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Hamiltonian for this problem is therefore H = H0 + Hdet , where Hdet represents the
interaction with the detectors only. The unperturbed Hamiltonian is H0 = HD +
Hem + H1 , where HD is the detector Hamiltonian and Hem is the ¬eld Hamiltonian.
The remaining term, H1 , describes the interaction of the ¬eld with the passive linear
optical devices, e.g. lenses, mirrors, beam splitters, etc., that direct the light to the
detectors.

A Single-photon detection
The simplest possible photon detector consists of a single atom interacting with the
¬eld. In the interaction picture, Hdet = ’d (t) · E (r, t) describes the interaction of the
¬eld with the detector atom located at r. The initial state is |˜ (t0 ) = |φγ , ¦e =
|φγ |¦e , where |φγ is the atomic ground state and |¦e is the initial state of the
radiation ¬eld, which is, for the moment, assumed to be pure. According to eqns
(4.95) and (4.103) the initial state vector evolves into
t
i
|˜ (t) = |˜ (t0 ) ’ dt1 Hdet (t1 ) |˜ (t0 ) + · · · , (9.1)
t0

so the ¬rst-order probability amplitude that a joint measurement at time t ¬nds the
atom in an excited state |φ and the ¬eld in the number state |n is
t
i
φ , n |˜ (t) = ’ dt1 φ , n |Hdet (t1 )| ˜ (t0 ) , (9.2)
t0


where |φ , n = |φ |n . Only the Rabi operator „¦ (+) in eqn (4.149) can contribute
to an absorptive transition, so the matrix element and the probability amplitude are
respectively given by
¾
Primary photon detection


φ , n |Hdet (t1 )| ˜ (t0 ) = ’eiω · n E(+) (r, t1 ) ¦e
γ t1
d (9.3)
γ

and
t
i
φ , n |˜ (t) = · n E(+) (r, t1 ) ¦e ,
dt1 eiω γ t1
d (9.4)
γ
t0

where d γ = φ d φγ is the dipole matrix element for the transition γ ’ .
The conditional probability for ¬nding |φ , n , given |φγ , ¦e , is therefore
2
t
i
· nE
iω γ t1 (+)
p (φ , n : φγ , ¦e ) = dt1 e d (r, t1 ) ¦e
γ
t0
d—γ t t
(d γ )j
γ (t2 ’t1 )
i
dt2 eiω
= dt1
2
t0 t0

(+) (+)
— n Ei (r, t1 ) ¦e n Ej (r, t2 ) ¦e . (9.5)

(+) (’)
The relation E(’) = E(+)† implies n Ei (r, t1 ) ¦e = ¦e Ei (r, t1 ) n , so that
eqn (9.5) can be rewritten as
d—γ t t
(d γ )j
γ (t2 ’t1 )
i
dt2 eiω
p (φ , n : φγ , ¦e ) = dt1
2
t0 t0
(’) (+)
— ¦e Ei (r, t1 ) n n Ej (r, t2 ) ¦e . (9.6)

Since the ¬nal state of the radiation ¬eld is not usually observed, the relevant quantity
is the sum of the conditional probabilities p (φ , n : φγ , ¦e ) over all ¬nal ¬eld states
|n :
p (φ : φγ , ¦e ) = p (φ , n : φγ , ¦e ) . (9.7)
n

The completeness identity (3.67) for the number states, combined with eqn (9.6) and
eqn (9.7), then yields
d—γ t t
(d γ )j
γ (t2 ’t1 )
i
dt2 eiω
p (φ : φγ , ¦e ) = dt1
2
t0 t0
(’) (+)
— ¦e Ei (r, t1 ) Ej (r, t2 ) ¦e . (9.8)

This result is valid when the radiation ¬eld is known to be initially in the pure state
|¦e . In most experiments all that is known is a probability distribution Pe over an
ensemble {|¦e } of pure initial states, so it is necessary to average over this ensemble
to get
p (φ : φγ , ¦e ) Pe
p (φ : φγ ) =
e
d—γ t t
(d γ )j
γ (t2 ’t1 ) (’) (+)
i
dt2 eiω
= dt1 Tr ρEi (r, t1 ) Ej (r, t2 ) ,
2
t0 t0
(9.9)
¾¼ Photon detection

where
Pe |¦e ¦e |
ρ= (9.10)
e

is the density operator de¬ned by the distribution Pe .
So far it has been assumed that the ¬nal atomic state |φ can be detected with
perfect accuracy, but of course this is never the case. Furthermore, most detection
schemes do not depend on a speci¬c transition to a bound level; instead, they involve
transitions into excited states lying in the continuum. The atom may be directly
ionized, or the absorption of the photon may lead to a bound state that is subject
to Stark ionization by a static electric ¬eld. The ionized electrons would then be
accelerated, and thereby produce further ionization by secondary collisions. All of
these complexities are subsumed in the probability D ( ) that the transition γ ’
occurs and produces a macroscopically observable event, e.g. a current pulse. The
overall probability is then

D ( ) p (φ : φγ ) .
p (t) = (9.11)

It should be understood that the -sum is really an integral, and that the factor
D ( ) includes the density of states for the continuum states of the atom. Putting this
together with the expression (9.9) leads to
t t
(1)
dt2 Sji (t1 ’ t2 ) Gij (r, t1 ; r, t2 ) ,
p (t) = dt1 (9.12)
t0 t0

where the sensitivity function
1
D ( ) d—γ (d γ )j e’iω
Sji (t) = γt
(9.13)
2 i


is determined solely by the properties of the atom, and the ¬eld“¬eld correlation
function
(1) (’) (+)
Gij (r, t1 ; r, t2 ) = Tr ρEi (r1 , t1 ) Ej (r, t2 ) (9.14)
is determined solely by the properties of the ¬eld.
Since D ( ) is real and positive, the sensitivity function obeys

S— (t) = Sij (’t) , (9.15)
ji

and other useful properties are found by studying the Fourier transform

Sji (ω) = dtSji (t) eiωt

D ( ) d—γ (d γ )j δ (ω ’ ω γ ) .
= (9.16)
2 i


The -sum is really an integral over the continuum of excited states, so Sji (ω) is a
smooth function of ω. This explicit expression shows that the 3 — 3 matrix S (ω),
¾½
Primary photon detection


with components Sji (ω), is hermitian”i.e. Sji (ω) = Sij (ω) ”and positive-de¬nite,
since


