ńņš. 11 |

interaction with the detectors only. The unperturbed Hamiltonian is H0 = HD +

Hem + H1 , where HD is the detector Hamiltonian and Hem is the ļ¬eld Hamiltonian.

The remaining term, H1 , describes the interaction of the ļ¬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

ļ¬eld. In the interaction picture, Hdet = ā’d (t) Ā· E (r, t) describes the interaction of the

ļ¬eld with the detector atom located at r. The initial state is |Ī˜ (t0 ) = |ĻĪ³ , Ī¦e =

|ĻĪ³ |Ī¦e , where |ĻĪ³ is the atomic ground state and |Ī¦e is the initial state of the

radiation ļ¬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

|Ī˜ (t) = |Ī˜ (t0 ) ā’ dt1 Hdet (t1 ) |Ī˜ (t0 ) + Ā· Ā· Ā· , (9.1)

t0

so the ļ¬rst-order probability amplitude that a joint measurement at time t ļ¬nds the

atom in an excited state |Ļ and the ļ¬eld in the number state |n is

t

i

Ļ , n |Ī˜ (t) = ā’ dt1 Ļ , n |Hdet (t1 )| Ī˜ (t0 ) , (9.2)

t0

where |Ļ , n = |Ļ |n . Only the Rabi operator ā„¦ (+) in eqn (4.149) can contribute

to an absorptive transition, so the matrix element and the probability amplitude are

respectively given by

Ā¾

Primary photon detection

Ļ , n |Hdet (t1 )| Ī˜ (t0 ) = ā’eiĻ Ā· n E(+) (r, t1 ) Ī¦e

Ī³ t1

d (9.3)

Ī³

and

t

i

Ļ , n |Ī˜ (t) = Ā· n E(+) (r, t1 ) Ī¦e ,

dt1 eiĻ Ī³ t1

d (9.4)

Ī³

t0

where d Ī³ = Ļ d ĻĪ³ is the dipole matrix element for the transition Ī³ ā’ .

The conditional probability for ļ¬nding |Ļ , n , given |ĻĪ³ , Ī¦e , is therefore

2

t

i

Ā· nE

iĻ Ī³ t1 (+)

p (Ļ , n : ĻĪ³ , Ī¦e ) = dt1 e d (r, t1 ) Ī¦e

Ī³

t0

dā—Ī³ t t

(d Ī³ )j

Ī³ (t2 ā’t1 )

i

dt2 eiĻ

= dt1

2

t0 t0

ā—

(+) (+)

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

ā—

(+) (ā’)

The relation E(ā’) = E(+)ā implies n Ei (r, t1 ) Ī¦e = Ī¦e Ei (r, t1 ) n , so that

eqn (9.5) can be rewritten as

dā—Ī³ t t

(d Ī³ )j

Ī³ (t2 ā’t1 )

i

dt2 eiĻ

p (Ļ , n : ĻĪ³ , Ī¦e ) = dt1

2

t0 t0

(ā’) (+)

Ć— Ī¦e Ei (r, t1 ) n n Ej (r, t2 ) Ī¦e . (9.6)

Since the ļ¬nal state of the radiation ļ¬eld is not usually observed, the relevant quantity

is the sum of the conditional probabilities p (Ļ , n : ĻĪ³ , Ī¦e ) over all ļ¬nal ļ¬eld states

|n :

p (Ļ : ĻĪ³ , Ī¦e ) = p (Ļ , n : ĻĪ³ , Ī¦e ) . (9.7)

n

The completeness identity (3.67) for the number states, combined with eqn (9.6) and

eqn (9.7), then yields

dā—Ī³ t t

(d Ī³ )j

Ī³ (t2 ā’t1 )

i

dt2 eiĻ

p (Ļ : ĻĪ³ , Ī¦e ) = dt1

2

t0 t0

(ā’) (+)

Ć— Ī¦e Ei (r, t1 ) Ej (r, t2 ) Ī¦e . (9.8)

This result is valid when the radiation ļ¬eld is known to be initially in the pure state

|Ī¦e . In most experiments all that is known is a probability distribution Pe over an

ensemble {|Ī¦e } of pure initial states, so it is necessary to average over this ensemble

to get

p (Ļ : ĻĪ³ , Ī¦e ) Pe

p (Ļ : ĻĪ³ ) =

e

dā—Ī³ t t

(d Ī³ )j

Ī³ (t2 ā’t1 ) (ā’) (+)

i

dt2 eiĻ

= dt1 Tr ĻEi (r, t1 ) Ej (r, t2 ) ,

2

t0 t0

(9.9)

Ā¾Ā¼ Photon detection

where

Pe |Ī¦e Ī¦e |

Ļ= (9.10)

e

is the density operator deļ¬ned by the distribution Pe .

So far it has been assumed that the ļ¬nal atomic state |Ļ can be detected with

perfect accuracy, but of course this is never the case. Furthermore, most detection

schemes do not depend on a speciļ¬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 ļ¬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 Ī³ ā’

occurs and produces a macroscopically observable event, e.g. a current pulse. The

overall probability is then

D ( ) p (Ļ : ĻĪ³ ) .

