“12 ’ i∆0 “12 ’ i∆0

We have to warn the reader that this procedure is something of a swindle, since

ξ12 (z, t) is not a slowly-varying function of t. Fortunately, the result can be justi¬ed”

see Exercise 16.3”by interpreting δ (t ’ t ) in eqn (16.67) as an even coarser-grained

delta function, that only acts on test functions that vary slowly on the dephasing time

scale T12 = 1/“12 .

In the linear approximation for eqn (16.68), the operator S 22 (z, t) ’ S 11 (z, t) can

be simpli¬ed in two ways. The ¬rst is to neglect the small quantum ¬‚uctuations,

i.e. to replace the operator by its average S 22 (z, t) ’ S 11 (z, t) . The next step is to

solve the averaged form of the operator Bloch equations (14.174)“(14.177), with the

approximation that HS1 = 0. The result is

S 22 (z, t) ’ S 11 (z, t) ≈ S 22 (z, t) ’ S 11 (z, t) = nat σD , (16.69)

where D is the steady-state inversion for a single atom. With these approximations,

the propagation equation (16.63) becomes

f—

2

|f| nat σD

‚ ‚

+ vg0 φ (z, t) = φ (z, t) + ξ12 (z, t) . (16.70)

“12 ’ i∆0 “12 ’ i∆0

‚t ‚z

This equation is readily solved by transforming to the wave coordinates:

„ = t ’ z/vg0 (the retarded time for the signal wave) ,

(16.71)

z = z,

Linear optical ampli¬ers—

½

to get

f—

‚

φ (z, „ ) = gφ (z, „ ) + ξ12 (z, „ ) , (16.72)

vg0 (“12 ’ i∆0 )

‚z

where

|f|2 nat σD k0 |d21 · e0 |2 nat D

g= = (16.73)

vg0 [“12 ’ i∆0 ] 2 0 [“12 ’ i∆0 ]

is the (complex) small-signal gain. The retarded time „ can be treated as a parameter

in eqn (16.72), so the solution is

z

f—

gz

dz1 eg(z’z1 ) ξ12 (z1 , „ ) ,

φ (z, „ ) = φ (0, „ ) e + (16.74)

vg0 (“12 ’ i∆0 ) 0

which has the form

f— z ’ z1

z

φ (z, t) = φ (0, t ’ z/vg0 ) e dz1 eg(z’z1 ) ξ12 z1 , t ’

gz

+

vg0 (“12 ’ i∆0 ) vg0

0

(16.75)

in the laboratory coordinates (z, t).

Setting z = LS and letting t ’ t + LS /vg0 gives the ¬eld value at the output face:

f— LS

z1

gLS

dz1 eg(z’z1 ) ξ12 z1 , t +

φ (LS , t + LS /vg0 ) = φ (0, t) e + .

vg0 (“12 ’ i∆0 ) vg0

0

(16.76)

In order to recover the standard form for input“output relations we introduce the

representations

1 dω

bin (ω) e’iω(t’z/vg0 ) for z < 0

φ (z, t) = √ (16.77)

vg0 2π

and

1 dω

bout (ω) e’iω(t’z/vg0 ) for z > LS ,

φ (z, t) = √ (16.78)

vg0 2π

√

for the solutions of eqn (16.64) outside the crystal. The factor 1/ vg0 is inserted to

guarantee that the commutation relation (16.55) for φ (z, t) and the standard input“

output commutation relations

bγ (ω) , b† (ω ) = 2πδ (ω ’ ω ) (γ = in, out) (16.79)

γ

are both satis¬ed. Substituting eqns (16.77) and (16.78) into eqn (16.76) and carrying

out a Fourier transform produces the input“output relation

bout (ω) = egLS bin (ω) + · (ω) , (16.80)

where

f— LS

dz1 eg(z’z1 ) e’iωz1 /vg0 ξ12 (z1 , ω)

· (ω) = √ (16.81)

vg0 (“12 ’ i∆0 ) 0

½

Traveling-wave ampli¬ers

is the ampli¬er noise operator. By using the frequency-domain form of eqn (16.67),

one can show that the noise correlation function is

1

· (ω) · † (ω ) + · † (ω) · (ω )

Kamp (ω, ω ) =

2

= Namp 2πδ (ω ’ ω ) , (16.82)

where the noise strength is

2

nat k0 |d21 · e0 | e2gLS ’ 1 1

Namp = (C12,12 + C21,21 ) . (16.83)

2 0 (“2 + ∆2 ) 2g 2

12 0

Comparing eqn (16.80) to eqn (16.46) shows that P (ω) = egLS and C (ω) = 0. Con-

sequently, this ampli¬er is phase insensitive and phase transmitting.

16.3.2 Traveling-wave OPA

For this example, we return to the down-conversion technique by removing the mir-

rors from the phase-sensitive design shown in Fig. 16.2. Even without the mirrors,

appropriately cutting the ends of the crystal will guarantee that the pump beam and

the degenerate signal and idler beams copropagate along the length of the crystal. We

assume a Gaussian pump beam with spot size w0 and Rayleigh range ZR focussed on

a nonlinear crystal with radius RS and length LS .

If w0 > 2RS and ZR LS , the e¬ects of di¬raction are negligible; consequently,

the problem is e¬ectively one-dimensional. In this limit, the classical pump ¬eld can

be expressed as

E P (r, t) = eP |EP 0 | eiθP fP (t ’ z/vgP ) ei(kP z’ωP t) , (16.84)

where we have assumed that the medium outside the crystal is linearly index matched.

