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)