tive; and (2) the noise operator is missing! In other words, the degenerate, traveling-

wave, parametric ampli¬er is intrinsically noiseless. This does not mean that the

right-to-left propagating vacuum ¬‚uctuations entering port 2 have been magically

eliminated; rather, they do not contribute to the output noise because they are not

scattered into the left-to-right propagating signal“idler mode.

16.4 General description of linear ampli¬ers

We now turn from the examples considered above to a general description of the

class of single-input, single-output linear ampli¬ers introduced at the beginning of

this chapter. This will be a black box treatment with no explicit assumptions about

the internal structure of the ampli¬er.

½

General description of linear ampli¬ers

At a given time t, bin (t) and bout (t) are annihilation operators for photons in the

input and output modes respectively. The basic assumption for linear ampli¬ers is

that bout (t) can be expressed as a linear combination of the input-mode creation and

annihilation operators b† (t ) and bin (t ), for times t < t, plus an operator representing

in

additional noise contributed by the ampli¬er. The mathematical statement of this

physical assumption is

dt C (t ’ t ) b† (t ) + · (t) .

dt P (t ’ t ) bin (t ) +

bout (t) = (16.94)

in

Carrying out a Fourier transform, and combining the convolution theorem (A.55)

with the representation (14.114) leads to the frequency-domain form

†

bout (ω) = P (ω) bin (ω) + C (ω) bin (’ω) + · (ω) . (16.95)

†

Since the right side involves both bin (ω) and bin (’ω), the adjoint equation

† †

bout (’ω) = C — (’ω) bin (ω) + P — (’ω) bin (’ω) + · † (’ω) (16.96)

is also required. This construction guarantees that the ampli¬er noise operator · (ω)

only depends on the internal modes of the ampli¬er.

The input“output relations (16.46) for the phase-sensitive ampli¬er described in

Section 16.2.2 have the form of eqns (16.95) and (16.96), except for the explicit phase

factor exp (iθP ) associated with the particular pumping mechanism for that example.

This blotch can be eliminated by carrying out the uniform phase transformation

bout (ω) = bout (ω) eiθP /2 , bin (ω) = bin (ω) eiθP /2 , b2,in (ω) = b2,in (ω) eiθP /2 .

(16.97)

When expressed in terms of the transformed (primed) operators the input“output

relation (16.46) is scrubbed clean of the o¬ending phase factor.

This kind of maneuver is usually expressed in a condensed form something like this:

let bout (ω) ’ bout (ω) exp (iθP /2), etc. This is all very well, except for the following

puzzle: What has happened to the reference phase that was supposed to be provided by

the pump? The answer is that the input“output relation is only half the story. The rest

is provided by the density operator ρ = ρin ρamp . For the ampli¬er of Section 16.2.2,

ρamp is assumed to be a phase-insensitive noise reservoir, so the phase transformation

of b2,in (ω) is not a problem. On the other hand, the input signal state ρin is not”one

hopes”pure noise; therefore, more care is needed.

To illustrate this point, consider the opposite extreme ρin = β β , where β is

a multimode coherent state de¬ned in Section 5.5.1. In the present case, this means

b„¦ (t0 ) β = β„¦ β , (16.98)

which in turn yields

bin (t) β = βin (t) β , (16.99)

where

Linear optical ampli¬ers—

½

∞

d„¦

β„¦ e’i(„¦’ω0 )(t’t0 ) .

βin (t) = (16.100)

2π

’∞

This coherent state is de¬ned with respect to the original in-¬eld operators; conse-

quently, the action of the transformed operators is given by

bin (ω) β = e’iθP /2 βin (t) β . (16.101)

Thus the pump phase removed from the input“output relation is not lost; it reappears

in the calculation of the ensemble averages that are to be compared to experimental

results.

The same trick works for the examples of phase-insensitive ampli¬ers in Sections

16.2.1-A and 16.2.1-B. With this reassurance, we can assume that the most general

input“output equation can be written in the form of eqns (16.95) and (16.96).

16.4.1 The input“output equation

The linearity assumption embodied in eqns (16.95) and (16.96) does not in itself impose

any additional conditions on the coe¬cients P (ω) and C (ω), but in all the three of

the examples given above the explicit expressions for these functions satisfy the useful

symmetry condition

P — (’ω) = P (ω) , C — (’ω) = C (ω) . (16.102)

It is worthwhile to devote some e¬ort to ¬nding out the source of this property. The

¬rst step is to recall that the Langevin equations for the sample and reservoir modes

are derived from the Heisenberg equations for the ¬elds. In the three examples con-

sidered above, a (t) is the only sample operator; and the internal sample interaction

Hamiltonian HSS is a quadratic function of a (t) and a† (t). The Heisenberg equation

for a (t) is therefore linear. The equations for the reservoir variables are also linear

by virtue of the general assumption, made in Section 14.1.1-A, that the interaction

Hamiltonian is linear in the reservoir operators.

