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By virtue of the equal spacing of the energy levels”a unique property of the harmonic
oscillator”the wave function is periodic, with period T = 2π/ω. This in turn implies
that the time-dependent width,

2
ψ (t) |P 2 | ψ (t) ’ ψ (t) |P | ψ (t) ,
∆p (t) = (15.6)

will exhibit the same periodicity. In other words, ψ (P, t) is a breathing Gaussian wave
packet which expands in size”as measured by ∆p (t)”from its minimum initial value
to a maximum size half a period later, and then contracts back to its initial size. This
periodic cycling from minimum to maximum spread repeats inde¬nitely. We recall
from eqns (2.99) and (2.100) that the operators p and q respectively correspond to
the electric and magnetic ¬elds. According to Section 2.5 this means that the variance
in the electric ¬eld for the squeezed vacuum state (15.3) is smaller than the vacuum-
¬‚uctuation variance.
The Hamiltonian for a radiation oscillator is unchanged by the (unitary) parity
transformation p ’ ’p, q ’ ’q on the operators p and q; therefore the energy
eigenstates, e.g. the momentum-space eigenfunctions ¦n (P ), are also eigenstates of
parity:
¦n (P ) ’ (’1)n ¦n (P ) for P ’ ’P .
An immediate consequence of this fact is that an initial state having de¬nite parity,
i.e. a superposition of eigenstates which all have the same parity, will evolve into
a state with the same parity at all times. Inspection of eqn (15.3) shows that this
initial Gaussian state is an even function of P ; consequently, the coe¬cients Cn in the
expansion (15.5) must vanish for all odd integers n. In other words, the evolution of the
squeezed vacuum state can only involve even-parity eigenfunctions for the radiation
oscillators. Since these eigenfunctions represent number states, an equivalent statement
is that only even integer number states can be involved in the production and the time
evolution of a squeezed vacuum state. Thus we arrive at the important conclusion that
the simplest elementary process leading to such a state is photon pair production.
For production of photons in pairs one needs to look to nonlinear optical inter-
actions, such as those provided by χ(2) and χ(3) media. The ¬rst experiment demon-
strating a squeezed state of light was performed by Slusher et al. (1985), who used
four-wave mixing in an atomic-vapor medium with a χ(3) nonlinearity. More strongly
¾ Nonclassical states of light

squeezed states of light were subsequently generated in χ(2) crystals by Kimble and co-
workers (Wu et al., 1986). In both cases the internal interaction in the sample induced
by the external classical ¬eld has the form

HSS = i„¦P a†2 ’ HC , (15.7)

for some c-number, phenomenological coupling constant „¦P . Long before these exper-
iments were performed, squeezed states were discovered theoretically by Stoler (1970),
in a study of minimum-uncertainty wave packets that are unitarily equivalent to co-
herent states. Yuen (1976) introduced squeezed states into quantum optics through
the notion of two-photon coherent states. He also made the important observation
that squeezed states would lead to the possibility of quantum noise reduction. Caves
(1981) studied squeezed states in the context of possible improvements in the fun-
damental sensitivity of gravitational-wave detectors based on optical interferometers
that use squeezed light.
But how are squeezed states of light to be detected? If there is a synchronous
experimental method to measure p (t), i.e. the electric ¬eld, just at the integer multiples
of the period”when the p-noise, ∆p (t), is at a minimum”it is plausible that one can
observe p-noise that is less than the standard quantum limit. The price we pay for
reduced p-noise at integer multiples of the period (t = 0, T, 2T, . . .) is an increased
p-noise at odd multiples of a half-period (t = T /2, 3T /2, 5T /2, . . .). This increase must
be such that the product of the alternating deviations, e.g. ∆p (T ) ∆p (3T /2), remains
larger than /2. An equivalent argument is based on the fact that q = p, so that

