electrode, labeled as the contact region and degenerate substrate.

¾ Photon detection

9.2.4 Coincidence counting

As we have already seen in Section 1.1.4, one of the most important experimental

techniques in quantum optics is coincidence counting, in which the output signals

of two independent single-photon detectors are sent to a device”the coincidence

counter”that only emits a signal when the pulses from the two detectors both arrive

during a narrow gate window Tgate . For simplicity, we will only consider idealized,

broadband, point detectors equipped with polarization ¬lters. This means that the

detectors can be treated as though they were single atoms, with the understanding

that the locations of the ˜atoms™ are to be treated classically. The detector Hamiltonian

is then

2

Hdet (t) = Hdn (t) , (9.72)

n=1

Hdn (t) = ’ dn (t) · en En (t) , (9.73)

where rn , dn , en , and En are respectively the location; the dipole operator; the po-

larization admitted by the ¬lter; and the corresponding ¬eld component

En (t) = en · E (rn , t) (9.74)

for the nth detector. In the following discussion we will show that coincidence count-

ing can be interpreted as a measurement of the second-order correlation function,

G(2) (r1 , t1 , r2 , t2 ; r3 , t3 , r4 , t4 ), introduced in Section 4.7.

Since a general initial state of the radiation ¬eld is described by a density matrix,

i.e. an ensemble of pure states, we can begin by assuming that the radiation ¬eld is

described a pure state |¦e and that both atoms are in the ground state. The initial

state of the total system is then

|˜i = |φγ , φγ , ¦e = |φγ (1) |φγ (2) |¦e , (9.75)

where |φγ (n) denotes the ground state of the atom located at rn . For coincidence

counting, it is su¬cient to consider the ¬nal states,

|˜f = |φ 1 , φ 2 , n = |φ 1 (1) |φ 2 (2) |n , (9.76)

where |φ (n) denotes a (continuum) excited state of the atom located at rn and |n

is a general photon number state. The probability amplitude for this transition is

Af i = ˜f |V (t)| ˜i = δf i + ˜f V (1) (t) ˜i + ˜f V (2) (t) ˜i + · · · , (9.77)

where the evolution operator V (t) is given by eqn (4.103), with Hint replaced by

Hdet . Both atoms must be raised from the ground state to an excited state, so the

lowest-order contribution to Af i comes from the cross terms in V (2) (t), i.e.

2 t t1

i

Af i = ’ dt2 ˜f |Hd1 (t1 ) Hd2 (t2 ) + Hd2 (t1 ) Hd1 (t2 )| ˜i . (9.78)

dt1

t0 t0

The excitation of the two atoms requires the annihilation of two photons; conse-

quently, in evaluating Af i the operator En (t) in eqn (9.73) can be replaced by the

¾

Postdetection signal processing

(+)

positive-frequency part En (t). The detectors are normally located in a passive linear

medium, so one can use eqn (3.102) to show that [Hd1 (t1 ) , Hd2 (t2 )] = 0 for all (t1 , t2 ).

This guarantees that the integrand in eqn (9.78) is a symmetrical function of t1 and

t2 , so that eqn (9.78) can be written as

2 t t

i

Af i = ’ dt2 ˜f |Hd1 (t1 ) Hd2 (t2 )| ˜i .

dt1 (9.79)

t0 t0

Finally, substituting the explicit expression (9.73) for the interaction Hamiltonian

yields

2 t t

i

Af i = ’ d d dt1 dt2 exp (iω t1 ) exp (iω t2 )

1γ 2γ 1γ 2γ

t0 t0

(+) (+)

— n E1 (t1 ) E2 (t2 ) ¦e , (9.80)

where we have used the relation between the interaction and Schr¨dinger pictures to

o

get

dn (t) · en φγ = exp (iω dn · en φγ = exp (iω

φ t1 ) φ t1 ) d . (9.81)

1γ 1γ nγ

n n

In a coincidence-counting experiment, the ¬nal states of the atoms and the radia-

2

tion ¬eld are not observed; therefore, the transition probability |Af i | must be summed

over 1 , 2 , and n. This result must then be averaged over the ensemble of pure states

de¬ning the initial state ρ of the radiation ¬eld. Thus the overall probability, p (t, t0 ),

that both detectors have clicked during the interval (t0 , t) is

2

D1 ( 1 ) D2 ( 2 ) Pe |Af i | .

