Vol. 6, No. 5/May 1989/J. Opt. Soc. Am. B
H. Fearn and R. Loudon
917
Theory of two-photon interference
H. Fearn and R. Loudon
Department of Physics,Essex University, Colchester C04 3SQ, England
Received September 13, 1988; accepted December 14, 1988
The interference effects that can be observed in the two output arms of a lossless beam splitter are calculated for in-
cident light in the form of a photon-pair excitation in the two input arms. The output state that occurs when the
photon pair is excited in a single input arm resembles that expected for independent classical particles, whereas
quantum interference effects occur when the photon pair is divided between the two input arms. Detailed output
photocount correlation functions are calculated for two-photon input states produced by a two-atom light source,a
degenerate or nondegenerate
1.
parametric oscillator in a high-Q cavity, and an atomic cascade emission light source.
INTRODUCTION
Fundamental properties of the Bose-Einstein statistics can
be studied experimentally with the aid of a beam splitter and
a two-photon light source. Thus, with the notation illustrated in Fig. 1, the photon outputs in arms 3 and 4 can be
measured for input states in which a pair of photons enters
the beam splitter through the samel"2 input arm 1 or through
different3' 4 arms, with one photon in each of the inputs 1 and
2. It is found that the output photon distribution resembles
that of independent particles when they arrive in the same
arm but that the quantum-mechanical interference effects,
characteristic of Bose-Einstein statistics, are observed when
the two photons arrive in different arms.
It is not difficult to calculate the expected output photon
distribution on the basis of a simple single-mode theory with
discrete input and output quantization, 5-8 and the results
are shown in Table 1. Here t and t are the complex reflection
and transmission coefficients of the beam splitter; the columns correspond to the three possible input states In,,n2),
and the rows correspond to the probabilities P(n3 , n 4 ) for the
three possible output states. The coefficients of a lossless
beam splitter satisfy
II2 + ItI 2
=
1,
tL* + ,to* = 0,
(1)
and the probability distributions in Table 1 are therefore
normalized. It is seen that the output probability has the
binomial form characteristic of classical particles when the
input photons are in the same arm (second and fourth columns). A strikingly different output probability is obtained
when the input photons are separated in arms 1 and 2 (third
column), with twice the classical probability for output photons in the same arm and a probability for one photon in
each output arm that vanishes in the case of a 50/50 beam
splitter with 1n1 = ItI.
The single-discrete-mode theory mentioned above is incapable of describing time-dependent correlations between
the input photons, and the aim of the present paper is a more
complete description of two-photon interference effects by
means of a continuous-mode theory for various types of light
source. The most elementary source has only two atoms,
whose excitation at appropriate times can provide input
photons with arbitrary time separation. Although the twoatom source is not practical in terms of realistic experiments,
0740-3224/89/050917-11$02.00
consideration of the interference effects produced with its
emitted photons clarifies the role of the Bose-Einstein statistics, particularly in regard to the distinction between the
results obtained for both photons in the same arm and the
results obtained for each photon in a different arm.
Practical two-photon light sources are based on the parametric downconverter or on atomic cascade emission. With
suitably low levels of excitation, the emissions from both of
these sources consist of highly correlated photon pairs, with
a relatively low correlation between the photons emitted in
different pair events, as was first shown experimentally by
Burnham and Weinberg9 and by Clauser.10 Consideration
of the two-photon interference observable with parametric
and atomic-cascade sources further clarifies the physical
natures of these effects. We treat the parametric oscillator
in an optical cavity, in both its degenerate and nondegenerate modes of operation, together with an incoherently driven
three-level atomic cascade and evaluate their potentials as
light sources for two-photon interference experiments.
The experiments performed to dates have used parametric downconverters
in free space as light sources, for which,
in contrast to the cavity oscillator and atomic cascade
sources, theory1 l12 and experiment'3 indicate negligible
time separations between the two photons in an emitted
pair. The detailed theoretical descriptions3"14 of the interference effects in these experiments must include the filters
that are used to limit the optical bandwidths. The light
sources treated below, on the other hand, have bandwidths
that are limited by their intrinsic properties, and there is no
need to include the effects of external filters.
2. TWO-ATOM
LIGHT SOURCE
In this section we derive the characteristics of the two-photon state emitted by a pair of atoms excited at arbitrary
times. Such a source is unrealistic in terms of practical
experiments, and the calculation below is somewhat artificial, but the results help in understanding the natures of the
two-photon interference effects.
Consider the far field produced by spontaneous emission
from an atom of transition frequency w and decay rate 2-yj
put into its excited state at time ti. The single-photon
electric-field matrix element at distance x from the atom is15
© 1989 Optical Society of America
918
J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989
H. Fearn and R. Loudon
a(w) = (yl/7r)1/2 exp[ic(tl + x/c)]/(w
+ i-yl).
-
(8)
If an associated time-dependent function is defined by
a(t) = (2r)-/
Source
2
J
dw a (co)exp(-iwt),
(9)
then the insertion of Eq. (8) produces the result
Detector
a(t) = -i(2yl)"/ 2 expl-(iw1 + yl)(t- t, - x/c)J0(t - t, - x/c),
1
X
4~~~~~I
(10)
when -y,is again assumed to be much smaller than w, so that
the range of integration can be extended to -a. This timedependent function satisfies the normalization condition
X2
Jdtla(t)1
Source
2
2
Fig. 1. Beam-splitter configuration showingnotation for input and
output arms.
(11)
= 1,
and a(t) is indeed a normalized replica of the electric-field
matrix element [Eq. (2)].
We now consider the use of pairs of atoms for the light
sources in the experiment represented in Fig. 1. It is neces-
(01l(x, t)II)
=
ewl2D sin
sary to add subscripts to the above notation to take into
account the two beam-splitter input arms. With independent boson operators, al(X)and &2(M, for the two inputs, the
0
4rEoc 2x
X expl-(iol + yl)(t - t,
-
xc)J0(t - t, - xc), (2)
where ao is the angle between the direction of the transition
dipole moment D and the observation direction x, in the
plane of polarization of the electric field; is the usual unit
step function; and the decay rate has been assumed to be
much smaller than the transition frequency, 7y << i. We
consider experiments in which a parallel beam is formed by
suitable collection of the light from a small section of the
spherical wave front.