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 ā’ t2 ) Gij (r, t1 ; r, t2 ) ,

p (t) = dt1 (9.12)

t0 t0

where the sensitivity function

1

D ( ) dā—Ī³ (d Ī³ )j eā’iĻ

Sji (t) = Ī³t

(9.13)

2 i

is determined solely by the properties of the atom, and the ļ¬eldā“ļ¬eld correlation

function

(1) (ā’) (+)

Gij (r, t1 ; r, t2 ) = Tr ĻEi (r1 , t1 ) Ej (r, t2 ) (9.14)

is determined solely by the properties of the ļ¬eld.

Since D ( ) is real and positive, the sensitivity function obeys

Sā— (t) = Sij (ā’t) , (9.15)

ji

and other useful properties are found by studying the Fourier transform

Sji (Ļ) = dtSji (t) eiĻt

2Ļ

D ( ) dā—Ī³ (d Ī³ )j Ī“ (Ļ ā’ Ļ Ī³ ) .

= (9.16)

2 i

The -sum is really an integral over the continuum of excited states, so Sji (Ļ) is a

smooth function of Ļ. This explicit expression shows that the 3 Ć— 3 matrix S (Ļ),

Ā¾Ā½

Primary photon detection

ā—

with components Sji (Ļ), is hermitianā”i.e. Sji (Ļ) = Sij (Ļ) ā”and positive-deļ¬nite,

since

2Ļ

ā—

D ( ) |vā— Ā· d Ī³ |2 Ī“ (Ļ ā’ Ļ Ī³ ) > 0

vj Sji (Ļ) 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 (Ļ) = Tr [S (Ļ)] , (9.18)

of the dipole transitions can be used to deļ¬ne averages over frequency by

dĻT (Ļ) f (Ļ)

f = . (9.19)

T

dĻT (Ļ)

The width āĻS of the sensitivity function is then deļ¬ned as the rms deviation

2

ā’Ļ

Ļ2

āĻS = . (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 ā’ t) Gij (r, t ; r, t) ,

(1)

w (t) = = 2 Re (9.21)

dt t0

where the ļ¬nal form comes from combining eqn (9.15) with the symmetry property

(1)ā— (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

dĻ

Sji (Ļ) Xij (Ļ, t) ,

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

2Ļ

where

t

dt eiĻ(tā’t ) Gij (r, t ; r, t) .

(1)

Xij (Ļ, 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 (Ļ, t), considered as a function of Ļ with t ļ¬xed. One

way to get this information is to use eqn (9.24) to evaluate the transform

dĻ iĻt

Xij (t , t) = e Xij (Ļ, t)

2Ļ

(1)

= Īø (t ) Īø (t ā’ t0 ā’ t ) Gij (r, t ā’ t ; r, t) . (9.25)

The step functions in this expression guarantee that Xij (t , t) vanishes outside the

t ā’ t0 . On the other hand, the correlation function vanishes for

interval 0 t

Tc , where Tc is the correlation time. The observation time t ā’ 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 Ļ-width of Xij (Ļ, t) is āĻX ā¼ 1/Tc = āĻG ,

(1)

where āĻG is the bandwidth of the correlation function Gij .

Ā¾Ā¾ Photon detection

B Broadband detection

The detector is said to be broadband if the bandwidth āĻS of the sensitivity function

satisļ¬es āĻS āĻG = 1/Tc. For a broadband detector, Xij (Ļ) is sharply peaked

compared to the sensitivity function; therefore, Sji (Ļ) can be treated as a constantā”

Sji (Ļ) ā Sji ā”and taken outside the integral. This is formally equivalent to setting

Sji (t ā’ t) = Sji Ī“ (t ā’ 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 Fermiā™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 satisļ¬ed: (1) single-atom absorption is the dominant

process; (2) interactions between the atoms can be ignored. These conditions will be

satisļ¬ed for atoms in a tenuous vapor or in an atomic beamā”see item (5) in Section

9.1.1ā”and they are also satisļ¬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 deļ¬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 modiļ¬ed sensitivity factor.

The sensitivity factor, deļ¬ned by eqn (9.16), is a 3 Ć— 3 hermitian matrix which has

the useful representation

3

Sa eai eā— ,

Sij = (9.30)

aj

a=1

where the eigenvalues, Sa , are real and the eigenvectors, ea , are orthonormal: eā— Ā· ea =

b

Ī“ab . Substituting this representation into eqn (9.26) produces

Ā¾Āæ

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 ĻEa (r, t) Ea (r, t) ,

(ā’) (+)

(9.32)

a

(ā’)

are deļ¬ned in terms of the scalar ļ¬eld operators Ea (r, t) = ea Ā· E(ā’) (r, t). This

form is useful for imposing special conditions on the detector. For example, a detector

equipped with a polarization ļ¬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 ļ¬lter. In this situation, eqn (9.29) becomes

w(1) (t) = S G(1) (r, t; r, t)

(ā’) (+)

= S Tr ĻE1 (r, t) E1 (r, t) , (9.33)

where E1 (r, t) = eā— Ā· 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 diļ¬erent frequencies that

may be contained in the incident ļ¬eld, so they do not determine the spectral func-

tion of the ļ¬eld. For this purpose, one needs narrowband detection, which is

usually achieved by passing the light through a narrowband ļ¬lter before it falls

on a broadband detector. The ļ¬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 (ā’) (t), the ļ¬ltered signal at Ļā”i.e. the part of the signal

corresponding to a narrow band of frequencies around Ļā”is deļ¬ned by

ā

(t ā’ t) eiĻ(t ā’t) X (+) (t )

(+)

X (Ļ; t) = dt

ā’ā

ā

(t ) eiĻt X (+) (t + t) ,

= dt (9.34)

ā’ā

where the factor exp [iĻ (t ā’ t)] serves to pick out the desired frequency. The weighting

function (t) has the following properties.

