This technique combines heterodyne detection with the properties of the ideal bal-

anced beam splitter discussed in Section 8.4. A strong quasiclassical ¬eld (the LO) is

injected into port 2, and a weak signal with the same frequency is injected into port

1 of a balanced beam splitter, as shown in Fig. 9.8. In practice, it is convenient to

generate both ¬elds from a single master oscillator. Note, however, that the signal and

local oscillator mode functions are orthogonal, because the plane-wave propagation

vectors are orthogonal. If the beam splitter is balanced, and the rest of the system is

designed to be as bilaterally symmetric as possible, this device is called a balanced

homodyne detector. In particular, the detectors placed at the output ports 1 and

2 are required to be identical within close tolerances. In practice, this is made possible

by the high reproducibility of semiconductor-based photon detectors fabricated on the

same homogeneous, single-crystal wafer using large-scale integration techniques.

The di¬erence between the outputs of the two identical detectors is generated by

means of a balanced, di¬erential electronic ampli¬er. Since the two input transistors of

the di¬erential ampli¬er”whose noise ¬gure dominates that of the entire postdetection

electronics”are themselves semiconductor devices fabricated on the same wafer, they

can also be made identical within close tolerances. The symmetry achieved in this way

guarantees that the technical noise in the laser source”from which both the signal

¿¼½

Heterodyne and homodyne detection

SA

D2

2'

+

1 ’

-s

Signal D1

1'

2

-L

Local oscillator (LO)

Fig. 9.8 Schematic of a balanced homodyne detector. Detectors D1 and D2 respectively

collect the output of ports 1 and 2 . The outputs of D2 and D1 are respectively fed into the

non-inverting input (+) and the inverting input (’) of a di¬erential ampli¬er. The output of

the di¬erential ampli¬er, i.e. the di¬erence between the two detected signals, is then fed into

a radio-frequency spectrum analyzer SA.

and the local oscillator are derived”will produce essentially identical ¬‚uctuations in

the outputs of detectors D1 and D2. These common-mode noise waveforms will cancel

out upon subtraction in the di¬erential ampli¬er. This technique can, therefore, lead

to almost ideal detection of purely quantum statistical properties of the signal. We will

encounter this method of detection later in connection with experiments on squeezed

states of light.

A Classical analysis of homodyne detection

It is instructive to begin with a classical analysis for general values of the re¬‚ection and

transmission coe¬cients r and t before specializing to the balanced case. The classical

amplitudes at detectors D1 and D2 are related to the input ¬elds by

ED1 = r EL + t Es ,

(9.148)

ED2 = t EL + r Es ,

and the di¬erence in the outputs of the square-law detectors is proportional to the

di¬erence in the intensities, so the homodyne signal is

2 2

Shom = |ED2 | ’ |ED1 |

—

2 2 2 2

= 1 ’ 2 |r| |EL | ’ 1 ’ 2 |r| |Es | + 4 |t r| Im [EL Es ] , (9.149)

where we have used the Stokes relations (8.7) and set r— t = i |r t| (this is the +-sign

in eqn (9.105)) to simplify the result. The ¬rst term on the right side is not sensitive

to the phase θL of the local oscillator, so it merely provides a constant background

¿¼¾ Photon detection

for measurements of the homodyne signal as a function of θL . By design, the signal

2

intensity is small compared to the local oscillator intensity, so the |Es | -term can be

neglected altogether. As mentioned in Section 9.3.2, the local oscillator amplitude

is subject to technical ¬‚uctuations δEL ”e.g. variations in the laser power due to

acoustical-noise-induced changes in the laser cavity dimensions”which in turn produce

phase-sensitive ¬‚uctuations in the output,

— —

δShom = ’ 1 ’ 2 |r|2 2 Re [EL δEL ] + 4 |t r| Im [δEL Es ] . (9.150)

2 2

The ¬‚uctuations associated with the direct detection signal, 1 ’ 2 |r| |EL | , for the

local oscillator are negligible compared to the ¬‚uctuations in the Es contribution if

|Es |

2

1 ’ 2 |r| , (9.151)

|EL |

2

and this is certainly satis¬ed for an ideal balanced beam splitter, for which |r| =

|t|2 = 1/2, and

—

Shom = 2 Im [EL Es ] . (9.152)

B Quantum analysis of homodyne detection

We turn now to the quantum analysis of homodyne detection, which is simpli¬ed by the

fact that the local oscillator and the signal have the same frequency. The complications

associated with the image band modes are therefore absent, and the in-¬eld is simply

Ein (r, t) = ies aL eiks y e’iωs t + ies as eiks x e’iωs t .