In all three examples, these properties allow the Langevin equations for a (t) and

†

a (t) to be written in the form

d

•S (t) = ’W •S (t) + F (t) , (16.103)

dt

where

a (t) ξ (t)

, F (t) =

•S (t) = , (16.104)

a† (t) ξ † (t)

W is a 2 — 2 hermitian matrix, and the noise operator ξ (t) is a linear combination of

reservoir operators. Solving eqn (16.103) via a Fourier transform leads to

•S (ω) = V (ω) F (ω) , (16.105)

where the 2 — 2 matrix

’1

V (ω) = (W ’ iω) (16.106)

½

General description of linear ampli¬ers

satis¬es

V † (’ω) = V (ω) . (16.107)

Substituting this solution into an input“output relation, such as eqn (14.109), produces

coe¬cients that have the symmetry property (16.102).

This analysis raises the following question: How restrictive is the assumption that

the sample operators satisfy linear equations of motion? To address this question, let us

assume that HSS contains terms that are more than quadratic in the sample operators,

so that the equations of motion are nonlinear. The solution would then express the

sample operators as nonlinear functions of the noise operators. This situation raises

two further questions, one physical and the other mathematical.

The physical question concerns the size of the higher-order terms in HSS . If they

are small, then HSS can be approximated by a quadratic form, and the linear model

is regained. If the higher-order terms cannot be neglected, then the sample must be

experiencing large amplitude excitations. Under these circumstances it is di¬cult to

see how the overall ampli¬cation process could be linear.

The mathematical issue is that nonlinear di¬erential equations for the sample op-

erators cannot readily be solved by the Fourier transform method. This makes it hard

to see how a frequency-domain relation like (16.95) could be derived.

These arguments are far from conclusive, but they do suggest that imposing the

assumption of weak sample excitations will not cause a signi¬cant loss of generality.

We will therefore extend the de¬nition of linear ampli¬ers to include the assumption

that the internal modes all satisfy linear equations of motion. This in turn implies that

the symmetry property (16.102) can be applied in general.

The necessity of working with the pair of input“output equations (16.95) and

(16.96) suggests that a matrix notation would be useful. The input“output equations

can be written as

•out (ω) = R (ω) •in (ω) + ζ (ω) , (16.108)

where

bγ (ω) · (ω)

•γ (ω) = (γ = in, out) , ζ (ω) = , (16.109)

† †

· (’ω)

bγ (’ω)

and

P (ω) C (ω)

R (ω) = (16.110)

C (ω) P (ω)

is the input“output matrix. In this notation the symmetry condition (16.102) is

R† (’ω) = R (ω) . (16.111)

The matrix R (ω) is neither hermitian nor unitary, but it does commute with

its adjoint, i.e. R† (ω) R (ω) = R (ω) R† (ω). Matrices with this property are called

Linear optical ampli¬ers—

¾¼

normal, and all normal matrices have a complete, orthonormal set of eigenvectors. An

explicit calculation yields the eigenvalue“eigenvector pairs

1 1

z1 (ω) = P (ω) + C (ω) , ˜1 = √ ,

1

2

(16.112)

i 1

z2 (ω) = P (ω) ’ C (ω) , ˜2 = √ .

’1

2

It is instructive to express the input“output equation in the basis {˜1 , ˜2 }. By writing

the expansion for the in-operator •in as

√ √

•in (ω) = 2Xin (ω) ˜1 + 2Yin (ω) ˜2 , (16.113)

one ¬nds the operator-valued coe¬cients to be

1 1 †

Xin (ω) = √ ˜† •in (ω) = bin (ω) + bin (’ω) = Xβ=0,in (ω) ,

1

2

2

(16.114)

1 1 †

Yin (ω) = √ ˜† •in (ω) = bin (ω) ’ bin (’ω) = Yβ=0,in (ω) .

2

2i

2

The special value, β = 0, of the quadrature angle is an artefact of the phase transfor-

mation trick”explained at the beginning of Section 16.4”used to ensure the absence

of explicit phase factors in the general input“output equation (16.95).