—¦
the deviation in displacement, ∆q (t), is 90 out of phase (in quadrature) with ∆p (t).
Consequently ∆q (t) is a maximum when ∆p (t) is a minimum, and the uncertainty
relation is maintained at all times. A synchronous measurement method is provided
by balanced homodyne detection, as discussed in Section 9.3.3. This kind of detection
scheme has blind spots precisely at those times when the p-noise is at a maximum, and
sensitive spots at the intermediate times when the p-noise is at a minimum. In this
way, the signal-to-noise ratio of a synchronous measurement scheme for the electric
¬eld can, in principle, be increased over the prediction of the standard quantum limit
associated with a coherent state.
The theory required to describe the generation of squeezed states is signi¬cantly
more complex than the discussion showing that coherent states are generated by clas-
sical currents. For this reason, we will follow the historical sequence outlined above,
by ¬rst studying the formal properties of squeezed states. This background is quite
useful for the analysis of experiments, even in the absence of a detailed model of the
source. In subsequent sections we will present the theory of squeezed-light generation,
and ¬nally describe an actual experiment.

15.1.2 General properties of squeezed states
A Quadrature operators
In place of eqn (15.3) we could equally well consider a squeezed vacuum for which
the deviation in the magnetic ¬eld (i.e. ∆q (t)) periodically achieves minimum values
less than the vacuum ¬‚uctuation value B0 . This can be done by using the coordinate
¿
Squeezed states

representation, and replacing P by Q everywhere in the discussion. More generally,
there is no reason to restrict attention to purely electric or purely magnetic ¬‚uctua-
tions; we could, instead, decide to measure any linear combination of the two. For this
discussion, let us ¬rst introduce the dimensionless canonical operators
1† q ω
X0 ≡ a +a = = q,
2 2∆q0 2
(15.8)
i† p 1
Y0 ≡ a ’a = = p,
2 2∆p0 2ω
which satisfy the commutation relation
i
[X0 , Y0 ] = . (15.9)
2
Comparing this to the canonical relations [q, p] = i and ∆q∆p /2 shows that the
corresponding uncertainty product is
1
∆X0 ∆Y0 . (15.10)
4
The solution a (t) = a exp (’iωt) of the free-¬eld Heisenberg equations yields the
time evolution of X0 and Y0 :
X0 (t) = X0 cos (ωt) + Y0 sin (ωt) ,
(15.11)
Y0 (t) = ’X0 sin (ωt) + Y0 cos (ωt) ,
which describes a rotation in the phase plane. It is often useful to generalize the
conventional choice, t = 0, of the reference time to t = t0 , so that the annihilation
operator is given by
a (t) = ae’iω(t’t0 ) = ae’iβ e’iωt , (15.12)
where β = ’ωt0 . In a mechanical context, choosing t0 amounts to setting a clock; but
in the optical context, the preceding equation shows that choosing the reference time t0
is equivalent to choosing the reference phase β. In the homodyne detection experiments
to be described later on, the phase β can be controlled by means of changes in the
relative phase between a local oscillator beam and the squeezed light which is being
measured. With this choice of reference phase, the time evolution of the magnetic and
electric ¬elds is given by
X0 (t) = X cos (ωt) + Y sin (ωt) ,
(15.13)
Y0 (t) = ’X sin (ωt) + Y cos (ωt) ,
where
1
ae’iβ + a† eiβ = X0 cos (β) + Y0 sin (β) ,
X=
2 (15.14)
1 ’iβ † iβ
’a e = ’X0 sin (β) + Y0 cos (β) .
Y= ae
2i
These are the same quadrature operators introduced in the analysis of heterodyne
and homodyne detection in Section 9.3; they are related to the canonical operators
Nonclassical states of light