The temporal shape of the pump pulse is described by the function fP („ ), with max-

imum value fP (0) = 1 and pulse duration „P . In the long-pulse limit, „P ’ ∞, the

problem is further simpli¬ed by setting fP (t ’ z/vgP ) = 1.

In the 1D limit the signal“idler mode is described by a paraxial state, so the

¬eld can again be represented by eqn (16.54), with ω0 = ωP /2. The polarization of

the signal“idler mode is ¬xed, relative to that of the pump, by the phase-matching

conditions in the nonlinear crystal. Applying the 1D approximation and the long-pulse

limit to the expressions (7.39) and (13.30) yields the e¬ective ¬eld Hamiltonian

∞ LS

†

dz e’iθP φ2 (z, t) + HC . (16.85)

dzφ (z, t) vg0 ∇z φ (z, t) + g (3)

Hem =

i 2

’∞ 0

The special form of the interaction Hamiltonian”which represents the pair-produc-

tion aspect of down-conversion”produces a propagation equation

‚ ‚

φ (z, t) = ’ig (3) eiθP φ† (z, t)

+ vg0 (16.86)

‚t ‚z

Linear optical ampli¬ers—

½

that couples the ¬eld φ (z, t) to its adjoint φ† (z, t). This means that the propagation

equation and its adjoint must be solved together. In the wave coordinates de¬ned by

eqn (16.71) the equations to be solved are

‚

φ (z, „ ) = ’igeiθP φ† (z, „ ) , (16.87)

‚z

‚†

φ (z, „ ) = ige’iθP φ (z, „ ) , (16.88)

‚z

where g = g (3) /vg0 is the weak signal gain. Since the retarded time „ only appears as

a parameter, these equations can be solved by standard techniques to ¬nd

φ (z, t) = φ (0, t ’ z/vg0 ) cosh (gz) ’ ieiθP φ† (0, t ’ z/vg0 ) sinh (gz) , (16.89)

where we have reverted to the original (z, t)-variables. Thus the solution at z, t is

expressed in terms of the ¬eld operators evaluated at the input face, z = 0, and the

retarded time „ = t ’ z/vg0 .

The time-domain, input“output relation for the traveling-wave ampli¬er is obtained

by evaluating this solution at the output face, z = LS , and letting t ’ t + LS /vg0 :

φ (LS , t + LS /vg0 ) = φ (0, t) cosh (gLS ) ’ ieiθP φ† (0, t) sinh (gLS ) . (16.90)

Fourier transforming this equation yields

e’iLS /vg0 φ (LS , ω) = φ (0, ω) cosh (gLS ) ’ ieiθP φ† (0, ’ω) sinh (gLS ) , (16.91)

which can be brought into the standard form for input“output relations by using the

representations (16.77) and (16.78) to ¬nd

bout (ω) = P bin (ω) ’ ieiθP Cb† (0, ’ω) , (16.92)

in

with

P = cosh (gLS ) and C = sinh (gLS ) . (16.93)

Comparing this to eqn (16.46) reveals two things: (1) the ampli¬er is phase sensi-

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—

¾

phase insensitive, so Kamp (ω, ω ) satis¬es eqn (16.131). Substituting this form into

the de¬nition (16.32) then leads to

1

· (ω) · † (ω ) + · † (ω ) · (ω)

Namp (ω) 2πδ (ω ’ ω ) =

2

1

= · † (ω ) · (ω) + · (ω) , · † (ω )

2

1

= · † (ω ) · (ω) + {1 “ G (ω)} 2πδ (ω ’ ω ) .

2

(16.146)

Since · † (ω ) · (ω) is a positive-de¬nite integral kernel, we see that

1

Namp (ω) |1 “ G (ω)| . (16.147)

2

Thus a phase-insensitive ampli¬er with G (ω) > 1 necessarily adds noise to any input

signal.

For phase-insensitive input noise, the output noise is also phase insensitive; and

eqn (16.132) can be rewritten as

Nout (ω) = G (ω) Nin (ω) + Namp (ω) . (16.148)

For some purposes it is useful to treat the ampli¬er noise as though it were due to the

ampli¬cation of a ¬ctitious input noise A (ω). This additional input noise”which is

called the ampli¬er noise number”is de¬ned by

Namp (ω)

A (ω) = . (16.149)

G (ω)

With this notation, the relation (16.148) and the inequality (16.147) are respectively

replaced by

Nout (ω) = G (ω) [Nin (ω) + A (ω)] (16.150)

and

1 1

A (ω) 1“ . (16.151)

2 G (ω)

Applying this inequality to eqn (16.150) yields a lower bound for the output noise:

1 1

Nout (ω) G (ω) Nin (ω) + 1“ . (16.152)

2 G (ω)

If the input noise is due to a heat bath at temperature T , the continuum versions

of eqns (14.28) and (2.177) combine to give the noise strength,

1 1

Nin (ω) = +

exp [β (ω0 + ω)] ’ 1 2

1 (ω0 + ω)

= coth . (16.153)

2 2kB T

This result suggests a more precise de¬nition of the e¬ective noise temperature, ¬rst

discussed in Section 9.3.2-B. The idea is to ask what increase in temperature (T ’ T +

¾

Noise limits for linear ampli¬ers

Tamp ) is required to blame the total pre-ampli¬cation noise on a ¬ctitious thermal

reservoir. A direct application of this idea leads to

1 (ω0 + ω) 1 (ω0 + ω)

+ A (ω) ,

coth = coth (16.154)

2 2kB (T + Tamp ) 2 2kB T

but this would make Tamp depend on the input-noise temperature T and on the fre-

quency ω. A natural way to get something that can be regarded as a property of the

ampli¬er alone is to impose eqn (16.154) only for the case T = 0 and for the resonance

frequency ω = 0. This yields the ampli¬er noise temperature

ω0

kB Tamp = . (16.155)

ln (1 + 1/A (0))

For G (0) = G (ω0 ) > 1, eqns (16.155) and (16.151) provide the lower bound

ω0 ω0

’

kB Tamp (16.156)

ln (3)

3G(ω0 )“1

ln G(ω0 )“1

on the noise temperature. The ¬nal form is the limiting value for high gains, i.e.

