Squeezed states

For a ¬xed value of the squeezing parameter ζ, the squeezed coherent states have

the same orthogonality and completeness properties as the coherent states. The or-

thogonality property follows from the unitary relation (15.47), which shows that the

inner product of two squeezed coherent states is

ζ; β |ζ; ± = β S † (ζ) S (ζ) ± = β |± . (15.49)

The resolution of the identity follows in the same way, since combining eqn (15.47)

with eqn (5.69) gives us

d2 ± d2 ±

|± ±| S † (ζ) = 1 .

|ζ; ± ζ; ±| = S (ζ) (15.50)

π π

An alternative family of states is de¬ned by the displaced squeezed states

|±; ζ ≡ D (±) |ζ = D (±) S (ζ) |0 , (15.51)

which are constructed by displacing a squeezed vacuum state. An idealized physical

model for this is to inject the output of a squeezed vacuum generator into a laser

ampli¬er for the squeezed mode. The squeezed vacuum is the simplest example of a

squeezed state, so the displaced squeezed states are also called ideal squeezed states

(Caves, 1981).

The states |ζ; ± and |±; ζ are quite di¬erent, since the operators S (ζ) and D (±)

do not commute. For this reason it is important to remember that ζ is the squeezing

parameter and ± is the displacement parameter. Despite their di¬erences, these two

states are both normalized, so there must be a unitary transformation connecting

them. Indeed it is not di¬cult to show that they are related by

|ζ; ± = |±’ ; ζ (15.52)

and

|±; ζ = |ζ; ±+ , (15.53)

where

±± = µ± ± ν±—

= ± cosh r ± ±— e2iφ sinh r . (15.54)

According to eqn (15.53) the displaced squeezed state |±; ζ is also an eigenvector of

a (ζ),

a (ζ) |±; ζ = ±+ |±; ζ , (15.55)

but the eigenvalue is ±+ rather than ±.

The relation (15.53) allows us to transfer the orthogonality and completeness re-

lations for squeezed coherent states to the displaced squeezed states. Applying eqn

(15.53) to eqns (15.49) and (15.50) yields

β; ζ |±; ζ = ζ; β+ |ζ; ±+ = β+ |±+ , (15.56)

¼ Nonclassical states of light

and

d2 β d2 β

|β’ ; ζ β’ ; ζ| = |ζ; β ζ; β| = 1 . (15.57)

π π

The general result (15.39) shows that squeezing any minimum-uncertainty state

produces the quadrature variances V (X) = e’r /4 and V (Y ) = er /4. For the case of

the squeezed coherent state |ζ; ± , with ± = |±| eiθ , the quadrature averages are given

by

ζ; ± |X| ζ; ± = |±| e’r cos (θ ’ φ) ,

(15.58)

ζ; ± |Y | ζ; ± = |±| er sin (θ ’ φ) .

For the special choice θ = φ one ¬nds ζ; ± |Y | ζ; ± = 0 and

ζ; ± |X| ζ; ± = |±| e’r , (15.59)

so the squeezed quadrature X represents the amplitude of the coherent state. Con-

sequently this process is called amplitude squeezing. This example has led to the

frequent use of the names amplitude quadrature and phase quadrature for X

and Y respectively.

Of course, the roles of X and Y can always be changed by making a di¬erent phase

choice. If we choose θ ’ φ = π/2, then ζ; ± |X| ζ; ± = 0 and ζ; ± |Y | ζ; ± = |±| er .

The amplitude of the coherent state is now carried by the stretched quadrature Y ,

and the squeezed quadrature X is conjugate to Y . Roughly speaking, the operator

conjugate to the amplitude is related to the phase; consequently, this process is called

phase squeezing.

Photon-counting statistics—

15.1.5

The variances and averages of the quadrature operators were used in the interpretation

of the homodyne detection scheme discussed in Section 9.3.3, but photon-counting

experiments are related to the average and variance of the photon number operator.

For the special squeezed states de¬ned by eqns (15.47) and (15.51), the most direct

way to calculate these quantities is ¬rst to use eqn (15.37) to express the operators N

and N 2 in terms of the transformed operators a (ζ) and a† (ζ), and then to rearrange

these expressions in normal-ordered form with respect to a (ζ) and a† (ζ). Finally, the

eigenvalue equations (15.48) and (15.55), together with their adjoints, can be used to

get the expectation values of N and N 2 as explicit functions of ζ and ±.

