radio engineering, we refer to this mode as the image band, and set M = IB and

(+)

ωIB = ωL ’ ωIF . The relevant terms in Ein (r, t) are thus

Ein (r, t) = ieL aL2 eikL y e’iωL t + ieL aL1 eikL x e’iωL t

(+)

+ ies as1 eiks x e’iωs t + ies as2 eiks y e’iωs t

+ ieIB aIB2 eikIB y e’iωIB t

+ ieIB aIB1 eikIB x e’iωIB t . (9.109)

A The heterodyne signal

(+)

The scattered ¬eld operator Eout (r, t) is split into two parts, which respectively de-

scribe propagation along the 2 ’ 2 arm and the 1 ’ 1 arm in Fig. 9.6. The latter

(+)

part”which we will call Eout,D (r, t)”is the one driving the detector. The spatial

(+)

modes in Eout,D (r, t) are all of the form exp (ikx), for various values of k. Since we

only need to evaluate the ¬eld at the detector location xD , the calculation is simpli¬ed

by choosing the coordinates so that xD = 0. In this way we ¬nd the expression

Eout,D (t) = ieL aL1 e’iωL t + ies as1 e’iωs t + ieIB aIB1 e’iωIB t .

(+)

(9.110)

The scattered annihilation operators are obtained by applying the beam-splitter scat-

tering matrix in eqn (8.63) to the incident annihilation operators. This simply amounts

to working out how each incident classical mode is scattered into the 1 ’ 1 arm, with

the results

as1 = t as1 + r as2 , aL1 = t aL1 + r aL2 , aIB1 = t aIB1 + r aIB2 . (9.111)

The ¬nite e¬ciency of the detector can be taken into account by using the technique

(+)

discussed in Section 9.1.4 to modify Eout,D (t).

¾ Photon detection

Applying eqn (9.33), for the total single-photon counting rate, to this case gives

(’) (+)

w(1) (t) ∝ Eout,D (t) Eout,D (t) , (9.112)

and the intermediate frequency part of this signal comes from the beat-note terms

between the local oscillator part of eqn (9.110)”or rather its conjugate”and the

signal and image band parts. This procedure leads to the operator expression

(’) (+)

Shet = Eout,D (t) Eout,D (t) = F cos (ωIF t) + G sin (ωIF t) , (9.113)

IF

where the operators F and G”which correspond to the classical quantities F and G

respectively”have contributions from both the signal and the image band, i.e.

F = Fs + FIB , G = Gs + GIB , (9.114)

where

†

Fs = eL es aL1 as1 + HC , (9.115)

†

FIB = eL eIB aL1 aIB1 + HC , (9.116)

†

Gs = ’ieL es aL1 as1 ’ HC , (9.117)

and

†

GIB = ’ieL eIB aL1 aIB1 ’ HC . (9.118)

By assumption, the density operator ρin describing the state of the incident light

is the vacuum for all annihilation operators other than aL2 and as1 , i.e.

aΛ ρin = ρin a† = 0 , Λ = s2, L1, IB1, IB2 . (9.119)

Λ

These conditions immediately yield

†

aL1 aIB1 = 0 , (9.120)

and

†

a† as1 .

aL1 as1 = r— t (9.121)

L2

Furthermore, the independently generated signal and local oscillator ¬elds are uncor-

related, so the total density operator can be written as a product

ρin = ρL ρs , (9.122)

where ρL and ρs are respectively the density operators for the local oscillator and the

signal. This leads to the further simpli¬cation

a† as1 = a† as1 . (9.123)

L2 L2 s

L

¾

Heterodyne and homodyne detection

From eqn (9.120) we see that the expectation values of the operators F and G are

completely determined by Fs and Gs , and eqn (9.123) allows the ¬nal result to be

written as

F = eL es 2 Re r— t a† as1 s , (9.124)

L2

L

a†

G = eL es 2 Im r— t as1 , (9.125)

L2 s

L

which suggests de¬ning e¬ective ¬eld amplitudes

EL = eL aL2 , Es = es as1 . (9.126)

L s

With this notation, the expectation values of the operators F and G have the same

form as the classical quantities F and G:

—

F = ±2 |EL Es | |r| |t| sin (θL ’ θs ) ,

(9.127)

—

G = ±2 |EL Es | |r| |t| cos (θL ’ θs ) .

This formal similarity becomes an identity, if both the signal and the local oscillator

are described by coherent states, i.e. aL2 ρin = ±L ρin and as1 ρin = ±s ρin .

The result (9.127) is valid for any state, ρin , that satis¬es the factorization rule

(9.122). Let us apply this to the extreme quantum situation of the pure number state

ρs = |ns ns |. In this case Es = es as1 s = 0, and the heterodyne signal vanishes. This

re¬‚ects the fact that pure number states have no well-de¬ned phase. The same result

holds for any density operator, ρs , that is diagonal in the number-state basis. On the

other hand, for a superposition of number states, e.g.

|ψ = C0 |0 + C1 |1s , (9.128)

the e¬ective ¬eld strength for the signal is

—

Es = es ψ |as1 | ψ = es C0 C1 . (9.129)

Consequently, a nonvanishing heterodyne signal can be measured even for superposi-

tions of states containing at most one photon.

B Noise in heterodyne detection

In the previous section, we carefully included all the relevant vacuum ¬‚uctuation terms,

only to reach the eminently sensible conclusion that none of them makes any contribu-

tion to the average signal. This was not a wasted e¬ort, since we saw in Section 8.4.2

that vacuum ¬‚uctuations will add to the noise in the measured signal. We will next

investigate the e¬ect of vacuum ¬‚uctuations in heterodyne detection by evaluating the

variance,

2

V (F ) = F 2 ’ F , (9.130)

of the operator F in eqn (9.114).

¾ Photon detection

Since the calculation of ¬‚uctuations is substantially more complicated than the

calculation of averages, it is a good idea to exploit any simpli¬cations that may turn

up. We begin by using eqn (9.114) to write F 2 as

F 2 = Fs + Fs FIB + FIB Fs + FIB .

2 2

(9.131)

The image band vacuum ¬‚uctuations and the signal are completely independent, so

there should be no correlations between them, i.e. one should ¬nd

Fs FIB = Fs FIB = FIB Fs . (9.132)

Since the density operator is the vacuum for the image band modes, the absence of

correlation further implies

Fs FIB = FIB Fs = 0 . (9.133)

This result can be veri¬ed by a straightforward calculation using eqn (9.119) and the

commutativity of operators for di¬erent modes.

At this point we have the exact result

2

V (F ) = Fs + FIB ’ Fs

2 2

= V (Fs ) + V (FIB ) , (9.134)

where we have used FIB = 0 again to get the ¬nal form. A glance at eqns (9.115)

and (9.116) shows that this is still rather complicated, but any further simpli¬cations

must be paid for with approximations. Since the strong local oscillator ¬eld is typically

generated by a laser, it is reasonable to model ρL as a coherent state,

aL2 ρL = ±L ρL , ρL a† = ±— ρL , (9.135)

L

L2

with

±L = |±L | eiθL . (9.136)

The variance V (FIB ) can be obtained from V (Fs ) by the simple expedient of replacing

the signal quantities {as1 , as2 , es } by the image band equivalents {aIB1 , aIB2 , eIB }, so we

begin by using eqns (9.111), (9.119), and (9.135) to evaluate V (Fs ). After a substantial

amount of algebra”see Exercise 9.5”one ¬nds

e’2iθL V (as1 ) + CC

2 2

V (Fs ) = ’e2 |r t| |EL |

s

a† as1 ’ | as1 |

2 2 2

+ 2e2 |r t| |EL |

s s1

a† as1 ,

2 2 2 2

+ e2 |r| |EL | + (eL es ) |t| (9.137)

s s1

where |EL | = eL |±L | is the laser amplitude. We may not appear to be achieving very

much in the way of simpli¬cation, but it is too soon to give up hope.

