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”