√

√

a1 = 1 ’ Ra1 ’ Ra2 ,

√ √

Ra1 + 1 ’ Ra2 ,

a2 =

¾¿

Exercises

2

where R = |r| is the re¬‚ectivity and = ±1. This form will prove useful in Section

20.5.3.

8.2 Single-frequency, two-photon state incident on a beam splitter

(1) Treat the coe¬cients Cmn in eqn (8.68) as a symmetric matrix and show that

C = SCS T ,

where S is given by eqn (8.63) and S T is its transpose.

√ √

(2) Evaluate eqn (8.70) for a balanced beam splitter (r = i/ 2, t = 1/ 2). If there

are detectors at both output ports, what can you say about the rate of coincidence

counting?

(3) Consider the initial state |Ψ = N0 cos θ a†2 + sin θ a†2 |0 .

1 2

(a) Evaluate the normalization constant N0 , calculate the matrices C and C , and

then calculate the scattered state |Ψ .

(b) For a balanced beam splitter, explain why the values θ = ±π/4 are especially

interesting.

8.3 Two-frequency state incident on a beam splitter

(1) For the initial state |Ψ = a† (ω1 ) a† (ω2 ) |0 , calculate the scattered state for the

1 2

case of a balanced beam splitter, and comment on the di¬erence between this

result and the one found in part (2) of Exercise 8.2.

(2) For the initial state |Ψ no photons of frequency ω2 are found in channel 1, but

they are present in the scattered solution. Where do they come from?

(3) According to the de¬nition in Section 6.5.3, the two states

1

|˜± (0) = √ a† (ω1 ) a† (ω2 ) ± a† (ω2 ) a† (ω1 ) |0

1 2 1 2

2

are dynamically entangled. Evaluate the scattered states for the case of a balanced

beam splitter, and compare the di¬erent experimental outcomes associated with

these examples and with the initial state |Ψ from part (1).

8.4 Two-polarization state falling on a beam splitter

Consider the initial state |Ψ de¬ned by eqn (8.77).

(1) Calculate the scattered state for a balanced beam splitter.

(2) Now calculate the scattered state for the alternative initial state

1

|Ψ = √ a† a† + a† a† |0 .

2v 1v 2h

2 1h

Comment on the di¬erence between the results.

¾ Linear optical devices

8.5 Symmetric Y-junction scattering matrix

Consider the symmetric Y-junction discussed in Section 8.5.

(1) Use the symmetry of the Y-junction to derive eqn (8.88).

(2) Evaluate the upper and lower bounds on |y11 | imposed by the unitarity condition

on Y .

8.6 Added noise at a Y-junction

Consider the case that photons are incident only in channel 1 of the symmetric Y-

junction.

(1) Verify conservation of average photon number, i.e. N1 + N2 + N3 = N1 .

(2) Evaluate the added noise in output channel 2 by expressing the normalized vari-

ance V (N2 ) in terms of the normalized variance V (N1 ) in the input channel 1.

What is the minimum value of the added noise?

8.7 The optical circulator

For a wave entering port 1 of the circulator depicted in Fig. 8.4(b), paths ± and β

lead to destructive interference at the mouth of port 3, under the choice of conditions

given by eqns (8.94) and (8.98).

(1) What conditions lead to constructive interference at the mouth of port 2?

(2) Show that the scattering matrix given by eqn (8.99) is unitary.

(3) Consider an experimental situation in which a perfect, lossless, retrore¬‚ecting

mirror terminates port 2. Show that the variance in photon number in the light

emitted through port 3 is exactly the same as the variance of the input light

entering through port 1.

9

Photon detection

Any experimental measurement sensitive to the discrete nature of photons evidently

requires a device that can detect photons one by one. For this purpose a single photon

must interact with a system of charged particles to induce a microscopic change, which

is subsequently ampli¬ed to the macroscopic level. The irreversible ampli¬cation stage

is needed to raise the quantum event to the classical level, so that it can be recorded.