In this basis the input“output relations have the diagonal form

Xout (ω) = [P (ω) + C (ω)] Xin (ω) + ζ1 (ω) ,

(16.115)

Yout (ω) = [P (ω) ’ C (ω)] Yin (ω) + ζ2 (ω) ,

where

1 1

ζ1 (ω) = √ ˜† ζ (ω) = †

· (ω) + · † (’ω) = ζ1 (’ω) ,

1

2

2

(16.116)

1 1

ζ2 (ω) = √ ˜† ζ (ω) = †

· (ω) ’ · † (’ω) = ζ2 (’ω) .

2

2i

2

We will refer to Xout (ω) and Yout (ω) as the principal quadratures.

The ensemble average of eqn (16.108) is

¦out (ω) = R (ω) ¦in (ω) , (16.117)

where

bγ (ω) bγ (ω)

¦γ (ω) = •γ (ω) = = (γ = in, out) . (16.118)

† —

bγ (’ω) bγ (’ω)

Subtracting eqn (16.117) from eqn (16.108) produces the input“output equation

δ•out (ω) = R (ω) δ•in (ω) + ζ (ω) , (16.119)

¾½

General description of linear ampli¬ers

where δ•in (ω) = •in (ω) ’ ¦in (ω) and δ•out (ω) = •out (ω) ’ ¦out (ω) are respectively

the ¬‚uctuation operators for the input and output. In the principal quadrature basis

this becomes

δXout (ω) = [P (ω) + C (ω)] δXin (ω) + ζ1 (ω) ,

(16.120)

δYout (ω) = [P (ω) ’ C (ω)] δYin (ω) + ζ2 (ω) .

The diagonalized form (16.115) of the input“output relation suggests two natural

de¬nitions for gain in a general linear ampli¬er. These are the principal gains de¬ned

by

| Xout (ω) |2 2

2 = |P (ω) + C (ω)| ,

G1 (ω) = (16.121)

| Xin (ω) |

2

| Yout (ω) | 2

= |P (ω) ’ C (ω)| .

G2 (ω) = (16.122)

2

| Yin (ω) |

The principal gains can also be de¬ned as the eigenvalues of the gain matrix

G (ω) = R† (ω) R (ω) , (16.123)

which has the same eigenvectors as the input“output matrix. For phase-insensitive

ampli¬ers, the gain matrix is diagonal, and the principal gains are the same: G1 (ω) =

G2 (ω).

The complex functions P (ω) ± C (ω) that appear in eqn (16.115) are expressed in

terms of the principal gains as

G1 (ω) ei‘1 (ω) ,

P (ω) + C (ω) = (16.124)

P (ω) ’ C (ω) = G2 (ω) ei‘2 (ω) , (16.125)

so that the symmetry condition (16.102) becomes

Gj (ω) = Gj (’ω) ,

(j = 1, 2) . (16.126)

‘j (ω) = ’‘j (’ω) mod 2π

With this notation eqn (16.115) is replaced by

G1 (ω) ei‘1 (ω) Xin (ω) + ζ1 (ω) ,

Xout (ω) =

(16.127)

G2 (ω) ei‘2 (ω) Yin (ω) + ζ2 (ω) .

Yout (ω) =

16.4.2 Conditions for phase insensitivity

According to eqn (16.51), imposing condition (i) of Section 16.1.1 requires

e2iθ ’ 1 P (ω) C — (ω) bin (ω)

2 Re bin (’ω) = 0. (16.128)

This is supposed to hold as an identity in θ for all input values bin (ω) ; consequently,

the coe¬cients must satisfy

P (ω) C — (ω) = 0 . (16.129)

Thus all phase-insensitive ampli¬ers fall into one of the two classes illustrated in Sec-

tion 16.2.1: (1) phase-transmitting ampli¬ers, with C (ω) = 0 and P (ω) = 0; or (2)

phase-conjugating ampli¬ers, with P (ω) = 0 and C (ω) = 0.

Linear optical ampli¬ers—

¾¾

Turning next to condition (ii), we recall that the ¬‚uctuation operators satisfy the

input“output relation (16.95), and that the ampli¬er noise operator is not correlated

with the input ¬elds. Combining these observations with the condition (16.129) yields

Kout (ω, ω ) = P (ω) P — (ω ) Kin (ω, ω ) + C (ω) C — (ω ) Kin (’ω , ’ω) + Kamp (ω, ω ) .