(+)

(9.153)

In this case all relevant vacuum ¬‚uctuations are dealt with by the operators aL and

(+)

as , so the operator Evac,in (r, t) will not contribute to either the signal or the noise.

The homodyne signal. The out-¬eld is

(+) (+) (+)

Eout (r, t) = ED1 (r, t) + ED2 (r, t) , (9.154)

where the ¬elds

ED1 (r, t) = ies as eiks x e’iωs t

(+)

(9.155)

and

ED2 (r, t) = ies aL eiks y e’iωs t

(+)

(9.156)

drive the detectors D1 and D2 respectively, and the scattered annihilation operators

satisfy the operator analogue of (9.148):

a L = t aL + r as ,

(9.157)

a s = r aL + t a s .

The di¬erence in the two counting rates is proportional to

¿¼¿

Heterodyne and homodyne detection

(’) (+) (’) (+)

Shom = ED2 (r, t) ED2 (r, t) ’ ED1 (r, t) ED1 (r, t)

= e2 N21 , (9.158)

s

where

†

N21 = aL aL ’ as† as

a† aL ’ 1 ’ 2 |r| a† as ’ 2i |r t| a† as ’ a† aL

2 2

= 1 ’ 2 |r| (9.159)

s s

L L

is the quantum analogue of the classical result (9.149). For a balanced beam splitter,

this simpli¬es to

N21 = ’i a† as ’ a† aL ; (9.160)

s

L

consequently, the balanced homodyne signal is

Shom = 2e2 Im a† as . (9.161)

s L

If we again assume that the signal and local oscillator are statistically independent,

then a† as = a† as , and

L L

—

Shom = 2 Im (EL Es ) , (9.162)

where the e¬ective ¬eld amplitudes are again de¬ned by

EL = es aL = es | aL | eiθL , (9.163)

and

Es = es as . (9.164)

Just as for heterodyne detection, the phase sensitivity of homodyne detection guaran-

tees that the detection rate vanishes for signal states described by density operators

that are diagonal in photon number. Alternatively, for the calculation of the signal we

can replace the di¬erence of number operators by

—

†

aL aL ’ as† as ’ ’i a s ’ a † aL = 2 | aL | Y ,

aL (9.165)

s

where Y is the quadrature operator de¬ned by eqn (9.143). This gives the equivalent

result

Shom = 2 |EL | es Y (9.166)

for the homodyne signal.

Noise in homodyne detection. Just as in the classical analysis, the ¬rst term in

the expression (9.159) for N21 would produce a phase-insensitive background, but for

2

|r| signi¬cantly di¬erent from the balanced value 1/2, the variance in the homodyne

output associated with technical noise in the local oscillator could seriously degrade

the signal-to-noise ratio. This danger is eliminated by using a balanced system, so that

N21 is given by eqn (9.160). The calculation of the variance V (N21 ) is considerably

¿¼ Photon detection

simpli¬ed by the assumption that the local oscillator is approximately described by a

coherent state with ±L = |±L | exp (iθL ). In this case one ¬nds

V (N21 ) = |±L | + a† as + 2 |±L | V a† , as ’ |±L | V e’iθL as ’ |±L | V e’iθL a† .

2 2 2 2

s s s

(9.167)

Expressing this in terms of the quadrature operator Y gives the simpler result

V (N21 ) = 4 |±L | V (Y ) + a† as

2 2

4 |±L | V (Y ) , (9.168)

s

where the last form is valid in the usual case that the input signal ¬‚ux is negligible

compared to the local oscillator ¬‚ux.