¾

Heterodyne and homodyne detection

The ¬rst promising sign comes from the simple result

2 2

V (FIB ) = e2 |r| |EL | . (9.138)

IB

This represents the ampli¬cation”by beating with the local oscillator”of the vacuum

¬‚uctuation noise at the image band frequency. With our normalization conventions,

the energy density in these vacuum ¬‚uctuations is

ωIB

uIB = 2 0 e2 = . (9.139)

IB

V

In Section 1.1.1 we used equipartition of energy to argue that the mean thermal energy

for each radiation oscillator is kB T , so the thermal energy density would be uT =

kB T /V . Equating the two energy densities de¬nes an e¬ective noise temperature

ωIB ωL

≈

Tnoise = . (9.140)

kB kB

This e¬ect will occur for any of the phase-insensitive linear ampli¬ers studied in Chap-

ter 16, including masers and parametric ampli¬ers (Shimoda et al., 1957; Caves, 1982).

With this encouragement, we begin to simplify the expression for V (Fs ) by intro-

ducing the new creation and annihilation operators

b† (θL ) = eiθL a† , bs (θL ) = e’iθL as1 . (9.141)

s s1

This eliminates the explicit dependence on θL from eqn (9.137), but the new oper-

ators are still non-hermitian. The next step is to consider the observable quantities

represented by the hermitian quadrature operators

e’iθL as1 + eiθL a†

bs (θL ) + b† (θL )

s s1

X (θL ) = = (9.142)

2 2

and

bs (θL ) ’ b† (θL ) e’iθL as1 ’ eiθL as1

s

Y (θL ) = = . (9.143)

2i 2i

These operators are the hermitian and anti-hermitian parts of the annihilation oper-

ator:

bs (θL ) = X (θL ) + iY (θL ) , (9.144)

and the canonical commutation relations imply

i

[X (θL ) , Y (θL )] = . (9.145)

2

By writing the de¬ning equations (9.142) and (9.143) as

X (θL ) = X (0) cos θL + Y (0) sin θL ,

(9.146)

Y (θL ) = X (0) sin θL ’ Y (0) cos θL ,

the quadrature operators can be interpreted as a rotation of the phase plane through

the angle θL , given by the phase of the local oscillator ¬eld. In the calculations to

follow we will shorten the notation by X (θL ) ’ X, etc.

¿¼¼ Photon detection

After substituting eqns (9.141) and (9.144), into eqn (9.134), we arrive at

1

+ |r| |EL | e2 + e2 + |t| e2 a† as1 e2 .

2 2 2 2 2

V (F ) = 4 |r t| |EL | e2 V (Y ) ’

s s IB s L

s1

4

(9.147)

The combination V (Y ) ’ 1/4 vanishes for any coherent state, in particular for the

vacuum, so it represents the excess noise in the signal. It is important to realize that

the excess noise can be either positive or negative, as we will see in the discussion of

squeezed states in Section 15.1.2. The ¬rst term on the right of eqn (9.147) represents

the ampli¬cation of the excess signal noise by beating with the strong local oscillator

¬eld. The second term represents the ampli¬cation of the vacuum noise at the signal

and the image band frequencies. Finally, the third term describes ampli¬cation”by

beating against the signal”of the vacuum noise at the local oscillator frequency. The

2 2 2

strong local oscillator assumption can be stated as |r| |±L | |t| , so the third term is

negligible. Neglecting it allows us to treat the local oscillator as an e¬ectively classical

¬eld.

The noise terms discussed above are fundamental, in the sense that they arise

directly from the uncertainty principle for the radiation oscillators. In practice, exper-

imentalists must also deal with additional noise sources, which are called technical in

order to distinguish them from fundamental noise. In the present context the primary

technical noise arises from various disturbances”e.g. thermal ¬‚uctuations in the laser

cavity dimensions, Johnson noise in the electronics, etc.”a¬ecting the laser providing

the local oscillator ¬eld. By contrast to the fundamental vacuum noise, the technical

noise is”at least to some degree”subject to experimental control. Standard practice

is therefore to drive the local oscillator by a master oscillator which is as well controlled

as possible.

9.3.3 Balanced homodyne detection

This technique combines heterodyne detection with the properties of the ideal bal-

anced beam splitter discussed in Section 8.4. A strong quasiclassical ¬eld (the LO) is

injected into port 2, and a weak signal with the same frequency is injected into port

1 of a balanced beam splitter, as shown in Fig. 9.8. In practice, it is convenient to

generate both ¬elds from a single master oscillator. Note, however, that the signal and

local oscillator mode functions are orthogonal, because the plane-wave propagation

vectors are orthogonal. If the beam splitter is balanced, and the rest of the system is

designed to be as bilaterally symmetric as possible, this device is called a balanced

homodyne detector. In particular, the detectors placed at the output ports 1 and

2 are required to be identical within close tolerances. In practice, this is made possible

by the high reproducibility of semiconductor-based photon detectors fabricated on the

same homogeneous, single-crystal wafer using large-scale integration techniques.

The di¬erence between the outputs of the two identical detectors is generated by

means of a balanced, di¬erential electronic ampli¬er. Since the two input transistors of

the di¬erential ampli¬er”whose noise ¬gure dominates that of the entire postdetection

electronics”are themselves semiconductor devices fabricated on the same wafer, they

can also be made identical within close tolerances. The symmetry achieved in this way

guarantees that the technical noise in the laser source”from which both the signal

¿¼½

Heterodyne and homodyne detection

SA

D2

2'

+

1 ’

-s

Signal D1

1'

2

-L

Local oscillator (LO)

Fig. 9.8 Schematic of a balanced homodyne detector. Detectors D1 and D2 respectively

collect the output of ports 1 and 2 . The outputs of D2 and D1 are respectively fed into the

non-inverting input (+) and the inverting input (’) of a di¬erential ampli¬er. The output of

the di¬erential ampli¬er, i.e. the di¬erence between the two detected signals, is then fed into

a radio-frequency spectrum analyzer SA.

and the local oscillator are derived”will produce essentially identical ¬‚uctuations in

the outputs of detectors D1 and D2. These common-mode noise waveforms will cancel

out upon subtraction in the di¬erential ampli¬er. This technique can, therefore, lead

to almost ideal detection of purely quantum statistical properties of the signal. We will

encounter this method of detection later in connection with experiments on squeezed

states of light.

A Classical analysis of homodyne detection

It is instructive to begin with a classical analysis for general values of the re¬‚ection and

transmission coe¬cients r and t before specializing to the balanced case. The classical

amplitudes at detectors D1 and D2 are related to the input ¬elds by

ED1 = r EL + t Es ,

(9.148)

ED2 = t EL + r Es ,

and the di¬erence in the outputs of the square-law detectors is proportional to the

di¬erence in the intensities, so the homodyne signal is

2 2

Shom = |ED2 | ’ |ED1 |

—

2 2 2 2

= 1 ’ 2 |r| |EL | ’ 1 ’ 2 |r| |Es | + 4 |t r| Im [EL Es ] , (9.149)

where we have used the Stokes relations (8.7) and set r— t = i |r t| (this is the +-sign

in eqn (9.105)) to simplify the result. The ¬rst term on the right side is not sensitive

to the phase θL of the local oscillator, so it merely provides a constant background

¿¼¾ Photon detection

for measurements of the homodyne signal as a function of θL . By design, the signal

2

intensity is small compared to the local oscillator intensity, so the |Es | -term can be

neglected altogether. As mentioned in Section 9.3.2, the local oscillator amplitude

is subject to technical ¬‚uctuations δEL ”e.g. variations in the laser power due to

acoustical-noise-induced changes in the laser cavity dimensions”which in turn produce

phase-sensitive ¬‚uctuations in the output,

— —

δShom = ’ 1 ’ 2 |r|2 2 Re [EL δEL ] + 4 |t r| Im [δEL Es ] . (9.150)

2 2

The ¬‚uctuations associated with the direct detection signal, 1 ’ 2 |r| |EL | , for the

local oscillator are negligible compared to the ¬‚uctuations in the Es contribution if

|Es |

2

1 ’ 2 |r| , (9.151)

|EL |

2

and this is certainly satis¬ed for an ideal balanced beam splitter, for which |r| =

|t|2 = 1/2, and

—

Shom = 2 Im [EL Es ] . (9.152)

B Quantum analysis of homodyne detection

We turn now to the quantum analysis of homodyne detection, which is simpli¬ed by the

fact that the local oscillator and the signal have the same frequency. The complications

associated with the image band modes are therefore absent, and the in-¬eld is simply

Ein (r, t) = ies aL eiks y e’iωs t + ies as eiks x e’iωs t .