One-dimensional optical excitations can be described in
general' 6 by a continuous-mode field whose creation and
destruction operators have the commutation relation
[a(U), at(e')] = (
-
').
(3)
commutator
[Eq. (3)] is generalized to
[ai()), &jt((o')]= 8b jb(c -
i, i = 1, 2.
'),
(12)
Creation operators Ait(a) and A2t(a) are defined by relations similar to Eq. (4) in terms of the input-arm creation
operators
dlt(co) and
2 t(C).
Thus a state with a single
excitation in arm 1 and none in arm 2 is denoted
1l, 02) = Alt(a)101, 02),
(13)
where the state on the right-hand side is the joint vacuum
state.
For two atoms in the light source of arm 1, in general
producing excitations with different frequency functions
al(w) and a 2 (M, the field state is represented by
A single-photon excitation is defined by means of the creation operator
121,02)
=
NA t(a)A 1t(a2 )101,
02),
(14)
where N is a normalization constant. The normalization
At(-) = fo dica()t(c),
(4)
where a(cX)is a function that describes the frequency distribution of the excitation, normalized according to
J
dwla(v)12 = 1.
(5)
It is easily shown with the use of Eqs. (3) and (5) that the
single-photon state
11) = At(a)Io)
(6)
is normalized and that the destruction operator matrix ele-
condition gives
(21, 02121,02) = 1.= N2[1 +
(7)
The far-field matrix element [Eq. (2)] can be Fourier
transformed and normalized to obtain the corresponding
frequency function in the form
a*(M)a 2 (G)
21 (15)
where the normalization poperty [Eq. (5)] and the commutation relation [Eq. (12)] have been used. Thus the usual
boson normalization constant 2-1/2is recovered for identical
excitation functions, al (co)= a 2(0, but N otherwise takes on
a larger value, and it becomes equal to unity for sufficiently
different excitation times t and t 2 of the two atoms in the
source. Frequency functions of the form of Eq. (8), with
different atomic parameters c, and 1 and 2 and 72, give an
overlap integral
ment is
(0I&(c)I1) = a(U).
Jdc
J
dw
ae1*(w)a
2 (C0) =
2i(^y1 y2 )"12 exp[(iwl - Y1 )(t
2 -
tJ
W - W2 + i(7 l + 2)
for t2 > tl
= same with 1
2
for t > t 2 . (16)
Vol. 6, No. 5/May 1989/J. Opt. Soc. Am. B
H. Fearn and R. Loudon
Thus, for the special case of two identical
JT
atoms in the
~~~~(mi(T)
) = i dt Oit(t)&#~)),
source, the field normalization constant is
N = [1 + exp(-2yltl
-
2,
t 2 1)]"
(17)
where 27yis the common decay rate.
A two-photon excitation with one photon in each of the
two input arms is correspondingly represented by the state
(18)
11i, 12) = Alt(aci)A 2t(a 2 )101,02)-
The state is normalized as it stands because of the independence of the operators for the two input arms and the normalization property of the single-photon states [Eq. (6)].
3. PHOTODETECTION
represented
The time-dependent operators defined in Eq. (21) have
properties that mimic those of their frequency-dependent
counterparts. Thus if the scope of the definition [Eq. (21)]
is extended to include the input-mode operators, it is not
difficult to prove the commutation properties
[ai(t), ajt(t/)] = bij(t
a3(t) = ,&1(t)+ t& 2 (t),
in Fig. 1. We need to express the
a4(t) = tal(t) + ta 2(t)
rT
i = 3,4,
(19)
where the angle brackets on the left-hand side denote an
average over repeated counting periods with the same integration time T and the electric-field operators in the quantum-mechanical expectation value on the right-hand side
refer to the fields at the two detectors. The positive-frequency part of the electric-field operator for a plane parallel
(24)
and the properties [Eqs. (1)] of the coefficients. The singlephoton creation operator [Eq. (4)] can be written in the
equivalent form
photocounts registered by the detectors in arms 3 and 4 are
given by
J| dt(2i-(t)i+(t)),
(23)
i, j = 1, 2 or 3, 4,
t'),
-
where the consistency of the input and output commutators
is ensured by the standard beam-splitter input-output relations
THEORY
various measurable quantities in terms of the properties of
the two-photon input states.
According to standard photodetection theory,'7 the mean
(mi(T))
(22)
where the ni are detector quantum efficiencies.
In this section we consider the nature of the measurements
that can be made by the detectors in the beam-splitter experiment
i = 3, 4,
919
Ait(a) =
f
dt a(t)ait(t),
(25)
where a(t) is defined by Eq. (9). The analog of the matrix
element [Eq. (7)] is
(26)
(01e(t)I1) = a,
and the overlap integral in Eq. (15) or Eq. (16) is equal to its
time-dependent twin
Jdt a1*(t)a (t) Jdco
2
=
(27)
v1*(0)a2(W)-
polarized light beam propagated parallel to axis xi is
P+(xi,t) = if
dw(hw/47re0cA)"2 a(c)exp[-i(t
-xic)],
i = 3, 4,
where A is a beam quantization
(20)
cross section and the &i(c)
are continuous-mode field operators for the two output
beams. The negative-frequency part of the field operator is
It should be emphasized again that the approximations inherent in the use of these time-dependent operators are
valid only for narrow-bandwidth excitations and photodetection.
Higher-order moments of the photocount distribution can
be treated in a similar fashion. Thus the second moments of
the photocounts in each detector are given by
given by the Hermitian conjugate of Eq. (20).
In the calculations that follow the integration time T in
expression (19) will be assumed to be much longer than any
correlation or coherence times of the input light and the
optical travel times from the beam splitter to the detectors.