p (t, t0 ) = (9.82)

n e

1 2

A calculation similar to the one-photon case shows that p (t, t0 ) can be written as

t t t t

dt2 S1 (t1 ’ t1 ) S2 (t2 ’ t2 )

p (t, t0 ) = dt1 dt2 dt1

t0 t0 t0 t0

— G(2) (r1 , t1 , r2 , t2 ; r1 , t1 , r2 , t2 ) , (9.83)

where the sensitivity functions are de¬ned by

1 2

Sn (t) = Dn ( ) |d · en | eiω γt

(n = 1, 2)

γ

2

= e— enj Snij (t) , (9.84)

ni

and G(2) is a special case of the scalar second-order correlation function de¬ned by

eqn (4.77). The assumption that the detectors are broadband allows us to set Sn (t) =

Sn δ (t) , and thus simplify eqn (9.83) to

t t

dt2 p(2) (t1 , t2 ) ,

p (t) = dt1 (9.85)

t0 t0

¾ Photon detection

where

p(2) (t1 , t2 ) = S1 S2 G(2) (r1 , t1 , r2 , t2 ; r1 , t1 , r2 , t2 ) . (9.86)

Since p (t, t0 ) is the probability that detections have occurred at r1 and r2 sometime

during the observation interval (t0 , t), the di¬erential probability that the detections

at r1 and r2 occur in the subintervals (t1 , t1 + dt1 ) and (t2 , t2 + dt2 ) respectively is

p(2) (t1 , t2 ) dt1 dt2 . The signal pulse from detector n arrives at the coincidence counter

at time tn +Tn , where Tn is the signal transit time from the detector to the coincidence

counter. The general condition for a coincidence count is

|(t2 + T2 ) ’ (t1 + T1 )| < Tgate , (9.87)

where Tgate is the gate width of the coincidence counter. The gate is typically triggered

by one of the signals, for example from the detector at r1 . In this case the coincidence

condition is

t1 + T1 < t2 + T2 < t1 + T1 + Tgate , (9.88)

and the coincidence count rate is

T12 +Tgate

w(2) = d„ p(2) (t1 , t1 + „ )

T12

T12 +Tgate

= S1 S 2 d„ G(2) (r1 , t1 , r2 , t1 + „ ; r1 , t1 , r2 , t1 + „ ) , (9.89)

T12

where T12 = T1 ’ T2 is the o¬set time for the two detectors. By using delay lines

to adjust the signal transit times, coincidence counting can be used to study the

correlation function G(2) (r1 , t1 , r2 , t2 ; r1 , t1 , r2 , t2 ) for a range of values of (r1 , t1 ) and

(r2 , t2 ).

In order to get some practice with the use of the general result (9.89) we will revisit

the photon indivisibility experiment discussed in Section 1.4 and preview a two-photon

interference experiment that will be treated in Section 10.2.1. The basic arrangement

for both experiments is shown in Fig. 9.5.

,

,

Fig. 9.5 The photon indivisibility and

two-photon interference experiments both use

this arrangement. The signals from detectors

D1 and D2 are sent to a coincidence counter.

¾

Postdetection signal processing

For the photon indivisibility experiment, we consider a general one-photon input

state ρ, i.e. the only condition is N ρ = ρN = ρ, where N is the total number operator.

Any one-photon density operator ρ can be expressed in the form

|1κ ρκ» 1» | ,

ρ= (9.90)

κ,»

where κ and » are mode labels. The identity aκ a» ρ = 0 = ρa† a† ”which holds for any

»κ

pair of annihilation operators”implies that

(’) (’) (+) (+)

ρE2 (r2 , t2 ) E1 (r1 , t1 ) = 0 = E1 (r1 , t1 ) E2 (r2 , t2 ) ρ . (9.91)

The coincidence count rate is determined by the second-order correlation function

(’) (’)

G(2) (r2 , t2 , r1 , t1 ; r2 , t2 , r1 , t1 ) = Tr ρE2 (r2 , t2 ) E1 (r1 , t1 )

(+) (+)

— E1 (r1 , t1 ) E2 (r2 , t2 ) , (9.92)

but eqn (9.91) clearly shows that the general second-order correlation function for a

one-photon state vanishes everywhere:

G(2) (r1 , t1 , r2 , t2 ; r1 , t1 , r2 , t2 ) ≡ 0 . (9.93)

The zero coincidence rate in the photon indivisibility experiment is an immediate

consequence of this result.