(+)

(9.153)

In this case all relevant vacuum ¬‚uctuations are dealt with by the operators aL and

(+)

as , so the operator Evac,in (r, t) will not contribute to either the signal or the noise.

The homodyne signal. The out-¬eld is

(+) (+) (+)

Eout (r, t) = ED1 (r, t) + ED2 (r, t) , (9.154)

where the ¬elds

ED1 (r, t) = ies as eiks x e’iωs t

(+)

(9.155)

and

ED2 (r, t) = ies aL eiks y e’iωs t

(+)

(9.156)

drive the detectors D1 and D2 respectively, and the scattered annihilation operators

satisfy the operator analogue of (9.148):

a L = t aL + r as ,

(9.157)

a s = r aL + t a s .

The di¬erence in the two counting rates is proportional to

¿¼¿

Heterodyne and homodyne detection

(’) (+) (’) (+)

Shom = ED2 (r, t) ED2 (r, t) ’ ED1 (r, t) ED1 (r, t)

= e2 N21 , (9.158)

s

where

†

N21 = aL aL ’ as† as

a† aL ’ 1 ’ 2 |r| a† as ’ 2i |r t| a† as ’ a† aL

2 2

= 1 ’ 2 |r| (9.159)

s s

L L

is the quantum analogue of the classical result (9.149). For a balanced beam splitter,

this simpli¬es to

N21 = ’i a† as ’ a† aL ; (9.160)

s

L

consequently, the balanced homodyne signal is

Shom = 2e2 Im a† as . (9.161)

s L

If we again assume that the signal and local oscillator are statistically independent,

then a† as = a† as , and

L L

—

Shom = 2 Im (EL Es ) , (9.162)

where the e¬ective ¬eld amplitudes are again de¬ned by

EL = es aL = es | aL | eiθL , (9.163)

and

Es = es as . (9.164)

Just as for heterodyne detection, the phase sensitivity of homodyne detection guaran-

tees that the detection rate vanishes for signal states described by density operators

that are diagonal in photon number. Alternatively, for the calculation of the signal we

can replace the di¬erence of number operators by

—

†

aL aL ’ as† as ’ ’i a s ’ a † aL = 2 | aL | Y ,

aL (9.165)

s

where Y is the quadrature operator de¬ned by eqn (9.143). This gives the equivalent

result

Shom = 2 |EL | es Y (9.166)

for the homodyne signal.

Noise in homodyne detection. Just as in the classical analysis, the ¬rst term in

the expression (9.159) for N21 would produce a phase-insensitive background, but for

2

|r| signi¬cantly di¬erent from the balanced value 1/2, the variance in the homodyne

output associated with technical noise in the local oscillator could seriously degrade

the signal-to-noise ratio. This danger is eliminated by using a balanced system, so that

N21 is given by eqn (9.160). The calculation of the variance V (N21 ) is considerably

¿¼ Photon detection

simpli¬ed by the assumption that the local oscillator is approximately described by a

coherent state with ±L = |±L | exp (iθL ). In this case one ¬nds

V (N21 ) = |±L | + a† as + 2 |±L | V a† , as ’ |±L | V e’iθL as ’ |±L | V e’iθL a† .

2 2 2 2

s s s

(9.167)

Expressing this in terms of the quadrature operator Y gives the simpler result

V (N21 ) = 4 |±L | V (Y ) + a† as

2 2

4 |±L | V (Y ) , (9.168)

s

where the last form is valid in the usual case that the input signal ¬‚ux is negligible

compared to the local oscillator ¬‚ux.

Corrections for ¬nite detector e¬ciency—

C

So far we have treated the detectors as though they were 100% e¬cient, but perfect

detectors are very hard to ¬nd. We can improve the argument given above by using

the model for imperfect detectors described in Section 9.1.4. Applying this model to

detector D1 requires us to replace the operator as ”describing the signal transmitted

through the beam splitter in Fig. 9.8”by

1 ’ ξcs ,

as = ξas + i (9.169)

where the annihilation operator cs is associated with the mode exp [i (ks y ’ ωs t)] en-

tering through port 2 of the imperfect-detector model shown in Fig. 9.1. A glance

at Fig. 9.8 shows that this is also the mode associated with aL . Since the quantiza-

tion rules assign a unique annihilation operator to each mode, things are getting a

bit confusing. This di¬culty stems from a violation of Einstein™s rule caused by an

uncritical use of plane-wave modes. For example, the local oscillator entering port 2

of the homodyne detector, as shown in Fig. 9.8, should be described by a Gaussian

wave packet wL with a transverse pro¬le that is approximately planar at the beam

splitter and e¬ectively zero at the detector D1. Correspondingly, the operator cs , rep-

resenting the vacuum ¬‚uctuations blamed for the detector noise, should be associated

with a wave packet that is approximately planar at the ¬ctitious beam splitter of the

imperfect-detector model and e¬ectively zero at the real beam splitter in Fig. 9.8. In

other words, the noise in detector D1 does not enter the beam splitter. All of this can

be done precisely by using the wave packet quantization methods developed in Section

3.5.2, but this is not necessary as long as we keep our wits about us. Thus we impose

cs ρ = 0, aL ρ = 0, and a† , cs = 0, even though”in the oversimpli¬ed plane-wave

L

picture”both operators cs and aL are associated with the same plane-wave mode.

In the same way, the noise in detector D2 is simulated by replacing the transmitted

LO-¬eld aL with

aL = ξaL + i 1 ’ ξcL , (9.170)

where cL ρ = 0, and cL , a† = 0.

s

Continuing in this vein, the di¬erence operator N21 is replaced by

N21 = aL† aL ’ as † as

= ξN21 + δN21 . (9.171)

¿¼

Exercises

Each term in δN21 contains at least one creation or annihilation operator for the vac-

uum modes discussed above. Since the vacuum operators commute with the operators

for the signal and local oscillator, the expectation value of δN21 vanishes, and the

homodyne signal is

—

Shom = e2 N21 = ξe2 N21 = 2ξ Im (EL Es ) . (9.172)

s s

As expected, the signal from the imperfect detector is just the perfect detector result

reduced by the quantum e¬ciency.

We next turn to the noise in the homodyne signal, which is proportional to the

variance V (N21 ). It is not immediately obvious how the extra partition noise in each

detector will contribute to the overall noise, so we ¬rst use eqn (9.171) again to get

2 2 2

= ξ 2 (N21 )

(N21 ) + ξ N21 δN21 + ξ δN21 N21 + (δN21 ) . (9.173)

There are no correlations between the vacuum ¬elds cL and cs entering the imperfect

detector and the signal and local oscillator ¬elds, so we should expect to ¬nd that the

second and third terms on the right side of eqn (9.173) vanish. An explicit calculation

shows that this is indeed the case. Evaluating the fourth term in the same way leads

to the result

†

aL aL + as† as

V (N21 ) = ξ 2 V (N21 ) + ξ (1 ’ ξ) . (9.174)

Comparing this to the single-detector result (9.57) shows that the partition noises

at the two detectors add, despite the fact that N21 represents the di¬erence in the

photon counts at the two detectors. After substituting eqn (9.168) for V (N21 ); using

the scattering relations (9.157); and neglecting the small signal ¬‚ux, we get the ¬nal

result

2 2

V (N21 ) = ξ 2 4 |±L | V (Y ) + ξ (1 ’ ξ) |±L | . (9.175)

9.4 Exercises

9.1 Poissonian statistics are reproduced

’1

nn exp (’n) for the incident photons in eqn

Use the Poisson distribution p(n) = (n!)