G (ω0 ) 1.

16.5.2 Phase-sensitive ampli¬ers

The de¬nition of a phase-sensitive ampli¬er is purely negative. An ampli¬er is phase

sensitive if it is not phase insensitive. One consequence of this broad de¬nition is that

explicit constraints”such as the special form imposed on the noise correlation func-

tion by eqn (16.131)”are not available for phase-sensitive ampli¬ers. In the general

case, e.g. when considering broadband ampli¬ers, further restrictions on the family of

ampli¬ers are used to make up for the absence of constraints (Caves, 1982). For the

narrowband ampli¬ers we are studying, an alternative approach will be described be-

low. It is precisely the presence of the constraint (16.131) which makes the alternative

approach unnecessary for the noise analysis of phase-insensitive ampli¬ers.

The basic idea of the alternative approach is to treat narrow frequency bands of the

input and output as though they were discrete modes. For this purpose, let ∆ω be a

frequency interval that is small compared to the characteristic widths of the functions

Gj (ω) and ‘j (ω)”or P (ω) and C (ω)”and de¬ne coarse-grained quadratures and

noise operators by

ω+∆ω/2

dω1

√

c

F (ω) = F (ω1 ) , (16.157)

2π∆ω

ω’∆ω/2

where F stands for any of the operators in the set

F = {Xin (ω) , Yin (ω) , Xout (ω) , Yout (ω) , ζ1 (ω) , ζ2 (ω)} . (16.158)

All of these operators satisfy F † (ω) = F (’ω), and this property is inherited by the

coarse-grained versions: F c† (ω) = F c (’ω). From the experimental point of view, the

coarse-graining operation is roughly equivalent to the use of a narrowband-pass ¬lter.

Linear optical ampli¬ers—

¾

The noise strength for the non-hermitian operator F c (ω) is

1

2

[∆F c (ω)] = δF c (ω) δF c† (ω) + δF c† (ω) δF c (ω) , (16.159)

2

but this general result can be simpli¬ed by using the special properties of the oper-

ators in F. The commutation relations (16.140) and (16.142) guarantee that all the

operators in F satisfy [F (ω) , F (ω )] = 0, and the property F † (ω) = F (’ω) shows

that this is equivalent to F (ω) , F † (ω ) = 0. Averaging ω and ω over the interval

(ω ’ ∆ω/2, ω + ∆ω/2) in the latter form yields the coarse-grained relation

F c (ω) , F c† (ω) = 0 , (16.160)

and this allows eqn (16.159) to be replaced by

2

[∆F c (ω)] = δF c† (ω) δF c (ω) = δF c (ω) δF c† (ω) . (16.161)

The output noise strength can be related to the input noise strength and the

ampli¬er noise by means of the coarse-grained input“output equations:

c

G1 (ω) ei‘1 (ω) Xin (ω) + ζ1 (ω) ,

c c

Xout (ω) =

(16.162)

c i‘2 (ω) c c

Yout (ω) = G2 (ω) e Yin (ω) + ζ2 (ω) .

These relations are obtained by applying the averaging procedure (16.157) to eqn

(16.127), and using the assumption that the gain functions are essentially constant over

the interval (ω ’ ∆ω/2, ω + ∆ω/2). The lack of correlation between the in-¬elds and

the ampli¬er noise implies that the output noise strength in each principal quadrature

is the sum of the ampli¬ed input noise and the ampli¬er noise in that quadrature:

2 2 2

c c c

[∆Xout (ω)] = G1 (ω) [∆Xin (ω)] + [∆ζ1 (ω)] ,

(16.163)

2 2 2

c c c

[∆Yout (ω)] = G2 (ω) [∆Yin (ω)] + [∆ζ2 (ω)] .

In this situation there is an ampli¬er noise number for each principal quadrature:

2

c

∆ζj (ω)

Aj (ω) = (j = 1, 2) , (16.164)

Gj (ω)

so that eqn (16.163) can be written as

2 2

[∆Xout (ω)] = G1 (ω) [∆Xin (ω)] + A1 (ω) ,

c c

(16.165)

2 2

(ω)] + A2 (ω) .

c c

[∆Yout (ω)] = G2 (ω) [∆Yin

The signal-to-noise ratios for the principal quadratures are de¬ned by

2

c

Xγ (ω)

[SNR (X)]γ = (γ = in, out) ,

2

c

∆Xγ (ω)

(16.166)

2

Yγc (ω)

[SNR (Y )]γ = (γ = in, out) .

2

c

∆Yγ (ω)

¾

Exercises

Input“output relations for the signal-to-noise ratios follow by combining the ensemble

average of the operator input“output equation, eqn (16.162), with eqn (16.165) to get

[SNR (X)]in

[SNR (X)]out = ,

2

1 + A1 (ω) / [∆Xin (ω)]

c

(16.167)

[SNR (Y )]in

[SNR (Y )]out = .