(1) It is even and positive,

(t) = (ā’t) 0 . (9.35)

(2) It is normalized by

ā

dt (t) = 1 . (9.36)

ā’ā

Ā¾ Photon detection

(3) It is peaked at t = 0.

The weighting function is therefore suitable for deļ¬ning averages, e.g. the tempo-

ral width āT :

ā 1/2

< ā.

2

āT = dt (t) t (9.37)

ā’ā

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

for ā’ āT

1 āT

t ,

āT 2 2 (9.38)

(t) =

0 otherwise .

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

X (+) (Ļ ; Ļ) = F (Ļ ā’ Ļ) X (+) (Ļ) , (9.39)

where the ļ¬lter function F (Ļ) is the Fourier transform of (t). Since the normal-

ization condition (9.36) implies F (0) = 1, the ļ¬ltered signal is essentially identical to

the original signal in the narrow band deļ¬ned by the width āĻF ā¼ 1/āT of the ļ¬lter

function; but, it is strongly suppressed outside this band.

The frequency Ļ selected by the ļ¬lter varies continuously, so the interesting quan-

tity is the spectral density S (Ļ), which is deļ¬ned as the counting rate per unit

frequency interval. Applying the broadband result (9.33) to the ļ¬ltered ļ¬eld operators

yields

w(1) (Ļ, t)

S (Ļ, t) =

āĻF

S (ā’) (+)

= E1 (r, t; Ļ) E1 (r, t; Ļ) . (9.40)

āĻF

For the following argument, we choose the simple form (9.38) for the averaging function

to calculate the ļ¬ltered operator:

āT /2

1

(+) (+)

dt eiĻt E1

E1 (r, t; Ļ) = (r, t + t) . (9.41a)

āT ā’āT /2

Substituting this result into eqn (9.40) and combining āĻF = 1/āT with the deļ¬nition

of the ļ¬rst-order correlation function yields

S āT /2 āT /2

dt2 eiĻ(t1 ā’t2 ) G(1) (r, t2 + t; r, t1 + t) .

S (Ļ, t) = dt1 (9.41b)

āT ā’āT /2 ā’āT /2

In almost all applications, we can assume that the correlation function only depends

on the diļ¬erence in the time arguments. This assumption is rigorously valid if the

Ā¾

Primary photon detection

density operator Ļ is stationary, and for dissipative systems it is approximately satisļ¬ed

for large t. Given this property, we set

dĻ (1)

G (r, Ļ ; t) eā’iĻ (t1 ā’t2 )

G(1) (r, t2 + t; r, t1 + t) = , (9.42)

2Ļ

and get

sin2 [(Ļ ā’ Ļ ) āT /2]

dĻ (1)

S (Ļ) = S G (r, Ļ ; t) . (9.43)

2

[(Ļ ā’ Ļ ) /2] āT

2Ļ

In this case, the width of the ļ¬lter is assumed to be very small compared to the width

of the correlation function, i.e. āĻS āĻG (āT Tc ). By means of the general

identity (A.102), one can show that

sin2 [Ī½āT /2]

lim = ĻĪ“ (Ī½/2) = 2ĻĪ“ (Ī½) , (9.44)

2

āT ā’ā [Ī½/2] āT

and substituting this result into eqn (9.43) leads to

dĻ„ eā’iĻĻ„ G(1) (r, Ļ„ + t; r, t) .

S (Ļ) = SG(1) (r, Ļ; t) = S (9.45)

In other words, the spectral density is proportional to the Fourier transform, with

respect to the diļ¬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 ļ¬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 diļ¬erent frequency components. One

technique is to use a diļ¬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 ļ¬xed. The spectrometer thus acts as a continuously tunable ļ¬lter, with

bandwidth determined by the width of the slits. Higher resolution can be achieved

by using a Fabryā“Perot spectrometer (Hecht, 2002, Secs 9.6.1 and 9.7.3) with an

adjustable spacing between the plates. A diļ¬erent approach is to use a heterodyne

spectrometer, in which the signal is mixed with a local oscillatorā”usually a laserā”

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 ļ¬eld, such as the Poissonian

statistics predicted for the coherent state |Ī± ? 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.

Ā¾ Photon detection

Consider a light beamā”produced, for example, by passing the output of a laser

operating far above threshold through an attenuatorā”that falls on the photocathode

surface of a photomultiplier tube. The amplitude of the attenuated coherent state

is Ī± = exp(ā’ĻL/2)Ī±0 , where Ļ is the absorption coeļ¬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, Ī¾) of the detection of m photoelectrons found in this

way is

ā

nm nā’m

Ī¾ (1 ā’ Ī¾)

P (m, Ī¾) = p(n) , (9.46)

m

n=m

where Ī¾ is the probability that the interaction of a given photon with the atoms in the

detector will produce a photoelectron. This quantityā”which is called the quantum

eļ¬ciencyā”is given by

cĻ

Ī¾=Ī¶ T, (9.47)

V

where Ī¶ (which is proportional to the sensitivity function S) is the photoelectron

ejection probability per unit time per unit light intensity, (c Ļ/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 Ī¶

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

coeļ¬cient m (the number of ways of distributing n photons among m photoelectron

ejections); the probability Ī¾ m that m photons are converted into photoelectrons; and

nā’m

the probability (1 ā’ Ī¾) that the remaining n ā’ m photons are not detected at all.