D ( ) |v— · d γ |2 δ (ω ’ ω γ ) > 0
vj Sji (ω) vi = 2 (9.17)

for any complex vector v. These properties in turn guarantee that the eigenvalues are
real and positive, so the power spectrum,
T (ω) = Tr [S (ω)] , (9.18)
of the dipole transitions can be used to de¬ne averages over frequency by
dωT (ω) f (ω)
f = . (9.19)
T
dωT (ω)
The width ∆ωS of the sensitivity function is then de¬ned as the rms deviation
2
’ω
ω2
∆ωS = . (9.20)
T T

The single-photon counting rate w(1) (t) is the rate of change of the probability:
t
dp (1)
dt Sji (t ’ t) Gij (r, t ; r, t) ,
(1)
w (t) = = 2 Re (9.21)
dt t0

where the ¬nal form comes from combining eqn (9.15) with the symmetry property
(1)— (1)
Gij (r1 , t1 ; r2 , t2 ) = Gji (r2 , t2 ; r1 , t1 ) , (9.22)
that follows from eqn (9.14). For later use it is better to express the counting rate as

Sji (ω) Xij (ω, t) ,
w(1) (t) = 2 Re (9.23)

where
t
dt eiω(t’t ) Gij (r, t ; r, t) .
(1)
Xij (ω, t) = (9.24)
t0
The value of the frequency integral in eqn (9.23) depends on the relative widths of
the sensitivity function and Xij (ω, t), considered as a function of ω with t ¬xed. One
way to get this information is to use eqn (9.24) to evaluate the transform
dω iωt
Xij (t , t) = e Xij (ω, t)

(1)
= θ (t ) θ (t ’ t0 ’ t ) Gij (r, t ’ t ; r, t) . (9.25)
The step functions in this expression guarantee that Xij (t , t) vanishes outside the
t ’ t0 . On the other hand, the correlation function vanishes for
interval 0 t
Tc , where Tc is the correlation time. The observation time t ’ t0 is normally
t
much longer than the correlation time, so the t -width of Xij (t , t) is approximately
Tc . By the uncertainty principle, the ω-width of Xij (ω, t) is ∆ωX ∼ 1/Tc = ∆ωG ,
(1)
where ∆ωG is the bandwidth of the correlation function Gij .
¾¾ Photon detection

B Broadband detection
The detector is said to be broadband if the bandwidth ∆ωS of the sensitivity function
satis¬es ∆ωS ∆ωG = 1/Tc. For a broadband detector, Xij (ω) is sharply peaked
compared to the sensitivity function; therefore, Sji (ω) can be treated as a constant”
Sji (ω) ≈ Sji ”and taken outside the integral. This is formally equivalent to setting
Sji (t ’ t) = Sji δ (t ’ t) in eqn (9.21), and the result
(1)
w(1) (t) = Sji Gij (r, t; r, t) (9.26)

is obtained by combining the end-point rule (A.98) for delta functions with the sym-
metries (9.15) and (9.22). Consequently, the broadband counting rate is proportional
to the equal-time correlation function. The argument leading to eqn (9.26) is similar
to the derivation of Fermi™s golden rule in perturbation theory. In practice, nearly all
detectors can be treated as broadband.
The analysis of ideal single-atom detectors can be extended to realistic many-atom
detectors when two conditions are satis¬ed: (1) single-atom absorption is the dominant
process; (2) interactions between the atoms can be ignored. These conditions will be
satis¬ed for atoms in a tenuous vapor or in an atomic beam”see item (5) in Section
9.1.1”and they are also satis¬ed by many solid-state detectors. For atoms located
at positions r1 , . . . , rN , the total single-photon counting rate is the average of the
counting rates for the individual atoms:
N
1 (A) (1)
Sji Gij (rA , t; rA , t) .
(1)
w (t) = (9.27)
N
A=1

It is often convenient to use a coarse-grained description which replaces the last equa-
tion by
1 (1)
d3 r n (r) Sji (r) Gij (r, t; r, t) ,
w(1) (t) = (9.28)
nVD
where n (r) is the density of atoms, Sji (r) is the sensitivity function at r, n is the mean
density of atoms, and VD is the volume occupied by the detector. A point detector
is de¬ned by the condition that the correlation function is essentially constant across
the volume of the detector. In this case, the counting rate is
(1)
w(1) (t) = Sji Gij (r, t; r, t) , (9.29)

where Sji is the average sensitivity function and r is the center of mass of the detector.
Comparing this to eqn (9.26) shows that a point detector is like a single-atom detector
with a modi¬ed sensitivity factor.
The sensitivity factor, de¬ned by eqn (9.16), is a 3 — 3 hermitian matrix which has
the useful representation
3
Sa eai e— ,
Sij = (9.30)
aj
a=1
where the eigenvalues, Sa , are real and the eigenvectors, ea , are orthonormal: e— · ea =
b
δab . Substituting this representation into eqn (9.26) produces
¾¿
Primary photon detection

3
Sa G(1) (r, t; r, t) ,
(1)
w (t) = (9.31)
a
a=1

where the new correlation functions,

G(1) (r, t; r, t) = Tr ρEa (r, t) Ea (r, t) ,
(’) (+)
(9.32)
a

(’)
are de¬ned in terms of the scalar ¬eld operators Ea (r, t) = ea · E(’) (r, t). This
form is useful for imposing special conditions on the detector. For example, a detector
equipped with a polarization ¬lter is described by the assumption that only one of the
eigenvalues, say S1 , is nonzero. The corresponding eigenvector e1 is the polarization
passed by the ¬lter. In this situation, eqn (9.29) becomes

w(1) (t) = S G(1) (r, t; r, t)
(’) (+)
= S Tr ρE1 (r, t) E1 (r, t) , (9.33)

where E1 (r, t) = e— · E(+) (r, t), e is the transmitted polarization, and S is the
(+)

sensitivity factor. As promised, the counting rate is the product of the sensitivity
factor S and the correlation function G(1) . Thus the broadband counting rate provides
a direct measurement of the equal-time correlation function G(1) (r, t; r, t).