by a rotation through the angle β in the phase plane. The cases considered previously
correspond to β = ’π/2 and β = 0 for the electric and magnetic ¬elds respectively.
For any value of β, the quadrature operators satisfy eqns (15.9) and (15.10). Conse-
quently, for any coherent state |± ”in particular for the vacuum state”the variances
of the quadrature operators are
1
V (X) = V (Y ) = , (15.15)
4
and the uncertainty product ∆X∆Y = 1/4 has the minimum possible value at all
times. Fig. 5.1 shows that the phase space portrait of the coherent state in the dimen-
sionless variables (X0 , Y0 ) consists of a circular quantum fuzzball, which surrounds
the tip of the coherent-state phasor ±. The rotation to X and Y amounts to choos-
ing the X-axis along the phasor. The isotropic quantum fuzzball corresponds to a
quasi-probability distribution which has the form of an isotropic Gaussian in phase
space.
A state ρ is said to be squeezed along the quadrature X, if the variance
V (X) = X 2 ’ X 2 satis¬es V (X) < 1/4, where Z = Tr (ρZ), for any operator Z.
This condition can be expressed more conveniently in terms of the normal-ordered
2
variance VN (X) ≡ : X 2 : ’ : X : , where : Z : is the normal-ordering operation
de¬ned by eqn (2.107). Since X is a linear function of the creation and annihilation
operators, : X : = X, but : X 2 : = X 2 . An explicit calculation leads to the relation
1
VN (X) = V (X) ’ . (15.16)
4
With this notation, the squeezing condition becomes VN (X) < 0 and perfect squeez-
ing, i.e. V (X) = 0, corresponds to VN (X) = ’1/4.
The straightforward calculation suggested in Exercise 15.1 establishes the relations
1 1
Re e’2iβ V (a) + V a† , a ,
VN (X) = (15.17)
2 2
1 1
VN (Y ) = ’ Re e’2iβ V (a) + V a† , a , (15.18)
2 2
between the normal quadrature variances and variances of the annihilation opera-
tors. The quantity V a† , a = a† a ’ a† a is an example of the joint vari-
ance, V (F, G) = F G ’ F G , introduced in Section 5.1.1. It is easy to see that
V a† , a 0; therefore, necessary conditions for squeezing along X or Y are

Re e’2iβ V (a) < 0 (15.19)

and
Re e’2iβ V (a) > 0 (15.20)
respectively. Thus a state for which V (a) = 0 is not squeezed along any quadra-
ture. This fact excludes both number states and coherent states from the category of
squeezed states.
Squeezed states

B The squeezing operator
As an aid to understanding how single-mode squeezing is generated by the interaction
Hamiltonian (15.7), let us recall the argument used in Section 5.4.1 to guess the form
of the displacement operator that generates coherent states from the vacuum. The
Hamiltonian Hint describing the interaction of a classical current with a single mode
of the radiation ¬eld is linear in the creation and annihilation operators. For the mode
exactly in resonance with a purely sinusoidal current, the time evolution of the state
vector in the interaction picture is represented by the unitary operator exp (’itHint / ),
which leads to the form D (±) = exp ±a† ’ ±— a for the displacement operator.
By analogy with this argument, the quadratic interaction Hamiltonian (15.7) sug-
gests that the squeezing operator should be de¬ned by
—2
a ’ζa†2 )
1
S (ζ) = e 2 (ζ , (15.21)

where the c-number ζ = r exp (2iφ) is called the complex squeeze parameter. The
modulus r = |ζ| describes the amount of squeezing, and the phase 2φ determines the
angle of the squeezing axis in phase space.
The unitary squeezing operator applied to a pure state |Ψ de¬nes the squeezing
transformation,
|Ψ (ζ) = S (ζ) |Ψ , (15.22)
for states. It is also useful to de¬ne squeezed operators by

X (ζ) = S (ζ) XS † (ζ) , (15.23)

so that expectation values are preserved, i.e.