2

1 + A2 (ω) / [∆Yin (ω)]

c

c c

Lower bounds on the ampli¬er noise strengths ∆ζ1 (ω) and ∆ζ2 (ω) can be derived

by applying the coarse-graining operation to the commutation relation (16.143) to get

i

c†

ζ1 (ω) , ζ2 (ω) = 1 ’

c

G1 (ω) G2 (ω)ei‘12 (ω) . (16.168)

2

This looks like the commutation relations between a canonical pair, except for the fact

c†

c

that the operators ζ1 (ω) and ζ2 (ω) are not hermitian. This ¬‚aw can be circumvented

by applying the generalized uncertainty relation, 2∆C∆D | [C, D] |, that is derived

in Appendix C.3.7. This result is usually quoted only for hermitian operators, but it

is actually valid for any pair of normal operators C and D, i.e. operators satisfying

C, C † = D, D† = 0. By virtue of eqn (16.160), ζ1 (ω) and ζ2 (ω) are both normal

c†

c

operators; therefore, the product of the noise strengths in the principal quadratures

satis¬es the ampli¬er uncertainty principle:

1

1’

c c

G1 (ω) G2 (ω) ei‘12 (ω) .

∆ζ1 (ω) ∆ζ2 (ω) (16.169)

4

This can be expressed in terms of the ampli¬er noise numbers as

1 1

e’i‘12 (ω) .

A1 (ω) A2 (ω) 1’ (16.170)

4 G1 (ω) G2 (ω)

At the carrier frequency, ω = 0, the symmetry condition (16.126) only allows the

values ‘12 (0) = 0, π, and the general ampli¬er uncertainty principle is replaced by

1 1

A1 (0) A2 (0) 1“ , (16.171)

4 G1 (0) G2 (0)

where the upper and lower signs correspond to ‘12 (0) = 0 and ‘12 (0) = π respectively.

16.6 Exercises

16.1 Quadrature gain

(1) Show that the frequency-domain form of eqn (16.3) is

1 ’iβ †

e bin (ω) + eiβ bin (’ω) ,

Xβ,in (ω) =

2

1 ’iβ †

e bin (ω) ’ eiβ bin (’ω) .

Yβ,in (ω) =

2i

Linear optical ampli¬ers—

¾

† †

(2) Show that the frequency-domain operators satisfy Xβ,in (ω) = Xβ,in (’ω), Yβ,in (ω)

= Yβ,in (’ω), and

†

Xβ,in (ω) , Xβ,in (ω ) = [Xβ,in (ω) , Xβ,in (’ω )] = 0 ,

†

Yβ,in (ω) , Yβ,in (ω ) = [Yβ,in (ω) , Yβ,in (’ω )] = 0 .

(3) Use the input“output relation (16.24) and its adjoint to conclude that the output

quadrature is related to the input quadrature by

1 ’iβ

e · (ω) + eiβ · † (’ω) .

Xβ,out (ω) = P (ω) Xβ,in (ω) +

2

(4) De¬ne the gain for this quadrature by

| Xβ,out (ω) |2

Gβ (ω) = ,

2

| Xβ,in (ω) |

and show that the gain is the same for all quadratures.

16.2 Phase-insensitive traveling-wave ampli¬er

(1) Work out the coarse-grained version of eqns (14.174)“(14.177), and then use eqn

(16.60) for HS1 to derive the reduced Langevin equations for the ampli¬er.

(n)

(2) Use the properties of ξ12 (t) to derive eqn (16.67).

(3) Show that

S qp (z, t) , S kl (z , t) = δpk S ql (z, t) ’ δlq S kp (z, t) δ (z ’ z ) .

(4) Show that S 22 (z, t) ’ S 11 (z, t) ≈ nat σD.

(5) Derive eqns (16.82) and (16.83).

16.3 Colored noise

Reconsider the use of adiabatic elimination to solve eqn (16.65).

(1) Use the formal solution of eqn (16.65) to conclude that the noise term on the right

side of eqn (16.68) should be replaced by

t

dt1 e(i∆0 ’“12 )(t’t1 ) ξ12 (z, t1 ) .

ζ12 (z, t) =

t0

(2) Use the properties of ξ12 (z, t1 ) to show that

e’“12 |t’t |

†

ζ12 (z, t) ζ12 (z , t ) = nat σC12,12 δ (z ’ z ) ei∆0 (t’t ) .

2“12

(3) Justify eqn (16.68) by evaluating

†

ζ12 (z, t) ζ12 (z , t ) f (t ) ,

dt

where f (t ) is slowly varying on the scale T12 = 1/“12 .

17

Quantum tomography

Classical tomography is an experimental method for examining the interior of a phys-

ical object by scanning a penetrating beam of radiation, for example, X-rays, through

its interior. In medicine, the density pro¬le of the interior of the body is reconstructed

by using the method of CAT scans (computer-assisted tomographic scans). This pro-

cedure allows a high-resolution image of an interior section of the human body to be

formed, and is therefore very useful for diagnostics.

In quantum tomography, the subject of interest is not the density distribution

inside a physical object, but rather the Wigner distribution describing a quantum state.

By exploiting the mathematical similarity between a physical density distribution and

the quasiprobability distribution W (±), the methods of tomography can be applied to

perform a high-resolution determination of a quantum state of light. We begin with a

review of the mathematical techniques used in classical tomography, and then proceed

to the application of these methods to the Wigner function and the description of a

representative set of experiments.

17.1 Classical tomography

Classical tomography consists of a sequence of measurements, called scans, of the

detected intensity of an X-ray beam at the end of a given path through the object.

The fraction of the intensity absorbed in a small interval ∆s is κρ∆s, where κ is

the opacity and ρ the density of the material. For the usual case of uniform opacity,

the ratio of the detected intensity to the source intensity is proportional to the line

integral of the density along the path. After a scan of lateral displacements through

the object is completed, the angle of the X-ray beam is changed, and a new sequence

of lateral scans is performed. When these lateral scans are completed, the angle is

then again incremented, etc. Thus a complete set of data for X-ray absorption can

be obtained by translations and rotations of the path of the X-ray beam through the

object. The density pro¬le is then recovered from these data by the mathematical

technique described below.