One can showā”see Exercise 9.1ā”that a Poissonian initial photon distribution, with

average photon number n, results in a Poissonian photoelectron distribution,

m m ā’m

P (m, Ī¾) = e , (9.48)

m!

where m = Ī¾ n is the average ejected photoelectron number. In the special case Ī¾ = 1,

there is a oneā“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 heliumā“neon laser operating

far above threshold is described in Section 5.3.2.

Quantum eļ¬ciencyā—

9.1.4

The quantum eļ¬ciency Ī¾ introduced in eqn (9.47) is a phenomenological parameter

that can represent any of a number of possible failure modes in photon detection:

reļ¬‚ection from the front surface of a cathode; a mismatch between the transverse

Ā¾

Primary photon detection

proļ¬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

ļ¬‚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 eļ¬ciency, as shown in Fig. 9.1.

The transmission and reļ¬‚ection coeļ¬cients of the ļ¬ctitious beam splitter must

be adjusted to obey the unitarity condition (8.7) and to account for the quantum

eļ¬ciency of the real detector. These requirements are satisļ¬ed by setting

t= Ī¾, r= i 1ā’Ī¾. (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 Ļ that is the vacuum for

all modes other than the signal mode. In this case we can specialize eqn (8.28) for the

in-ļ¬eld to

Ein (r, t) = iE0s a1 eiks x eā’iĻs t + Evac,in (r, t) ,

(+) (+)

(9.50)

where we have chosen the x- and y-axes along the 1 ā’ 1 and 2 ā’ 2 arms of the

device respectively, E0s = Ļs /2 0 V is the vacuum ļ¬‚uctuation ļ¬eld strength for a

plane wave with frequency Ļs , and a1 is the annihilation operator for the plane-wave

(+)

mode exp [i (ks x ā’ Ļs 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 ā’ Ļs t)] entering through port 2. This

leaves us with the simpliļ¬ed in-ļ¬eld

Ein (r, t) = iE0s a1 eiks x eā’iĻs t + iE0s a2 eiks y eā’iĻs t .

(+)

(9.51)

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

Ī¾a1 + i 1 ā’ Ī¾a2 ,

a1 =

(9.52)

1 ā’ Ī¾a1 +

a2 = i Ī¾a2 ,

and the corresponding out-ļ¬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 ļ¬‚uctuations (at the signal fre-

1 detector

quency) entering port 2, ED is the eļ¬ective sig-

2

Beam nal entering the detector, and Elost describes

splitter the part of the signal lost due to ineļ¬ciencies.

Ā¾ Photon detection

(+) (+) (+)

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

where

ED (r, t) = iE0s a1 eiks x eā’iĻs t

(+)

(9.54)

and

Elost (r, t) = iE0s a2 eiks y eā’iĻs t .

(+)

(9.55)

The counting rate of the imperfect detector is by deļ¬nition the counting rate of the

perfect detector viewing port 1 of the beam splitter, soā”for the simple case of a

broadband detectorā”eqn (9.33) gives

(ā’) (+)

w(1) (t) = S ED (rD , t) ED (rD , t)

= S E0s a1ā a1

2

= Ī¾ S E0s aā a1 ,

2

(9.56)

1

where (Ā· Ā· Ā· ) = Tr [Ļ (Ā· Ā· Ā· )], rD is the location of the detector, and we have used

a2 Ļ = Ļaā = 0. The operator Elost represents the part of the signal lost by scattering

(+)

2

into the 2 ā’ 2 channel.

As expected, the counting rate of the imperfect detector is reduced by the quantum

eļ¬ciency Ī¾; and the vacuum ļ¬‚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 ļ¬‚uctuations will add to the variance of the scattered number operator

N1 = a1ā 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 ) = Ī¾ 2 V (N1 ) + Ī¾ (1 ā’ Ī¾) N1 . (9.57)

The ļ¬rst term on the right represents the variance in photon number for the inci-

dent ļ¬eld, reduced by the square of the quantum eļ¬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 Ī¾ or no

click with probability 1 ā’ Ī¾.

The Mandel Q-parameter

9.1.5

Most photon detectors are based on the photoelectric eļ¬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 ļ¬eld operators. In the example of a single

mode, this leads to averages of normal-ordered products of the general form aā n an .

As seen in Section 5.6.3, the most useful quasi-probability distribution for the de-

scription of such measurements is the Glauberā“Sudarshan function P (Ī±). If this dis-

tribution function is non-negative everywhere on the complex Ī±-plane, then there is

a classical modelā”described by stochastic c-number phasors Ī± with the same P (Ī±)

distributionā”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

Ā¾

Primary photon detection

photoelectric eļ¬ect can distinguish between a quantum state and a classical stochastic

model that share the same P (Ī±) distribution.