(9.46) to derive eqn (9.48).

m-fold coincidence counting

9.2

Generalize the two-detector version of coincidence counting to any number m. Show

that the m-photon coincidence rate is

m

2 T12 +Tgate T1m +Tgate

1

Sn d„2 · · ·

(m)

w = d„m

m! T12 T1m

n=1

G(m) (r1 , t1 , . . . , rm , tm + „m ; r1 , t1 , . . . , rm , tm + „m ) ,

where the signal from the ¬rst detector is used to gate the coincidence counter and

T1n = T1 ’ Tn .

¿¼ Photon detection

9.3 Super-Poissonian statistics

2 2

Consider the state |Ψ = ± |n + β |n + 1 , with |±| + |β| = 1. Show that |Ψ is a

nonclassical state that exhibits super-Poissonian statistics.

9.4 Alignment in heterodyne detection

For the heterodyne scheme shown in Fig. 9.6, assume that the re¬‚ected LO beam has

the wavevector kL = kL cos •ux + kL sin •uy . Rederive the expression for Shet and

show that averaging over the detector surface wipes out the heterodyne signal.

9.5 Noise in heterodyne detection

Use eqn (9.111), eqn (9.119), and eqn (9.135) to derive eqn (9.137).

10

Experiments in linear optics

In this chapter we will study a collection of signi¬cant experiments which were carried

out with the aid of the linear optical devices described in Chapter 8 and the detection

techniques discussed in Chapter 9.

10.1 Single-photon interference

The essential features of quantum interference between alternative Feynman paths

are illustrated by the familiar Young™s arrangement”sketched in Fig. 10.1”in which

there are two pinholes in a perfectly re¬‚ecting screen. The screen is illuminated by a

plane-wave mode occupied by a single photon with energy ω, and after many suc-

cessive photons have passed through the pinholes the detection events”e.g. spots on

a photographic plate”build up the pattern observed in classical interference experi-

ments.

An elementary quantum mechanical explanation of the single-photon interference

pattern can be constructed by applying Feynman™s rules of interference (Feynman

et al., 1965, Chaps 1“7).

(1) The probability of an event in an ideal experiment is given by the square of the

absolute value of a complex number A which is called the probability amplitude:

P = probability ,

A = probability amplitude , (10.1)

2

P = |A| .

L

2 2' Fig. 10.1 A two-pinhole interferometer. The

arrows represent an incident plane wave. The

L four ports are de¬ned by the surfaces P1, P1 ,

P2, P2 , and the path lengths from the pin-

2 2' holes 1 and 2”bracketed by the ports (P1, P1 )

and (P2, P2 ) respectively”to the interference

point are L1 and L2 .

¿¼ Experiments in linear optics

(2) When an event can occur in several alternative ways, the probability amplitude

for the event is the sum of the probability amplitudes for each way considered

separately; i.e. there is interference between the alternatives:

A = A1 + A2 ,

(10.2)

P = |A1 + A2 |2 .

(3) If an experiment is performed which is capable of determining whether one or

another alternative is actually taken, the probability of the event is the sum of the

probabilities for each alternative. In this case,

P = P1 + P2 , (10.3)

and there is no interference.

In applying rule (2) it is essential to be sure that the situation described in rule (3)

is excluded. This means that the experimental arrangement must be such that it is

impossible”even in principle”to determine which of the alternatives actually occurs.

In the literature”and in the present book”it is customary to refer to the alternative

ways of reaching the ¬nal event as Feynman processes or Feynman paths.

In the two-pinhole experiment, the two alternative processes are passage of the

photon through the lower pinhole 1 or the upper pinhole 2 to arrive at the ¬nal event:

detection at the same point on the screen. In the absence of any experimental procedure

for determining which process actually occurs, the amplitudes for the two alternatives

must be added. Let Ain be the quantum amplitude for the incoming wave; then the

amplitudes for the two processes are A1 = Ain exp (ikL1 ) and A2 = Ain exp (ikL2 ),

where k = ω/c. The probability of detection at the point on the screen (determined

by the values of L1 and L2 ) is therefore

2 2 2

|A1 + A2 | = 2 |Ain | + 2 |Ain | cos [k (L2 ’ L1 )] , (10.4)

which has the same form as the interference pattern in the classical theory.

This thought experiment provides one of the simplest examples of wave“particle

duality. The presence of the interference term in eqn (10.4) exhibits the wave-aspect

of the photon, while the detection of the photon at a point on the screen displays

its particle-aspect. Arguments based on the uncertainty principle (Cohen-Tannoudji

et al., 1977a, Complement D1; Bransden and Joachain, 1989, Sec. 2.5) show that any

experimental procedure that actually determines which pinhole the photon passed

through”this is called which-path information”will destroy the interference pat-

tern. These arguments typically involve an interaction with the particle”in this case

a photon”which introduces uncontrollable ¬‚uctuations in physical properties, such as

the momentum. The arguments based on the uncertainty principle show that which-

path information obtained by disturbing the particle destroys the interference pattern,

but this is not the only kind of experiment that can provide which-path information.

In Section 10.3 we will describe an experiment demonstrating that single-photon inter-

ference is destroyed by an experimental arrangement that merely makes it possible to

obtain which-path information, even if none of the required measurements are actually

made and there is no interaction with the particle.

¿¼

Single-photon interference

The description of the two-pinhole experiment presented above provides a simple

physical model which helps us to understand single-photon interference, but a more

detailed analysis requires the use of the scattering theory methods developed in Sec-

tions 8.1 and 8.2. For the two-pinhole problem, the e¬ects of di¬raction cannot be

ignored, so it will not be possible to con¬ne attention to a small number of plane

waves, as in the analysis of the beam splitter and the stop. Instead, we will use the

general relations (8.29) and (8.27) to guide a calculation of the ¬eld operator in po-

sition space. This is equivalent to using the classical Green function de¬ned by this

boundary value problem to describe the propagation of the ¬eld operator through the

pinhole.

In the plane-wave basis the positive frequency part of the out-¬eld is given by

(+)

iωk aks es ei(k·r’ωk t) ,

Eout (r, t) = (10.5)

2 0 cV

ks

where the scattered annihilation operators obey

aks = Sks,k s ak s . (10.6)

ks

If the source of the incident ¬eld is on the left (z < 0), then the problem is to calculate

the transmitted ¬eld on the right (z > 0). The ¬eld will be observed at points r lying

on a detection plane at z = L. The plane waves that impinge on a detector at r must

have kz > 0, and the terms in eqn (10.6) can be split into those with kz > 0 (forward

waves) and kz < 0 (backwards waves). The contribution of the forward waves to eqn

(10.5) represents the part of the incident ¬eld transmitted through the pinholes, while

the backward waves”vacuum ¬‚uctuations in this case”scatter into forward waves by

re¬‚ection from the screen. The total ¬eld in the region z > 0 is then the sum of three

terms:

(+) (+) (+) (+)

Eout (r, t) = E1 (r, t) + E2 (r, t) + E3 (r, t) , (10.7)

(+) (+)

where E1 and E2 are the ¬elds coming from pinholes 1 and 2 respectively, and the

¬eld resulting from re¬‚ections of backwards waves at the screen is

(+)

iωk aks es ei(k·r’ωk t) ,

<

E3 (r, t) = (10.8)

2 0 cV

ks,kz >0

where

<

aks = Sks,k s ak s . (10.9)

k s ,kz <0

(+) (+)

In the absence of the re¬‚ected vacuum ¬‚uctuations, E3 , the total ¬eld Eout would

not satisfy the commutation relation (3.17), and this would lead to violations of the

uncertainty principle, as shown in Exercise 10.1.

