describing this situation is therefore an example

of a multimode coherent state. From this point of view, the multimode coherent state

|± is actually no more complicated than a single-mode coherent state (Deutsch, 1991).

This is a linguistic paradox caused by the various meanings assigned to the word

˜mode™. This term normally describes a solution of Maxwell™s equations with some

additional properties associated with the boundary conditions imposed by the problem

at hand. Examples are the modes of a rectangular cavity or a single plane wave.

General classical ¬elds are linear combinations of the mode functions, and they are

called wave packets rather than modes. Let us now return to eqn (5.82) which gives

a constructive de¬nition of the multimode state |± . Since the operators ±κ a† ’ ±— aκ

κ κ

and ±κ a† ’±— aκ commute for κ = κ , the product of unitary operators in eqn (5.82)

κ

κ

can be rewritten as a single unitary operator,

±κ a† ’ ±— aκ

|± = exp |0

κ κ

κ

†

= exp a [±] ’ a [±] |0 , (5.88)

where

±— aκ

a [±] = (5.89)

κ

κ

is an example of the general de¬nition (3.191). In other words the multimode coherent

state |± is a coherent state for the wave packet

w (r) = ±κ wκ (r) , (5.90)

κ

½

Multimode coherent states

where the wκ (r)s are mode functions. The wave packet w(r) de¬nes a point in the

classical phase space, so it represents one degree of freedom of the ¬eld. This suggests

changing the notation by

|± ’ |w = D [w] |0 , (5.91)

where

D [w] = exp a† [w] ’ a [w] (5.92)

is the wave packet displacement operator, and a [w] is simply another notation for

a [±].

The displacement rule,

D† [w] a [v] D [w] = a [v] + (v, w) , (5.93)

and the product rule,

D [v] D [w] = D [v + w] exp {i Im (w, v)} , (5.94)

are readily established by using the commutation relations (3.192), the interpolating

operator method outlined in Section 5.4.1, and the Campbell“Baker“Hausdor¬ formula

(C.66). The displacement rule (5.93) immediately yields the eigenvalue equation

a [v] |w = (v, w) |w . (5.95)

This says that the coherent state for the wave packet w is also an eigenstate”with

the eigenvalue (v, w)”of the annihilation operator for any other wave packet v. To

recover the familiar single-mode form, a |± = ± |± , simply set w = ±w0 , where w0

is normalized to unity, and v = w0 ; then eqn (5.95) becomes a [w0 ] |±w0 = ± |±w0 .

The inner product of two multimode (wave packet) coherent states is obtained from

(5.91) by calculating

v |w = 0 D† [v] D [w] 0

= exp {i Im (v, w)} 0 |D [w ’ v]| 0

1 2

= exp {i Im (v, w)} exp ’ w ’ v , (5.96)

2

where u = (u, u) is the norm of the wave packet u.

Sources of multimode coherent states—

5.5.3

In Section 5.2 we saw that a monochromatic classical current serves as the source for a

single-mode coherent state. This demonstration is readily generalized as follows. The

total Hamiltonian in the hemiclassical approximation is the sum of eqns (3.40) and

(5.36),

d3 rA(’) (r, t) · ’∇2 A(+) (r, t) ’ d3 r J (r, t) · A (r, t) .

H = 2 0 c2 (5.97)

The corresponding Heisenberg equation for A(+) ,

½¼ Coherent states

‚A(+) (r, t) 1 ’1/2

1/2

= c ’∇2 A(+) (r, t) ’ ’∇2 J (r, t) ,

i (5.98)

‚t 2 0c

has the formal solution

1/2

A(+) (r, t) = exp ’i (t ’ t0 ) c ’∇2 A(+) (r, t0 ) + w (r, t) , (5.99)

where

t

i ’1/2

1/2

dt exp ’i (t ’ t ) c ’∇2 ’∇2 J (r, t ) ,

w (r, t) = (5.100)

2 0c t0

and the Schr¨dinger and Heisenberg pictures coincide at the time, t0 , when the current

o

is turned on. The classical ¬eld w (r, t) satis¬es the c-number version of eqn (5.98),

‚w (r, t) 1 ’1/2

1/2

= c ’∇2 w (r, t) ’ ’∇2 J (r, t) .

i (5.101)

‚t 2 0c

Applying this solution to the vacuum gives A(+) (r, t) |0 = w (r, t) |0 in the Heisen-

berg picture, and A(+) (r) |w, t = w (r, t) |w, t in the Schr¨dinger picture. The time-

o

dependent coherent state |w, t evolves from the vacuum state (|w, t0 = |0 ) under

the action of the Hamiltonian given by eqn (5.97).

Completeness and representation of operators—

5.5.4

The issue of completeness for the multimode coherent states is (in¬nitely) more com-

plicated than in the single-mode case. Since we are considering all modes on an equal

footing, the identity (5.69) for a single mode must be replaced by

d2 ±κ

|±κ ±κ | = Iκ , (5.102)

π

where Iκ is the identity operator for the single-mode subspace Hκ . The resolution of

the identity on the entire space HF is given by

d2 ±κ

|± ±| = IF . (5.103)

π

κ

The mathematical respectability of this in¬nite-dimensional integral has been estab-

lished for basis sets labeled by a discrete index (Klauder and Sudarshan, 1968, Sec.

