Although we tend to think of optical interference as a classical wave phenomenon, recent experiments have revealed a number of effects that are not describable in classical terms. This is particularly true of interference effects involving the detection of a photon pair. We shall refer to them as fourth order interference, on the grounds that the joint probability density for the detection of one photon at r sub 1 at time t and another r sub 2 at time t is proportional to the fourth order correlation function of the field. This probability is readily measured when two photodetectors are positioned at r sub 1 and r sub 2 and the signals from the two detectors are fed to a coincidence counter that registers 'simultaneous' detections by the two detectors in coincidence. The topics covered include: fourth order interference measurements; the Franson experiment; and experimental test of the de Broglie guided wave theory

Mandel, L.

English

NASA Technical Reports Server

N92-22046
NONCLASSICAL AND NONLOCAL EFFECTS IN THE INTERFERENCE OF LIGHT
L. Mandel
Department of Physics and Astronomy
University of Rochester
Rochester, New York 14627
INTRODUCTION
Although we tend to think of
optical interference as a classical
wave phenomenon, recent experiments
have revealed a number of effects that
are not describable in classical
terms. This is particularly true of
interference effects involving the
detection of a photon pair. We shall
refer to them as fourth order inter-
ference, on the grounds that the Joint
probability density for the detection
of one photon at _ at time t and
another at _2 at time t is propor-
tional to the fourth order correlation
function of the field (Ref. I)
^(-)(r2t )F(2'2)(rlt,r2 t) . <E(i-)(rlt)E j _
x E(+)(r2t)E(i-)j_ (rlt)> . (1)
This probability is readily measured
when two photodetectors are positioned
at _, and _2 and the signals from the
two detectors are fed to a coincidence
counter that registers 'simultaneous'
detections by the two detectors in
coincidence.
4'th ORDER INTERFERENCE MEASUREMENTS
In the special case in which the
two points x_,x 2 lie on a llne, and
This research was supported by the
National Science Foundation and by the
U.S. Office of Naval Research.
the light is produced by two sources
A,B on a parallel line such that A
emits one photon and B emits one
photon, it can be shown that (Refs.
2,3)
2_(Xl-X 2 )
F(2,2)
[1+COS L ] , (2)
where L = A/8. 8 is the small angle
subtended by the two points A,B at x,
or x2 and A is the wavelength. L is
the same fringe spacing that is
encountered in the more usual second
order interference. According to Eq.
(2) the visibility of the fourth order
interference effect can be 100%,
despite the absence of phase correla-
tion between the two sources. By
contrast a classical field that ex-
hibits 4'th order interference cannot
achieve a visibility higher than 50%.
(Refs. 2-4)
W
L
_o M _o M
Fig. I Experimental results showing
4'th order interference. [Reproduced
from Ref. 6.]
5
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We have observed greater than 50%
visibility in several recent inter-
ference experiments,(Refs. 5,6) in
which the two photons were generated
together in the process of spontaneous
parametric down-conversion in a non-
linear crystal. (Ref. 7) It is
convenient to produce the interference
pattern by mixing the two incoming
photons with the help of a 50%:50%
beam splitter with a photodetector at
each output port. Figure I shows the
experimental results when the rate of
coincidence counting, after some
corrections are applied, is plotted
against the position of one detector,
while the other detector remains
fixed. The interference pattern has
the expected periodicity L, and the
observed 75% visibility shows that we
are dealing with a quantum phenomenon,
because there is no classical field
that can give rise to more than 50%
visibility.
The same mixing technique has
been applied to the measurement of the
time separation between the two
photons on a femtosecond time scale,
and to study violations of locality.
In order to understand the principle
of the method, let us consider the
symmetric beam splitter with intensity
transmissivity T and reflecticlty R
(R+T = I), shown in Fig. 2. Let a,B
label the two input ports and U,_ the
two output ports. Suppose that the two
photons enter in the state I1a,IB>, in
which each photon is in the form of a
I
Fig. 2 The beam splitter.
short wave packet and the two wave
packets are identical and overlap
completely in time. In order to arrive
at the output state l_0out > we first
note that there are three
possibilities: (a) one photon appears
at each output (11 ,I_>); (b) both
photons appear at output port
(12 ,0 >); (c) both photons appear at
U
output port _ I lO 2 >). It can beW'
shown (Ref. 8) that I_0out> is given by
the linear superposition
l_out > = (T-R)11 I >+i 24_
U' '_
" (12 %>+ 2>)
_' I0_ ' , (3)
from which it follows that when T =
I/2 = R, both photons always appear
together at one or the other output.