C Narrowband detection
Broadband detectors do not distinguish between photons of di¬erent frequencies that
may be contained in the incident ¬eld, so they do not determine the spectral func-
tion of the ¬eld. For this purpose, one needs narrowband detection, which is
usually achieved by passing the light through a narrowband ¬lter before it falls
on a broadband detector. The ¬lter is a linear device, so its action can be repre-
sented mathematically as a linear operation applied to the signal. For a real signal,
X (t) = X (+) (t) + X (’) (t), the ¬ltered signal at ω”i.e. the part of the signal
corresponding to a narrow band of frequencies around ω”is de¬ned by

(t ’ t) eiω(t ’t) X (+) (t )
(+)
X (ω; t) = dt
’∞

(t ) eiωt X (+) (t + t) ,
= dt (9.34)
’∞

where the factor exp [iω (t ’ t)] serves to pick out the desired frequency. The weighting
function (t) has the following properties.
(1) It is even and positive,
(t) = (’t) 0 . (9.35)
(2) It is normalized by

dt (t) = 1 . (9.36)
’∞
¾ Photon detection

(3) It is peaked at t = 0.
The weighting function is therefore suitable for de¬ning averages, e.g. the tempo-
ral width ∆T :
∞ 1/2
< ∞.
2
∆T = dt (t) t (9.37)
’∞

A simple example of an averaging function satisfying eqns (9.35)“(9.37) is

for ’ ∆T
1 ∆T
t ,
∆T 2 2 (9.38)
(t) =
0 otherwise .

The meaning of ¬ltering can be clari¬ed by Fourier transforming eqn (9.34) to get

X (+) (ω ; ω) = F (ω ’ ω) X (+) (ω) , (9.39)

where the ¬lter function F (ω) is the Fourier transform of (t). Since the normal-
ization condition (9.36) implies F (0) = 1, the ¬ltered signal is essentially identical to
the original signal in the narrow band de¬ned by the width ∆ωF ∼ 1/∆T of the ¬lter
function; but, it is strongly suppressed outside this band.
The frequency ω selected by the ¬lter varies continuously, so the interesting quan-
tity is the spectral density S (ω), which is de¬ned as the counting rate per unit
frequency interval. Applying the broadband result (9.33) to the ¬ltered ¬eld operators
yields

w(1) (ω, t)
S (ω, t) =
∆ωF
S (’) (+)
= E1 (r, t; ω) E1 (r, t; ω) . (9.40)
∆ωF

For the following argument, we choose the simple form (9.38) for the averaging function
to calculate the ¬ltered operator:

∆T /2
1
(+) (+)
dt eiωt E1
E1 (r, t; ω) = (r, t + t) . (9.41a)
∆T ’∆T /2


Substituting this result into eqn (9.40) and combining ∆ωF = 1/∆T with the de¬nition
of the ¬rst-order correlation function yields

S ∆T /2 ∆T /2
dt2 eiω(t1 ’t2 ) G(1) (r, t2 + t; r, t1 + t) .
S (ω, t) = dt1 (9.41b)
∆T ’∆T /2 ’∆T /2


In almost all applications, we can assume that the correlation function only depends
on the di¬erence in the time arguments. This assumption is rigorously valid if the
¾
Primary photon detection

density operator ρ is stationary, and for dissipative systems it is approximately satis¬ed
for large t. Given this property, we set
dω (1)
G (r, ω ; t) e’iω (t1 ’t2 )
G(1) (r, t2 + t; r, t1 + t) = , (9.42)

and get
sin2 [(ω ’ ω ) ∆T /2]
dω (1)
S (ω) = S G (r, ω ; t) . (9.43)
2
[(ω ’ ω ) /2] ∆T

In this case, the width of the ¬lter is assumed to be very small compared to the width
of the correlation function, i.e. ∆ωS ∆ωG (∆T Tc ). By means of the general
identity (A.102), one can show that

sin2 [ν∆T /2]
lim = πδ (ν/2) = 2πδ (ν) , (9.44)
2
∆T ’∞ [ν/2] ∆T
and substituting this result into eqn (9.43) leads to

d„ e’iω„ G(1) (r, „ + t; r, t) .
S (ω) = SG(1) (r, ω; t) = S (9.45)

In other words, the spectral density is proportional to the Fourier transform, with
respect to the di¬erence of the time arguments, of the two-time correlation function
G(1) (r, t2 + t; r, t1 + t).
It is often useful to have a tunable ¬lter, so that the selected frequency can be
swept across the spectral region of interest. The main methods for accomplishing this
employ spectrometers to spatially separate the di¬erent frequency components. One
technique is to use a di¬raction grating spectrometer (Hecht, 2002, Sec. 10.2.8) placed
on a mount that can be continuously swept in angle, while the input and output
slits remain ¬xed. The spectrometer thus acts as a continuously tunable ¬lter, with
bandwidth determined by the width of the slits. Higher resolution can be achieved
by using a Fabry“Perot spectrometer (Hecht, 2002, Secs 9.6.1 and 9.7.3) with an
adjustable spacing between the plates. A di¬erent approach is to use a heterodyne
spectrometer, in which the signal is mixed with a local oscillator”usually a laser”
which is close to the signal frequency. The beat signal oscillates at an intermediate
frequency which is typically in the radio range, so that standard electronics techniques
can be used. For example, the radio frequency signal is analyzed by a radio frequency
spectrometer or a correlator. The Fourier transform of the correlator output signal
yields the radio frequency spectrum of the beat signal.