This naturally suggests dividing the treatment of photon detection into several sec-

tions. In Section 9.1 we consider the process of primary detection of the incoming

photon or photons, and in Section 9.2 we study postdetection signal processing, in-

cluding the quantum methods of ampli¬cation of the primary photon event. Finally in

Section 9.3 we study the important techniques of heterodyne and homodyne detection.

9.1 Primary photon detection

In the ¬rst section below, we describe six physical mechanisms commonly employed

in the primary process of photon detection, and in the second section we present a

theoretical analysis of the simplest detection scheme, in which individual atoms are

excited by absorption of a single photon. The remaining sections are concerned with

the relation of incident photon statistics to the statistics of the ejected photoelectrons,

the ¬nite quantum e¬ciency of detectors, and some general statistical features of the

photon distribution.

9.1.1 Photon detection methods

Photon detection is currently based on one of the following physical mechanisms.

(1) Photoelectric detection. These detectors fall into two main categories:

(i) vacuum tube devices, in which the incident photon ejects an electron, bound

to a photocathode surface, into the vacuum;

(ii) solid-state devices, in which absorption of the incident photon deep within the

body of the semiconductor promotes an electron from the valence band to the

conduction band (Kittel, 1985).

In both cases the resulting output signal is proportional to the intensity of the

incident light, and thus to the time-averaged square of the electric ¬eld strength.

This method is, accordingly, also called square-law detection.

There are several classes of vacuum tube devices”for example, the photomultiplier

tubes and channeltrons described in Section 9.2.1”but most modern photoelectric

detectors are based on semiconductors. The promotion of an electron from the

valence band to the conduction band”which is analogous to photoionization of an

¾ Photon detection

atom”leaves behind a positively charged hole in the valence band. Both members

of the electron“hole pair are free to move through the material.

The energy needed for electron“hole pair production is substantially less than the

typical energy”of the order of electron volts”needed to eject a photoelectron

into the vacuum outside a metal surface; consequently, semiconductor devices can

detect much lower energy photons. Thus the sensitivity of semiconductor detectors

extends into the infrared and far-infrared parts of the electromagnetic spectrum.

Furthermore, the photon absorption length in the semiconductor material is so

small that relatively thin detectors will absorb almost all the incident photons. This

means that quantum e¬ciencies are high (50“90%). Semiconductor detectors are

very fast as well as very sensitive, with response times on the scale of nanoseconds.

These devices, which are very important for quantum optics, are also called single-

photon counters.

Solid-state detectors are further divided into two subcategories: photoconduc-

tive and photovoltaic. In photoconductive devices, the photoelectrons are re-

leased into a homogeneous semiconducting material, and a uniform internal elec-

tric ¬eld is applied across the material to accelerate the released photoelectrons.

Thus the current in the homogeneous material is proportional to the number of

photo-released carriers, and hence to the incident intensity of the light beam falling

on the semiconductor. In photovoltaic devices, photons are absorbed and photo-

electrons are released in a highly inhomogeneous region inside the semiconductor,

where there is a large internal electric ¬eld, viz., the depletion range inside a p“n

or p“i“n junction. The large internal ¬elds then accelerate the photoelectrons to

create a voltage across the junction, which can drive currents in an external cir-

cuit. Devices of this type are commonly known as photodiodes (Saleh and Teich,

1991, Chap. 17).

(2) Rectifying detection. The oscillating electric ¬eld of the electromagnetic wave

is recti¬ed, in a diode with a nonlinear I“V characteristic, to produce a direct-

current signal which is proportional to the intensity of the wave. The recti¬ca-

tion e¬ect arises from a physical asymmetry in the structure of the diode, for

example, at the p“n junction of a semiconductor diode device. Such detectors

include Schottky diodes, consisting of a small metallic contact on the surface

of a semiconductor, and biased superconducting“insulator“superconducting

(SIS) electron tunneling devices. These rectifying detectors are used mainly in the

radio and microwave regions of the electromagnetic spectrum, and are commonly

called square-law or direct detectors.

