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to be exactly equal to the number of holes”collected on the right by the grounded
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.

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