9.1.3 Photoelectron counting statistics
How does one measure the photon statistics of a light ¬eld, such as the Poissonian
statistics predicted for the coherent state |± ? In practice, these statistics must be in-
ferred from photoelectron counting statistics which, fortunately, often faithfully repro-
duce the counting statistics of the photons. For example, in the case of light prepared
in a coherent state, both the incident photon and the detected photoelectron statistics
turn out to be Poissonian.
¾ Photon detection

Consider a light beam”produced, for example, by passing the output of a laser
operating far above threshold through an attenuator”that falls on the photocathode
surface of a photomultiplier tube. The amplitude of the attenuated coherent state
is ± = exp(’χL/2)±0 , where χ is the absorption coe¬cient and L is the length of
the absorber. The photoelectron probability distribution can be obtained from the
probability distribution for the number of incident photons, p(n), by folding it into the
Bernoulli distribution function using the standard classical technique (Feller, 1957a,
Chap. VI). The probability P (m, ξ) of the detection of m photoelectrons found in this
way is

nm n’m
ξ (1 ’ ξ)
P (m, ξ) = p(n) , (9.46)
m
n=m

where ξ is the probability that the interaction of a given photon with the atoms in the
detector will produce a photoelectron. This quantity”which is called the quantum
e¬ciency”is given by

ξ=ζ T, (9.47)
V
where ζ (which is proportional to the sensitivity function S) is the photoelectron
ejection probability per unit time per unit light intensity, (c ω/V ) is the intensity
due to a single photon, and T is the integration time of the photon detector. The
integration time is usually the RC time constant of the detection system, which in
the case of photomultiplier tubes is of the order of nanoseconds. The parameter ζ
can be calculated quantum mechanically, but is usually determined empirically. The
factors in the summand in eqn (9.46) are: the photon distribution p(n); the binomial
n
coe¬cient m (the number of ways of distributing n photons among m photoelectron
ejections); the probability ξ m that m photons are converted into photoelectrons; and
n’m
the probability (1 ’ ξ) that the remaining n ’ m photons are not detected at all.
One can show”see Exercise 9.1”that a Poissonian initial photon distribution, with
average photon number n, results in a Poissonian photoelectron distribution,
m m ’m
P (m, ξ) = e , (9.48)
m!
where m = ξ n is the average ejected photoelectron number. In the special case ξ = 1,
there is a one“one correspondence between an incident photon and a single ejected
photoelectron. In this case, the Bernoulli sum in eqn (9.46) consists of only the single
term n = m, so that the photon and photoelectron distribution functions are identical.
Thus the photoelectron statistics will faithfully reproduce the photon statistics in the
incident light beam, for example, the Poissonian statistics of the coherent state dis-
cussed above. An experiment demonstrating this fact for a helium“neon laser operating
far above threshold is described in Section 5.3.2.

Quantum e¬ciency—
9.1.4
The quantum e¬ciency ξ introduced in eqn (9.47) is a phenomenological parameter
that can represent any of a number of possible failure modes in photon detection:
re¬‚ection from the front surface of a cathode; a mismatch between the transverse
¾
Primary photon detection

pro¬le of the signal and the aperture of the detector; arrival of the signal during a
dead time of the detector; etc. In each case, there is some scattering or absorption
channel in addition to the one that yields the current pulse signaling the detection
event. We have already seen, in the discussion of beam splitters in Section 8.4, that
the presence of additional channels adds partition noise to the signal, due to vacuum
¬‚uctuations entering through an unused port. This generic feature allows us to model
an imperfect detector as a compound device composed of a beam splitter followed by
an ideal detector with 100% quantum e¬ciency, as shown in Fig. 9.1.
The transmission and re¬‚ection coe¬cients of the ¬ctitious beam splitter must
be adjusted to obey the unitarity condition (8.7) and to account for the quantum
e¬ciency of the real detector. These requirements are satis¬ed by setting

t= ξ, r= i 1’ξ. (9.49)

The beam splitter is a linear device, so no generality is lost by restricting attention to
monochromatic input signals described by a density operator ρ that is the vacuum for
all modes other than the signal mode. In this case we can specialize eqn (8.28) for the
in-¬eld to
Ein (r, t) = iE0s a1 eiks x e’iωs t + Evac,in (r, t) ,
(+) (+)
(9.50)
where we have chosen the x- and y-axes along the 1 ’ 1 and 2 ’ 2 arms of the
device respectively, E0s = ωs /2 0 V is the vacuum ¬‚uctuation ¬eld strength for a
plane wave with frequency ωs , and a1 is the annihilation operator for the plane-wave
(+)
mode exp [i (ks x ’ ωs t)]. In principle, the operator Evac,in (r, t) should be a sum over
all modes orthogonal to the signal mode, but the discussion in Section 8.4.1 shows
that we need only consider the mode exp [i (ks y ’ ωs t)] entering through port 2. This
leaves us with the simpli¬ed in-¬eld

Ein (r, t) = iE0s a1 eiks x e’iωs t + iE0s a2 eiks y e’iωs t .
(+)
(9.51)

An application of eqn (8.63) yields the scattered annihilation operators

ξa1 + i 1 ’ ξa2 ,
a1 =
(9.52)
1 ’ ξa1 +
a2 = i ξa2 ,

and the corresponding out-¬eld

2
1
-lost
Fig. 9.1 An imperfect detector modeled by
-sig -, combining an ideal detector with a beam split-
ter. Esig is the signal entering port 1, Evac rep-
Ideal
-vac resents vacuum ¬‚uctuations (at the signal fre-
1 detector
quency) entering port 2, ED is the e¬ective sig-
2
Beam nal entering the detector, and Elost describes
splitter the part of the signal lost due to ine¬ciencies.
¾ Photon detection

(+) (+) (+)
Eout (r, t) = ED (r, t) + Elost (r, t) , (9.53)

where
ED (r, t) = iE0s a1 eiks x e’iωs t
(+)
(9.54)
and
Elost (r, t) = iE0s a2 eiks y e’iωs t .
(+)
(9.55)
The counting rate of the imperfect detector is by de¬nition the counting rate of the
perfect detector viewing port 1 of the beam splitter, so”for the simple case of a
broadband detector”eqn (9.33) gives
(’) (+)
w(1) (t) = S ED (rD , t) ED (rD , t)

= S E0s a1† a1
2


= ξ S E0s a† a1 ,
2
(9.56)
1


where (· · · ) = Tr [ρ (· · · )], rD is the location of the detector, and we have used
a2 ρ = ρa† = 0. The operator Elost represents the part of the signal lost by scattering
(+)
2
into the 2 ’ 2 channel.
As expected, the counting rate of the imperfect detector is reduced by the quantum
e¬ciency ξ; and the vacuum ¬‚uctuations entering through port 2 do not contribute
to the average detector output. From our experience with the beam splitter, we know
that the vacuum ¬‚uctuations will add to the variance of the scattered number operator
N1 = a1† a1 . Combining the canonical commutation relations for the creation and
annihilation operators with the scattering equation (9.52) and a little algebra gives us

V (N1 ) = ξ 2 V (N1 ) + ξ (1 ’ ξ) N1 . (9.57)

The ¬rst term on the right represents the variance in photon number for the inci-
dent ¬eld, reduced by the square of the quantum e¬ciency. The second term is the
contribution of the extra partition noise associated with the random response of the
imperfect detector, i.e. the arrival of a photon causes a click with probability ξ or no
click with probability 1 ’ ξ.