The medical motivation for this procedure is the desire to locate a single lump

of matter”such as a tumor which possesses a density di¬ering from that of normal

tissue”in the interior of a body. The source and the detector straddle the body in

such a way that the line of sight connecting them can be stepped through lateral

displacements, and then stepped through di¬erent angles with respect to the body. One

can thereby determine”in fact, overdetermine”the location of the lump by observing

which of the translational and rotational data sets yield the maximum absorption.

¿¼ Quantum tomography

17.1.1 Procedure for classical tomography

Consider an object whose density pro¬le ρ(x, y, z) we wish to map by probing its

interior with a thin beam of X-rays directed from the source S to the detector D, as

shown in Fig. 17.1. We place the origin O of coordinates near the center of the object,

and choose a plane containing the source and the detector as the (x, y)-plane, i.e. the

(z = 0)-plane. The line SD that joins the source to the detector is traditionally called

the line of sight. For a given line of sight, we introduce a rotated coordinate system

(x , y ), where the y -axis is parallel to the line of sight, the x -axis is perpendicular

to it, and θ is the rotation angle between the x - and x-axes. Two lines of sight that

di¬er only by interchanging the source and detector are redundant, since they provide

the same information; consequently, the rotation angle θ can be restricted to the range

0 < θ < π.

The intensity ratio measured by passing the X-ray beam along the line of sight SD

is proportional to the line integral

Pθ (x ) = ρ(x, y, 0) ds , (17.1)

SD

where s is a coordinate measured along the line of sight. We will call this line integral

the projection of the density along the SD direction. It is also commonly called a line-

out of the density. Incrementing the x -value, while keeping the line of sight parallel

to the y -axis, generates a set of data which yields information about the integrated

column density of the object as a function of x . After a sequence of scans at di¬erent

x -values has been completed, a new set of line-outs can be generated by incrementing

the rotation angle θ. For applications of classical tomography to real three-dimensional

objects, data for slices at z = 0 can be obtained by translating the source“detector

system in the z-direction, and then repeating the steps listed above. This part of the

procedure will not be relevant for the application to quantum tomography, so from

now on we only consider z = 0 and replace ρ (x, y, z = 0) by ρ (x, y).

From the above considerations, we formulate the following (not necessarily optimal)

procedure for collecting tomographic data.

Fig. 17.1 Coordinate system used in tomog-

raphy.

¿½

Classical tomography

(1) Collect the projections for lines of sight at a ¬xed angle θ, while scanning the

coordinate x from one side of O to the other.

(2) Repeat this procedure after incrementing the angle θ by a small amount.

(3) Repeat steps (1) and (2), collecting data for Pθ (x ) for ’∞ < x < ∞ and 0 <

θ < π.

(4) Determine the original density ρ(x, y) by means of the inverse Radon transform

described below.

17.1.2 The Radon transform

The rotated coordinates (x , y ) are related to the ¬xed coordinates by

x = x cos θ + y sin θ , y = ’x sin θ + y cos θ , (17.2)

and the inverse relation is

x = x cos θ ’ y sin θ , y = x sin θ + y cos θ . (17.3)

The projection Pθ (x ) de¬nes the forward Radon transform:

Pθ (x ) = ρ(x, y) ds (17.4)

SD

+∞

ρ(x cos θ ’ y sin θ, x sin θ + y cos θ) dy .

= (17.5)

’∞

For the application at hand, the convention for Fourier transforms used in the other

parts of this book can lead to confusion; therefore, we revert to the usual notation in

which f denotes the Fourier transform of f . Let us then consider the one-dimensional

Fourier transform of the projection Pθ (x ),

+∞

dx Pθ (x )e’ikx ,

Pθ (k) ≡ (17.6)

’∞

and the two-dimensional Fourier transform of the density,

+∞ +∞

dy ρ(x, y)e’i(xu+yv) .

ρ(u, v) ≡ dx (17.7)

’∞ ’∞

The Fourier slice theorem states that

Pθ (k) = ρ(k cos θ, k sin θ) . (17.8)

The proof proceeds as follows. Inspection of Fig. 17.1 shows that the two-dimensional

wavevector

k = (k cos θ, k sin θ) (17.9)

is directed along the line OP, i.e. the x -axis. For any point on the line of sight, with

coordinates r = (x, y), one ¬nds k · r = kx cos θ + ky sin θ = kx . Substituting this

¿¾ Quantum tomography

relation and the de¬nition (17.5) of the forward Radon transform into eqn (17.6) then

leads to

+∞ +∞

dy e’ik·r ρ(x(x , y ), y(x , y ))

Pθ (k) = dx

’∞ ’∞

+∞ +∞

dy e’ik·r ρ(x, y) ,

= dx (17.10)

’∞ ’∞

where the last form follows by changing integration variables and using the fact that

the transformation linking (x , y ) to (x, y) has unit Jacobian. This result is just the

de¬nition of the Fourier transform of the density, so we arrive at eqn (17.8).

For the ¬nal step, we ¬rst express the density in physical space, ρ(x, y), as the

inverse Fourier transform of the density in reciprocal space:

∞ ∞

1

dv ρ(u, v)ei(xu+yv) .

ρ(x, y) = du (17.11)

4π 2 ’∞ ’∞

In order to use the Fourier slice theorem, we identify (u, v) with the two-dimensional

vector k, de¬ned in eqn (17.9), so that u = k cos θ and v = k sin θ. This resembles the

familiar transformation to polar coordinates, but one result of Exercise 17.1 is that

the restriction 0 < θ < π requires k to take on negative as well as positive values, i.e.