If there is a photodetector at each
output, there will be no coincidence
detections (other than accidentals),
because the corresponding two-photon
probability amplitude vanishes by
destructive interference. But if one
photon is delayed slightly relative to
the other one by some amount T, the
destructive interference is no longer
complete, and the coincidence prob-
ability P(:) rises from zero with
increasing _. When T exceeds the
duration of the wave packet and the
two wave packets no longer overlap,
P(T) becomes constant and independent
of :.
•J!l
Position of beam llplitter (/J.rrO
Fig. 3 Measured coincidence rate as a
function of time delay in fsec between
the two photons. [Reproduced from Ref.
9.]
Figure 3 shows the result of such
a coincidence counting experiment
(Ref. 9) in which each photon wave
packet had a length of about 100 fsec.
It can be seen that the observed
probability P(_) is close to zero for
= O, and rises to become constant
when _+_ equals or exceeds about 100
fsec. We therefore have a technique
for measuring the time separation
between two pulses of light and the
length of the pulse, when each pulse
consists of a single photon. The time
resolution achieved in this experiment
was about 3 fsec, which is about a
million times shorter than the resolv-
ing time of the detectors and the
associated electronics. In some later
experiments (Ref. 10) the resolution
was further measured to about I fsec,
which is less than half an optical
period.
THE FRANSON EXPERIMENT
A number of experiments have also
been performed for which there is no
adequate classical model to explain
the 4'th order interference.(Refs. 11-
14) Let us consider the experimental
situation illustrated in Fig. 4, which
Fig. 4 The principle of the Franson
4'th order interference experiment.
[Reproduced from Ref. 13.]
was first proposed and discussed by
Franson.(Ref. 15) A source emits two
photons A and B simultaneously, each
with some center frequency _A,mB and
bandwidth Am. The photons emerge in
two different directions and fall on
two photodetectors D A and DB without
ever coming together. Some beam split-
ters and mirrors forming two similar
interferometers are introduced, so as
to provide two alternative paths for
each photon, as shown: a direct path
and a longer indirect path. Let the
propagation time difference between
the two paths be T+_ A in channel A and
T+_ in channel B, with T >> I/Am,
B
<< I/Am.
_A,_B
Because the path difference in
each of the two interferometers
greatly exceeds the coherence length
c/A_ of the light, no second order
interference is expected. The prob-
ability that a photon is detected by
DA does not change significantly when
_A is changed slightly, and similarly
for DB. However, if we calculate the
joint probability PAB that a photon is
detected by D A and by D B in
coincidence, which can be measured
with a coincidence counter, we find
that it exhibits interference of the
form
PAB _ [I+_ cos{mA_A+mB_B+const.)]. (4)
can be 100% if the coincidence
resolving time T R is sufficiently
short, and it is about 50% when TR >>
T >> I/A_.
This result is best understood as
an interference of a photon pair.
There are several different ways in
which a coincidence can occur: (a)both
photons follow the short inter-
ferometer paths and arrive
simultaneously at the two detectors;
(b)both photons follow the long inter-
f erome t er path5 and arrive
simultaneously at the detectors;
(c)one photon follows the long path
and one follows the short path but the
7
time difference T+_A (say) lies within
the coincidence resolving time TR, so
that the photons are deemedto arrive
'simultaneously'. As these probabil-
ities are intrinsically
indistinguishable, we have to add the
corresponding probability amplitudes
and then square in order to arrive at
the probability PAB" This leads to the
result in Eq. (4). The interference
exhibits non-local features, because
the outcome of a measurement
registered by DA depends not only on
_A but also on TB, even though the
interferometer in channel B cannot
influence what happens in channel A.
This interference effect has
recently been observed (Refs. 13,14)
in experiments in which the two
photons were produced by down-
conversion in a non-linear crystal.
Figure 5 shows the results of such an
experiment in which one mirror was
movedpiezoelectrically and the two-
photon coincidence rate was measured.