7-4). Fortunately, the Hilbert spaces of interest for quantum theory are separable, i.e.

they can always be represented by discrete basis sets. In most applications only a few

modes are relevant, so the necessary integrals are approximately ¬nite dimensional.

Combining the multimode completeness relation (5.103) with the fact that op-

erators for orthogonal modes commute justi¬es the application of the arguments in

Sections 5.4.3 and 5.6.3 to obtain the multimode version of the diagonal expansion for

the density operator:

d2 ± |± P (±) ±| ,

ρ= (5.104)

where

d2 ±κ

d2 ± = . (5.105)

π

κ

½½

Multimode coherent states

Applications of multimode states—

5.5.5

Substituting the relation

2 0c 2 0c 1/2

d3 rw— (r) · ’∇2

A(+) [w] = A(+) (r)

a [w] = (5.106)

into eqn (5.95) provides the r-space version of the eigenvalue equation:

A(+) (r) |w = w (r) |w . (5.107)

2 0c

For many applications it is more useful to use eqn (3.15) to express this in terms of

the electric ¬eld,

E(+) (r) |w = E (r) |w , (5.108)

where

c 1/2

E (r) = i ’∇2 w (r) (5.109)

2 0

is the positive-frequency part of the classical electric ¬eld corresponding to the wave

packet w. The result (5.108) can be usefully applied to the calculation of the ¬eld

correlation functions for the coherent state described by the density operator ρ =

|w w|. For example, the equal-time version of G(2) , de¬ned by setting all times to

zero in eqn (4.77), factorizes into

— —

G(2) (x1 , x2 ; x3 , x4 ) = E1 (r1 ) E2 (r2 ) E3 (r3 ) E4 (r4 ) , (5.110)

where Ep (r) =s— · E (r). In fact, correlation functions of all orders factorize in the same

p

way.

Now let us consider an experimental situation in which the classical current is

turned on at some time t0 < 0 and turned o¬ at t = 0, leaving the ¬eld prepared in a

coherent state |w . The time at which the Schr¨dinger and Heisenberg pictures agree

o

is now shifted to t = 0, and we assume that the ¬elds propagate freely for t > 0. The

Schr¨dinger-picture state vector |w, t evolves from its initial value |w, 0 according to

o

the free-¬eld Hamiltonian, while the operators remain unchanged.

In the Heisenberg picture the state vector is always |w and the operators evolve

freely according to eqn (3.94). This guarantees that

E(+) (r, t) |w = E (r, t) |w , (5.111)

where E (r, t) is the freely propagating positive-frequency part that evolves from the

initial (t = 0) function given by eqn (5.109). According to eqn (5.110) the correlation

function factorizes at t = 0, and by the last equation each factor evolves independently;

therefore, the multi-time correlation function for the wave packet coherent state |w

factorizes according to

— —

G(2) (x1 , x2 ; x3 , x4 ) = E1 (r1 , t1 ) E2 (r2 , t2 ) E3 (r3 , t3 ) E4 (r4 , t4 ) . (5.112)

½¾ Coherent states

5.6 Phase space description of quantum optics

The set of all classical ¬elds obtained by exciting a single mode is described by a two-

dimensional phase space, as shown in eqn (5.1). The set of all quasiclassical states for

the same mode is described by the coherent states {|± }, that are also labeled by a

two-dimensional space. This correspondence is the basis for a phase-space-like descrip-

tion of quantum optics. This representation of states and operators has several useful

applications. The ¬rst is a precise description of the correspondence-principle limit.

The relation between coherent states and classical ¬elds also provides a quantitative

description of the departure from classical behavior. Finally, as we will see in Section

18.5, the phase space representation of the density operator ρ gives a way to convert

the quantum Liouville equation for the operator ρ into a c-number equation that can

be used in numerical simulations.

In Section 9.1 we will see that the results of photon detection experiments are ex-

pressed in terms of expectation values of normal-ordered products of ¬eld operators. In

this way, counting experiments yield information about the state of the electromagnetic

¬eld. In order to extract this information, we need a general scheme for representing

the density operators describing the ¬eld states. The original construction of the elec-

tromagnetic Fock space in Chapter 3 emphasized the role of the number states. Every

density operator can indeed be represented in the basis of number states, but there are

many situations for which the coherent states provide a more useful representation.

For the sake of simplicity, we will continue to emphasize a single classical ¬eld mode

for which the phase space “em can be identi¬ed with the complex plane.