¿½¼ Experiments in linear optics

If the distance to the observation point r is large compared to the sizes of the

pinholes and to the distance between them”this is called Fraunhofer di¬raction or

the far-¬eld approximation”the ¬elds due to the two pinholes are given by

E(+) (r, t) = iDp E(+) (rp , t ’ Lp /c) (p = 1, 2) , (10.10)

p

where Lp is the distance from the pth pinhole to the observation point r, and Dp is

a real coe¬cient that depends on the pinhole geometry. For simplicity we will assume

that the pinholes are identical, D1 = D2 = D, and that the incident radiation is

monochromatic. If the direction of the incident beam and the vectors r ’ r1 and

r ’ r2 are approximately orthogonal to the screen, then Dp ≈ σ/ (»0 L), where σ is

the common area of the pinholes and »0 is the average wavelength in the incident ¬eld

(Born and Wolf, 1980, Sec. 8.3). This is the standard classical expression, except for

replacing the classical ¬eld in the pinhole by the quantum ¬eld operator. The average

intensity in a de¬nite polarization e at a detection point r is proportional to

(’) (+)

Itot = Eout (r, t) Eout (r, t)

3 3

(’) (+)

= Eq (r, t) Ep (r, t) , (10.11)

q=1 p=1

(’) (’) (’) (’)

where Eout = e · Eout , Eq = e · Eq , and the indices p and q represent the three

terms in eqn (10.7). The density operator, ρ, that de¬nes the ensemble average, · · · ,

contains no backwards waves, since it represents the ¬eld generated by a source to the

(+)

left of the screen. According to eqn (10.8) and eqn (10.9) the operator E3 is a linear

combination of annihilation operators for backwards waves, therefore

(+) (’)

E3 ρ = 0 = ρE3 . (10.12)

By using this fact, plus the cyclic invariance of the trace, it is easy to show that eqn

(10.11) reduces to

2 2

(’) (+)

Itot = Eq (r, t) Ep (r, t)

q=1 p=1

= I1 + I2 + I12 , (10.13)

where Ip is the intensity due to the pth pinhole alone,

2

Ip = |D| E (’) (rp , t ’ Lp /c) E (+) (rp , t ’ Lp /c) (p = 1, 2) , (10.14)

I12 is the interference term,

(’) (+)

I12 = 2 Re E1 (r, t) E2 (r, t)

= 2D2 Re E (’) (r1 , t ’ L1 /c) E (+) (r2 , t ’ L2 /c) , (10.15)

and E (’) (r, t) = e · E(’) (r, t) .

¿½½

Single-photon interference

The expectation values appearing in these expressions are special cases of the ¬rst-

order ¬eld correlation function G(1) de¬ned by eqn (4.76). In this notation, the results

are

Ip = |D|2 G(1) (rp , t ’ Lp /c; rp , t ’ Lp /c) (p = 1, 2) , (10.16)

and

I12 = 2D2 Re G(1) (r1 , t ’ L1 /c; r2 , t ’ L2 /c) . (10.17)

From the classical theory of two-pinhole interference we know that high visibility

interference patterns are obtained with monochromatic light. In quantum theory this

means that the power spectrum a† aks is strongly peaked at |k| = k0 = ω0 /c. If

ks

the density operator ρ satis¬es this condition, then the plane-wave expansion for E (+)

implies that the temporal Fourier transform of Tr E (+) (r, t) ρ is strongly peaked at

(+)

ω0 . This means that the envelope operator E de¬ned by

(+)

(r, t) = E (+) (r, t) eiω0 t

E (10.18)

can be treated as slowly varying”on the time scale 1/ω0 ”provided that it is applied

to the monochromatic density matrix ρ. In this case, the correlation functions can be

written as

(’) (+)

(r2 , t2 ) e’iω0 (t2 ’t1 )

G(1) (r1 , t1 ; r2 , t2 ) = Tr ρE (r1 , t1 ) E

(1)

(r1 , t1 ; r2 , t2 ) e’iω0 (t2 ’t1 ) ,

≡G (10.19)

(1)

where G (r1 , t1 ; r2 , t2 ) is a slowly-varying function of t1 and t2 . For su¬ciently long

pulses, the incident radiation is approximately stationary, so the correlation functions

are unchanged by a time translation tp ’ tp + „ . In other words they only depend on

the time di¬erence t1 ’ t2 , so the direct terms become

(1)

Ip = D2 G (rp , 0; rp , 0) (p = 1, 2) , (10.20)

while the interference term reduces to

(1)

I12 = 2D2 Re G (r1 , „ ; r2 , 0) eiω0 „ , (10.21)

where „ = (L2 ’ L1 ) /c is the di¬erence in the light travel time for the two pinholes.

All three terms are independent of the time t. The direct terms only depend on the

average intensities at the pinholes, but the factor

eiω0 „ = eik0 (L2 ’L1 ) (10.22)

in the interference term produces rapid oscillations along the detection plane. This is

(1)

explicitly exhibited by expressing G in terms of its amplitude and phase:

(1) (1)

(r1 , t1 ; r2 , t2 ) ei¦(r1 ,t1 ;r2 ,t2 ) ,

G (r1 , t1 ; r2 , t2 ) = G (10.23)

so that I12 is given by

¿½¾ Experiments in linear optics

(1)

I12 = 2D2 G (r1 , „ ; r2 , 0) cos [¦ (r1 , „ ; r2 , 0) + ω0 „ ] . (10.24)

The interference pattern is modulated by slow variations in the amplitude and phase

(1)

of G due to the ¬nite length of the pulse. When these modulations are ignored, the

interference maxima occur at the path length di¬erences

¦»0

L2 ’ L1 = c„ = n»0 ’ , n = 0, ±1, ±2, . . . . (10.25)

2π

The interference pattern calculated from the ¬rst-order quantum correlation function

is identical to the classical interference pattern. Since this is true even if the ¬eld state

contains only one photon, ¬rst-order interference is also called one-photon interference.

An important quantity for interference experiments is the fringe visibility

’I

I

V≡ max min

, (10.26)

I max + I min

where I and I are respectively the maximum and minimum values of the

max min

(1)

total intensity on the detection plane. If the slow variations in G are neglected, then

one ¬nds

(1) (1) (1)

= D2 G

I (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0) + 2 G (r1 , „ ; r2 , 0) , (10.27)

max

(1) (1) (1)

(r2 , 0; r2 , 0) ’ 2 G

= D2 G

I (r1 , 0; r1 , 0) + G (r1 , „ ; r2 , 0) , (10.28)

min

so the visibility is

(1)

2G (r1 , „ ; r2 , 0)

V= . (10.29)

(1) (1)

G (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0)

(1)

The ¬eld“¬eld correlation function G (r1 , „ ; r2 , 0) is therefore a measure of the coher-

ence of the signals from the two pinholes. There are no fringes (V = 0) if the correlation

function vanishes. On the other hand, the inequality (4.85) shows that the visibility is

bounded by

(1) (1)

2 G (r1 , 0; r1 , 0) G (r2 , 0; r2 , 0)

V 1, (10.30)

(1) (1)

G (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0)

where the maximum value of unity occurs when the intensities at the two pinholes

are equal. This suggests introducing a normalized correlation function, the mutual

coherence function,

(1)

G (x; x )

g (1) (x; x ) = , (10.31)

(1) (1)

G (x; x) G (x ; x )

¿½¿

Single-photon interference

which satis¬es g (1) (x; x ) 1. In these terms, perfect coherence corresponds to

(1)

g (x; x ) = 1, and the fringe visibility is

(1) (1)

(r2 , 0; r2 , 0) g (1) (r1 , „ ; r2 , 0)

2 G (r1 , 0; r1 , 0) G

V= . (10.32)

(1) (1)

G (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0)

Thus measurements of the intensity at each pinhole, the fringe visibility, and the fringe

spacing completely determine the complex mutual coherence function g (1) (x; x ). This

means that the correlation function G(1) (x; x ) or g (1) (x; x ) can always be interpreted

in terms of a Young™s-style interference experiment.

10.1.1 Hanbury Brown“Twiss e¬ect

We have just seen that ¬rst-order interference, e.g. in Young™s experiment or in the

Michelson interferometer, is described by the ¬rst-order ¬eld correlation function G(1) .