The di¬erence between the photon indivisibility and two-photon interference ex-

periments lies in the choice of the initial state. For the moment, we consider a general

incident state which contains at least two photons. This state will be used in the

evaluation of the correlation function de¬ned by eqn (9.92). In addition, the original

plane-wave modes will be replaced by general wave packets wκ (r). The ¬eld operator

produced by scattering from the beam splitter can then be written as

ωκ ’iωκ t

E(+) (r, t) = i e wκ (r) aκ . (9.94)

20

κ

(2)

Substituting this expansion into the general de¬nition (4.75) for Gijkl yields

2

√ — —

(2)

({x} ; {x}) =

Gijkl ωµ ωκ ω» ων wµi (r ) wκj (r) w»k (r) wνl (r )

2 0

µκ»ν

— ei(ωµ ’ων )t ei(ωκ ’ω» )t Tr ρaµ aκ a» aν ,

††

(9.95)

where {x} = {r , t , r, t}, but using this in eqn (9.92) would be wrong. The problem

is that the last optical element encountered by the ¬eld is not the beam splitter, but

rather the collimators attached to the detectors. The ¬eld scattered from the beam

splitter is further scattered, or rather ¬ltered, by the collimators. To be completely

precise, we should work out the scattering matrix for the collimator and use eqn (9.94)

¾¼ Photon detection

as the input ¬eld. In practice, this is rarely necessary, since the e¬ect of these ¬lters is

well approximated by simply omitting the excluded terms when the ¬eld is evaluated

at a detector location. In this all-or-nothing approximation the explicit use of the

collimator scattering matrix is replaced by imposing the following rule at the nth

detector:

wκ (rn ) = 0 if wκ is blocked by the collimator at detector n . (9.96)

We emphasize that this rule is only to be used at the detector locations. For other

points, the expression (9.95) must be evaluated without restrictions on the mode func-

tions.

A more realistic description of the incident light leads to essentially the same

conclusion. In real experiments, the incident modes are not plane waves but beams

(Gaussian wave packets), and the widths of their transverse pro¬les are usually small

compared to the distance from the beam splitter to the detectors. For the two modes

pictured in Fig. 9.5, this implies w2 (r1 ) ≈ 0 and w1 (r2 ) ≈ 0. In other words, the

beam w2 misses detector D1 and w1 misses detector D2 . This argument justi¬es the

rule (9.96) even if the collimators are ignored.

For the initial state, ρ = |¦in ¦in |, with |¦in = a† a† |0 , each mode sum in eqn

21

(9.95) is restricted to the values κ = 1, 2. If the rule (9.96) were ignored there would

be sixteen terms in eqn (9.95), corresponding to all normal-ordered combinations of

a1† and a2† with a1 and a2 . Imposing eqn (9.96) reduces this to one term, so that

2

ω

|w2 (r2 )|2 |w1 (r1 )|2 ¦in a2† a1† a1 a2 ¦in ,

({x} ; {x}) =

(2)

G (9.97)

2 0

where ω2 = ω1 = ω. Thus the counting rate is proportional to the average of the

product of the intensity operators at the two detectors. Combining eqn (9.89) with

eqn (8.62) and the relation r = ±i |t| gives the coincidence-counting rate

2 2

ω 2 2 2 2

= S2 S1 Tgate |w2 (r2 )| |w1 (r1 )| |r| ’ |t|

(2)

w . (9.98)

2 0

The combination of eqn (9.95) and eqn (9.96) yields the correct expression for any

choice of the incident state. This allows for an explicit calculation of the coincidence

rate as a function of the time delay between pulses.

9.3 Heterodyne and homodyne detection

Heterodyne detection is an optical adaptation of a standard method for the detection

of weak radio-frequency signals. For almost a century, heterodyne detection in the

radio region has been based on square-law detection by diodes, in nonlinear devices

known as mixers. After the invention of the laser, this technique was extended to the

optical and infrared regions using square-law detectors based on the photoelectric ef-

fect. We will ¬rst give a brief description of heterodyne detection in classical optics,

and then turn to the quantum version. Homodyne detection is a special case of

¾½

Heterodyne and homodyne detection

heterodyne detection in which the signal and the local oscillator have the same fre-

quency, ωL = ωs . One variant of this scheme (Mandel and Wolf, 1995, Sec. 21.6) uses

the heterodyne arrangement shown in Fig. 9.6, but we will describe a di¬erent method,

called balanced homodyne detection, that employs a balanced beam splitter and

two identical detectors at the output ports. This technique is especially important at

the quantum level, since it is one of the primary tools of measurement for nonclas-

sical states of light, e.g. squeezed states. More generally, it is used in quantum-state

tomography”described in Chapter 17”which allows a complete characterization of

the quantum state of the light entering the signal port.