Direct experimental veriļ¬cation of the condition P (Ī±) 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 = aā a, where (Ā· Ā· Ā· ) = Tr [Ļ (Ā· Ā· Ā· )], and Ļ

is the density operator for the state under consideration. We will study the second

2

moment, or rather the variance, V (N ) = N 2 ā’ N , which is a measure of the

noise in the light. In Section 5.1.3 we found that a coherent state Ļ = |Ī± Ī±| exhibits

Poissonian statistics, i.e. for a coherent state the variance in photon number is equal

to the average number: V (N ) = N 2 ā’ 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 Ļ 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 ) ā’ N

Qā” , (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 deļ¬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 Ļ. 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

ā’1 Q < ā. (9.59)

A very useful property of the Q-parameter can be derived by ļ¬rst expressing the

numerator in eqn (9.58) as

2

V (N ) ā’ N = N 2 ā’ N ā’N

2

= aā 2 a2 ā’ aā a , (9.60)

where the last line follows from another application of the commutation relations

a, aā = 1. Since all the operators are now in normal-ordered form, we may use the

P -representation (5.168) to get

2

d2 Ī± 4 d2 Ī± 2

V (N ) ā’ N = |Ī±| P (Ī±) ā’ |Ī±| P (Ī±) . (9.61)

Ļ Ļ

By using the fact that P (Ī±) is normalized to unity, the ļ¬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.

Ā¾Ā¼ Photon detection

d2 Ī± 4 d2 Ī±

V (N ) ā’ N = |Ī±| P (Ī±) P (Ī± )

Ļ Ļ

d2 Ī± 2 d2 Ī± 2

ā’ |Ī±| P (Ī±) |Ī± | P (Ī± ) . (9.62)

Ļ Ļ

The ļ¬nal step is to interchange the dummy integration variables Ī± and Ī± in the ļ¬rst

term, and then to average the two equivalent expressions; this yields the ļ¬nal result:

d2 Ī± d2 Ī± 22

1 2

V (N ) ā’ N = |Ī±| ā’ |Ī± | P (Ī±) P (Ī± ) . (9.63)

2 Ļ Ļ

The right side is positive for P (Ī±) 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 suļ¬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 ļ¬‚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 ļ¬‚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 deļ¬ned by using the total

number operator,

aā aM ,

N= (9.64)

M

M

in eqn (9.58). The deļ¬nition of a classical state is P (Ī±) 0, where P (Ī±) is the

multimode P -function deļ¬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 ļ¬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 reļ¬‚ecting walls. The atom can absorb the photon and enter an

excited state, butā”with equal facilityā”the excited atom can return to the ground

state by emitting the photon. The photonā”none the worse for its adventureā”can

then initiate the process again. We will see in Chapter 12 that this dance can go

on indeļ¬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.

Ā¾Ā½

Postdetection signal processing

The electron can be excited to the conduction bandā”leaving a hole in the valence

bandā”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 = Ī± (t) |photon |valence-band-electron

+ Ī² (t) |vacuum |electronā“hole-pair

= Ī± (t) |photon-not-detected + Ī² (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 deļ¬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 beā”indeed it has

beenā”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 SchrĀØdingerā™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 deļ¬nitely dead

or deļ¬nitely aliveā”and not some spooky superposition of |dead cat and |live cat ā”

we can also be assured that an irreversible interaction with a macroscopic system will

yield a deļ¬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 ampliļ¬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 processingā”which bring quantum measurements to a

close by processes involving irreversible ampliļ¬cationā”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 intensiļ¬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, ampliļ¬cation is required to raise the micro-

scopic energy released in the primary photodetection eventā” Ļ ā¼ 10ā’19 J for a typical

Ā¾Ā¾ Photon detection

visible photonā”to a macroscopic value much larger than the typical thermal noiseā”

kB T ā¼ 10ā’20 Jā”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 ļ¬rst-stage electronic ampliļ¬cation process. The

typical electron multiplication factor in these postdetection mechanisms is between

104 to 106 .

One ampliļ¬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 ļ¬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 ļ¬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 intensiļ¬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

ļ¬eld applied along the length of each channel induces electron multiplication on the

interior surface, which is coated with a thin, conducting ļ¬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 electronā“hole pair inside the solid-state

material, as shown in Fig. 9.3. When a static electric ļ¬eld is applied, the initial electron

and hole are accelerated in opposite directions, in the so-called Geiger mode of

operation. For a suļ¬ciently large ļ¬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 pulseā”like the single click of a Geiger counterā”that signals the arrival of

a single photon. In this strong-ļ¬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 deļ¬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.

Ā¾Āæ

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 electronā“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 ļ¬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 + āz) is

Ī± (z) āz, where the gain, Ī± (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 + āz) is 1ā’Ī± (z) āz. Thus the probability of failure in the next z-interval

is

p (1, z + āz) = (1 ā’ Ī± (z) āz) p (1, z) . (9.66)

Take the limit āz ā’ 0, or Taylor-series expand the left side, to get the diļ¬erential

equation

ā‚p (1, z)

= ā’Ī± (z) p (1, z) , (9.67)

ā‚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 + āz could come from n ā’ k carriers at z by production of k carriers, where

k = 0, 1, . . . , n ā’ 1. Adding up the possible processes gives

n

p (n, z + āz) = (1 ā’ Ī± (z) āz) p (n, z) + (n ā’ 1) (Ī± (z) āz) p (n ā’ 1, z)