5.6.1 The Wigner distribution

The earliest”and still one of the most useful”representations of the density operator

was introduced by Wigner (1932) in the context of elementary quantum mechanics. In

classical mechanics the most general state of a single particle moving in one dimension

is described by a normalized probability density f (Q, P ) de¬ned on the classical phase

space “mech = {(Q, P )}, i.e. f (Q, P ) dQdP is the probability that the particle has

position and momentum in the in¬nitesimal rectangle with area dQdP centered at the

point (Q, P ) and

dQ dP f (Q, P ) = 1 . (5.113)

In classical probability theory it is often useful to represent a distribution in terms of

its Fourier transform,

dP f (Q, P ) e’i(uP +vQ) ,

χ (u, v) = dQ (5.114)

which is called the characteristic function (Feller, 1957b, Chap. XV). In some ap-

plications it is easier to evaluate the characteristic function, and then construct the

probability distribution itself from the inverse transformation:

du dv

χ (u, v) ei(uP +vQ) .

f (Q, P ) = (5.115)

2π 2π

½¿

Phase space description of quantum optics

An example of the utility of the characteristic function is the calculation of the mo-

ments of the distribution, e.g.

‚nχ

2

2

Q = (i) ,

‚v 2 (u,v)=(0,0)

‚nχ

2 (5.116)

QP = (i) ,

‚v‚u (u,v)=(0,0)

.

.

.

A The Wigner distribution in quantum mechanics

In quantum mechanics, a phase space description like f (Q, P ) is forbidden by the

uncertainty principle. Wigner™s insight can be interpreted as an attempt to ¬nd a

quantum replacement for the phase space integral in eqn (5.114). Since the integral is

a sum over all classical states, it is natural to replace it by the sum over all quantum

states, i.e. by the quantum mechanical trace operation. The role of the classical dis-

tribution is naturally played by the density operator ρ, and the classical exponential

exp [’i (uP + vQ)] can be replaced by the unitary operator exp [’i (up + vq)]. In this

way one is led to the de¬nition of the Wigner characteristic function

χW (u, v) = Tr ρe’i(up+vq) , (5.117)

which is a c-number function of the real variables u and v. The classical de¬nition

(5.114) of the characteristic function by a phase space integral is meaningless for

quantum theory, but the inverse transformation (5.115) still makes sense when applied

to χW . This suggests the de¬nition of the Wigner distribution,

dv

du

χW (u, v) ei(uP +vQ) ,

W (Q, P ) = (5.118)

2π 2π

where the normalization has been chosen to make W (Q, P ) dimensionless. The Wigner

distribution is real and normalized by

dQdP

W (Q, P ) = 1 , (5.119)

but”as we will see later on”there are physical states for which W (Q, P ) assumes

negative values in some regions of the (Q, P )-plane. For these cases W (Q, P ) cannot be

interpreted as a probability density like f (Q, P ); consequently, the Wigner distribution

is called a quasiprobability density.

Substituting eqn (2.116) for the density operator into eqn (5.117) leads to the

alternative form

Pe Ψe e’i(up+vq) Ψe

χW (u, v) =

e

Ψe e’ivq e’iup Ψe ,

Pe e i uv/2

= (5.120)

e

½ Coherent states

where the last line follows from the identity (C.67). Since exp (’iup) is the spatial

translation operator, the expectation value can be expressed as

dQ Ψ— (Q ) e’iv(Q + u) Ψe (Q + u) .

Ψe e’ivq e’iup Ψe = (5.121)

e

Substituting these results into eqn (5.118) ¬nally leads to

dX 2iXP/

Ψe (Q + X) Ψ— (Q ’ X) ,

Pe

W (Q, P ) = e (5.122)

e

π

e

which is the de¬nition used in Wigner™s original paper. Thus the ˜momentum™ depen-

dence of the Wigner distribution comes from the Fourier transform with respect to the

relative coordinate X. Integrating out the momentum dependence yields the marginal

distribution in Q:

dP 2

Pe |Ψe (Q)| .

W (Q, P ) = (5.123)

e

Despite the fact that W (Q, P ) can have negative values, the marginal distribution in

Q is evidently a genuine probability density.

B The Wigner distribution for quantum optics

In the transition to quantum optics the mechanical operators q and p are replaced

by the operators q and p for the radiation oscillator. In agreement with our earlier

experience, it turns out to be more useful to use the relations (2.66) to rewrite the

unitary operator exp [’i (up + vq)] as exp ·a† ’ · — a , where

ω

u’i v, (5.124)

·=

2 2ω

so that eqn (5.117) is replaced by

†

’· — a

χW (·) = Tr ρe·a . (5.125)

The characteristic function χW (·) has the useful properties χW (0) = 1 and χ— (·) =

W

χW (’·). The Wigner distribution is then de¬ned (Walls and Milburn, 1994, Sec.

4.2.2) as the Fourier transform of χW (·):

1 —

±’·±—

d2 ·e·

W (±) = χW (·) . (5.126)

π2

After verifying the identity

d2 ± ·— ±’·±—

e = δ2 (·) , (5.127)

π2

where δ2 (·) ≡ δ (Re ·) δ (Im ·), one ¬nds that the Wigner function W (±) is normalized

by

½

Phase space description of quantum optics