The Hanbury Brown“Twiss e¬ect (Hanbury Brown, 1974) was one of the earliest ob-

servations that demonstrated optical interference in the intensity“intensity correlation

function G(2) . This observation was interpreted as a measurement of photon“photon

correlation, so it eventually led to the founding of the ¬eld of quantum optics. The

e¬ect was originally discovered in a simple laboratory experiment in which light from

a mercury arc lamp passes through an interference ¬lter that singles out a strong green

line of the mercury atom at a wavelength of 546.1 nm. The spectrally pure green light

is split by means of a balanced beam splitter into two beams, which are detected by

square-law detectors placed at the output ports of the beam splitter. The experimental

arrangement is shown in Fig. 10.2. The output current I (t) from each detector is a

measure of the intensity in that arm of the beam splitter. The intensities are slowly

varying on the optical scale, with typical Fourier components in the radio range. The

outputs of the two detectors are fed into a radio-frequency mixer that accumulates the

time integral of the product of the two signals. By sending the signal from one of the

detectors through a variable delay line the intensity“intensity correlation,

∞

I(t)I(t ’ „ )dt ,

f („ ) = (10.33)

’∞

DB RF

mixer

B Fig. 10.2 Experimental arrangement for ob-

serving the Hanbury Brown“Twiss e¬ect. The

A

ωsig signal is split by a 50/50 beam splitter and

Signal the split ¬elds enter detectors at B and C. The

BS

output of the detectors is fed into a radio-fre-

DC

C quency (RF) mixer which integrates the prod-

D uct of the two signals.

¿½ Experiments in linear optics

is measured as a function of the delay time „ . The data (Hanbury Brown and Twiss,

1957) show a peak in the intensity“intensity correlation function f („ ) near „ = 0.

Hanbury Brown and Twiss interpreted this as a photon-bunching e¬ect explained

by the fact that the Bose character of photons enhances the probability that two

photons will arrive simultaneously at the two detectors. However, Glauber showed

that classical intensity ¬‚uctuations in the thermal light emitted by the mercury arc

lamp yield a completely satisfactory description, so that there is no need to invoke the

Bose statistics of photons.

The experimental technique for measuring the intensity“intensity correlation was

later changed from simple square-law detection to coincidence detection based on

a photoelectron counting technique using photomultipliers. Since this technique can

register clicks associated with the arrival of individual photons, it would seem to be

closer to a measurement of a photon“photon correlation function.

For the thermal light source which was used in this experiment, this hope is un-

justi¬ed, because we can explain the results on the basis of classical-¬eld notions by

using the semiclassical theory of the photoelectric e¬ect. A quantum description of this

experiment, to be presented later on, employs an expansion of the density operator

in the basis of coherent states. We will see that the radiation emitted by the thermal

source is described by a completely positive quasi-probability distribution function

P (±), which is consistent with a semiclassical explanation in terms of ¬‚uctuations in

the intensity of the classical electromagnetic ¬eld.

On the other hand, for a pure coherent state the Hanbury Brown“Twiss e¬ect per se

does not exist. Thus if we were to replace the mercury arc lamp by a laser operating far

above threshold, the photon arrivals would be described by a pure Poissonian random

process, with no photon-bunching e¬ect.

This intensity“intensity correlation method was applied to astrophysical stellar in-

terferometry to measure stellar diameters (Hanbury Brown and Twiss, 1956). Stellar

interferometry depends on the di¬erence in path lengths to the telescope from points

on opposite limbs of the star. For example, Michelson stellar interferometry (Born and

Wolf, 1980, Sec. 7.3.6) is based on ¬rst-order interference”i.e. on the ¬eld“¬eld corre-

lation function”so the optical path lengths must be equalized to high precision. This

is done by adjusting the positions of the interferometer mirrors attached to the tele-

scope so that all wavelengths of light interfere constructively in the ¬eld of view. Under

these conditions, white light entering the telescope will result in a bright white-light

fringe. The white-light fringe condition must be met before attempting to measure a

stellar diameter by this method.

By contrast, the beauty of the intensity stellar interferometer is that one can com-

pensate for the delays corresponding to the di¬erence in path lengths in the radio-

wavelength region after detection, rather than in the optical-wavelength region before

detection. Compensating the optical delay by an electronic delay produces a maximum

in the intensity“intensity correlation function of the optical signals.

Furthermore, the optical quality of the telescope surfaces for the intensity inter-

ferometer can be much lower than that required for Michelson stellar interferometry,

so that one can use the re¬‚ectors of searchlights as light buckets, rather than astro-

nomical telescopes with optically perfect surfaces. However, the disadvantage of the

¿½

Two-photon interference

intensity interferometer is that it requires higher intensity sources than the Michelson

stellar interferometer. Thus intensity interferometry can only be used to measure the

diameters of the brightest stars.

10.2 Two-photon interference

The results in Section 10.1 provide support for Dirac™s dictum that each photon inter-

feres with itself, but he went on to say (Dirac, 1958, Sec. I.3)

Each photon then interferes only with itself. Interference between two di¬erent pho-

tons never occurs.

This is one of the very few instances in which Dirac was wrong. Further experimental

progress in the generation of states containing exactly two photons has led to the

realization that di¬erent photons can indeed interfere. These phenomena involve the

second-order correlation function G(2) , de¬ned in Section 4.7, so they are sometimes

called second-order interference. Another terminology calls them fourth-order

interference, since G(2) is an average over the product of four electric ¬eld operators.

We will study two important examples of two-photon interference: the Hong“Ou“

Mandel interferometer, in which interference between two photons occurs locally at

a single beam splitter, and the Franson interferometer, where the interference occurs

between two photons falling on spatially-separated beam splitters.

10.2.1 The Hong“Ou“Mandel interferometer

The quantum property of photon indivisibility was demonstrated by allowing a single

photon to enter through one port of a beam splitter. In an experiment performed

by Hong, Ou, and Mandel (Hong et al., 1987), interference between two Feynman

processes was demonstrated by illuminating a beam splitter with a two-photon state

produced by pumping a crystal of potassium dihydrogen phosphate (KDP) with an

ultraviolet laser beam, as shown in Fig. 10.3. In a process known as spontaneous

down-conversion”which will be discussed in Section 13.3.2”a pump photon with

frequency ωp splits into a pair of lower frequency photons, traditionally called the

Fig. 10.3 The Hong“Ou“Mandel interferometer illuminated by a two-photon state, produced

by spontaneous down-conversion in the crystal labeled SDC. The two photon wave packets

are re¬‚ected from mirrors M1 and M2 so that they meet at the beam splitter BS. The output

of detectors D1 and D2 are fed to the coincidence counter CC. (Adapted from Hong et al.

(1987).)

¿½ Experiments in linear optics

signal and idler.1 Since photons are indistinguishable, they cannot be assigned labels;

therefore, the traditional language must be used carefully and sparingly. The words

˜signal photon™ or ˜idler photon™ simply mean that a photon occupies the signal mode

or the idler mode. It is the modes, rather than the photons, that are distinguishable.

Prior to their arrival at the beam splitter, e.g. at the mirrors M1 and M2, the di¬raction

patterns of the signal and idler modes do not overlap.

In the following discussion, the production process can be treated as a black box;

we only need to know that one pump photon enters the crystal and that two (down-

converted) photons are produced simultaneously and leave the crystal as wave packets

with widths of the order of 15 fs. In the notation used in Fig. 8.2, the signal mode

(ksig , ssig ) enters through port 1 and the idler mode (kidl , sidl ) enters through port 2

of the beam splitter BS.

A Degenerate plane-wave model

It is instructive to analyze this situation in terms of interference between Feynman

processes. We begin with the idealized case of plane-wave modes”propagating from

the beam splitter to the detectors”with degenerate frequencies: ωidl = ωsig = ω0 =

ωp /2. The experimental feature of interest is the coincidence-counting rate. Since a

given photon can only be counted once, the events leading to coincidence counts are

those in which each detector receives one photon.

There are, consequently, two processes leading to coincidence events.

(1) The re¬‚ection“re¬‚ection (rr) process: both wave packets are re¬‚ected from the

beam splitter towards the two detectors.

(2) The transmission“transmission (tt) process: both wave packets are transmitted

through the beam splitter towards the two detectors.