The Mandel Q-parameter
9.1.5
Most photon detectors are based on the photoelectric e¬ect, and in Section 9.1.2
we have seen that counting rates can be expressed in terms of expectation values
of normally-ordered products of electric ¬eld operators. In the example of a single
mode, this leads to averages of normal-ordered products of the general form a†n an .
As seen in Section 5.6.3, the most useful quasi-probability distribution for the de-
scription of such measurements is the Glauber“Sudarshan function P (±). If this dis-
tribution function is non-negative everywhere on the complex ±-plane, then there is
a classical model”described by stochastic c-number phasors ± with the same P (±)
distribution”that reproduces the average values of the quantum theory. It is reason-
able to call such light distributions classical, because no measurements based on the
¾
Primary photon detection

photoelectric e¬ect can distinguish between a quantum state and a classical stochastic
model that share the same P (±) distribution.
Direct experimental veri¬cation of the condition P (±) 0 requires rather sophis-
ticated methods, which we will study in Chapter 17. A simpler, but still very useful,
distinction between classical and nonclassical states of light employs the global sta-
tistical properties of the state. Photoelectric counters can measure the moments N r
(r = 1, 2, . . .) of the number operator N = a† a, where (· · · ) = Tr [ρ (· · · )], and ρ
is the density operator for the state under consideration. We will study the second
2
moment, or rather the variance, V (N ) = N 2 ’ N , which is a measure of the
noise in the light. In Section 5.1.3 we found that a coherent state ρ = |± ±| exhibits
Poissonian statistics, i.e. for a coherent state the variance in photon number is equal
to the average number: V (N ) = N 2 ’ N 2 = N , which is the standard quantum
N , this is just another name for the shot noise1
limit. Since the rms deviation is
in the photoelectric detector. The coherent states are constructed to be as classical as
possible, so it is useful to compare the variance for a given state ρ with the variance
for a coherent state with the same average number of photons. The fractional excess
of the variance relative to that of shot noise,
V (N ) ’ N
Q≡ , (9.58)
N
is called the Mandel Q parameter (Mandel and Wolf, 1995, Sec. 12.10.3). This new
usage should not be confused with the Q-function de¬ned by eqn (5.154).
The Q-parameter vanishes for a coherent state, so it can be regarded as a measure
of the excess photon-number noise in the light described by the state ρ. Since the
operator N is hermitian, the variance V (N ) is non-negative, and it only vanishes for
number states. Consequently the range of Q-values is

’1 Q < ∞. (9.59)

A very useful property of the Q-parameter can be derived by ¬rst expressing the
numerator in eqn (9.58) as
2
V (N ) ’ N = N 2 ’ N ’N
2
= a†2 a2 ’ a† a , (9.60)

where the last line follows from another application of the commutation relations
a, a† = 1. Since all the operators are now in normal-ordered form, we may use the
P -representation (5.168) to get
2
d2 ± 4 d2 ± 2
V (N ) ’ N = |±| P (±) ’ |±| P (±) . (9.61)
π π
By using the fact that P (±) is normalized to unity, the ¬rst term can be expressed as
a double integral, so that
1 Shot noise describes the statistics associated with the random arrivals of discrete objects at a
detector, e.g. the noise associated with raindrops falling onto a tin rooftop.
¾¼ Photon detection

d2 ± 4 d2 ±
V (N ) ’ N = |±| P (±) P (± )
π π
d2 ± 2 d2 ± 2
’ |±| P (±) |± | P (± ) . (9.62)
π π
The ¬nal step is to interchange the dummy integration variables ± and ± in the ¬rst
term, and then to average the two equivalent expressions; this yields the ¬nal result:

d2 ± d2 ± 22
1 2
V (N ) ’ N = |±| ’ |± | P (±) P (± ) . (9.63)
2 π π
The right side is positive for P (±) 0; therefore, classical states always correspond
to non-negative Q values. An equivalent, but more useful statement, is that negative
values of the Q-parameter always correspond to nonclassical states. A point which
is often overlooked is that the condition Q < 0 is su¬cient but not necessary for a
nonclassical state. In other words, there are nonclassical states with Q > 0.
A coherent state has Q = 0 (Poissonian statistics for the vacuum ¬‚uctuations), so a
state with Q < 0 is said to be sub-Poissonian. These states are quieter than coherent
states as far as photon number ¬‚uctuations are concerned. We will see another example
later on in the study of squeezed states. By the same logic, super-Poissonian states,
with Q > 0, are noisier than coherent states. Thermal states, or more generally chaotic
states, are familiar examples of super-Poissonian statistics; and a nonclassical example
is presented in Exercise 9.3.
An overall Q-parameter for multimode states can be de¬ned by using the total
number operator,
a† aM ,
N= (9.64)
M
M

in eqn (9.58). The de¬nition of a classical state is P (±) 0, where P (±) is the
multimode P -function de¬ned by eqn (5.104). A straightforward generalization of the
single-mode argument again leads to the conclusion that states with Q < 0 are neces-
sarily nonclassical.