By virtue of the relation (15.52), it is enough to consider the expectation values

for the displaced squeezed state |±; ζ . Using eqn (15.37) produces the expression

N = a† a = µa† (ζ) ’ ν — a (ζ) µa (ζ) ’ νa† (ζ)

= µ2 + |ν|2 a† (ζ) a (ζ) ’ ν — µa (ζ)2 ’ νµa† (ζ) + |ν|2 (15.60)

for the number operator, so eqn (15.55) and its adjoint yield

N = ±; ζ |N | ±; ζ = µ±— ’ ν — ±+ µ±+ ’ ν±— + |ν|

2

+ +

2 2 2

= |±| + |ν| = |±| + sinh2 (r) . (15.61)

To get the ¬nal result we have used the solution ± = µ±+ ’ ν±— of eqn (15.54).

+

½

Squeezed states

For the calculation of N 2 , we ¬rst use the commutation relations to establish the

identity N 2 = a†2 a2 + N , which leads to

N 2 = a†2 a2 + N . (15.62)

The next step is to use eqn (15.37) to derive the normal-ordered expression”with

respect to the squeezed operators a (ζ) and a† (ζ)”for a2 :

a2 = µ2 a (ζ) ’ 2µνa† (ζ) a (ζ) + ν 2 a† (ζ) ’ µν .

2

(15.63)

This can be used in turn to derive the normal-ordered form for a†2 a2 and thus to

evaluate a†2 a2 in the same way as N . This calculation is straightforward but

rather lengthy. A somewhat more compact method is to use the completeness relation

(15.57) to get

a†2 a2 = ±; ζ a†2 a2 ±; ζ

d2 β

†2

|β’ ; ζ β’ ; ζ| a2 |±; ζ

= ±; ζ| a

π

d2 β 2

β’ ; ζ a2 ±; ζ

= . (15.64)

π

Applying the eigenvalue equation (15.55) to |±; ζ and the adjoint equation to β’ ; ζ|

produces a (ζ) |±; ζ = ±+ |±; ζ and β’ ; ζ| a† (ζ) = β’ ; ζ| β — , so the matrix element

in the integrand is given by

β’ ; ζ a2 ±; ζ = f (β — ) β’ ; ζ |±; ζ = f (β — ) β |±+ , (15.65)

where

f (β — ) = µ2 ±2 ’ 2µνβ — ±+ + ν 2 β —2 ’ µν . (15.66)

+

Substituting this result in eqn (15.64) and using the explicit formula (5.58) for the

inner product leaves us with

d2 β

†2 2

|f (β — )|2 e’|β’±+ |

2

aa =

π

d2 β 2 ’|β|2

f β — + ±—

= e , (15.67)

+

π

where the last line was obtained by the change of integration variables β ’ β + ±+ .

This rather elaborate preparation would be useless if the remaining integrals could

not be easily evaluated. Fortunately, the integrals can be readily done in polar co-

ordinates, β = b exp (i‘), as can be seen in Exercise 15.4. After a certain amount of

algebra, one ¬nds

a†2 a2 = |±| + µ2 |ν| ’ µ ±2 ν — + CC + 4 |±| |ν| + 2 |ν| .

4 2 2 2 4

(15.68)

Combining this result with eqns (15.36), (15.62), and (9.58) leads to the general ex-

pression

sinh2 r cosh 2r + 2 |±|2 sinh r [sinh r ’ cosh r cos (θ ’ φ)]

Q= (15.69)

2

|±| + sinh2 r

for the Mandel Q parameter.

¾ Nonclassical states of light

The Q parameter is positive (super-Poissonian statistics) for cos (θ ’ φ) 0, but

it can be negative (sub-Poissonian statistics) if cos (θ ’ φ) > 0. In the case θ = φ

we have amplitude squeezing (see eqn (15.59) for the squeezed quadrature X), so the

general result becomes

1 ’ e’2r

2

sinh2 r cosh 2r ’ |±|

Q= . (15.70)

2

|±| + sinh2 r

In the strong-¬eld limit |±| exp (4r), Q becomes

Q ≈ ’ 1 ’ e’2r . (15.71)

1), then Q ≈ ’1, i.e. there is negligible noise

If we also assume strong squeezing (r

in photon number. Consequently, amplitude squeezed states are also called number

squeezed states. This terminology is rather misleading, since eqn (15.19) shows that

a squeezed state can never be a number state.