In the absence of which-path information these processes are indistinguishable, since

they both lead to the same ¬nal state: one scattered photon is in the idler mode

and the other is in the signal mode. This results in simultaneous clicks in the two

detectors, and one cannot know, even in principle, which of the two processes actually

occurred. According to the Feynman rules of interference we must add the probability

amplitudes for the two processes, and then calculate the absolute square of the sum

to ¬nd the total probability. If the incident amplitude is set to one, the amplitudes

of the two processes are Arr = r2 and Att = t2 , where r and t are respectively the

complex re¬‚ection and transmission coe¬cients for the beam splitter; therefore, the

coincidence amplitude is

Acoinc = Arr + Att = r2 + t2 . (10.34)

According to eqn (8.8), r and t are π/2 out of phase; therefore the coincidence proba-

bility is

2

2 2 2

Pcoinc = |Acoinc | = |r| ’ |t| , (10.35)

1 These

names are borrowed from radio engineering, which in turn borrowed the ˜idler™ from the

mechanical term ˜idler gear™.

¿½

Two-photon interference

which, happily, agrees with the result (9.98) for the coincidence-counting rate. The

partial destructive interference between the rr- and tt-processes, demonstrated by the

expression for Pcoinc , becomes total interference for the special case of a balanced

beam splitter, i.e. the coincidence probability vanishes. We will refer to this as the

Hong“Ou“Mandel (HOM) e¬ect. This is a strictly quantum interference e¬ect

which cannot be explained by any semiclassical theory.

Another way of describing this phenomenon is that two photons, in the appropriate

initial state, impinging simultaneously onto a balanced beam splitter will pair o¬ and

leave together through one of the two exit ports, i.e. both photons occupy one of the

output modes, (ksig , ssig ) or (kidl , sidl ). This behavior is permitted for photons, which

are bosons, but it would be forbidden by the Pauli principle for electrons, which are

fermions. As a result of this pairing e¬ect, detectors placed at the two exit ports of a

balanced beam splitter will never register a coincidence count. The exit port used by

the photon pair varies randomly from one incident pair to the next.

The argument based on the Feynman rules very e¬ectively highlights the fundamen-

tal principles involved in two-photon interference, but it is helpful to derive the result

by using a Schr¨dinger-picture scattering analysis. The Schr¨dinger-picture state pro-

o o

† †

duced by degenerate, spontaneous down-conversion is asig aidl |0 , but the initial state

for the beam splitter scattering calculation is modi¬ed by the further propagation

from the twin-photon source to the beam splitter. According to eqn (8.1) the scatter-

ing matrix S for propagation through vacuum is simply multiplication by exp (ikL),

where k is the wavenumber and L is the propagation distance; therefore, the general

rule (8.44) shows that the state incident on the beam splitter is

|¦in = eik0 Lsig eik0 Lidl a† a† |0 , (10.36)

sig idl

where Lidl and Lsig are respectively the distances along the idler and signal arms

from the point of creation of the photon pair to the beam splitter. For the present

calculation this phase factor is not important; however, it will play a signi¬cant role

in Section 10.2.1-B. According to eqn (6.92), |¦in is an entangled state, and the ¬nal

state

2 2

a† + a†

|¦¬n = r t e’2iω0 t eik0 Lsig eik0 Lidl |0

sig

idl

+ eik0 Lsig eik0 Lidl r2 + t2 a† a† |0 , (10.37)

idl sig

obtained by using eqn (8.43), is also entangled. For a balanced beam splitter this

reduces to

2 2

i ’2iω0 t ik0 Lsig ik0 Lidl

a† + a†

|¦¬n = |0 ,

e e e (10.38)

sig

idl

2

which explicitly exhibits the ¬nal state as a superposition of paired-photon states.

Once again the conclusion is that the coincidence rate vanishes for a balanced beam

splitter.

¿½ Experiments in linear optics

The quantum nature of this result can be demonstrated by considering a semiclassi-

cal model in which the signal and idler beams are represented by c-number amplitudes

±sig and ±idl . The classical version of the beam splitter equation (8.62) is

±sig = t ±sig + r ±idl ,

(10.39)

±idl = r ±sig + t ±idl ,

and the singles counting rates at detectors D1 and D2 are respectively proportional

2 2

to |±idl |2 and ±sig . The coincidence-counting rate is proportional to |±idl |2 ±sig =

2

±idl ±sig , and eqn (10.39) yields

±idl ±sig = r t ±2 + ±2 + r2 + t2 ±sig ±idl

sig idl

i2

’ ±sig + ±2 , (10.40)

idl

2

where the last line is the result for a balanced beam splitter. This classical result

resembles eqn (10.38), but now the coincidence rate cannot vanish unless one of the

singles rates does. A more satisfactory model can be constructed along the lines of the

argument used for the discussion of photon indivisibility in Section 1.4. Spontaneous

emission is a real transition, while the down-conversion process depends on the virtual

excitation of the quantum states of the atoms in the crystal; nevertheless, spontaneous

down-conversion is a quantum event. A semiclassical model can be constructed by

assuming that the quantum down-conversion event produces classical ¬elds that vary

randomly from one coincidence gate to the next. With this model one can show, as in

Exercise 10.2, that

pcoinc 1

>, (10.41)

psig pidl 2

where pcoinc is the probability for a coincidence count, and psig and pidl are the prob-

abilities for singles counts”all averaged over many counting windows. This semiclas-

sical model limits the visibility of the interference minimum to 50%; the essentially

perfect null seen in the experimental data can only be predicted by using the complete

destructive interference between probability amplitudes allowed by the full quantum

theory. Thus the HOM null provides further evidence for the indivisibility of photons.

Nondegenerate wave packet analysis—

B

The simpli¬ed model used above su¬ces to explain the physical basis of the Hong“

Ou“Mandel interferometer, but it is inadequate for describing some interesting ap-

plications to precise timing, such as the measurement of the propagation velocity of

single-photon wave packets in a dielectric, and the nonclassical dispersion cancelation

e¬ect, discussed in Sections 10.2.2 and 10.2.3 respectively. These applications exploit

the fact that the signal and idler modes produced in the experiment are not plane

waves; instead, they are described by wave packets with temporal widths T ∼ 15 fs.

In order to deal with this situation, it is necessary to allow continuous variation of the

frequencies and to relax the degeneracy condition ωidl = ωsig , while retaining the sim-

ple geometry of the scattering problem. To this end, we ¬rst use eqn (3.64) to replace

¿½

Two-photon interference

the box-normalized operator aks by the continuum operator as (k), which obeys the

canonical commutation relations (3.26). In polar coordinates the propagation vectors

are described by k = (k, θ, φ), so the propagation directions of the modes (ksig , ssig )

and (kidl , sidl ) are given by (θσ , φσ ), where σ = sig, idl is the channel index. The as-

sumption of frequency degeneracy can be eliminated, while maintaining the scattering

geometry, by considering wave packets corresponding to narrow cones of propagation

directions. The wave packets are described by real averaging functions fσ (θ, φ) that

are strongly peaked at (θ, φ) = (θσ , φσ ) and normalized by

d„¦ |fσ (θ, φ)|2 = 1 , (10.42)

where d„¦ = d (cos θ) dφ. In practice the widths of the averaging functions can be made

so small that

d„¦fσ (θ, φ) fρ (θ, φ) ≈ δσρ . (10.43)

With this preparation, we de¬ne wave packet operators

ω d„¦

a† (ω) ≡ fσ (θ, φ) a†σ (k) , (10.44)

σ s

c3/2 2π

that satisfy

aσ (ω) , a† (ω ) = δσρ 2πδ (ω ’ ω ) ,

ρ

(10.45)

[aσ (ω) , aρ (ω )] = 0 .

For a given value of the channel index σ, the operator a† (ω) creates photons in a wave

σ

packet with propagation unit vectors clustered near the channel value kσ = kσ /kσ ,

and polarization sσ ; however, the frequency ω can vary continuously. These operators

are the continuum generalization of the operators ams (ω) de¬ned in eqn (8.71).