Ψ (ζ) |X (ζ)| Ψ (ζ) = Ψ |X| Ψ . (15.24)

Applying eqn (15.23) to the density operator describing a mixed state, as well as to
the observable X, shows that mixed-state expectation values are also preserved:

Tr [ρ (ζ) X (ζ)] = Tr [ρX] . (15.25)

The ¬rst example is the squeezed vacuum state ψ (P, 0) in eqn (15.3). With the
correct choice for ζ this can be expressed as

ψ (P, 0) = P |S (ζ)| 0 . (15.26)

In the limit of weak squeezing, i.e. |ζ| 1, the operator in eqn (15.22) can be expanded
to get
1 —2
ζ a ’ ζa†2 |0 + · · ·
S (ζ) |0 = |0 +
2
1 †2
= |0 ’ ζa |0 + · · · . (15.27)
2
The ¬rst-order term on the right side is the output state for the degenerate case of
the down-conversion process discussed in Section 13.3.2. Thus down-conversion rep-
resents incipient single-mode squeezing. The transformation of a single pump photon
Nonclassical states of light

of frequency ωP into a pair of photons, each with frequency ω0 = ωP /2, is the source
of the photons in the squeezed vacuum S (ζ) |0 . The general case of nondegenerate
down-conversion can similarly serve as the source of a two-mode squeezed state. In
this case, the nonlocal phenomena associated with entangled states would play an
important role.
For general squeezed states, the features of experimental interest are expressible
in terms of variances of the quadrature operators or other observables, such as the
number operator. For example, the variance, V (X), of X in the squeezed state is
2
V (X) = Tr ρ (ζ) X 2 ’ (Tr [ρ (ζ) X]) . (15.28)
The easiest way to evaluate these expressions is to use the relation (15.23) between the
original operators X and their squeezed versions X (ζ). Since all observables can be
expressed in terms of the creation and annihilation operators it is su¬cient to consider
a (ζ) = S (ζ) aS † (ζ) . (15.29)
The ¬rst step in evaluating the right side of this equation is to de¬ne the squeezing
generator K (ζ) by
i
K (ζ) = ’ ζ — a2 ’ ζa†2 , (15.30)
2
so that S (ζ) = exp [iK (ζ)]. The second step is to imitate eqn (5.49) by introducing
the interpolating operators
c („ ) = ei„ K(ζ) ae’i„ K(ζ) , (15.31)
where „ is a real variable in the interval (0, 1). The interpolation formula has the form
of a time evolution with Hamiltonian K, so the interpolating operators satisfy the
Heisenberg-like equations
d
i c („ ) = c („ ) , K , (15.32)
d„
where
i
K = ’ ζ — c2 („ ) ’ ζc†2 („ ) . (15.33)
2
If we identify ζ with ’2i„¦P , then K has the general form (15.7). This means that
we will be able to use the results obtained here to treat the model for squeezed-state
generation to be given in Section 15.2.
The explicit form (15.33), together with the canonical commutation relation
c („ ) , c† („ ) = 1, yields a pair of ¬rst-order equations for c and c† :
dc†
dc
= ζc† , = ζ—c , (15.34)
d„ d„
and eliminating c† produces a single second-order equation:
d2 c 2
= |ζ| c . (15.35)
2
d„
Since |ζ| = r2 is real and positive the fundamental solutions are e±r„ and the general
2

solution is c („ ) = C+ er„ +C’ e’r„ . Substituting this form into either of the ¬rst-order
Squeezed states

equations yields one relation between C+ and C’ , and the initial condition c (0) = a
gives another. The solution of this pair of algebraic equations provides the expression
a (ζ) = µa + νa† for the squeezed annihilation operator, where the coe¬cients

µ = cosh (r) , ν = e2iφ sinh (r) , (15.36)
2
satisfy the identity µ2 ’ |ν| = 1. The relation between a and a (ζ) is another example
of the Bogoliubov transformation. The inverse transformation,

a = µa (ζ) ’ νa† (ζ) , (15.37)

will be useful in subsequent calculations.
Let us ¬rst apply eqn (15.37) to express the quadrature operators, de¬ned by eqn
(15.14), as