Corrections for ¬nite detector e¬ciency—

C

So far we have treated the detectors as though they were 100% e¬cient, but perfect

detectors are very hard to ¬nd. We can improve the argument given above by using

the model for imperfect detectors described in Section 9.1.4. Applying this model to

detector D1 requires us to replace the operator as ”describing the signal transmitted

through the beam splitter in Fig. 9.8”by

1 ’ ξcs ,

as = ξas + i (9.169)

where the annihilation operator cs is associated with the mode exp [i (ks y ’ ωs t)] en-

tering through port 2 of the imperfect-detector model shown in Fig. 9.1. A glance

at Fig. 9.8 shows that this is also the mode associated with aL . Since the quantiza-

tion rules assign a unique annihilation operator to each mode, things are getting a

bit confusing. This di¬culty stems from a violation of Einstein™s rule caused by an

uncritical use of plane-wave modes. For example, the local oscillator entering port 2

of the homodyne detector, as shown in Fig. 9.8, should be described by a Gaussian

wave packet wL with a transverse pro¬le that is approximately planar at the beam

splitter and e¬ectively zero at the detector D1. Correspondingly, the operator cs , rep-

resenting the vacuum ¬‚uctuations blamed for the detector noise, should be associated

with a wave packet that is approximately planar at the ¬ctitious beam splitter of the

imperfect-detector model and e¬ectively zero at the real beam splitter in Fig. 9.8. In

other words, the noise in detector D1 does not enter the beam splitter. All of this can

be done precisely by using the wave packet quantization methods developed in Section

3.5.2, but this is not necessary as long as we keep our wits about us. Thus we impose

cs ρ = 0, aL ρ = 0, and a† , cs = 0, even though”in the oversimpli¬ed plane-wave

L

picture”both operators cs and aL are associated with the same plane-wave mode.

In the same way, the noise in detector D2 is simulated by replacing the transmitted

LO-¬eld aL with

aL = ξaL + i 1 ’ ξcL , (9.170)

where cL ρ = 0, and cL , a† = 0.

s

Continuing in this vein, the di¬erence operator N21 is replaced by

N21 = aL† aL ’ as † as

= ξN21 + δN21 . (9.171)

¿¼

Exercises

Each term in δN21 contains at least one creation or annihilation operator for the vac-

uum modes discussed above. Since the vacuum operators commute with the operators

for the signal and local oscillator, the expectation value of δN21 vanishes, and the

homodyne signal is

—

Shom = e2 N21 = ξe2 N21 = 2ξ Im (EL Es ) . (9.172)

s s

As expected, the signal from the imperfect detector is just the perfect detector result

reduced by the quantum e¬ciency.

We next turn to the noise in the homodyne signal, which is proportional to the

variance V (N21 ). It is not immediately obvious how the extra partition noise in each

detector will contribute to the overall noise, so we ¬rst use eqn (9.171) again to get

2 2 2

= ξ 2 (N21 )

(N21 ) + ξ N21 δN21 + ξ δN21 N21 + (δN21 ) . (9.173)

There are no correlations between the vacuum ¬elds cL and cs entering the imperfect

detector and the signal and local oscillator ¬elds, so we should expect to ¬nd that the

second and third terms on the right side of eqn (9.173) vanish. An explicit calculation

shows that this is indeed the case. Evaluating the fourth term in the same way leads

to the result

†

aL aL + as† as

V (N21 ) = ξ 2 V (N21 ) + ξ (1 ’ ξ) . (9.174)

Comparing this to the single-detector result (9.57) shows that the partition noises

at the two detectors add, despite the fact that N21 represents the di¬erence in the

photon counts at the two detectors. After substituting eqn (9.168) for V (N21 ); using

the scattering relations (9.157); and neglecting the small signal ¬‚ux, we get the ¬nal

result

2 2

V (N21 ) = ξ 2 4 |±L | V (Y ) + ξ (1 ’ ξ) |±L | . (9.175)

9.4 Exercises

9.1 Poissonian statistics are reproduced

’1

nn exp (’n) for the incident photons in eqn

Use the Poisson distribution p(n) = (n!)

(9.46) to derive eqn (9.48).

m-fold coincidence counting

9.2

Generalize the two-detector version of coincidence counting to any number m. Show

that the m-photon coincidence rate is

m

2 T12 +Tgate T1m +Tgate

1

Sn d„2 · · ·

(m)

w = d„m

m! T12 T1m

n=1

G(m) (r1 , t1 , . . . , rm , tm + „m ; r1 , t1 , . . . , rm , tm + „m ) ,

where the signal from the ¬rst detector is used to gate the coincidence counter and

T1n = T1 ’ Tn .

¿¼ Photon detection

9.3 Super-Poissonian statistics

2 2

Consider the state |Ψ = ± |n + β |n + 1 , with |±| + |β| = 1. Show that |Ψ is a

nonclassical state that exhibits super-Poissonian statistics.