’∞ < k < ∞. This transformation implies that du dv = dk |k| dθ, so that eqn (17.11)

becomes

∞ π

1

|k| dk dθ ρ (k cos θ, k sin θ) eik(x cos θ+y sin θ) ,

ρ(x, y) = (17.12)

4π 2 ’∞ 0

and the Fourier slice theorem allows this to be expressed as

∞ π

1

|k| dk dθ Pθ (k)eik(x cos θ+y sin θ) .

ρ(x, y) = 2 (17.13)

4π ’∞ 0

Substituting eqn (17.6) in this relation yields the inverse Radon transform:

∞ ∞

π

1

|k| dk dx Pθ (x )eik(x cos θ+y sin θ’x ) .

ρ(x, y) = dθ (17.14)

4π 2 ’∞ ’∞

0

This result reconstructs the density distribution ρ(x, y) from the measured data set

Pθ (x ).

17.2 Optical homodyne tomography

In eqn (5.126) we introduced a version of the Wigner distribution, W (±), that is

particularly well suited to quantum optics. The complex argument ±, which is the

amplitude de¬ning a coherent state, is equivalent to the pair of real variables x = Re ±

and y = Im ±; consequently, W (±) can equally well be regarded as a function of x and

y, as in Exercise 17.2. Expressing the Wigner distribution in this form suggests that

W (x, y) is an analogue of the density function ρ (x, y). With this interpretation, the

¿¿

Experiments in optical homodyne tomography

mathematical analysis used for classical tomography can be applied to recover W (x, y)

from an appropriate set of measurements. The objection that the quasiprobability

W (x, y) can be negative”as shown by the number-state example in eqn (5.153)”

does not pose a serious di¬culty, since negative absorption in the classical problem

would simply correspond to ampli¬cation.

In order to apply the inverse Radon transform (17.14) to quantum optics, we must

¬rst understand the physical signi¬cance of the projection Pθ (x ). In this context, the

parameter θ is not a geometrical angle; instead, it is the phase of the local oscillator

¬eld in a homodyne measurement scheme. As explained in Section 9.3, this parameter

labels the natural quadratures,

Xθ = X0 cos θ + Y0 sin θ , Yθ = X0 sin θ ’ Y0 cos θ , (17.15)

for homodyne measurement. Generalizing eqn (5.123) tells us that integrating the

Wigner distribution over one of the conjugate variables generates the marginal prob-

ability distribution for the other; so applying the forward Radon transform (17.5) to

the Wigner distribution leads to the conclusion that the projection,

+∞

W (x cos θ ’ y sin θ, x sin θ + y cos θ)dy ,

Pθ (x ) = (17.16)

’∞

is the probability distribution for measured values x of the operator Xθ .

The di¬erence between the physical interpretations of Pθ (x ) in classical and quan-

tum tomography requires corresponding changes in the experimental protocol. Setting

the phase of the local oscillator in a homodyne measurement scheme is analogous to

setting the angle θ of the X-ray beam, but there is no analogue for setting the lateral

position x . In the quantum optics application, the variable x is not under experimen-

tal control. Instead, it represents the possible values of the quadrature Xθ , which are

subject to quantum ¬‚uctuations.

In this situation, the procedure is to set a value of θ and then carry out many

homodyne measurements of Xθ . A histogram of the results determines the fraction of

the values falling in the interval x to x + ∆x , and thus the probability distribution

Pθ (x ). This is easier said than done, and it represents a substantial advance beyond

previous experiments, that simply measured the average and variance of the quadra-

ture. Once Pθ (x ) has been experimentally determined, the inverse Radon transform

yields the Wigner function as

∞ ∞

π

1

|k| dk dx Pθ (x )eik(x cos θ+y sin θ’x ) .

W (x, y) = dθ (17.17)

4π 2 ’∞ ’∞

0

As shown in Section 5.6.1, the Wigner distribution permits the evaluation of the av-

erage of any observable; consequently, this reconstruction of the Wigner distribution

provides a complete description of the quantum state of the light.

17.3 Experiments in optical homodyne tomography

The method of optical homodyne tomography sketched above is one example from a

general ¬eld variously called quantum-state tomography (Raymer and Funk, 2000)

¿ Quantum tomography

or quantum-state reconstruction (Altepeter et al., 2005). Techniques for recovering

the density matrix from measured values have been applied to atoms (Ashburn et al.,

1990), molecules (Dunn et al., 1995), and Bose“Einstein condensates (Bolda et al.,

1998). In the domain of quantum optics, Raymer and co-workers (Smithey et al.,

1993) studied the properties of squeezed states by using pulsed light for the signal and

the local oscillator. This is an important technique for obtaining time-resolved data

for various processes (Raymer et al., 1995), but the simple theory presented above is

more suitable for describing experiments with continuous-wave (cw) beams.

17.3.1 Optical tomography for squeezed states

Following Raymer™s pulsed-light, quantum-state tomography experiments, Mlynek and

his co-workers (Breitenbach et al., 1997) performed experiments in which they gener-

ated and then analyzed squeezed states. The description of the experiment is therefore

naturally divided into the generation and measurement steps.