1
cosh (r) ’ e’2i(φ’β) sinh (r) a (ζ) e’iβ + HC ,
X=
2 (15.38)
i
cosh (r) + e’2i(φ’β) sinh (r) a (ζ) e’iβ ’ HC .
Y=
2
For the quadrature angle β = φ this simpli¬es to X = e’r X (ζ) and Y = er Y (ζ), so
that
V (X) = V e’r X (ζ) = e’2r V (X (ζ)) = e’2r V0 (X) ,
(15.39)
V (Y ) = V (er Y (ζ)) = e2r V (Y (ζ)) = e2r V0 (Y ) ,
i.e. the X-quadrature is squeezed and the Y -quadrature is stretched, relative to the
variances V0 in the original state. The alternative choice β = φ ’ π/2 reverses the roles
of X and Y . For either choice, the deviations in the squeezed state satisfy

V (X) = e±r ∆0 X , ∆Y = V (Y ) = e“r ∆0 Y ,
∆X = (15.40)

which shows that the uncertainty product is unchanged by squeezing. In particular, if
|Ψ is a minimum-uncertainty state, then so is the squeezed state |Ψ (ζ) , i.e.

1
∆X∆Y = ∆0 X∆0 Y = . (15.41)
4
We now turn to the question of the classical versus nonclassical nature of squeezed
states. Suppose that ρ (ζ) is squeezed along X. The P -representation (5.168) can be
used to express the variance as

d2 ± 2
P (±) ± (X ’ X ) ±
V (X) =
π
d2 ± 2
P (±) ± X 2 ± ’ 2 X ± |X| ± + X
= , (15.42)
π

where P (±) is the P -function representing the squeezed state ρ (ζ) and X =
Tr [ρ (ζ) X]. The coherent-state expectation values can be evaluated by ¬rst using eqn
Nonclassical states of light

(15.14) and the commutation relations to express X 2 in normal-ordered form. After a
little further algebra one ¬nds that the normal-ordered variance is
2
±e’iβ + ±— eiβ
d2 ±
’X
VN (X) = P (±) . (15.43)
π 2

Now let us suppose that the squeezed state ρ (ζ) is classical, i.e. P (±) 0, then the
last result shows that VN (X) > 0. Since this contradicts the assumption that ρ (ζ) is
squeezed along X, we conclude that all squeezed states are nonclassical.

Multimode squeezed states—
15.1.3
A description of multimode squeezed states can be constructed by imitating the treat-
ment of multimode coherent states in Section 5.5.1. The single-mode squeezing oper-
ator can be applied to any member of a complete set of modes, e.g. the plane waves
of a box-quantized description; consequently, the simplest de¬nition of a multimode
squeezed state is
Ψ ζ = S ζ |Ψ , (15.44)
where
1 —2
ζks aks ’ ζks a†2
Sζ= exp (15.45)
ks
2
ks

is the multimode squeezing operator. Since the individual squeezing generators com-
mute, the de¬nition of S ζ can also be expressed as

1
ζks a2 ’ ζks a†2

S ζ = exp . (15.46)
ks ks
2
ks

Special squeezed states—
15.1.4
Coherent states are minimum-uncertainty states, so eqn (15.41) implies that the squee-
zed coherent states,

|ζ; ± ≡ S (ζ) |± = S (ζ) D (±) |0 , (15.47)

are also minimum-uncertainty states. In this notation, the squeezed vacuum state dis-
cussed previously is denoted by |ζ; 0 . The squeezed vacuum is generated by injecting
pump radiation into a nonlinear medium with an e¬ective interaction given by eqn
(15.7), and the more general squeezed coherent state can be obtained by simultane-
ously injecting the pump beam and the output of a laser matching the squeezed mode.
Furthermore, the squeezed coherent states are eigenstates of the transformed operator
a (ζ), since
a (ζ) |ζ; ± = S (ζ) a |± = ± |ζ; ± . (15.48)
The state |ζ; ± is therefore an analogue of the coherent state |± , but it is generated by
creating and annihilating pairs of photons. The squeezed coherent states are therefore

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