Evidently there is interference
despite the fact that the two photons
never mix and the path difference
exceeds the coherence length of the
light more than 100-fold.
i | i i
__t e _i (am)
Fig. 5 Results of the Franson-type
interference experiment. [Reproduced
from Ref. 13.]
The question whether a classical
field can give rise to this kind of
behavior has been discussed.(Refs. 16-
18) Let us attempt to describe the
experimental situation in Fig. 4 in
terms of classical waves. Let
VA(t),VB(t) be the complex analytic
signals representing the stationary
light field leaving the source. Then
the fields at the two detectors DA,D B
can be expressed in the form
WA(t) = aVA(t) + SVA[t+T+_A]
WB(t) : _VB(t) + BVB[t+T+_B) ,
(5)
where _,B are constants characteristic
of the beam splitters and mirrors. The
joint probability that a photoemission
occurs at D A at time t and at D B at
time t+_ is proportional to the two-
time correlation
PAB(_) = <]WA(t)IelWB(t+_)I2> (6)
The integral of PAB(_) with
respect to _ over the resolving time
TR of the coincidence counter yields
the coincidence counting rate, which
is proportional to
ITR/2 d_<IWA(t)I21WB(t+_)I2>. (7)
_c =_-TR/2
With the help of Eqs. (5) it may be
shown (Ref. 16) that _c contains an
interference term of the form
_. "2B2 ITR/2J interfer. _ a d_
_-TR/2
× <VA(t)VA(t+T)VB(t+_)VB(t+_+T)>
-i(_A_A+mB_B )
e + c.c., (8)
together with a somewhat similar
interference term involving
exp[i(WB%B-_A_A) ]. But _c also con-
tains a non-oscillatory or background
contribution
"Ybackground = [ TR/2 d_(l_I4+ISl 4)
"-TR/2
x <IA(t)IB(t+_)>+ I_121812
x (<IA(t)IB(t+_+T)>
+ <IA(t)IB(t+_-T)>) , (9)
which represents light background for
the interference. Here IA(t) =
I%[A (t) 12, etc. The presence of the
interference terms suggests that
certain classical fields can exhibit
the observed interference effect.
Let us now examine the
magnitudes. Whereas the integrand in
Eq. (8) tends to zero with increasing
3, that in Eq. (9) does not. We recall
that for any ergodic process correla-
tions must eventually die out. It
follows that for sufficiently long T
the terms in T are no longer corre-
lated with those without _, and
therefore for a stationary field,
<VA(t)VA(t+T)VB(t+T)VB(t+_+T)>
÷ <VA(t)VA(t+T)><VB(t)VB(t+T)>
- 0 , (10)
because T >> I/A_. The integrand in
Eq. (9), on the other hand, tends to
the constant value [I(,12+ISI2J2<IA><IB >
as _ _ _. Therefore if we integrate
with respect to _ over a sufficiently
long resolving time TR, the background
term will greatly exceed the inter-
ference terms, and the visibility of
the interference will become negli-
gibly small. In ref. 16 it was argued
that the integrand in Eq. (8) has a
range in _ of order I/A_. But even if
it has a longer range, so long as TR
is much longer than this range, the
visibility of the interference given
by Eqs. (8) and (9) would be very
small.
Actually, a classical model of
the light from a parametric down-
converter fails for other, more
compelling reasons. It can be shown
(Ref. 19) that for any classical field
whose correlation time is much shorter
than TR,
_AB-%B accid. < _AA -_A accid.' (11)
where _AB is the coincidence counting
rate when signal light falls on one
detector and idler light on the other,
and _AA is the self-coincidence rate
for the signal. Accidental coincidence
contributions are subtracted on both
sides. In practice, classical ine-
quality (11) is, however, found to be
violated by down-converted light by
several hundred standard
deviations.(Ref. 19)
EXPERIMENTAL TEST OF THE DE BROGLIE
GUIDED WAVE THEORY
Finally, we describe a recent
experiment to test a prediction of the
de Broglie guided wave theory relating
to interference. (Refs. 20,21 )
According to this theory, which is a
hybrid of classical and quantum con-
cepts, there exist both photons and
electromagnetic waves, with the latter
acting as guides for the former. But,
in addition to yielding the probabil-
ity for detecting a photon, the
electromagnetic wave is supposed to
have a physical reality that extends
beyond being a probability wave.