The term xi/c that occurs in the exponent of Eq. (20) can
thus be neglected in view of the occurrence of a conjugate
pair of field operators in expression (19). Furthermore, we
will be concerned only with input excitations of narrow frequency spread, where the matrix elements of the &e(cv),simi-
(m,(T)2)
oT
T
(mp(T)) + n,2J dt J dt'(eit(t)Wzt(t')eai(t)ai(t')),
=
i = 3, 4
(28)
and the correlation between the two detectors is
(m 3 (T)m
4
T
T
(T)) = n3 44 J" dt J| dt/(& 3 t(t)
4 t(t')a 3 (t)a 4
(t')).
(29)
lar to Eq. (7), are sharply peaked functions, similar to a(c)
given by Eq. (8). In this regime it is a good approximation to
set cvin the square-root factor of Eq. (20) equal to a mean
excitation frequency and to extend the range of integration
down to
--.
Then, the field operator [Eq. (20)] is approxi-
mately proportional to
2
ai(t) = (27r)-Y f
dcoai(c)exp(-iwt),
and the mean photocount [expression (19)] becomes
(21)
Table 1. Output Photon Probabilities for Two Input
Photonsa
Probability
21,02)
P(2, 0)
JtJ4
P(1, 1)
21,,121.t12
P(O,2)8.J4
aFrom
Ref. 8-
11, 12)
21,,121t12
2 2
(,LI2- Jti )
21tll
101,
22)
JtJ4
21r121.t12
JtJ4
920
4.
J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989
TWO-PHOTON
H. Fearn and R. Loudon
INTERFERENCE
(m 3 m 4 ) = 137 7 4P(1,1),
We now use the formalism of the preceding sections to treat
two-photon interference effects in the output arms of the
beam splitter.
A.
Both Incident Photons in Arm 1
The input state is that defined in Eq. (14), with the normalization constant given in Eq. (15). The mean photocount at
the detector in arm 3 obtained by evaluation of Eq. (22) with
this input state, using the relations of Eqs. (24) between
input and output operators and the commutator [Eq. (23)],
is
J
(m3 (T)) = n3t 2N2 J| dt
dt'Iel(t)a 2(t') + a1(t')a2 (t)12 ,
(30)
where the t integral is restricted to the photocounting period
but the t' integral is unrestricted.
It is assumed here and in the remainder of this section that
the experimental conditions are such that the photons arrive
at the detectors during the counting period. In other words,
the functions al(t) and a 2(t) have negligible amplitudes
when t lies outside the range 0-T. The range of the t integration in Eq. (30) is then effectively infinite, and with the
use of Eqs. (11), (15), and (27),
(m 3 (7))
= 2
2
3 k1 .
(31)
(32)
(m 4 (T)) = 2 4LtJ2 .
Similar calculations can be made for the second-order
photocount averages defined in Eqs. (28) and (29). Thus,
analogous to Eq. (30), the correlation between the two detectors is found to be
2
(m 3 (T)m 4 (T)) = 2 3n4lLI
j'tI2N2
J
dt
J
(m 4 (m4 - 1)) = 27742 P(0, 2),
(37)
where the spurious notational dependences of the photocount averages on the integration time T have been removed. Comparison of Eqs. (37) with Eqs. (34)-(36) produces expressions for the probabilities identical to the second column of Table 1. Therefore the present calculation
shows that, provided that both incident photons in arm
arrive at the detectors during their counting periods, it
makes no difference to the detection statistics whether the
photons are temporally coincident [al(t) = a2(t)] or completely separated [no overlap between al(t) and a 2(t)]. This
insensitivity to the forms of a,(t) and a2(t) occurs essentially
because the apparent dependence of the detection statistics
on photon overlap is canceled by an identical dependence on
photon overlap of the normalization constant of the incident
two-photon state. Lange et al.2 have shown both theoretically and experimentally that a similar cancellation effect
removes any dependence of detection statistics on the spatial overlap of the photon states. Thus in all cases the
beam-splitter statistics for photons incident in the same arm
follow the same binomial distribution as would be expected
for incident classical distinguishable particles.5 - 8
B.
One Incident Photon in Each Arm
The input state is that defined in Eq. (18), and the mean
photocount at the detector in arm 3, obtained in the same
With the same assumptions,
X
(m 3 (m 3 - 1)) = 2q3 2 P(2,0),
dt'[Ial(t)12 1a2 (t')12
+ al*(t)a
2 (t)al(t')a
manner as in Subsection 4.A above, is
rT
(m 3 (M)) =
73
J dt[IoIja,(t)j
2
2
2
+ t121ja
2 (t)1].
(38)
We again assume that al(t) and a 2(t) have negligible amplitudes outside the range 0-T, so that use of Eq. (11) simplifies
Eq. (38) to
(m 3(M))= 73.
2*(t')]-
(39)
With the same assumption,
(33)
Both time integrals are now restricted to the counting period, but the assumption of photon arrivals during this period
effectively converts the double integral to 1/N2 , and Eq. (33)
reduces to
(m 3 (O)m
=
4 (T))
2
237741,121[t
.
(34)
Similarly, evaluation of the second moments [Eq. (28)] for
the input two-photon state [Eq. (14)]gives
(m3(7)m 3(T) - 1]) = 2 321
4
(35)
(m4(T)) =
Similar calculations can again be made for the secondorder photocount averages, and the correlation between the
two detectors is found to be
2
(m3 (T)m 4 (T)) = 773774
f dt JTdt',[(Il4 + t14)IC,1(t)j
ja2 (t')j2
+ (t 2,t*2 +
2
t*
.t2,*2 + t2t*2
-
4
1]) = 277421t1
.
(36)
The probabilities P(n3, n4 ) for finding n3 and n4 photons at
the detectors in arms 3 and 4, respectively, according to the
discrete-mode theory of the beam splitting of photon number states,8 are listed in Table 1. The above results are
related to these probabilities by
2
)a*(t)a2(t)a1(t)a2*(t')].
(41)
It followsfrom Eqs. (1) that
and
(m 4 (T)[m 4 (T)
(40)
4.