1 2

+ (n ā’ 2) (n ā’ 3) (Ī± (z) āz) p (n ā’ 2, z) + Ā· Ā· Ā· . (9.68)

2

In the limit of small āz this leads to the diļ¬erential equation

ā‚p (n, z)

= ā’nĪ± (z) p (n, z) + (n ā’ 1) Ī± (z) p (n ā’ 1, z) , (9.69)

ā‚z

Ā¾ 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

ā’Ī¶(z)

p (1, z) = e , where Ī¶ (z) = dz Ī± (z ) . (9.70)

0

The recursive system of diļ¬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:

nā’1

eĪ¶(z) ā’ 1

p (n, z) = . (9.71)

enĪ¶(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 ampliļ¬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 diļ¬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

oddā“even eļ¬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).)

Ā¾

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 electronā“

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 ļ¬eld in the gain region is larger than in the intrinsic region, due to

the diļ¬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 ļ¬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-reļ¬‚ection coating applied to the transparent electrode

ensures that the incoming photon is admitted with high eļ¬ciency into the interior of

the silicon intrinsic region, so that the quantum eļ¬ciency of the device can be quite

high.

The primary hole propagates to the right and enters the gain region, whereupon

the higher electric ļ¬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 Franckā“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 ļ¬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 ampliļ¬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 ā” M 2 / M 2 , where M is the multiplication factor, show that F =

1.00 Ā± 0.05 for M between 1 Ć— 104 and 2 Ć— 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 ampliļ¬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-deļ¬ned, deterministic, nonstochastic

delay time between electron multiplication events. Note also that charge conservation

requires the number of electronsā”collected by the transparent electrode on the leftā”

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

electrode, labeled as the contact region and degenerate substrate.

Ā¾ 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 deviceā”the coincidence

counterā”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 ļ¬lters. This means that the

detectors can be treated as though they were single atoms, with the understanding

that the locations of the ā˜atomsā™ are to be treated classically. The detector Hamiltonian

is then

2

Hdet (t) = Hdn (t) , (9.72)

n=1

Hdn (t) = ā’ dn (t) Ā· en En (t) , (9.73)

where rn , dn , en , and En are respectively the location; the dipole operator; the po-

larization admitted by the ļ¬lter; and the corresponding ļ¬eld component

En (t) = en Ā· 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 ļ¬eld is described by a density matrix,

i.e. an ensemble of pure states, we can begin by assuming that the radiation ļ¬eld is

described a pure state |Ī¦e and that both atoms are in the ground state. The initial

state of the total system is then

|Ī˜i = |ĻĪ³ , ĻĪ³ , Ī¦e = |ĻĪ³ (1) |ĻĪ³ (2) |Ī¦e , (9.75)

where |ĻĪ³ (n) denotes the ground state of the atom located at rn . For coincidence

counting, it is suļ¬cient to consider the ļ¬nal states,

|Ī˜f = |Ļ 1 , Ļ 2 , n = |Ļ 1 (1) |Ļ 2 (2) |n , (9.76)

where |Ļ (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 = Ī˜f |V (t)| Ī˜i = Ī“f i + Ī˜f V (1) (t) Ī˜i + Ī˜f V (2) (t) Ī˜i + Ā· Ā· Ā· , (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 = ā’ dt2 Ī˜f |Hd1 (t1 ) Hd2 (t2 ) + Hd2 (t1 ) Hd1 (t2 )| Ī˜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

Ā¾

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 = ā’ dt2 Ī˜f |Hd1 (t1 ) Hd2 (t2 )| Ī˜i .

dt1 (9.79)

t0 t0

Finally, substituting the explicit expression (9.73) for the interaction Hamiltonian

yields

2 t t

i

Af i = ā’ d d dt1 dt2 exp (iĻ t1 ) exp (iĻ t2 )

1Ī³ 2Ī³ 1Ī³ 2Ī³

t0 t0

(+) (+)

Ć— n E1 (t1 ) E2 (t2 ) Ī¦e , (9.80)

where we have used the relation between the interaction and SchrĀØdinger pictures to

o

get

dn (t) Ā· en ĻĪ³ = exp (iĻ dn Ā· en ĻĪ³ = exp (iĻ

Ļ t1 ) Ļ t1 ) d . (9.81)

1Ī³ 1Ī³ nĪ³

n n

In a coincidence-counting experiment, the ļ¬nal states of the atoms and the radia-

2

tion ļ¬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

deļ¬ning the initial state Ļ of the radiation ļ¬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 ā’ t1 ) S2 (t2 ā’ t2 )

p (t, t0 ) = dt1 dt2 dt1

t0 t0 t0 t0

Ć— G(2) (r1 , t1 , r2 , t2 ; r1 , t1 , r2 , t2 ) , (9.83)

where the sensitivity functions are deļ¬ned by

1 2

Sn (t) = Dn ( ) |d Ā· en | eiĻ Ī³t

(n = 1, 2)

Ī³

2

= eā— enj Snij (t) , (9.84)

ni

and G(2) is a special case of the scalar second-order correlation function deļ¬ned by

eqn (4.77). The assumption that the detectors are broadband allows us to set Sn (t) =

Sn Ī“ (t) , and thus simplify eqn (9.83) to

t t

dt2 p(2) (t1 , t2 ) ,

p (t) = dt1 (9.85)

t0 t0

Ā¾ 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 diļ¬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 ) ā’ (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) = dĻ„ p(2) (t1 , t1 + Ļ„ )

T12

T12 +Tgate

= S1 S 2 dĻ„ G(2) (r1 , t1 , r2 , t1 + Ļ„ ; r1 , t1 , r2 , t1 + Ļ„ ) , (9.89)

T12

where T12 = T1 ā’ T2 is the oļ¬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.

