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and the corresponding out-¬eld

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-
-vac resents vacuum ¬‚uctuations (at the signal fre-
1 detector
quency) entering port 2, ED is the e¬ective sig-
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)

ED (r, t) = iE0s a1 eiks x e’iωs t
Elost (r, t) = iE0s a2 eiks y e’iωs t .
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

= ξ S E0s a† a1 ,

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
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
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
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
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)
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
V (N ) ’ N = N 2 ’ N ’N
= 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
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)

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.
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
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.


Laser beam
Fig. 9.2 Schematic of a laser beam incident Dynodes
upon a photomultiplier tube.
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

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
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
‚p (1, z)
= ’± (z) p (1, z) , (9.67)
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
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)
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)
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
p (1, z) = e , where ζ (z) = dz ± (z ) . (9.70)

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:

eζ(z) ’ 1
p (n, z) = . (9.71)

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).)
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
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”

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