=
-21
2
2
[1tJ
,
(42)
and with the same restrictions as above on the functions
al(t) and 2(t), the correlation becomes
(m3 (T)M4 ())
= 17377
+ ftI4 4 [I7LI4
2412 12 f dt al*(t)a 2(t) ]- (43)
Vol. 6, No. 5/May 1989/J. Opt. Soc. Am. B
H. Fearn and R. Loudon
The corresponding result for the second factorial moments
of the counts at the individual detectors, obtained by evaluation of Eq. (28), is
(Mi(y)[Mi(T)
-
J
dt a,*(t)a
1]) = 2nq,2j7,L2tj2[1 +
2 (t)
i = 3, 4.
921
1.0
21
l
/
/
(44)
For identical photon states in the two arms, with al(t) =
a2(t), these second moments simplify with the use of Eq.
(11), and the photon distribution in the two output arms,
obtained by comparison with Eq. (37), is
P(1, 1) = (1,2-
t12 ) 2 ,
P(2, 0)
=
2 2
[tJ, (45)
P(0, 2) = 21XI
in agreement with the results of the discrete-mode calculation 8 reproduced in the
Ii,, 12)
for photon states that have negligible overlap in time, the
output distribution is
P(1, 1) =
1,14 +
LtI4,
2 2
LtI,
P(2, 0) = P(0, 2) = I,1I
(46)
in agreement with the result expected for incident classical
distinguishable particles. An expression equivalent to Eq.
3
(43) has been derived by Hong et al.
The overlap integral that occurs in Eqs. (43) and (44) has
the explicit form given in Eq. (16). With a slight generaliza-
tion to allow for different distances xl and x2 from the source
atoms in arms 1 and 2 to the beam splitter, the correlation
[Eq. (43)] becomes
(m 3 (T)m4 (T))
=
4
1,14 + [t1 713174{
0
column of Table 1. However,
21t12717 exp[-27,(t 81'1
2
2
(cv-
2)
t + 60)]
+ (71 + 72)2
J
(47)
Y(t2-t + )
2
Fig. 2. The photocount second factorial moment (mi(mi-1) )/mi
(dashed curve) and the correlation
(m3m4 )/7 3
n4
(solid curve) for
excitation of each input arm by identical atoms.
5. TWO-PHOTON INTERFERENCE WITH
LIGHT FROM A PARAMETRIC OSCILLATOR
The theory of two-photon interference given in Section 4 is
somewhat unrealistic because it assumes the availability of
single-photon light sources. In practice the experiments are
done with continuous sources in which the light is emitted as
a stream of correlated photon pairs. In this section we
consider the theory of beam-splitter interference effects
with light from a parametric oscillator or downconverter.
The light may be emitted as photons of degenerate frequency, in which case the beam is assumed to enter the beam
splitter through a single arm, or as nondegenerate,
in which
case beams of different frequency xl and cv2 can be arranged
to enter the beam splitter through arms 1 and 2.
where
T
(48)
(X2 - X)/C.
This expression applies when the atomic excitation times
and distances from the beam splitter are such that t < t 2 +
6r; when the reverse inequality is satisfied,
y in the expo-
nent must be replaced by -72Figure 2 shows the forms of the second-order photocount
averages for identical source atoms, with ci = c2 and 71 = 72
= y, and for a 50/50 beam splitter with ,12 = ltl2 = 1/2. It is
seen that the correlation [Eq. (47)] vanishes in this case
when the atoms are excited at times t1 and t2 such that tj = t2
+ br. In other words there is a zero probability for finding
one photon in each of the output arms. This phenomenon is
a form of a quantum-mechanical
interference
effect that
arises from the Bose-Einstein commutation properties of
the photon-creation and -destruction operators. Feynman
et al.18 discussed the related doubling of the Bose-Einstein
probability for two photons in the same output arm shown in
Eq. (45), compared with the corresponding probability for
3
classical particles given in Eq. (46). Hong et al. measured
the variation of the correlation function [Eq. (43)] with the
degree of superposition of the input photon states, and they
observed the expected interference for complete superposition at a symmetrical beam splitter. The functions a, and
a2 in their experiment were determined by the passbands of
interference filters placed in the light beams from a parametric downconverter.
A. Degenerate Parametric Oscillator, Entry through a
Single Arm
We assume a parametric oscillator of the kind treated by
Collett and Gardiner,19 in which the parametric crystal,
driven at a rate denoted by a parameter e, is placed in a
single-ended cavity with mirror transmission rate 1/27. The
output properties of such an oscillator have been summarized by Collett and Loudon,20 and we use their notation and
results here. In the absence of any input excitation at the
degenerate frequency c, the output beam has a steady-state
flux
2
f = ( lt(t)a l(t) ) - /2 6 2 7Y/(1/
47 _-
)
(49)
where the oscillator is assumed to be operated below threshold (e < /2y) and the mode spacing within the cavity is larger
than y, so that only a single internal mode may be excited.
The subscript 1's in Eq. (49) signify that the light beam is
directed to input arm 1 of the beam splitter. The light is
squeezed with a squeezing parameter s given by
exp s =
(/2y + E)/('/
2
y-e),
(50)
and strong squeezing is obtained when the driving rate f is
increased toward its threshold value 1/2y. However, the twophoton interference effects are particularly marked for small
values of E,where Eq. (49) reduces to
922
J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989
f1 = 2 2/y
H. Fearn and R. Loudon
for e << 1/27y.
2/fI
(51)
The mean photocounts registered by the detectors in arms
3 and 4 are easily calculated from Eq. (22) with the use of
Eqs. (24) and (49), where the input vacuum state is assumed
for arm 2. Thus
(52)
2)
(/2E7)
2le)+
E)
+ '/2exp(-l7l[(l /l
('/27+xel)2
J'
(53)
where
T=
t - C.
