2π

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

2π

—

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

dω

Sji (ω) Xij (ω, t) ,

w(1) (t) = 2 Re (9.23)

2π

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)

2π

(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)

2π

and get

sin2 [(ω ’ ω ) ∆T /2]

dω (1)

S (ω) = S G (r, ω ; t) . (9.43)

2

[(ω ’ ω ) /2] ∆T

2π

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

cω

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 ,