+∞

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.