(54)
The correlation [Eq. (29)] between the two detectors, assumed to be equidistant from the beam splitter, is thus
readily evaluated with the use of Eq. (24), and in the limit of
a long integration time T we find that
2
tl 2 fT
L
'/2 7TI.(55)
4
2
X T+14Y(1/7EY)2
('/ 7+ E)2 j
{ fT +'/
'1/
7+
+
for yT >> 1. This result is particularly transparent in the
limit of a small parametric driving rate where it reduces to
(T))
4
2
2
l3 ?14 II ItI fT(fT
=
e <<1/2y, (56)
+ 1),
and the corresponding expressions for the second moments
of the photocounts for the individual detectors obtained
from Eq. (28) are
(m 3 (T)[m 3 (T)
-
1]) =
73
]=
4
(m 4 (T)[m 4 (T)
2
2
1'L4fT(fT + 1),
tI 4fT(fT
+ 1),
<< 1/27.
(57)
These expressions generalize the corresponding results of
Eqs. (34)-(36) obtained for a single pair of incident photons.
They can be rederived by a simple statistical calculation.
Suppose that the parametric oscillator emits n photons
during the integration time T. The integral of the correlation function [Eq. (53)] is then obtained from a statistical
T
T
exp(-(v))2v(2v
- 1)
= 2(v)(2(v) + 1)
+ 1).
(61)
This procedure thus reproduces the results of Eqs. (56) and
(57). The first and second terms in the last set of parentheses of Eq. (61) represent the contributions of photons within
different pairs and within the same pair, respectively. The
occurrence of these two contributions in the second-order
correlation for the output of a parametric oscillator has been
verified experimentally.2'
B. Nondegenerate
Both Arms
dt'(alt(t),lt(t')el(t)l(t'))
-
(nl(n - 1)). (58)
Parametric Oscillator, Entry through
Consider now a nondegenerate parametric oscillator as the
light source, with the two output beams, whose frequency
spectra are centered on cv and 2, directed to inputs 1 and 2,
respectively, of the beam splitter. It is convenient to scale
the parametric driving rate e by a factor of 1/2 for the nondegenerate case to maintain similarities with the equations for
the degenerate case. The identical fluxes in the output
beams are then the same as in Eqs. (49) and (51), given by
f f
=
2
=
2
/ 2 E y/('/
4
7
2
_
2
)
2E 2 /
for e << 1/27y. (62)
The mean photocounts at the two detectors are obtained
with the use of Eqs. (22) and (24) as
(m 3 (T)) = 3fT,
(m 4(T)) = n4fT,
(63)
similar to Eqs. (39) and (40) but with the single incident
photon in each input arm replaced by the mean integrated
photon number fT.
The interference experiment can be arranged so that the
optical path lengths x and 2 from the parametric oscillator
to the beam-splitter inputs are different, and it is then important to take account of retardation effects. We define
retarded times
average according to
fT dt
()
v0=
= fT(fT
[1/7Y2
(m3 (T)m 4 (T)) = M4
3 fl4 l
'
1)) =
-
2E
-
(m3(T)m
(nl(n
2
(t))=
(60)
The average required for expression (58) is therefore
similar to Eqs. (31) and (32) but with the pair of incident
photons replaced by the mean photon number fjT arriving
at the beam splitter during the integration time T.
The fourth-order correlation function of the light from the
parametric oscillator is2 0
(&j~te~t~')j
(P) = /2 (nj) = '12fT
v = /2 n,
(m 4 (T)) = n4flTtI ,
(59)
which is assumed much longer than the time separation, of
order 1/7, between photons in the same pair. Let v denote
the number of photon pairs emitted in the integration time
T. so that
(m 3 (T)) = n3fTj ;&,
2
= 7/E2,
t = t-x,/c,
t, = t -/c,
t 2 = t-X
t 2' = t -X 2 /C
2 /c,
(64)
and again use the notation
r = t - t,
3 = (X2
X)/C
(65)
The light beam is modeled as a random Poisson stream of
photon pairs, whose mean time separation is determined by
The required fourth-order correlation functions obtained by
Eq. (51) to be
then
generalization
and extension of results given in Ref. 20 are
Vol. 6, No. 5/May 1989/J. Opt. Soc. Am. B
H. Fearn and R. Loudon
2t(t 2 )a 2 t(t 2 1)&2
= (
(2
923
(t2 )a 2 (t2 1))
1
)\2(r472
e)2 + - exp(-1T1l)
Yi(1/72..
7)
_)exp-C|T) 2
[exp(d4l)
- E
1/ +e
2
1
1/
/1\2 rE
)=y7)
1(1/2 E22+
(66)
|
exp(7rT
+ rI)
exp(-edr +
+5,rl)
'/27 + C
+
Xexp(El
L/27-E
2rI)
J
(a2t(t2)lt(tl')2(t2)al(tl') = same with ,r- -
(67)
0
-4
-8
4
8
(68)
Tr,
Fig. 3. The contribution to the scaled photocount correlation
(m3 (T)m4 (T))/A 3 in4fT,given by Eq. (70), that is independent of T
and
for values of (w1
-
W2)/1Y
shown against the curves.
(ailt(tl)a'2t(t2')ea2(t2)l(tl)) = (2t(t2)allt(tl')el(tl)ea2(t2') )*
=
( e7y)2 exp[i(cv - c2)T]
4 /
X exp(_,Yrl)[exp(EiTl) exp(-f
1exP
L
7y(T+ &rI+
+ exp-
L
2
rexp(,El|+ 6-rb
X
T-
/2 -e
[exp(eIT-
2
1
+
/f
shows some representative curves of the photocount correla-
6TI)
exp(-,Elr
+21/27
X /27 - e
Ir)
plicated function than Eq. (16). The additional time dependence in Eqs. (70) and (71) has the effect of rounding the
sharp feature that occurs for single-atom light sources with
simultaneous photon arrivals at the beam splitter. Figure 3
1/2Y+
1 12/2-
L
timing of the two inputs is now represented by a more com-
+
+
tion for a 50/50 beam splitter and for different values of cv c2, where it has been assumed that fT <<1 and only the term
T|l
eThe
- 6I)T
3rl)+ exp(-EIr
.