9.3.1 Classical analysis of heterodyne detection

Classical heterodyne detection involves a strong monochromatic wave,

EL (r, t) = EL (t) wL (r) e’iωL t + CC , (9.99)

called the local oscillator (LO), and a weak monochromatic wave,

Es (r, t) = Es (t) ws (r) e’iωs t + CC , (9.100)

2'

1

IB

Signal -s -D

LO 1'

-L Fast detector

Beam

splitter

2

IB

Signal

Local oscillator (LO)

Fig. 9.6 Schematic for heterodyne detection. A strong local oscillator beam (the heavy solid

arrow) is combined with a weak signal beam (the light solid arrow) at a beam splitter, and

the intensity of the combined beam (light solid arrow) is detected by a fast photodetector.

The dashed arrows represent vacuum ¬‚uctuations.

¾¾ Photon detection

called the signal, where EL (t) and Es (t) are slowly-varying envelope functions. The

two waves are mixed at a beam splitter”as shown in Fig. 9.6”so that their combined

wavefronts overlap at a fast detector. In a realistic description, the mode functions

wL (r) and ws (r) would be Gaussian wave packets, but in the interests of simplicity

√

we will idealize them as S-polarized plane waves, e.g. wL = e exp (ikL y) / V and

√

ws = e exp (iks y) / V , where V is the quantization volume and e is the common

polarization vector. Since the output ¬elds will also be S-polarized, the polarization

vector will be omitted from the following discussion. The two incident waves have

di¬erent frequencies, so the beam-splitter scattering matrix of eqn (8.63) has to be

applied separately to each amplitude. The resulting wave that falls on the detector is

ED (r, t) = E D (r, t) + CC, where

1 1

E D (r, t) = EL (t) √ ei(kL x’ωL t) + Es (t) √ ei(ks x’ωs t) . (9.101)

V V

Since the detector surface lies in a plane xD = const, it is natural to choose coordinates

so that xD = 0. The scattered amplitudes are given by EL (t) = r EL (t) and Es =

t Es (t), provided that the coe¬cients r and t are essentially constant over the frequency

bandwidth of the slowly-varying amplitudes EL (t) and Es (t). Since the signal is weak,

it is desirable to lose as little of it as possible. This requires |t| ≈ 1, which in turn

implies |r| 1. The second condition means that only a small fraction of the local

oscillator ¬eld is re¬‚ected into the detector arm, but this loss can be compensated by

2

increasing the incident intensity |EL | . Thus the beam splitter in a heterodyne detector

should be highly unbalanced.

2

The output of the square-law detector is proportional to the average of |ED (r, t)|

over the detector response time TD , which is always much larger than an optical period.

On the other hand, the interference term between the local oscillator and the signal is

modulated at the intermediate frequency: ωIF ≡ ωs ’ ωL . In optical applications

the local oscillator ¬eld is usually generated by a laser, with ωL ∼ 1015 Hz, but ωIF

is typically in the radio-frequency part of the electromagnetic spectrum, around 106

to 109 Hz. The IF signal is therefore much easier to detect than the incident optical

signal. For the remainder of this section we will assume that the bandwidths of both

the signal and the local oscillator are small compared to ωIF . This assumption allows

us to treat the envelope ¬elds as constants.

1/ |ωIF |.

In this context, a fast detector is de¬ned by the conditions 1/ωL TD

This inequality, together with the strong-¬eld condition |EL | |Es |, allows the time

average over TD to be approximated by

TD /2

1

d„ |E D (r, t + „ )| ≈ |EL | + 2 Re EL Es e’iωIF t + · · · .

—

2

2

(9.102)

TD ’TD /2

The large ¬rst term |EL |2 can safely be ignored, since it represents a DC current

signal which is easily ¬ltered out by means of a high-pass, radio-frequency ¬lter. The

photocurrent from the detector is then dominated by the heterodyne signal

Shet (t) = 2 Re r— t EL Es e’iωIF t ,

—

(9.103)

¾¿

Heterodyne and homodyne detection

which describes the beat signal between the LO and the signal wave at the intermedi-

ate frequency ωIF . Optical heterodyne detection is the sensitive detection of the

heterodyne signal by standard radio-frequency techniques.

Experimentally, it is important to align the directions of the LO and signal beams

at the surface of the photon detector, since any misalignment will produce spatial

interference fringes over the detector surface. The fringes make both positive and neg-

ative contributions to Shet ; consequently”as can be seen in Exercise 9.4”averaging

over the entire surface will wash out the IF signal. Alignment of the two beams can

be accomplished by adjusting the tilt of the beam splitter until they overlap interfer-

ometrically.