A Squeezed state generation

The light used in this experiment is provided by an Nd: YAG (neodymium-doped,

yttrium“aluminum garnet) laser (1064 nm and 500 mW) operated in cw mode. As

shown in Fig. 17.2, the laser beam, at frequency ω, ¬rst passes through a mode clean-

ing cavity (a high ¬nesse Fabry“Perot resonator with a 170 kHz bandwidth) in order

to reduce technical noise arising from relaxation oscillations in the laser. The ¬ltered

beam is then split into three parts: the upper part is sent into a second-harmonic

generator (SHG); the middle part is sent into an electro-optic modulator (EOM)

Pump 2ω

Resonant

SHG

Output mirror

Filter HR

cavity

DM Phase

ω

Nd:YAG OPA

EOM

laser φ

Signal

Homodyne

ω

detector

Phase Local

θ

oscillator

„¦

(J)

E_

Phase Low-pass

ψ

filter

(J)

E„¦

Fig. 17.2 Experimental setup used for generating and detecting squeezed light in a tomo-

graphic scheme. (Reproduced from Mlynek et al. (1998).)

¿

Experiments in optical homodyne tomography

(Saleh and Teich, 1991, Sec. 18.1-B); and the lower part serves as the local oscillator

for the homodyne detector.

The resonant SHG”a χ(2) crystal placed inside a 2ω-resonator”produces a second-

harmonic pump beam that enters the OPA through the right-hand mirror. This mirror

also serves as the output port for the squeezed light near frequency ω. The OPA con-

sists of a χ(2) crystal coated on the left end with a mirror (HR) that is highly re¬‚ective

at both ω and 2ω and on the right end with the output mirror. The two mirrors de¬ne

a cavity that is resonant at both the ¬rst and second harmonics. For an unmodulated

input, e.g. vacuum ¬‚uctuations, this is a degenerate OPA con¬guration.

The down-converted photons in each pair share the same spatial mode, polar-

ization, and frequency. For a su¬ciently high transmission coe¬cient of the output

mirror at frequency ω, the OPA produces an intense, squeezed-light output signal in

the vicinity of ω. The parametric gain of the OPA at the pump frequency is maximized

by adjusting the temperature of the crystal. The dichroic mirror (DM)”located to

the right of the output mirror”transmits the incoming 2ω-pump beam toward the

OPA, but de¬‚ects the outgoing squeezed-light beam into the homodyne detector.

The EOM voltage is modulated at frequency „¦, where „¦/2π = 1.5 or 2.5 MHz.

This adds two side bands to the coherent middle beam, at frequencies ω ± „¦ that are

well within the cavity bandwidth, “/2π = 17 MHz. The OPA is operated in a dual port

con¬guration, i.e. the pump beam enters through the output mirror on the right and

the coherent signal is injected through the mirror HR on the left. The OPA cavity is

also highly asymmetric; the transmission coe¬cient at frequency ω is less than 0.1%

for the mirror HR, but about 2.1% for the output mirror.

Due to this high asymmetry, the transmitted sidebands and their quantum ¬‚uctu-

ations are strongly attenuated, as shown in Exercise 17.3, so that the squeezed output

comes primarily from the vacuum ¬‚uctuations at ω, entering through the output cou-

pler. The output of the OPA then consists of squeezed vacuum at ω together with

bright sidebands at ω ± „¦. If the output from the EOM is blocked, the OPA emits a

pure squeezed vacuum state. If the output from the SHG is blocked, the OPA emits a

coherent state.

B Tomographic measurements

The output of the OPA is sent into the homodyne detector, but this is a new way of

using homodyne methods. The usual approach, presented in Section 9.3, assumes that

the detectors are only sensitive to the overall energy ¬‚ux of the light; consequently,

the homodyne signal is de¬ned by averaging over the ¬eld state: Shom ∝ N21 , where

N21 represents the di¬erence in the ¬ring rates of the two detectors.

For photoemissive detectors, i.e. those with frequency-independent quantum ef-

¬ciency (Raymer et al., 1995), the quantum ¬‚uctuations represented by the operator

N21 are visible as ¬‚uctuations in the di¬erence between the output currents of the

detectors. In the present case, N21 = ’i ±— bout ’ b† ±L , where ±L = |±L | exp (’iθ)

out

L

is the classical amplitude of the local oscillator and bout describes the output ¬eld of

the OPA. Expressing N21 in terms of quadrature operators as

N21 ∝ X cos θ + Y sin θ = Xθ (17.18)

¿ Quantum tomography

shows that observations of the current ¬‚uctuations represent measurements of the

quadrature Xθ .

The data (about half a million points per trace) for the current i„¦ were taken with

a high-speed 12 bit analog-to-digital converter, as the phase of the local oscillator was

swept by 360—¦ in approximately 200 ms. Time traces of i„¦ for coherent states and for

squeezed states are shown in the left-most column of Fig. 17.3.

The top trace represents the coherent state output, which is obtained by blocking

the second-harmonic pump beam. This characterizes and calibrates the laser system

used for the local oscillator and the ¬rst-harmonic input into the resonant, second-

harmonic generator crystal. The next three traces represent squeezed coherent states.

The second trace is the waveform for a phase-squeezed state, where the noise is mini-

mum at the zero-crossings of the waveform. The third trace represents a state squeezed

along the φ = 48—¦ quadrature, where φ is the relative phase between the pump wave

and the coherent input wave. The fourth trace represents the waveform for amplitude-

1

2θ(N)

0.5

0

0

π

10

0

2π ’10

1

2θ(N)

0.5

0

Noise current E„¦ (arbitrary units)

0

π

10

0

2π ’10

1

2θ(N)

0.5

0

0

π

10

0

2π ’10

1

2θ(N)

0.5

0

0

π

10

0

2π ’10

1

2θ(N)

10

0.5

0

0

N0

0

’10

Ph

as π

0

0 100 200 ’10

ea 10

Nπ/2

0

ng 2π ’10 N

amplitude

Time (ms) le

θ Quadrature 10

Fig. 17.3 Data showing noise waveforms for a coherent state (top trace) and various kinds

of squeezed states (lower traces), along with their phase space tomographic portraits on the

right. (Reproduced from Mlynek et al. (1998).)