'/2 + e
6
(69)
The second moments [Eq. (28)] and the correlation [Eq.
(29)] can be obtained from these results with the use of Eq.
(24), but the expressions are quite complicated, and we
quote them only for the limit of a small parametric driving
«) when
rate (e <<
(m3 (T)m4 (T))
4
4
tl =3'04fT fT(+WI, ++ I t2
y sin[(wc,-cv
2 )ITI]\
cv1-cv2
X {7
(mi(T)[m(T)
2IxI2 [tI2 72 exp(-76ITI)
W2
)2 +
1
72
cv2 )2 +(cv,
1 -3n41k1
+ cos[(cv -
2)Ib-rI]jS
(70)
-1])
/f
2
2
2 + 2l)112[t121'
exp(-7IlrI)
= 17,2T fT + 21 2 1-t1
2
(cv - cv
2) + 72
1
2 1s
.
tzy sm[(@1-(o2)lat+
cv1-cv 2
t
cos[(Wj,-
2) 6TI]
J
3 4. (71)
i 4.=3, (71)
These results again apply in the limit of an integration time
T long compared with 1/y and I&I. They show two-photon
interference effects similar to those inherent in the corresponding Eqs. (43) and (44) for a single incident photon in
each input arm except that the dependence on the relative
i
linear in fT in Eq. (70) has been retained.
qualitative nature of the interference effect remains
much the same as for single-atom sources, with reduced
photocount correlation for simultaneous photon arrivals.
The temporal overlap of the photon pair emitted in a single
nondegenerate parametric generation event is determined
outside the cavity by the internal trapping time, of the order
of 1/y, consistent with the factor exp(-ylbl) in the interference terms in Eqs. (70) and (71). The model of a parametric
oscillator on which the above calculations are based has
essentially zero time delay between the emission of the two
photons of a pair inside the cavity, which is consistent with
experimental
observations.'
requirements
for parametric
3
It is seen in Fig. 3 that the
maximum interference should be observed only for c 1 = c 2 ,
but in this degenerate case the theory of the one-dimensional parametric oscillator in a cavity given in this subsection is
not strictly valid, and in practice there would be difficulties
in separating the two photons of a pair to route them into the
different beam-splitter input arms. The phase-matching
oscillators in free space pro-
duce photons in a pair traveling in slightly different directions; thus they can be separated into the two input arms
even in the degenerate case.3' 4
We note that Eqs. (70) and (71) contain terms proportion-
al to f 2T 2 in addition to the terms linear in fT discussed
above. These quadratic terms represent the contributions
of photons produced in different parametric generation
events, similar to the corresponding terms for the degenerate parametric oscillator considered in Subsection 5.A.
They tend to obscure the two-photon interference effect,
which is most clearly observable for conditions such that fT
<<1.
924
J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989
H. Fearn and R. Loudon
6. TWO-PHOTON INTERFERENCE WITH
LIGHT FROM ATOMIC CASCADE EMISSION
(*l(t)) = p21(t)exp(-iwjt),
(* 2 (t)) = p32 (t)exp(-i
An alternative source of light with strong two-photon correlations is provided by atomic cascade emission. The correlations were measured some time ago,' 0 and cascade emis-
sion light has been used2 2 to perform Hanbury Brown-Twiss
and Mach-Zehnder interference experiments under singlephoton conditions. In this section we evaluate the possibility of observing two-photon interference effects with this
kind of light source.
A.
Atomic Correlation Functions
We consider a light source consisting of identical atoms with
the three-level structure illustrated in Fig. 4. Each atom is
assumed to be incoherently excited from its ground state 11)
to an upper level 13)at a steady rate R. The atom returns to
its ground state through an intermediate level 12),emitting
photons of frequencies cv2 and cv in cascade sequence. The
relevant spontaneous emission rates are given by the usual
expressions
272 = e2 cv2 3 D2 3 2 /37re0 hc3,
27,j = e2cv13D122/37re 0 hc3,
('1t(t)'(t))
( *2t(t)*
= P22(01
t) ) = p 33 (t).
* = 1)(21,
~*rt = 12)(11,
12) (31
*r2t = 13)(21,
*r2 =
whose actions are represented
(73)
in Fig. 4.
The field correlation functions for the emitted light that
are needed to evaluate the outcome of a two-photon interference experiment are determined by the corresponding correlation functions of the atomic transition operators. We first
evaluate these atomic correlation functions by using the
equations of motion of the atomic density matrix ele-
expectation
values [Eqs. (75)].
(ijt(t)rj(t)
dp 22 /dt = 2Y2p 33 -
R,
dp 21/dt =
= R/2,yl,
( it(t,/)$lt(t,)f,(t),(t1)
(
2
dP 32 /dt = -Q(Y1+ 7Y
2)P32,
7ylp
22 ,
dp 33 /dt = R -
2
2P33'
(74)
The density matrix elements are related to expectation values of the transition operators according to6
2t(t)* 2 (t))
= R/2 72 .
(76)
_R2
)=
472(7Yl- 72)
exp(-2'y2l-I)]
-
( 2t(t2')2'(t2)t2(t2)*2(t2')
1Y1P21,
multitime
The transition operators *l and *2 given by Eqs. (73) do not
commute, and the values of the required correlation functions depend on their ordering. Time-ordered correlation
functions are needed to evaluate the detected photocount
distribution in the two-photon interference experiment
treated below. We accordingly introduce a time-ordering
operator T, which places the excitation operators to its left in
order of increasing time and the deexcitation operators to its
right in order of decreasing time. Then, with the notation
for times defined in Eqs. (64) and (65), the fourth-order
atomic correlation functions are
X 17,[1
P22 -
More-complex
expectation values can then be calculated with the use of the
quantum regression theorem.