(3) Photothermal detection. Light is directly converted into heat by absorption,

and the resulting temperature rise of the absorber is measured. These detectors

are also called bolometers. Since thermal response times are relatively long,

these detectors are usually slower than many of the others. Nevertheless, they are

useful for detection of broad-bandwidth radiation, in experiments allowing long

integration times. Thus they are presently being used in the millimeter-wave and

far-infrared parts of the electromagnetic spectrum as detectors for astrophysical

measurements, including measurements of the anisotropy of the cosmic microwave

background (Richards, 1994).

¾

Primary photon detection

(4) Photon beam ampli¬ers. The incoming photon beam is coherently ampli¬ed

by a device such as a maser or a parametric ampli¬er. These devices are primarily

used in the millimeter-wave and microwave region of the electromagnetic spectrum,

and play the same role as the electronic pre-ampli¬ers used at radio frequencies.

Rather than providing postdetection ampli¬cation, they coherently pre-amplify

the incoming electromagnetic wave, by directly providing gain at the carrier fre-

quency. Examples include solid-state masers, which amplify the incoming signal

by stimulated emission of radiation (Gordon et al., 1954), and varactor parametric

ampli¬ers (paramps), where a pumped, nonlinear, reactive element”such as a

nonlinear capacitance of the depletion region in a back-biased p“n junction”can

amplify an incoming signal. The nonlinear reactance is modulated by a strong,

higher-frequency pump wave which beats with the signal wave to produce an idler

wave at the di¬erence frequency between the pump and signal frequencies. The

idler wave reacts back via the pump wave to produce more signal wave, etc. This

causes a mutual reinforcement, and hence ampli¬cation, of both the signal and

idler waves, at the expense of power in the pump wave. The idler wave power is

dumped into a matched termination.

(5) Single-microwave-photon counters. Single microwave photons in a supercon-

ducting microwave cavity are detected by using atomic beam techniques to pass

individual Rydberg atoms through the cavity. The microwave photon can cause

a transition between two high-lying levels (Rydberg levels) of a Rydberg atom,

which is subsequently probed by a state-selective ¬eld ionization process. The re-

sult of this measurement indicates whether a transition has occurred, and therefore

provides information about the state of excitation of the microwave cavity (Hulet

and Kleppner, 1983; Raushcenbeutal et al., 2000; Varcoe et al., 2000).

(6) Quantum nondemolition detectors. The presence of a single photon is de-

tected without destroying it in an absorption process. This detection relies on

the phase shift produced by the passage of a single photon through a nonlinear

medium, such as a Kerr medium. Such detectors have recently been implemented

in the laboratory (Yamamoto et al., 1986).

The last three of these detection schemes, (4) to (6), are especially promising for

quantum optics. However, all the basic mechanisms (1) through (3) can be extended,

by a number of important auxiliary methods, to provide photon detection at the

single-quantum level.

9.1.2 Theory of photoelectric detection

The theory presented here is formulated for the simplest case of excitation of free

atoms by the incident light, and it is solely concerned with the primary microscopic

detection event. In situations for which photon counting is relevant, the ¬elds are

weak; therefore, the response of the atoms can be calculated by ¬rst-order perturbation

theory. As we will see, the ¬rst-order perturbative expression for the counting rate is

the product of two factors. The ¬rst depends only on the state of the atom, and the

second depends only on the state of the ¬eld. This clean separation between properties

of the detector and properties of the ¬eld will hold for any detection scheme that can

¾ Photon detection

be described by ¬rst-order perturbation theory. Thus the use of the independent atom

model does not really restrict the generality of the results. In practice, the sensitivity

function describing the detector response is determined empirically, rather than being

calculated from ¬rst principles.