9.2 Postdetection signal processing
In the preceding sections, we discussed several processes for primary photon detection.
Now we must study postdetection signal processing, which is absolutely necessary for
completing a measurement of the state of a light ¬eld. The problem that must be faced
in carrying out a measurement on any quantum system is that microscopic processes,
such as the events involved in primary photon detection, are inherently reversible.
Consider, for example, a photon and a ground-state atom, both trapped in a small
cavity with perfectly re¬‚ecting walls. The atom can absorb the photon and enter an
excited state, but”with equal facility”the excited atom can return to the ground
state by emitting the photon. The photon”none the worse for its adventure”can
then initiate the process again. We will see in Chapter 12 that this dance can go
on inde¬nitely. In a solid-state photon detector, the cavity is replaced by the crystal
lattice, and the ground-state atom is replaced by an electron in the valence band.
¾½
Postdetection signal processing

The electron can be excited to the conduction band”leaving a hole in the valence
band”by absorbing the photon. Just as for the atom, time-reversal invariance assures
us that the conduction band electron can return to the valence band by emitting the
photon, and so on. This behavior is described by the state vector

|photon-detector = ± (t) |photon |valence-band-electron
+ β (t) |vacuum |electron“hole-pair
= ± (t) |photon-not-detected + β (t) |photon-detected (9.65)

for the photon-detector system. As long as the situation is described by this entangled
state, there is no way to know if the photon was detected or not.
The purpose of a measurement is to put a stop to this quantum dithering by
perturbing the system in such a way that it is forced to make a de¬nite choice. An
interaction with another physical system having a small number of degrees of freedom
clearly will not do, since the reversibility argument could be applied to the enlarged
system. Thus the perturbation must involve coupling to a system with a very large
number of degrees of freedom, i.e. a macroscopic system. It could be”indeed it has
been”argued that this procedure simply produces another entangled state, albeit with
many degrees of freedom. While correct in principle, this line of argument brings us
back to Schr¨dinger™s diabolical machine and the unfortunate cat. Just as we can be
o
quite certain that looking into this device will reveal a cat that is either de¬nitely dead
or de¬nitely alive”and not some spooky superposition of |dead cat and |live cat ”
we can also be assured that an irreversible interaction with a macroscopic system will
yield a de¬nite answer: the photon was detected or it was not detected. In the words
of Bohr (1958, p. 88):
. . .every atomic phenomenon is closed in the sense that its observation is based on
registrations obtained by means of suitable ampli¬cation devices with irreversible
functioning such as, for example, permanent marks on the photographic plate,
caused by the penetration of electrons into the emulsion (emphasis added).

Thus postdetection signal processing”which bring quantum measurements to a
close by processes involving irreversible ampli¬cation”is an essential part of pho-
ton detection. In the following sections we will discuss several modern postdetection
processes: (1) electron multiplication in Markovian avalanche processes, e.g. in vacuum
tube photomultipliers, channeltrons, and image intensi¬ers; (2) solid-state avalanche
photodiodes, and solid-state multipliers with noise-free, non-Markovian avalanche elec-
tron multiplication. Finally we discuss coincidence detection, which is an important
application of postdetection signal processing.

9.2.1 Electron multiplication
We begin with a discussion of electron multiplication processes in photomultipliers,
channeltrons, and solid-state avalanche photodiodes. As pointed out above, postde-
tection gain mechanisms are not only a practical, but also a fundamental, component
of all photon detectors. They are necessary for the closing of the quantum process
of measurement. As a practical matter, ampli¬cation is required to raise the micro-
scopic energy released in the primary photodetection event” ω ∼ 10’19 J for a typical
¾¾ Photon detection

visible photon”to a macroscopic value much larger than the typical thermal noise”
kB T ∼ 10’20 J”in electronic circuits. From this point on, the signal processing can be
easily handled by standard electronics, since the noise in any electronic detection sys-
tem is determined by the noise in the ¬rst-stage electronic ampli¬cation process. The
typical electron multiplication factor in these postdetection mechanisms is between
104 to 106 .
One ampli¬cation mechanism is electron multiplication by secondary impact ion-
izations occurring at the surfaces of the dynode structures of vacuum-tube photomul-
tipliers. A large electric ¬eld is applied across successive dynode structures, as shown
in Fig. 9.2. The initial photoelectron released from the photocathode is thus acceler-
ated to such high energies that its impact on the surface of the ¬rst dynode releases
many secondary electrons. By repeated multiplications on successive dynodes, a large
electrical signal can be obtained.
In channeltron vacuum tubes, which are also called image intensi¬ers, the pho-
toelectrons released from various spots on a single photocathode are collected by a
bundle of small, hollow channels, each corresponding to a single pixel. A large electric
¬eld applied along the length of each channel induces electron multiplication on the
interior surface, which is coated with a thin, conducting ¬lm. Repeated multiplica-
tions by means of successive impacts of the electrons along the length of each channel
produce a large electrical signal, which can be easily handled by standard electronics.
There is a similar postdetection gain mechanism in solid-state photodiodes. The
primary event is the production of a single electron“hole pair inside the solid-state
material, as shown in Fig. 9.3. When a static electric ¬eld is applied, the initial electron
and hole are accelerated in opposite directions, in the so-called Geiger mode of
operation. For a su¬ciently large ¬eld, the electron and hole reach such high energies
that secondary pairs are produced. The secondary pairs in turn cause further pair
production, so that an avalanche breakdown occurs. This process produces a large
electrical pulse”like the single click of a Geiger counter”that signals the arrival of
a single photon. In this strong-¬eld limit, the secondary emission processes occur so
quickly and randomly that all correlations with previous emissions are wiped out. The
absence of any dependence on the previous history is the de¬ning characteristic of a
Markov process.

Photomultiplier
Photoelectron


Photon
Laser beam
Fig. 9.2 Schematic of a laser beam incident Dynodes
Photocathode
upon a photomultiplier tube.
¾¿
Postdetection signal processing



Fig. 9.3 In a semiconductor photodetection
device, photoionization occurs inside the body
of a semiconductor. In (a) the photon enters
the semiconductor. In (b) a photoionization
event produces an electron“hole pair inside the
semiconductor.