Full accounts of the application of these methods to the
three-level model atom assumed here are available,6 25 and
the required correlation functions are generalized versions
of previously published results. We therefore quote the
expressions needed without detailed derivations. It is assumed throughout that the atomic excitation mechanism
has been turned on for a sufficiently long time to establish
steady-state conditions. The excited-state populations are
then given by
ments 2 3 ' 2 4
dp1j/dt = 2
(75)
It is simple matter to solve the equations of motion (74) and
hence to determine the time dependence of the low-order
(72)
where D12 and D 2 3 are the appropriate dipole matrix elements for the two transitions. It is convenient to define
transition operators
2 t),
72[1
R2
-
-
2
exp(-2,yjIj-)]},
(77)
) = (R 2/4Y2 2 )[1 - exp(-2y 2 171)],
(78)
(7r2 t(t 2 9)7rlt(tl)trl(tl)1r2 (t2 ))
+
-
=
(R/272)exp[-2,yl( +
ar)]
-7,72(7
711
y - exp[-27'2( + r)]l
7211- exp[-27,( + &r)]j) for +
= (R /47 ly
7 2 )11 - exp[272 ( + r)]
for
T+
> 0
6r < 0.
(79)
Expressions equivalent to these have been given previous-
ly, 6 25 and their forms provide a good theoretical description
1
Fig. 4. Notation for atomic energylevels and transition operators
used in cascade emission theory.
of the measured'0 dependence of the correlation between
photodetections at frequencies w, and 2 on the time delay,
here denoted T + &r. The lack of symmetry between the two
expressions in Eq. (79) for positive and negative time delays
produces a striking asymmetry in the experimental observations, particularly for small excitation rates R, where the
term linear in R may greatly exceed the terms in R 2. The
linear term represents the contribution of a single cascade
emission event, and from the nature of this process it occurs
Vol. 6, No. 5/May 1989/J. Opt. Soc. Am. B
H. Fearn and R. Loudon
in which photon wcis emitted after
only for measurements
2
photon cv
2. The terms in R represent the contributions of
different cascade emissions.
Finally, we need a more general fourth-order correlation
function in which all the times are different, given by
2 t(t2 )*lt(tl')t~rl(tl)$r2 (t2 '
= exp[i(cv
c 2 )r -
-
- 72[1 -
The fluxes introduced
(2,y
+ y2 )II](R
2
>
/4,17
2
(86)
f
in this way are the input analogs of
the output fluxes that form the integrand in Eq. (22), and
the mean photocounts at the two detectors obtained with the
exp(-27 1 3r)]I)
for 5,r
-
an expression that could have been written down from ele-
h1 =
72
47,72(7 - 72)
X f'yJ1 - exp(-2y 2ar)]
c 2)r
(85)
set
(27 + 72)171]
\~~~~~
-
f = RA/47rx, ,
the two beams are collected from equal solid angles, we can
))*
R2
/~~~~~~~~~
+4yl2(Y
X• (R/2_Y2)exp(-2-yj6T)
= exp[i(v 1
For steady-state conditions for which the level populations
are given by Eqs. (76), Eq. (84) reduces to
mentary considerations of the nature of the light source. A
similar derivation applies to the light of frequency c2, and if
(*1t(t1)'2t(t2/)t'2(t2)'1(t1))
= (
925
use of Eq. (24) are
(m 3 (T)) = 3fT,
I|r
(m 4 (T)) =
4fT.
(87)
The correlation [Eq. (29)] between the photocounts at the
2)
(80)
two detectors is now obtained with further use of Eq. (24)
and conversion of expectation values of field operators to
This correlation function vanishes for other relative values
expectation values of atomic transition operators according
to the prescriptions
X [1 - exp(27 2 67)]
for,& < -ITI.
of 6-r and |T|l.The above expressions also show the effects of
single cascade events, proportional to R, and different cascade emissions, proportional to R 2 .
The other possible varieties of fourth-order correlation
function, not listed above, all vanish.
B. Two-Photon Interference
We now consider two-photon interference with a cascade
emission light source in which the light of frequency v
enters the beam splitter through arm 1 and the light of
frequency c2 enters through arm 2. The electric-field operators are related to atomic transition operators by the
source-field expression (see, for example, Ref. 6)
PI+(xj, t) =
D12 sin(aj)/47rE0 c2 x)*(t,),
2
-(ecv
(81)
where a1 is the angle between the transition dipole moment
and the observation direction. The field matrix element
[Eq. (2)] is obtained by a suitable application of this operator
relation. A similar expression relates the field operator .2+
for the light of frequency c2 to the transition operator r2.
The photon-number flux through a small section of the
spherical wave front of area Al is
(m3(T)m 4 (T)) =
3
4fT[fT + (f/R)(1L14+ t
2-,y +
72
4-
Thus, retaining only nonzero atomic correlation functions,
the photocount correlation is given by
(m,(T)m 4 (T)) = n04 (4f 2 /R 2 ) J dt Jdt'{l t2ltI2
X (* 2lt(tl)*rlt(tl')Trl(tl)l(tl'))
Y2([7*2t t(tl)*2(t'2)
+ ,1j2 t127 22 (*2 t(t2)j 2t(t 2')t* 2(t2) 2 (t2) ))}
(83)
use of Eq. (81) and insertion of the spontaneous emission
rate from Eqs. (72) gives
2
.
(89)
The correlation functions that appear in the integrand are
all obtainable from Eqs. (77)-(80), and it is straightforward
but tedious to evaluate the double integral. The general
result is complicated, and, analogous to the limits assumed
above for the nondegenerate parametric oscillator, we restrict our attention to integration times T that are much
longer than 1/7l, 1/72, and 6T. Again, analogous to the
parametric oscillator, we assume a small driving rate with R
<< and R <<72, so that the effects of photon pairs emitted
The photo-
count correlation [Eq. (89)] then reduces to
81,t2 jtj 2 7,exp(-2-yj5,r)
Thus, for an averaged atomic orientation with
7 1(*,t(tj)i(t))A,/2rx,
y12
402(42 ) + t2 '2(2)*1(tl )])
+ t*2*2t(tOijt(t1)t[7L2*r'
2
(Cvl - cv2) + (27y + 72 )2
f, =
+
- exp[-(27, + 72)6Th](27,+ 72 )cos[(W1- cv2)6Tr]- (Co - cv2 )sin[(coj -
sin2 a1 = 2/3,
(88)
i = 1, 2.