Are squeezed states robust?—

15.1.6

In Section 8.4.3 we saw that a coherent state |±1 incident on a beam splitter is

scattered into a two-mode coherent state |±1 , ±2 , where ±1 = t ±1 and ±2 = r ±1 . A

similar result would be found for any passive, linear optical element. An even more

impressive feature appears in Section 18.5.2, where it is shown that an initial coherent

state |±0 coupled to a zero-temperature reservoir evolves into the coherent state

±0 e’“t/2 e’iω0 t . In other words, the de¬ning statistical property, V a† , a = 0, of

the coherent state is unchanged by this form of dissipation. Only the amplitude of the

parameter ±0 is reduced. For these reasons the coherent state is regarded as robust.

The situation for squeezed states turns out to be a bit more subtle.

Let us ¬rst consider an experiment in which light in a squeezed state enters through

port 1 of a beam splitter, as shown in Fig. 8.2. The input state |Ψ is the vacuum for

the mode entering through the unused port 2, i.e.

a2 |Ψ = 0 , (15.72)

but it is squeezed along a quadrature

1

a1 e’iβ + a† eiβ

X1 = (15.73)

1

2

of the incident mode 1, i.e. VN (X1 ) < 0. According to eqn (8.62) the scattered oper-

ators a1 and a2 are related to the incident operators a1 and a2 by

a 2 = r a1 + t a 2 ,

(15.74)

a 1 = t a 1 + r a2 ,

2 2

where |t| +|r| = 1. We choose the phases of r and t so that the transmission coe¬cient

t is real and the re¬‚ection coe¬cient r is purely imaginary.

¿

Squeezed states

The question to be investigated is whether there is squeezing along any output

quadrature. We begin by examining general quadratures

1

a1 e’iβ1 + a1† eiβ1

X1 = (15.75)

2

and

1

a2 e’iβ2 + a2† eiβ2

X2 = (15.76)

2

for the transmitted and re¬‚ected modes respectively. Applying eqns (15.17), (15.72),

and (15.74) to the X1 -quadrature leads to

1 1

Re V a1 e’iβ1 + V a1† , a1

VN (X1 ) =

2 2

1 12

= Re t2 V a1 e’iβ1 + |t| V a† , a1

1

2 2

t2

= t VN (X1 ) + Re ei• ’ 1 V a1 e’iβ

2

, (15.77)

2

where • = 2 (β ’ β1 ). Squeezing along X1 means that VN (X1 ) < 0, but the second

term depends on the value of β1 . The simplest choice”β1 = β”leads to

VN (X1 ) = t2 VN (X1 ) , (15.78)

which shows that squeezing along X1 implies squeezing along X1 for the quadrature

angle β1 = β. As might be expected, the inescapable partition noise at the beam

splitter reduces the amount of squeezing by the intensity transmission coe¬cient t2 <

1. This particular choice of output quadrature does answer the squeezing question,

but it does not necessarily yield the largest degree of squeezing.

A similar argument applied to X2 begins with

1 1

+ V a2† , a2 ,

Re V a2 e’iβ2

VN (X2 ) = (15.79)

2 2

2

but the relation r2 = ’ |r| produces

Re V a2 e’iβ2 = Re r2 V a1 e’iβ2 = ’ |r| Re V a1 e’iβ2

2

. (15.80)

The ¬nal result in this case is

|r|2

ei• + 1 V a1 e’iβ

2

V (X2 ) = |r| VN (X1 ) ’ Re , (15.81)

2

where • = 2 (β ’ β2 ). For the re¬‚ected mode, the choice β2 = β ’ π/2 (• = π) shows

reduced squeezing along X2 . Alternatively, we can use the relation

X2 |β2 =β’π/2 = ’ Y2 |β2 =β (15.82)

to say that squeezing occurs along the conjugate quadrature Y2 for β2 = β.

Nonclassical states of light

We next consider the evolution of a squeezed state coupled to a zero-temperature

reservoir. For the quadrature

1

ae’iβ + a† eiβ ,

Xβ = (15.83)

2

eqn (15.43) gives us

2

±e’iβ + ±— eiβ

d2 ±

P (±, ±— ; t) ’ Xβ ; t

VN (Xβ ; t) = , (15.84)

π 2

where

±e’iβ + ±— eiβ

d2 ± —

Xβ ; t = P (±, ± ; t) . (15.85)

π 2

The assumption that the state is initially squeezed along Xβ means that

2

±e’iβ + ±— eiβ

d2 ±

P0 (±, ±— ) ’ Xβ

VN (Xβ ; 0) = < 0, (15.86)