9.4 Alignment in heterodyne detection

For the heterodyne scheme shown in Fig. 9.6, assume that the re¬‚ected LO beam has

the wavevector kL = kL cos •ux + kL sin •uy . Rederive the expression for Shet and

show that averaging over the detector surface wipes out the heterodyne signal.

9.5 Noise in heterodyne detection

Use eqn (9.111), eqn (9.119), and eqn (9.135) to derive eqn (9.137).

10

Experiments in linear optics

In this chapter we will study a collection of signi¬cant experiments which were carried

out with the aid of the linear optical devices described in Chapter 8 and the detection

techniques discussed in Chapter 9.

10.1 Single-photon interference

The essential features of quantum interference between alternative Feynman paths

are illustrated by the familiar Young™s arrangement”sketched in Fig. 10.1”in which

there are two pinholes in a perfectly re¬‚ecting screen. The screen is illuminated by a

plane-wave mode occupied by a single photon with energy ω, and after many suc-

cessive photons have passed through the pinholes the detection events”e.g. spots on

a photographic plate”build up the pattern observed in classical interference experi-

ments.

An elementary quantum mechanical explanation of the single-photon interference

pattern can be constructed by applying Feynman™s rules of interference (Feynman

et al., 1965, Chaps 1“7).

(1) The probability of an event in an ideal experiment is given by the square of the

absolute value of a complex number A which is called the probability amplitude:

P = probability ,

A = probability amplitude , (10.1)

2

P = |A| .

L

2 2' Fig. 10.1 A two-pinhole interferometer. The

arrows represent an incident plane wave. The

L four ports are de¬ned by the surfaces P1, P1 ,

P2, P2 , and the path lengths from the pin-

2 2' holes 1 and 2”bracketed by the ports (P1, P1 )

and (P2, P2 ) respectively”to the interference

point are L1 and L2 .

¿¼ Experiments in linear optics

(2) When an event can occur in several alternative ways, the probability amplitude

for the event is the sum of the probability amplitudes for each way considered

separately; i.e. there is interference between the alternatives:

A = A1 + A2 ,

(10.2)

P = |A1 + A2 |2 .

(3) If an experiment is performed which is capable of determining whether one or

another alternative is actually taken, the probability of the event is the sum of the

probabilities for each alternative. In this case,

P = P1 + P2 , (10.3)

and there is no interference.

In applying rule (2) it is essential to be sure that the situation described in rule (3)

is excluded. This means that the experimental arrangement must be such that it is

impossible”even in principle”to determine which of the alternatives actually occurs.

In the literature”and in the present book”it is customary to refer to the alternative

ways of reaching the ¬nal event as Feynman processes or Feynman paths.

In the two-pinhole experiment, the two alternative processes are passage of the

photon through the lower pinhole 1 or the upper pinhole 2 to arrive at the ¬nal event:

detection at the same point on the screen. In the absence of any experimental procedure

for determining which process actually occurs, the amplitudes for the two alternatives

must be added. Let Ain be the quantum amplitude for the incoming wave; then the

amplitudes for the two processes are A1 = Ain exp (ikL1 ) and A2 = Ain exp (ikL2 ),

where k = ω/c. The probability of detection at the point on the screen (determined

by the values of L1 and L2 ) is therefore

2 2 2

|A1 + A2 | = 2 |Ain | + 2 |Ain | cos [k (L2 ’ L1 )] , (10.4)

which has the same form as the interference pattern in the classical theory.

This thought experiment provides one of the simplest examples of wave“particle

duality. The presence of the interference term in eqn (10.4) exhibits the wave-aspect

of the photon, while the detection of the photon at a point on the screen displays

its particle-aspect. Arguments based on the uncertainty principle (Cohen-Tannoudji

et al., 1977a, Complement D1; Bransden and Joachain, 1989, Sec. 2.5) show that any

experimental procedure that actually determines which pinhole the photon passed

through”this is called which-path information”will destroy the interference pat-

tern. These arguments typically involve an interaction with the particle”in this case

a photon”which introduces uncontrollable ¬‚uctuations in physical properties, such as

the momentum. The arguments based on the uncertainty principle show that which-

path information obtained by disturbing the particle destroys the interference pattern,

but this is not the only kind of experiment that can provide which-path information.

In Section 10.3 we will describe an experiment demonstrating that single-photon inter-

ference is destroyed by an experimental arrangement that merely makes it possible to

obtain which-path information, even if none of the required measurements are actually

made and there is no interaction with the particle.

¿¼

Single-photon interference