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only depends on the ¬xed amplitude |±| = n, so the phase wander of the laser
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

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