(16.130)

Imposing eqn (16.9) on both Kout (ω, ω ) and Kin (ω, ω ) then implies that

Kamp (ω, ω ) = Namp (ω) 2πδ (ω ’ ω ) , (16.131)

where

Namp (ω) = Nout (ω) ’ G (ω) Nin (ω) , (16.132)

and

2

|P (ω)| (phase-transmitting ampli¬er) ,

G (ω) = (16.133)

2

|C (ω)| (phase-conjugating ampli¬er)

is the gain for the phase-insensitive ampli¬er. A similar calculation yields

δbout (ω) δbout (ω ) = P (ω) P (ω ) δbin (ω) δbin (ω )

† †

+ C (ω) C (ω ) δbin (’ω) δbin (’ω )

+ · (ω) · (ω ) . (16.134)

Imposing eqn (16.11) on the input and output ¬elds implies

· (ω) · (ω ) = 0 . (16.135)

In other words, the ampli¬er noise is itself phase insensitive, since it satis¬es eqns

(16.9) and (16.11).

16.4.3 Unitarity constraints

The derivation of the Langevin equation from the linear Heisenberg equations of mo-

tion imposes the symmetry condition (16.102) on the coe¬cients P (ω) and C (ω), but

the sole constraint on the ampli¬er noise is that it can only depend on the internal

modes of the ampli¬er. Additional constraints follow from the requirement that the

out-¬eld operators are related to the in-¬eld operators by a unitary transformation.

An immediate consequence is that the out-¬eld operators and the in-¬eld operators

satisfy the same canonical commutation relations:

«

†

bγ (ω) , bγ (ω ) = 2πδ (ω ’ ω ) ,¬

(γ = in, out) . (16.136)

⎭

b (ω) , b (ω ) = 0

γ γ

Substituting eqns (16.95) and (16.96) into eqn (16.136), with γ = out, imposes con-

†

ditions on the ampli¬er noise operator. Once again, we recall that bin (ω) and bin (ω)

are linear functions of the input ¬eld operators evaluated at the initial time t = t0 .

The ampli¬er noise operator depends on the noise reservoir operators evaluated at the

¾¿

Noise limits for linear ampli¬ers

same time t0 , e.g. see eqn (16.49). The equal-time commutators between the internal

mode operators and the input operators all vanish; therefore, the in-¬eld operators

†

bin (ω) and bin (ω) commute with the ampli¬er noise operators · (ω) and · † (ω).

With this simpli¬cation in mind, eqn (16.136) imposes two relations between the

ampli¬er noise operator and the c-number coe¬cients:

[· (ω) , · (ω )] = i G1 (ω) G2 (ω) sin [‘12 (ω)] 2πδ (ω + ω ) , (16.137)

and

· (ω) , · † (ω ) = 1 ’ G1 (ω) G2 (ω) cos [‘12 (ω)] 2πδ (ω ’ ω ) , (16.138)

where ‘12 (ω) = ‘1 (ω) ’ ‘2 (ω). The two kinds of phase-insensitive ampli¬ers corre-

spond to the values ‘12 (ω) = 0”the phase-transmitting ampli¬ers”and ‘12 (ω) =

π”the phase-conjugating ampli¬ers.

Combining the expression (16.114) for the input quadratures with eqn (16.136)

and the identities

† †

Xβ,in (ω) = Xβ,in (’ω) , Yβ,in (ω) = Yβ,in (’ω) (16.139)

yields the commutation relations

[Xin (ω) , Xin (ω )] = [Yin (ω) , Yin (ω )] = 0 , (16.140)

and

i

†

2πδ (ω ’ ω ) .

Xin (ω) , Yin (ω ) = (16.141)

2

The unitary connection between the in- and out-¬elds requires the output quadratures

to satisfy the same relations. Substituting the input“output equation (16.115) into eqns

(16.140) and (16.141) yields an equivalent form of the unitarity conditions:

[ζj (ω) , ζj (ω )] = 0 (j = 1, 2) , (16.142)

i

†

1’ G1 (ω) G2 (ω) ei‘12 (ω) 2πδ (ω ’ ω ) .

ζ1 (ω) , ζ2 (ω ) = (16.143)

2

16.5 Noise limits for linear ampli¬ers

The familiar uncertainty relations of quantum mechanics can be derived from the

canonical commutation relations by specializing the general result in Appendix C.3.7.

By a similar argument, the unitarity constraints on the noise operators impose lower

bounds on the noise added by an ampli¬er.

16.5.1 Phase-insensitive ampli¬ers

For phase-insensitive ampli¬ers, the commutation relations (16.137) and (16.138) re-

spectively reduce to

[· (ω) , · (ω )] = 0 , (16.144)

and

· (ω) , · † (ω ) = {1 “ G (ω)} 2πδ (ω ’ ω ) , (16.145)

where G (ω) is the gain. The upper and lower signs correspond respectively to phase-

transmitting and phase-conjugating ampli¬ers. In both cases, the ampli¬er noise is

Linear optical ampli¬ers—