0

π 2

where P0 (±, ±— ) = P (±, ±— ; t = 0). Anticipating the general solution (18.88) for dis-

sipation by interaction with a zero-temperature reservoir leads to

d2 ±

P0 e(“/2+iω0 )t ±, e(“/2’iω0 )t ±— e“t

VN (Xβ ; t) =

π

2

±e’iβ + ±— eiβ

— ’ Xβ ; t , (15.87)

2

and

d2 ±

P0 e(“/2+iω0 )t ±, e(“/2’iω0 )t ±— e“t

Xβ ; t =

π

±e’iβ + ±— eiβ

— . (15.88)

2

Our next step is to make the change of integration variables ± ’ ± exp [’ (“/2 + iω0 ) t]

in the last two equations. For eqn (15.88) the result is

±e’i(β+ω0 t) + ±— e’i(β+ω0 t)

d2 ±

’“t/2

P0 (±, ±— )

Xβ ; t = e

π 2

= e’“t/2 Xβ+ω0 t , (15.89)

0

and a similar calculation starting with eqn (15.87) yields

VN (Xβ ; t) = e’“t VN (Xβ+ω0 t ; 0) . (15.90)

Just as in the case of the beam splitter, we are free to choose new quadratures to

investigate, in this case at di¬erent times. At time t we take advantage of this freedom

to let β ’ β ’ ω0 t, so that

VN (Xβ’ω0 t ; t) = e’“t VN (Xβ ; 0) < 0 . (15.91)

Thus at any time t, there is a squeezed quadrature”with the amount of squeezing

reduced by exp (’“t)”but the required quadrature angle rotates with frequency ω0 .

Theory of squeezed-light generation—

With the results (15.78) and (15.91) in hand, we can now judge the robustness

of squeezed states. Let us begin by recalling that coherent states are regarded as

robust because the de¬ning property, V a† , a = 0, is strictly conserved by dissipa-

tive scattering”i.e. coupling to a zero-temperature reservoir”as well as by passage

through passive, linear devices. By contrast, dissipative scattering degrades the degree

of squeezing as well as the overall intensity of the squeezed input light, so that

|VN (X : t)| ’ 0 as t ’ ∞ . (15.92)

Even this result depends on the detection of a quadrature that is rotating at the

optical frequency ω0 . Detector response times are large compared to optical periods,

so even the reduced squeezing shown by eqn (15.91) would be extremely di¬cult to

detect. Passage through a linear optical device also degrades the degree of squeezing,

as shown by eqn (15.78). This combination of properties is the basis for the general

opinion that squeezed states are not robust.

Theory of squeezed-light generation—

15.2

The method used by Kimble and co-workers (Wu et al., 1986) to generate squeezed

states relies on the microscopic process responsible for the spontaneous down-conver-

sion e¬ect discussed in Section 13.3.2; but two important changes in the experimental

arrangement are shown in Fig. 15.1. The ¬rst is that the χ(2) crystal is cut so as

to produce collinear phase matching with degenerate pairs (ω1 = ω2 = ω0 = ωP /2) of

photons, and also anti-re¬‚ection coated for both the ¬rst- and the second-harmonic

frequencies ω0 and ωP = 2ω0 . In this con¬guration the down-converted photons have

identical frequencies and propagate in the same direction as the pump photons; in

other words, this is time-reversed second harmonic generation. The second change

is that the crystal is enclosed by a resonant cavity that is tuned to the degenerate

frequency ω0 = ωP /2 and, therefore, also to the pump frequency ωP .

The degeneracy conditions between the down-converted photons and the cavity

resonance frequency are maintained by a combination of temperature tuning for the

crystal and a servo control of the optical resonator length. This arrangement strongly

Mirrors for ω0 and ω2

Degenerate

Pump ω1

ω2

signal and idler

at ω2 at ω1 = ω2 = ω0 = ω2 /2

ω2

1 2

crystal

χ(2)

Fig. 15.1 A simpli¬ed schematic for the squeezed state generator employed in the experiment

of Kimble and co-workers (Wu et al., 1986).

Nonclassical states of light

favors the degenerate pairs over all other pairs of photons that are produced by down-

conversion. In this way, the crystal”pumped by the strong laser beam at the second-

harmonic frequency 2ω0 ”becomes an optical parametric ampli¬er1 (OPA) for the

degenerate photon pairs at the ¬rst-harmonic frequency ω0 .

This device can be understood at the classical level in the following way. The χ(2)

nonlinearity couples the two weak down-converted light beams to the strong-pump

laser beam, so that the weak light signals can be ampli¬ed by drawing energy from