The primary objective of the theory is therefore to exhibit the information on the

state of the ¬eld that the counting rate provides. As we will see below, this information

is naturally presented in terms of the ¬eld“¬eld correlation functions de¬ned in Section

4.7. In a typical experiment, light from an external source, such as a laser, is injected

into a sample of some interesting medium and extracted through an output port. The

output light is then directed to the detectors by appropriate linear optical elements. An

elementary, but nonetheless important, point is that the correlation function associated

with a detector signal is necessarily evaluated at the detector, which is typically not

located in the interior of the sample being probed. Thus the correlation functions

evaluated in the interior of the sample, while of great theoretical interest, are not

directly related to the experimental results. Information about the interaction of the

light with the sample is e¬ectively stored in the state of the emitted radiation ¬eld,

which is used in the calculation of the correlation functions at the detectors. Thus

for the analysis of photon detection per se we only need to consider the interaction

of the electromagnetic ¬eld with the optical elements and the detectors. The total

Hamiltonian for this problem is therefore H = H0 + Hdet , where Hdet represents the

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

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

The remaining term, H1 , describes the interaction of the ¬eld with the passive linear

optical devices, e.g. lenses, mirrors, beam splitters, etc., that direct the light to the

detectors.

A Single-photon detection

The simplest possible photon detector consists of a single atom interacting with the

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

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

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

radiation ¬eld, which is, for the moment, assumed to be pure. According to eqns

(4.95) and (4.103) the initial state vector evolves into

t

i

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

t0

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

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

t

i

φ , n |˜ (t) = ’ dt1 φ , n |Hdet (t1 )| ˜ (t0 ) , (9.2)

t0

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

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

respectively given by

¾

Primary photon detection

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

γ t1

d (9.3)

γ

and

t

i

φ , n |˜ (t) = · n E(+) (r, t1 ) ¦e ,

dt1 eiω γ t1

d (9.4)

γ

t0

where d γ = φ d φγ is the dipole matrix element for the transition γ ’ .

The conditional probability for ¬nding |φ , n , given |φγ , ¦e , is therefore

2

t

i

· nE

iω γ t1 (+)

p (φ , n : φγ , ¦e ) = dt1 e d (r, t1 ) ¦e

γ

t0

d—γ t t

(d γ )j

γ (t2 ’t1 )

i

dt2 eiω

= dt1

2

t0 t0

—

(+) (+)

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

—

(+) (’)

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

eqn (9.5) can be rewritten as

d—γ t t

(d γ )j

γ (t2 ’t1 )

i

dt2 eiω

p (φ , n : φγ , ¦e ) = dt1

2

t0 t0

(’) (+)

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

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

is the sum of the conditional probabilities p (φ , n : φγ , ¦e ) over all ¬nal ¬eld states

|n :

p (φ : φγ , ¦e ) = p (φ , n : φγ , ¦e ) . (9.7)

n

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

eqn (9.7), then yields

d—γ t t

(d γ )j

γ (t2 ’t1 )

i

dt2 eiω

p (φ : φγ , ¦e ) = dt1

2

t0 t0

(’) (+)

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

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

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

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

to get

p (φ : φγ , ¦e ) Pe

p (φ : φγ ) =

e

d—γ t t

(d γ )j

γ (t2 ’t1 ) (’) (+)

i

dt2 eiω

= dt1 Tr ρEi (r, t1 ) Ej (r, t2 ) ,

2

t0 t0

(9.9)

¾¼ Photon detection

where

Pe |¦e ¦e |

ρ= (9.10)

e

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

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

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

schemes do not depend on a speci¬c transition to a bound level; instead, they involve

transitions into excited states lying in the continuum. The atom may be directly

ionized, or the absorption of the photon may lead to a bound state that is subject

to Stark ionization by a static electric ¬eld. The ionized electrons would then be

accelerated, and thereby produce further ionization by secondary collisions. All of

these complexities are subsumed in the probability D ( ) that the transition γ ’

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

overall probability is then

D ( ) p (φ : φγ ) .

p (t) = (9.11)

It should be understood that the -sum is really an integral, and that the factor

D ( ) includes the density of states for the continuum states of the atom. Putting this

together with the expression (9.9) leads to

t t

(1)

dt2 Sji (t1 ’ t2 ) Gij (r, t1 ; r, t2 ) ,

p (t) = dt1 (9.12)

t0 t0

where the sensitivity function

1

D ( ) d—γ (d γ )j e’iω

Sji (t) = γt

(9.13)

2 i

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

function

(1) (’) (+)

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

is determined solely by the properties of the ¬eld.

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

S— (t) = Sij (’t) , (9.15)

ji

and other useful properties are found by studying the Fourier transform