9.2.2 Markovian model for avalanche electron multiplication
We now discuss a simple model (LaViolette and Stapelbroek, 1989) of electron multi-
plication, such as that of avalanche breakdown in the Geiger mode of silicon solid-state
avalanche photon detectors (APDs). This model is based on the Markov approxima-
tion; that is, the electron completely forgets all previous scatterings, so that its behav-
ior is solely determined by the initial conditions at each branch point of the avalanche
process. The model rests on two underlying assumptions.
(1) The initial photoelectron production always occurs at the same place (z = 0),
where z is the coordinate along the electric ¬eld axis.
(2) Upon impact ionization of an impurity atom, the incoming electron dies and two
new electrons are born. This is the Markov approximation. None of the electrons
recombine or otherwise disappear.
The probability that a new carrier is generated in the interval (z, z + ∆z) is
± (z) ∆z, where the gain, ± (z), is allowed to vary with z. The probability that n
carriers are present at z, given that one carrier is introduced at z = 0, is denoted by
p (n, z). There are two cases to examine p (1, z) (total failure) and p (n, z) for n > 1.
The probability that the incident carrier fails to produce a new carrier in the
interval (z, z + ∆z) is 1’± (z) ∆z. Thus the probability of failure in the next z-interval
is
p (1, z + ∆z) = (1 ’ ± (z) ∆z) p (1, z) . (9.66)
Take the limit ∆z ’ 0, or Taylor-series expand the left side, to get the di¬erential
equation
‚p (1, z)
= ’± (z) p (1, z) , (9.67)
‚z
with the initial condition p (1, 0) = 1.
For the successful case that n > 1, there are more possibilities, since n carriers
at z + ∆z could come from n ’ k carriers at z by production of k carriers, where
k = 0, 1, . . . , n ’ 1. Adding up the possible processes gives
n
p (n, z + ∆z) = (1 ’ ± (z) ∆z) p (n, z) + (n ’ 1) (± (z) ∆z) p (n ’ 1, z)
1 2
+ (n ’ 2) (n ’ 3) (± (z) ∆z) p (n ’ 2, z) + · · · . (9.68)
2
In the limit of small ∆z this leads to the di¬erential equation
‚p (n, z)
= ’n± (z) p (n, z) + (n ’ 1) ± (z) p (n ’ 1, z) , (9.69)
‚z
¾ Photon detection

with the initial condition p (n, z) = 0 for n > 1.
The solution of eqn (9.67) is easily seen to be
z
’ζ(z)
p (1, z) = e , where ζ (z) = dz ± (z ) . (9.70)
0


The recursive system of di¬erential equations in eqn (9.69) is a bit more complicated.
Perhaps the easiest way is to work out the explicit solutions for n = 2, 3 and use the
results to guess the general form:

n’1
eζ(z) ’ 1
p (n, z) = . (9.71)
enζ(z)

9.2.3 Noise-free, non-Markovian avalanche multiplication
One recent and very important development in postdetection gain mechanisms for
photon detectors is noise-free avalanche multiplication in silicon, solid-state photo-
multipliers (SSPMs) (Kim et al., 1997). Noise-free, postdetection ampli¬cation allows
the photon detector to distinguish clearly between one and two photons in the primary
photodetection event; i.e. the output electronic pulse heights can be cleanly resolved
as originating either from a one- or a two-photon primary event. This has led to the
direct detection, with high resolution, of the di¬erence between even and odd photon
numbers in an incoming beam of light. Applying this photodetection technique to a
squeezed state of light shows that there is a pronounced preference for the occupation
of even photon numbers; the odd photon numbers are essentially absent. This striking
odd“even e¬ect in the photon number distribution is not observed with a coherent
state of light, such as that produced by a laser.
A schematic of a noise-free avalanche multiplication device in a SSPM, also known
as a visible-light photon counter (VLPC), is shown in Fig. 9.4.




Fig. 9.4 Structure of a solid-state photomultiplier (SSPM) or a visible-light photon counter
(VLPC). (Reproduced from Kim et al. (1997).)
¾
Postdetection signal processing

In contrast to the APD, the SSPM is divided into two separate spatial regions: an
intrinsic region, inside which the incident photon is converted into a primary electron“
hole pair in an intrinsic silicon crystalline material; followed by a gain region, consisting
of n-doped silicon, inside which well-controlled, noise-free electron multiplication oc-
curs. The electric ¬eld in the gain region is larger than in the intrinsic region, due to
the di¬erence between the respective dielectric constants. The primary electron and
hole, produced by the incoming visible photon, are accelerated in opposite directions
by the local electric ¬eld in the intrinsic region. The primary electron propagates to
the left towards a transparent electrode (the transparent contact) raised to a modest
positive potential +V . An anti-re¬‚ection coating applied to the transparent electrode
ensures that the incoming photon is admitted with high e¬ciency into the interior of
the silicon intrinsic region, so that the quantum e¬ciency of the device can be quite
high.
The primary hole propagates to the right and enters the gain region, whereupon
the higher electric ¬eld present there accelerates it up to the energy (54 meV) required
to ionize an arsenic n-type donor atom. The ionization is a single quantum-jump
event (a Franck“Hertz-type excitation) in which the hole gives up its entire energy
and comes to a complete halt. However, the halted hole is immediately accelerated
by the local electric ¬eld towards the right, so that the process repeats itself, i.e.
the hole again acquires an ionization energy of 54 meV, whereupon it ionizes another
local arsenic atom and comes to a complete halt, and so on. In this start-and-stop
manner, the hole generates a discrete, deterministic sequence of secondary electrons
in a well-controlled manner, as indicated in Fig. 9.4 by the electron vertices inside
the gain region. In this way, a sequence of leftwards-propagating secondary electrons
is emitted in regular, deterministic manner by the rightwards-propagating hole. Each
ionized arsenic atom thus releases a single secondary electron into the conduction
band, whereupon it is promptly accelerated to the left towards the interface between
the gain and intrinsic region. The secondary electrons enter the intrinsic region, where
they are collected, along with the primary electron, by the +V transparent electrode.
The result is a noise-free avalanche ampli¬cation process, whose gain is given by the
number of starts-and-stops of the hole inside the gain region. Measurements of the
noise factor, F ≡ M 2 / M 2 , where M is the multiplication factor, show that F =
1.00 ± 0.05 for M between 1 — 104 and 2 — 104 (Kim et al., 1997). This constitutes
direct experimental evidence that there is essentially no shot noise in the postdetection
electron multiplication process.
Note that this description of the noise-free ampli¬cation process depends on the
assumption that the motions of the holes and electrons are ballistic, i.e. they propagate
freely between collision events. Also, it is assumed that only holes have large enough
cross-sections to cause impact ionizations of the arsenic atoms. The resulting process is
non-Markovian, in the sense that there is a well-de¬ned, deterministic, nonstochastic
delay time between electron multiplication events. Note also that charge conservation
requires the number of electrons”collected by the transparent electrode on the left”
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.
An important advantage of heterodyne detection is that Shet (t) is linear in the local
— —
oscillator ¬eld EL and in the signal ¬eld Es (t). Thus a large value for |EL | e¬ectively
ampli¬es the contribution of the weak optical signal to the low-frequency heterodyne