Ā¾

Postdetection signal processing

For the photon indivisibility experiment, we consider a general one-photon input

state Ļ, i.e. the only condition is N Ļ = ĻN = Ļ, where N is the total number operator.

Any one-photon density operator Ļ can be expressed in the form

|1Īŗ ĻĪŗĪ» 1Ī» | ,

Ļ= (9.90)

Īŗ,Ī»

where Īŗ and Ī» are mode labels. The identity aĪŗ aĪ» Ļ = 0 = Ļaā aā ā”which holds for any

Ī»Īŗ

pair of annihilation operatorsā”implies that

(ā’) (ā’) (+) (+)

ĻE2 (r2 , t2 ) E1 (r1 , t1 ) = 0 = E1 (r1 , t1 ) E2 (r2 , t2 ) Ļ . (9.91)

The coincidence count rate is determined by the second-order correlation function

(ā’) (ā’)

G(2) (r2 , t2 , r1 , t1 ; r2 , t2 , r1 , t1 ) = Tr ĻE2 (r2 , t2 ) E1 (r1 , t1 )

(+) (+)

Ć— 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 ) ā” 0 . (9.93)

The zero coincidence rate in the photon indivisibility experiment is an immediate

consequence of this result.

The diļ¬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 deļ¬ned by eqn (9.92). In addition, the original

plane-wave modes will be replaced by general wave packets wĪŗ (r). The ļ¬eld operator

produced by scattering from the beam splitter can then be written as

ĻĪŗ ā’iĻĪŗ t

E(+) (r, t) = i e wĪŗ (r) aĪŗ . (9.94)

20

Īŗ

(2)

Substituting this expansion into the general deļ¬nition (4.75) for Gijkl yields

2

ā ā— ā—

(2)

({x} ; {x}) =

Gijkl ĻĀµ ĻĪŗ ĻĪ» ĻĪ½ wĀµi (r ) wĪŗj (r) wĪ»k (r) wĪ½l (r )

2 0

ĀµĪŗĪ»Ī½

Ć— ei(ĻĀµ ā’ĻĪ½ )t ei(ĻĪŗ ā’ĻĪ» )t Tr ĻaĀµ aĪŗ aĪ» aĪ½ ,

ā ā

(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 ļ¬eld is not the beam splitter, but

rather the collimators attached to the detectors. The ļ¬eld scattered from the beam

splitter is further scattered, or rather ļ¬ltered, by the collimators. To be completely

precise, we should work out the scattering matrix for the collimator and use eqn (9.94)

Ā¾Ā¼ Photon detection

as the input ļ¬eld. In practice, this is rarely necessary, since the eļ¬ect of these ļ¬lters is

well approximated by simply omitting the excluded terms when the ļ¬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:

wĪŗ (rn ) = 0 if wĪŗ 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 proļ¬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 ) ā 0 and w1 (r2 ) ā 0. In other words, the

beam w2 misses detector D1 and w1 misses detector D2 . This argument justiļ¬es the

rule (9.96) even if the collimators are ignored.

For the initial state, Ļ = |Ī¦in Ī¦in |, with |Ī¦in = aā aā |0 , each mode sum in eqn

21

(9.95) is restricted to the values Īŗ = 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

a1ā and a2ā with a1 and a2 . Imposing eqn (9.96) reduces this to one term, so that

2

Ļ

|w2 (r2 )|2 |w1 (r1 )|2 Ī¦in a2ā a1ā a1 a2 Ī¦in ,

({x} ; {x}) =

(2)

G (9.97)

2 0

where Ļ2 = Ļ1 = Ļ. 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 = Ā±i |t| gives the coincidence-counting rate

2 2

Ļ 2 2 2 2

= S2 S1 Tgate |w2 (r2 )| |w1 (r1 )| |r| ā’ |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 ļ¬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

Ā¾Ā½

Heterodyne and homodyne detection

heterodyne detection in which the signal and the local oscillator have the same fre-

quency, ĻL = Ļs . 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 diļ¬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

tomographyā”described in Chapter 17ā”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) eā’iĻL t + CC , (9.99)

called the local oscillator (LO), and a weak monochromatic wave,

Es (r, t) = Es (t) ws (r) eā’iĻs 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 ļ¬‚uctuations.

Ā¾Ā¾ Photon detection

called the signal, where EL (t) and Es (t) are slowly-varying envelope functions. The

two waves are mixed at a beam splitterā”as shown in Fig. 9.6ā”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

ā

we will idealize them as S-polarized plane waves, e.g. wL = e exp (ikL y) / V and

ā

ws = e exp (iks y) / V , where V is the quantization volume and e is the common

polarization vector. Since the output ļ¬elds will also be S-polarized, the polarization

vector will be omitted from the following discussion. The two incident waves have

diļ¬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) ā ei(kL xā’ĻL t) + Es (t) ā ei(ks xā’Ļs 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 coeļ¬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| ā 1, which in turn

implies |r| 1. The second condition means that only a small fraction of the local

oscillator ļ¬eld is reļ¬‚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: ĻIF ā” Ļs ā’ ĻL . In optical applications

the local oscillator ļ¬eld is usually generated by a laser, with ĻL ā¼ 1015 Hz, but ĻIF

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 ĻIF . This assumption allows

us to treat the envelope ļ¬elds as constants.

