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