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so the only remaining possibility is ωM = ωL ’ ωIF . Again borrowing terminology from
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)

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

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