1/ |ĻIF |.

In this context, a fast detector is deļ¬ned by the conditions 1/ĻL TD

This inequality, together with the strong-ļ¬eld condition |EL | |Es |, allows the time

average over TD to be approximated by

TD /2

1

dĻ„ |E D (r, t + Ļ„ )| ā |EL | + 2 Re EL Es eā’iĻIF t + Ā· Ā· Ā· .

ā—

2

2

(9.102)

TD ā’TD /2

The large ļ¬rst term |EL |2 can safely be ignored, since it represents a DC current

signal which is easily ļ¬ltered out by means of a high-pass, radio-frequency ļ¬lter. The

photocurrent from the detector is then dominated by the heterodyne signal

Shet (t) = 2 Re rā— t EL Es eā’iĻIF t ,

ā—

(9.103)

Ā¾Āæ

Heterodyne and homodyne detection

which describes the beat signal between the LO and the signal wave at the intermedi-

ate frequency ĻIF . 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 ; consequentlyā”as can be seen in Exercise 9.4ā”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

ā— ā—

oscillator ļ¬eld EL and in the signal ļ¬eld Es (t). Thus a large value for |EL | eļ¬ectively

ampliļ¬es the contribution of the weak optical signal to the low-frequency heterodyne

ā—

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, ļ¬rst rewrite eqn (9.103) as Shet (t) = F cos (ĻIF t) +

G sin (ĻIF t), where the Fourier components are given by

F = 2 Re [rā— t EL Es ] , G = 2 Im [rā— t EL Es ] .

ā— ā—

(9.104)

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

rā— t = |r| |t| eĀ±iĻ/2 , (9.105)

to rewrite eqn (9.104) as

ā— ā—

F = Ā±2 |EL Es | |r| |t| sin (ĪøL ā’ Īøs ) , G = Ā±2 |EL Es | |r| |t| cos (ĪøL ā’ Īøs ) , (9.106)

where ĪøL and Īøs 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 90ā—¦ 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 ā˜IF mixerā™ denotes the

combination of a radio-frequency oscillatorā”conventionally called a 2nd LO ā” 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 diļ¬erence ĪøL ā’ Īøs relative

to the phase of the 2nd LO, since

F

= tan (ĪøL ā’ Īøs ) . (9.107)

G

The heterodyne signal corresponding to F is maximized when ĪøL ā’ Īøs = Ļ/2 and

minimized when ĪøL ā’ Īøs = 0, whereas the heterodyne signal corresponding to G is

Ā¾ Photon detection

Fig. 9.7 Schematic of an apparatus for two-quadrature heterodyne detection. The beam

splitters marked as ā˜High transā™ have |t| ā 1.

maximized when ĪøL ā’ Īøs = 0 and minimized when ĪøL ā’ Īøs = Ļ/2, where all the phases

are deļ¬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 diļ¬erences in the times of arrival

of two optical wavefronts; for example, in the diļ¬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; superconductorā“insulatorā“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 ļ¬eld operators are expressed in terms of classical mode functions and their

associated annihilation operators, we can retain the assumptionsā”i.e. plane waves, S-

polarization, etc.ā”employed in Section 9.3.1. This allows us to use a simpliļ¬ed form

of the general expression (8.28) for the in-ļ¬eld operator to replace the classical ļ¬eld

(9.101) by the Heisenberg-picture operator

Ein (r, t) = ieL aL2 eikL y eā’iĻL t + ies as1 eiks x eā’iĻs t + Evac,in (r, t) ,

(+) (+)

(9.108)

where eM = ĻM /2 0 V is the vacuum ļ¬‚uctuation ļ¬eld strength for a plane wave

with frequency ĻM . This is an extension of the method used in Section 9.1.4 to model

Ā¾

Heterodyne and homodyne detection

imperfect detectors. The annihilation operators aL2 and as1 respectively represent the

local oscillator ļ¬eld, entering through port 2, and the signal ļ¬eld, entering through

port 1; and, we have again assumed that the bandwidths of the signal and local os-

cillator ļ¬elds are small compared to ĻIF . If this assumption has to be relaxed, then

the SchrĀØdinger-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 ļ¬eld is necessarily paired with vacuum ļ¬‚uctuations of the

same frequencyā”indicated by the dashed arrows in Fig. 9.6ā”entering through the

(+)

other input port. Thus Evac,in (r, t) must include the operators aL1 and as2 describing

vacuum ļ¬‚uctuations with frequencies ĻL and Ļs entering through ports 1 and 2 respec-

tively. It should also include any other vacuum ļ¬‚uctuations that could combine with

the local oscillator to yield terms at the intermediate frequency, i.e. modes satisfying

ĻM = ĻL Ā±ĻIF . The +-choice yields the signal frequency Ļs , which is already included,

ńņš. 11 |