¿

Exercises

squeezed light, where the noise is minimum at the maxima of the waveform. Finally,

the ¬fth trace represents the squeezed vacuum state, where the coherent state input

to the parametric ampli¬er has been completely blocked, so that only vacuum ¬‚uctu-

ations are admitted into the OPA. Ones sees that the noise vanishes periodically at

the zero-crossings of the noise envelope.

The middle column of Fig. 17.3 depicts the tomographic projections Pθ (x ), which

are substituted into the inverse Radon transform (17.17) to generate the portraits of

Wigner functions depicted in the third column of the ¬gure. Numerical analysis of

the distributions for the second through the ¬fth traces shows that they all have the

Gaussian shape predicted for squeezed coherent states.

17.4 Exercises

17.1 Radon transform

(1) For the transformation u = k cos θ, v = k sin θ, with the restriction 0 < θ < π,

work out the inverse transformation expressing k and θ as functions of u and v,

and thus show that k must have negative as well as positive values.

(2) Derive the relation du dv = |k| dk dθ by evaluating the Jacobian or else by just

drawing the appropriate picture.

17.2 Wigner distribution

Starting from the de¬nition (5.126), show that W (±) = W (x, y) can be written in the

form

d2 k ik·r

W (x, y) = 2e χW (k) ,

(2π)

where k = (k1 , k2 ) and r = (x, y). Derive the explicit form of the Wigner characteristic

function in terms of the density operator ρ and the quadrature operators X0 and Y0 .

What normalization condition does W (x, y) satisfy?

Dual port OPA—

17.3

Model the dual port OPA discussed in Section 17.3.1 by identifying the input and

output ¬elds as bout = b1,out and bin = b2,in , where the notation is taken from Fig.

16.2.

(1) Use eqn (15.117) to work out the coe¬cients P and C in the input“output relations

for this ampli¬er.

(2) Explicitly evaluate the ampli¬er noise operator.

κ1 show that the incident ¬eld is strongly attenuated

(3) For the unbalanced case κ2

and that the primary source of the squeezed output is the vacuum ¬‚uctuations

entering through the mirror M1.

18

The master equation

In this chapter we will study the time evolution of an open system”the sample dis-

cussed in Chapter 14”by means of the quantum Liouville equation for the world

density operator. This approach, which employs the interaction-picture description of

the density operator, is complementary to the Heisenberg-picture treatment presented

in Chapter 14. The physical ideas involved in the two methods are, however, the same.

The equation of motion of the reduced sample density operator is derived by an

approximate elimination of the environment degrees of freedom that depends crucially

on the Markov approximation. This approximate equation of motion for the sample

density operator is mainly used to derive c-number equations that can be solved by

numerical methods. In this connection, we will discuss the Fokker“Planck equation in

the P -representation and the method of quantum Monte Carlo wave functions.

18.1 Reduced density operators

As explained in Section 14.1.1, the world”the composite system of the sample and

the environment”is described by a density operator ρW acting on the Hilbert space

HW = HS — HE . The application of the general de¬nition (6.21) of the reduced density

operator to ρW produces two reduced density operators: S = TrE (ρW ) and E =

TrS (ρW ), that describe the sample and the environment respectively. For example,

the rule (6.26) for partial traces shows that the average of a sample operator Q,

Q = TrW [ρW (Q — IE )] = TrS ( S Q) , (18.1)

is entirely determined by the reduced density operator for the sample.

For an open system, the reduced density operator S will always describe a mixed

state. According to Theorem 6.1 in Section 6.4.1, the reduced density operators for the

sample and the environment can only describe pure states if the world density operator

describes a separable pure state, i.e. ρW = |„¦W „¦W |, and |„¦W = |ΨS |¦E . Even

if this were initially the case, the interaction between the sample and the environment

would inevitably turn the separable pure state |„¦W into an entangled pure state. The

reduced density operators for an entangled state necessarily describe mixed states, so

the sample state will always evolve into a mixed state.

18.2 The environment picture

In the Schr¨dinger picture, the world density operator satis¬es the quantum Liouville

o

equation

¿

Averaging over the environment

‚

i ρW (t) = [HW (t) , ρW (t)]

‚t

= [HS (t) + HE + HSE , ρW (t)] , (18.2)

where the terms in HW are de¬ned in eqns (14.6)“(14.11). Since the sample“environ-

ment interaction is assumed to be weak, it is natural to regard HW 0 (t) = HS (t) + HE

as the zeroth-order part, and HSE (t) as the perturbation. This allows us to introduce

an interaction picture, through the unitary transformation,

†

|Ψenv (t) = UW 0 (t) |Ψ (t) , (18.3)

where UW 0 (t) satis¬es

‚

i UW 0 (t) = HW 0 (t) UW 0 (t) , UW 0 (0) = 1 . (18.4)

‚t

We will call this interaction picture the environment picture, since it plays a special

role in the theory.

The di¬erential equation (18.4) has the same form as eqn (4.90), but now the

ordering of the operators UW 0 (t) and HW 0 (t) is important, since the time-dependent

Hamiltonian HW 0 (t) will not in general commute with UW 0 (t). If this warning is kept

¬rmly in mind, the formal procedure described in Section 4.8 can be used again to

¬nd the Schr¨dinger equation

o

‚

|Ψenv (t) = HSE (t) |Ψenv (t) ,

env

i (18.5)

‚t

and the quantum Liouville equation