(2-yf/R)ijti,
in the same cascade emission are enhanced.
(82)
t)P,+(xl, t))A1/hv.
f, = 2ec(j-(xj,
lit-
(84)
W 2 )6tr]00,)l
(90)
This result has a similar overall form to the photocount
correlation [Eq. (70)] obtained for the nondegenerate parametric light source, and the analogous result for the second
moment of the photocounts in each detector, which we do
not reproduce here, is similar to Eq. (71). The term proportional to f 2T 2 in Eq. (90) represents the contributions of
photons produced in different cascade emission events, and
926
J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989
H. Fearn and R. Loudon
produced in the two input arms. This feature contrasts with
the property of the two-atom and parametric light sources
that identical excitations can be generated in the two arms,
and it limits the amount of output interference that can be
-05
observed with a cascade source.
0
7.
2
3
Fig. 5. The contribution to the scaled photocount correlation
R(m3 (T)m 4 (T))/tnaM 4j 2T, given by Eq. (91), that is independent of T
for values of
shown against the curves.
Y2/'Yl
for a light source that contains N identical atoms this term
scales with N2. The remaining terms represent the contributions of photons emitted in the same cascade emission
events, and they scale with N. These terms show independent-particle behavior when 6,r is negative, and the step
function O(63i)
removes the negative contribution in Eq. (90).
CONCLUSIONS
We have calculated the photocount second factorial moments and interbeam correlation function for the two output
arms of a lossless beam splitter illuminated with input light
that shows strong two-photon correlations. For arrangements in which the photons in a pair enter the beam splitter
through the two different input arms, output states for
which one photon is detected in each arm are strongly inhibited for simultaneous arrival of the input photons. The
inhibition is displayed by the behavior of the output interbeam photocount correlation, which may vanish for appropriate values of the parameters for input light from twoatom and parametric oscillator sources, and can be reduced
to one half of its independent-particle
cascade emission source.
value for light from a
positive 63T,where the photon cv emitted second in the cascade has a shorter distance to travel so that both photons
may arrive simultaneously at the beam splitter. The nega-
For the calculations performed here the dependence of
output correlation on simultaneous illumination of the two
input arms, examples of which are shown in Figs. 2, 3, and 5,
are controlled by intrinsic parameters of the sources, namely, spontaneous emission rates for the atomic sources and
cavity Q for the parametric oscillator. However, similar
interference effects occur for broadband parametric oscillators in which the detection bandwidth is restricted by filters.
The dependence of output correlation on simultaneous
beam-splitter illumination is then determined by the filter
bandwidth. Striking two-photon interference effects, analogous to those derived above, have been observed in this
way.34 Oscillatory behavior of the output correlation, akin
to that apparent in Fig. 3 but much more pronounced, has
been detected with the use of narrowband filters centered on
different frequencies.2 6 Two-photon experiments involving
tive contribution in Eq. (90) is maximized for cv close to
spatial interference
A negative 3r corresponds to an input arm 1 that has a longer
path x, from source to beam splitter than the corresponding
path x2 for input arm 2. Thus the photon c1 , which is
emitted second in the cascade, is delayed further by its
additional path length, and there is no possibility of both
photons c1 and c2 arriving simultaneously at the beam splitter.
Two-photon
interference
effects do occur, however, for
2,
and we illustrate the interference by considering the case cv
= cv2,where Eq. (90) reduces to
(m 3 (T)m
4 (T))
X exp(-2y
= l3n4fT(fT +
1
br)
2
{1,I4 + [t14 - 81,121t1
71
1 - exp[-(27 1
2,y
+ 72)'7-]
+ 72
(
)1
(91)
Figure 5 shows the part of this correlation function that is
linear in T for a 50/50 beam splitter and for two relative
values of the spontaneous emission rates. It is seen that, in
contrast to the results found for the two-atom light source
and the parametric oscillator, as represented in Figs. 2 and 3,
respectively, there is now no value of the relative time delay
63T for which the photocount correlation vanishes. The
minimum value of the normalized correlation shown in Fig. 5
is 0.25, and it occurs at ylb = 0.347, in the limit whereya >>
72.
This limit corresponds physically to a cascade sequence
in which the emission of photon cv follows as rapidly as
possible after the emission of photon cv2. However, there are
no values of the parameters for a cascade emission source
such that identical and simultaneous photon states can be
27
28 show quantum
effects of a similar
nature to those considered here.
The calculations have assumed photodetection integration times T much longer than any correlation times that
characterize the input light. This assumption allows the
approximate photodetection theory of Section 3 to be used,
in conjunction with the specification of narrow-bandwidth
input excitations. However, these restrictions make the
theory given here inapplicable to studies of the intrinsic
localization properties of photons, for which it is essential to
measure the broadband characteristics of the field excitations. Such measurements may be feasible with the twophoton correlated light produced by a parametric oscillator.
29 30
,
In summary, the two-photon interference that occurs
when single-photon excitations are superposed by a beam
splitter provides a further example of nonclassical effects in
the statistical properties of light.2 529 It illustrates a textbook18 property of particles that obey Bose-Einstein statistics, and it clearly shows the continuous transition from
boson to independent-particle behavior as the temporal
overlap of the excitations is removed by a steady increase in
their relative time delay.
Vol. 6, No. 5/May 1989/J. Opt. Soc. Am. B
H. Fearn and R. Loudon
15. P. L. Knight and L. Allen, "Interference and spontaneous emis-
ACKNOWLEDGMENTS
H. Fearn thanks the Science and Engineering Research
Council for financial support in the form of a research studentship.
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16. M. J. Collett, Department of Physics, University of Waikato,
Hamilton, New Zealand, and T. J. Shepherd, Royal Signals and
Radar Establishment, Malvern, England (personal communication).
17. R. J. Glauber, "The quantum theory of optical coherence,"
Phys. Rev. 130, 2529-2539 (1963).
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