signal. For instance, doubling the size of EL , doubles the size of the heterodyne signal
for a given signal amplitude Es . Furthermore, the relative phase between the linear
oscillator and the incident signal is faithfully preserved in the heterodyne signal. To
make this point more explicit, ¬rst rewrite eqn (9.103) as Shet (t) = F cos (ωIF t) +
G sin (ωIF t), where the Fourier components are given by

F = 2 Re [r— t EL Es ] , G = 2 Im [r— t EL Es ] .
— —
(9.104)

We use the Stokes relation (8.7), in the form

r— t = |r| |t| e±iπ/2 , (9.105)

to rewrite eqn (9.104) as
— —
F = ±2 |EL Es | |r| |t| sin (θL ’ θs ) , G = ±2 |EL Es | |r| |t| cos (θL ’ θs ) , (9.106)

where θL and θs are respectively the phases of the local oscillator EL and the signal
Es .
The quantities F and G can be separately measured. For example, F and G can
be simultaneously determined by means of the apparatus sketched in Fig. 9.7. Note
that the insertion of a 90—¦ phase shifter into one of the two local-oscillator arms
allows the measurement of both the sine and cosine components of the intermediate-
frequency signals at the two photon detectors. Each box labeled ˜IF mixer™ denotes the
combination of a radio-frequency oscillator”conventionally called a 2nd LO ” that
operates at the IF frequency, with two local radio-frequency diodes that mix the 2nd
LO signal with the two IF signals from the photon detectors. The net result is that
these IF mixers produce two DC output signals proportional to the IF amplitudes F
and G. The ratio of F and G is a direct measure of the phase di¬erence θL ’ θs relative
to the phase of the 2nd LO, since
F
= tan (θL ’ θs ) . (9.107)
G
The heterodyne signal corresponding to F is maximized when θL ’ θs = π/2 and
minimized when θL ’ θs = 0, whereas the heterodyne signal corresponding to G is
¾ Photon detection




Fig. 9.7 Schematic of an apparatus for two-quadrature heterodyne detection. The beam
splitters marked as ˜High trans™ have |t| ≈ 1.

maximized when θL ’ θs = 0 and minimized when θL ’ θs = π/2, where all the phases
are de¬ned relative to the 2nd LO phase. The optical phase information in the signal
waveform is therefore preserved through the entire heterodyne process, and is stored
in the ratio of F to G. This phase information is valuable for the measurement of
small optical time delays corresponding to small di¬erences in the times of arrival
of two optical wavefronts; for example, in the di¬erence in the times of arrival at
two telescopes of the wavefronts emanating from a single star. Such optical phase
information can be used for the measurement of stellar diameters in infrared stellar
interferometry with a carbon-dioxide laser as the local oscillator (Hale et al., 2000).
This is an extension of the technique of radio-astronomical interferometry to the mid-
infrared frequency range.
Examples of important heterodyne systems include: Schottky diode mixers in the
radio and microwave regions; superconductor“insulator“superconductor (SIS) mixers,
for radio astronomy in the millimeter-wave range; and optical heterodyne mixers,
using the carbon-dioxide lasers in combination with semiconductor photoconductors,
employed as square-law detectors in infrared stellar interferometry (Kraus, 1986).

9.3.2 Quantum analysis of heterodyne detection
Since the ¬eld operators are expressed in terms of classical mode functions and their
associated annihilation operators, we can retain the assumptions”i.e. plane waves, S-
polarization, etc.”employed in Section 9.3.1. This allows us to use a simpli¬ed form
of the general expression (8.28) for the in-¬eld operator to replace the classical ¬eld
(9.101) by the Heisenberg-picture operator

Ein (r, t) = ieL aL2 eikL y e’iωL t + ies as1 eiks x e’iωs t + Evac,in (r, t) ,
(+) (+)
(9.108)

where eM = ωM /2 0 V is the vacuum ¬‚uctuation ¬eld strength for a plane wave
with frequency ωM . This is an extension of the method used in Section 9.1.4 to model
¾
Heterodyne and homodyne detection

imperfect detectors. The annihilation operators aL2 and as1 respectively represent the
local oscillator ¬eld, entering through port 2, and the signal ¬eld, entering through
port 1; and, we have again assumed that the bandwidths of the signal and local os-
cillator ¬elds are small compared to ωIF . If this assumption has to be relaxed, then
the Schr¨dinger-picture annihilation operators must be replaced by slowly-varying en-
o
(+)
velope operators aL2 (t) and as1 (t). In principle, the operator Evac,in (r, t) includes
all modes other than the signal and local oscillator, but most of these terms will
not contribute in the subsequent calculations. According to the discussion in Section
8.4.1, each physical input ¬eld is necessarily paired with vacuum ¬‚uctuations of the
same frequency”indicated by the dashed arrows in Fig. 9.6”entering through the
(+)
other input port. Thus Evac,in (r, t) must include the operators aL1 and as2 describing
vacuum ¬‚uctuations with frequencies ωL and ωs entering through ports 1 and 2 respec-
tively. It should also include any other vacuum ¬‚uctuations that could combine with
the local oscillator to yield terms at the intermediate frequency, i.e. modes satisfying
ωM = ωL ±ωIF . The +-choice yields the signal frequency ωs , which is already included,

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