With this machinery in place, we next look for the appropriate generalization of

the incident state in eqn (10.36). Since the frequencies of the emitted photons are not

¬xed, we assume that the source generates a state

dω dω

C (ω, ω ) a† (ω) a† (ω ) |0 , (10.46)

sig idl

2π 2π

describing a pair of photons, with one in the signal channel and the other in the

idler channel. As discussed above, propagation from the source to the beam splitter

multiplies the state a† (ω) a† (ω ) |0 by the phase factor exp (ikLsig ) exp (ik Lidl ). It

sig idl

is more convenient to express this as

eikLsig eik Lidl = ei(k+k )Lidl eik∆L , (10.47)

where ∆L = Lsig ’ Lidl is the di¬erence in path lengths. Consequently, the initial state

for scattering from the beam splitter has the general form

dω dω

C (ω, ω ) eik∆L a† (ω) a† (ω ) |0 ,

|¦in = (10.48)

sig idl

2π 2π

where we have absorbed the symmetrical phase factor exp [i (k + k ) Lidl ] into the

coe¬cient C (ω, ω ).

¿¾¼ Experiments in linear optics

By virtue of the commutation relations (10.45), every two-photon state

a† (ω) a† (ω ) |0 satis¬es Bose symmetry; consequently, the two-photon wave packet

sig idl

state |¦in satis¬es Bose symmetry for any choice of C (ω, ω ). However, not all states

of this form will exhibit the two-photon interference e¬ect. To see what further restric-

tions are needed, we consider the balanced case ∆L = 0, and examine the e¬ects of the

alternative processes on |¦in . In the transmission“transmission process the directions

of propagation are preserved, but in the re¬‚ection“re¬‚ection process the directions of

propagation are interchanged. Thus the actions on the incident state are respectively

given by

1 dω dω

tt

C (ω, ω ) a† (ω) a† (ω ) |0 ,

|¦in ’ |¦in = (10.49)

tt sig idl

2 2π 2π

and

1 dω dω

rr

C (ω, ω ) a† (ω) a† (ω ) |0

|¦in ’ |¦in =’

rr sig

idl

2 2π 2π

1 dω dω

C (ω , ω) a† (ω) a† (ω ) |0 .

=’ (10.50)

sig idl

2 2π 2π

For interference to take place, the ¬nal states |¦in tt and |¦in rr must agree up

to a phase factor, i.e. |¦in tt = exp (iΛ) |¦in rr . This in turn implies C (ω, ω ) =

’ exp (iΛ) C (ω , ω), and a second use of this relation shows that exp (2iΛ) = 1. Con-

sequently the condition for interference is

C (ω, ω ) = ±C (ω , ω) . (10.51)

We will see below that the (+)-version of this condition leads to the photon pairing

e¬ect as in the degenerate case. The (’)-version is a new feature which is possible

only in the nondegenerate case. As shown in Exercise 10.5, it leads to destructive

interference for the emission of photon pairs.

In order to see what happens when the interference condition is violated, consider

the function

2

C (ω, ω ) = (2π) C0 δ (ω ’ ω1 ) δ (ω ’ ω2 ) (10.52)

describing the input state a† (ω1 ) a† (ω2 ) |0 , where ω1 = ω2 . In this situation pho-

sig idl

tons entering through port 1 always have frequency ω1 and photons entering through

port 2 always have frequency ω2 ; therefore, a measurement of the photon energy at ei-

ther detector would provide which-path information by determining the path followed

by the photon through the beam splitter. This leads to a very striking conclusion:

even if no energy determination is actually made, the mere possibility that it could be

made is enough to destroy the interference e¬ect.

The input state de¬ned by eqn (10.52) is entangled, but this is evidently not enough

to ensure the HOM e¬ect. Let us therefore consider the symmetrized function

C (ω, ω ) = (2π)2 C0 [δ (ω ’ ω1 ) δ (ω ’ ω2 ) + δ (ω ’ ω1 ) δ (ω ’ ω2 )] , (10.53)

which does satisfy the interference condition. The corresponding state

¿¾½

Two-photon interference

|¦in = C0 a† (ω1 ) a† (ω2 ) |0 + a† (ω2 ) a† (ω1 ) |0 (10.54)

sig sig

idl idl

is not just entangled, it is dynamically entangled, according to the de¬nition in Section

6.5.3. Thus dynamical entanglement is a necessary condition for the photon pairing

or antipairing e¬ect associated with the ± sign in eqn (10.51). This feature plays an

important role in quantum information processing with photons.

In the experiments to be discussed below, the two-photon state is generated by the

spontaneous down-conversion process in which momentum and energy are conserved:

ωp = ω + ω ,

(10.55)

kp = k + k ,

where ( ωp , kp ) is the energy“momentum four-vector of the parent ultraviolet photon,

and ( ω, k) and ( ω , k ) are the energy“momentum four-vectors for the daughter

photons. The energy conservation law allows C (ω, ω ) to be written as

C (ω, ω ) = 2πδ (ω + ω ’ ωp ) g (ν) , (10.56)

where

ω’ω

, ω = ω0 + ν , ω = ω0 ’ ν .

ν= (10.57)

2

The interference condition (10.51), which ensures that the two Feynman processes lead

to the same ¬nal state, becomes g (ν) = ±g (’ν).

The conservation rule (10.55) tells us that the down-converted photons are anti-

correlated in energy. A bluer photon (ω > ω0 ) is always associated with a redder photon

(ω < ω0 ). Furthermore, the photons are produced with equal amplitudes on either

side of the degeneracy value, ω = ω0 = ωp /2, i.e. g (ν) = g (’ν). Thus the coe¬cient

function C (ω, ω ) for down-conversion satis¬es the (+)-version of eqn (10.51). The

2

width, ∆ν, of the power spectrum |g (ν)| is jointly determined by the properties of

the KDP crystal and the ¬lters that select out a particular pair of conjugate photons.

The two-photon coherence time corresponding to ∆ν is

1

„2 ∼ . (10.58)

∆ν

We are now ready to carry out a more realistic analysis of the Hong“Ou“Mandel

experiment in terms of the interference between the tt- and rr-processes. For a given

value of ν = (ω ’ ω ) /2, the amplitudes are

1

Att (ν) = t2 g (ν) ei¦tt (ν) ’ g (ν) ei¦tt (ν) (10.59)

2

and

1

Arr (ν) = r2 g (ν) ei¦rr (ν) ’ ’ g (ν) ei¦rr (ν) , (10.60)

2

where the ¬nal forms hold for a balanced beam splitter and ¦tt (ν) and ¦rr (ν) are the

phase shifts for the rr- and tt-processes respectively. The total coincidence probability

is therefore

¿¾¾ Experiments in linear optics

2

dν |Att (ν) + Arr (ν)|

Pcoinc =

∆¦ (ν)

2

dν |g (ν)| sin2

= , (10.61)

2

where

∆¦ (ν) = ¦tt (ν) ’ ¦rr (ν) . (10.62)

The phase changes ¦tt (ν) and ¦rr (ν) depend on the frequencies of the two photons

and the geometrical distances involved. The distances traveled by the idler and signal

wave packets in the tt-process are

Ltt = Lidl + L1 ,

idl

(10.63)

Ltt = Lsig + L2 ,

sig

where L1 (L2 ) is the distance from the beam splitter to the detector D1 (D2). The

corresponding distances for the rr-process are

Lrr = Lidl + L2 ,

idl

(10.64)

rr

Lsig = Lsig + L1 .

In the tt-process the idler (signal) wave packet enters detector D1 (D2), so the phase

change is

ω ω

¦tt (ν) = Ltt + Ltt . (10.65)

idl

c sig

c

According to eqn (10.50), ω and ω switch roles in the rr-process; consequently,

ω rr ω

Lidl + Lrr .

¦rr (ν) = (10.66)

c sig

c

Substituting eqns (10.63)“(10.66) into eqn (10.62) leads to the simple result

∆L

. (10.67)

∆¦ (ν) = 2ν

c

Since the two photons are created simultaneously, the di¬erence in arrival times of the

signal and idler wave packets is

∆L

∆t = . (10.68)

c

The resulting form for the coincidence probability,

2

dν |g (ν)| sin2 (ν∆t) ,

Pcoinc (∆t) = (10.69)

2

has a width determined by |g (ν)| and a null at ∆t = 0, as shown in Exercise 10.3.

As expected, the null occurs for the balanced case,

Lsig = Lidl = L0 . (10.70)

In this argument, we have replaced the plane waves of Section 10.2.1-A with