Figure 6 shows the essential
features of the experlment.(Ref. 22)
Three 50%:50% beam splitters
BS_,BS2,BS 3 form a Michelson type of
interferometer, and BS 2 can be ad-
justed piezoelectrically to move
through one or two microns. Any light
that penetrates BS, and BS 2 falls on
detector D_ and D2, respectively. The
counting rates R_,R 2 of the two detec-
D,,,_
Fig. 6 Outline of the interference
experiment to test the de Broglie
guided wave theory. [Reproduced from
Ref. 22.]
tors are measured as a function of
beam splitter BS 2 displacement Ax,
together with the coincidence counting
rate R_2. The interferometer is fed
with the signal (s) and idler (i)
light produced by down-conversion in
the non-linear crystal NLC, as shown,
and it is balanced so that the paths
of i from NLC to BS 3 and of s from NLC
to BS_ to BS 3 are equal.
Reference to Fig. 6 shows that
the idler can only reach detector D_.
On the other hand, the signal can
reach both detector D 2 and detector
D_, and moreover it can reach D_ via
the two different paths NLC to BS_ to
BS_ to BS_ to D_ and NLC to BS I to BS 2
to BS_ to D_. If the distances BS_ to
BS 3 and BS_ to BS 2 are nearly equal,
these two paths interfere, so that
counting rate R_ of D_, which is given
by the expectation of the square of
the wave function _ at D_, depends on
Ax. On the other hand, the counting
rate R 2 of D2, which is given by
< I_2 I 2> is independent of Ax.
According to the guided wave theory,
(Ref. 21) the counting rate R_2 of D_
and D2 in coincidence is proportional
to the expectation <I_i121_212>, and
since I_212 is constant and independ-
ent of Ax, whereas I_i I2 shows
interference, this would be expected
to exhibit much the same interference
as R l.
Let us compare that prediction
with the quantum mechanical one. As
there is only one signal and one idler
photon emitted at one time, and be-
cause the idler can only reach D2, it
follows that whenever a coincidence is
registered, D_ must have detected the
idler photon and D 2 the signal photon.
But reference to Fig. 6 shows that, in
that case, there is no ambiguity in
the photon paths, because the wave
function _ collapses along the two
paths s to BS_ to BS 3 to BS_ and s to
BS_ to BS_ to BS_ that interfere.
Therefore R_2 should exhibit no inter-
ference or dependence on gx. A similar
conclusion is reached by a mathemati-
cal treatment of the problem.(Ref. 22)
The results of the experiment are
shown in Figs. 7 and 8. Figure 7 gives
1o'
8
Fig. 7
I:_ent of _ # _m0 0.4 0.8
t 2 3 4,
Pha._ in multiples of n
The measured photon counting
rate R_ as a function of the displace-
ment of BS 2. [Reproduced from Ref.
22.]
D_plac_ment of _ in _m
0.2 0.4, 0.6 OJI
I.
o
i ± ! - I
Fig. 8
, i , i , i , i ,
t _, 3 • 5
Lr,. multiples of 71"
The measured two-photon
coincidence counting rate as a func-
fromti°nRef.°fBS_22._isplacement. [Reproduced
I0
the measured photon counting rate R I
as a function of the displacement Ax
of BS 2. As expected, this exhibits
interference attributable to the two
alternative paths of s to D I. But this
is predicted by all theories, by
quantum mechanics, by classical wave
theory and by the guided wave theory.
Figure 8 gives the measured two-
photon coincidence rate R12 , after
subtraction of accidental counts, as a
function of BS 2 displacement. This
time there is no evidence of any
interference, in agreement with quan-
tum mechanics, but in violation of the
guided wave theory. We have therefore
disproved one prediction of the guided
wave theory. Needless to say, this
conclusion applies only to the par-
ticular form of the theory described
above, in which probabilities are
calculated very much as in semiclassi-
cal radiation theory.
The fourth order interference
technique is capable not only of very
high accuracy, such as the measurement
of the time separation between two
photons to I fsec accuracy, but it
also lends itself to the exploration
of quite fundamental questions about
our quantum world•
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Nonclassical and nonlocal effects in the interference of light

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