Classical treatment of the Franson two-photon correlation experiment

Abstract

A two-photon correlation experiment that was proposed by Franson [ Phys. Rev. Lett. 62, 2205 ( 1989)] and was recently carried out is analyzed in terms of electromagnetic waves in order to see whether the observed fourth-order interference effects can be explained classically. The conclusion is that, although a classical field can give rise to such interference effects, no classical field can actually account for the observations reported.

Full text

Vol.
7,
No.
10/October 1990/J.
Opt.
Soc.
Am.
B
2127
Classical
treatment
of the
Franson
two-photon
correlation
experiment
Z.
Y.
Ou and
L.
Mandel
Department
of
Physics
and
Astronomy,
University
of
Rochester,
Rochester,
New
York
14627
Received
January
1,
1990;
accepted
March
4,
1990
A
two-photon
correlation
experiment
that
was
proposed
by
Franson
[Phys.
Rev.
Lett.
62,
2205
(1989)]
and
was
recently
carried
out
is
analyzed
in
terms
of
electromagnetic
waves
in order
to
see
whether
the
observed
fourth-
order
interference
effects
can
be
explained
classically.
The
conclusion
is
that,
although
a
classical
field
can
give
rise
to
such
interference
effects,
no
classical field
can
actually
account
for
the
observations
reported.
1.
INTRODUCTION
A
number
of
experiments
have
recently
been reported1
2
and
proposed
37
in
which
optical
interference
effects
were
used
to
look
for
quantum-mechanical
violations
of
locality.
In particular,
Franson
8has
proposed
an
especially
simple
and
elegant
experiment
to
exhibit
nonclassical
and
nonlo-
cal
behavior
in
two-photon
interference.
Two
versions
of
such
an
experiment
were
recently
reported
in
which
fourth-order interference
was
observed
with
two
photons
that
are
never
mixed.91
0
The
basic
idea
is
illustrated
in
Fig.
1.
Two
photons
are
emitted
simultaneously
from
a common
source
and are
di-
rected
to
two
separate
photodetectors,
D,
and
D2, without
being mixed
in any
way.
Beam
splitters
and
mirrors
forming
a
Mach-Zehnder
type
of
interferometer
are
in-
serted
in
each
arm,
so
that
the
light
reaching
any
one
de-
tector
arrives
via
two
possible
paths.
When
the
time
delay
between
the
longer
path
and
the
shorter path
is
much
greater
than
the
coherence
time
of
the
incident
light,
one
would
normally
not expect
to
observe
any
inter-
ference
either in
channel
1
or
in
channel
2.
Indeed,
it
has
been
confirmed
that
the
probability
of
a
photoelectric
detection
registered
by
detector
D,
is
independent
of
any small change
of
optical
path-length
difference
in
channel 1,
and
similarly
for
detector
D2.
Nevertheless,
when
the
joint
probability
of
simultaneous
detections
by
both
detectors
D
1
and
D2
is
measured,
it
is
found
to
de-
pend
sensitively
on
the
optical
path
differences in
both
channels 1
and
2.
The
probability
of
a
photodetection
by
Di
then
is
influenced
by
changes of
optical
path
difference
in
channel
2,
even
though
the
two
channels
may
be
far
apart.
The reason
is
that
coincident photon
detections
can
result
when
both
photons
follow
the
two
short
paths
through
the
interferometers
and
also
when
both
photons
follow
the
two
long
paths
through
the
interferometers
(or,
if
the
coincidence-resolving
time
is
sufficiently
great,
even
when
one
photon
follows
the
short
path
while
the
other
follows
the
long
one).
Because
the detectors
are un-
able
to
distinguish
among
these
various possibilities,
we
have
to add
the
corresponding
probability amplitudes
to
calculate
the
coincidence
detection
probability. This
leads
to
the
observed
interference
effects,
even
though
the
two
emitted
photons
never
come
together
again,
in
agree-
ment
with
quantum-mechanical predictions.
It
is
just
because
the
two photons
are
never
mixed
that
this
ex-
periment
differs
from
previous
two-photon
interference
experiments.
11
"1
2
Recently
the
question
has been
raised
whether
two clas-
sical
light
waves
emitted
by
the
source
in
Fig. 1
might
also
exhibit
such
fourth-order interference
effects
if
the
waves
had
certain
special
correlation
properties.
This
question
is obviously
of
considerable
fundamental
interest
because,
if
the
observed
effects
R
0
can
be
described in
terms
of
clas-
sical
light
waves,
the
experiments
lose
their
significance
as
indicators
of
nonclassical
and
nonlocal
behavior.
In
the
following,
we
therefore
analyze
the
experiment
shown
schematically
in
Fig.
1 carefully
on
the
basis of
er-
godic
classical
wave
theory.
We
show
that
some
fourth-
order
interference
effects
with
the
observed
periodicities
can indeed
be
produced
by
classical
light
waves
but
not
under
the
conditions
of
the
reported
experiments.R'
0
The
visibility
is
so
small
as
to
be effectively
unobservable.
It
follows
that
ergodic
classical
waves
cannot
account
for
the
observations.
However,
the
possibility
of
generating
in-
terference
with
nonergodic
optical fields
is
briefly
dis-
cussed in Section
5.
2.
CLASSICAL
THEORY
OF
THE EXPERIMENT
Let
the
two
light
waves
emitted
by
the
source
be
repre-
sented
by
the
complex
analytic
signals
V,(t)
and
V2(t)
(see
Fig.
1).
Similarly,
let
W,(t)
and
W2
(t)
represent
the
ampli-
tudes
of
the
waves
falling
upon
detectors
D1
and
D2,
re-
spectively.
We
shall
not assume
stationarity
in
general,
but
we
assume
that
we
are
dealing
with
ergodic,
or
physi-
cal,
fields,
in
the
sense
that
any
one
long
realization
of
the
ensemble
fluctuates
in time
in
a
way
that
is
characteristic
of
the
ensemble as
a
whole.
We
suppose
that
V,(t)
and
V
2
(t)
have
a coherence
time,
or field
amplitude
correla-
tion
time,
Tc
and
a field
intensity
correlation
time
TcI.
Thus
V1
(t)
or
V1
*(t)
is
not correlated
with
V1
(t
+
T)
when
ITI
>>
Tc,
and
similarly
for
V
2
(t).
Also,
if
V1
(t)
or
V,*(t) is
correlated
with
V
2
(t
+
X),
then
it
is
not correlated with
V
2
(t
+
X
+
T)
when
[TI
>>
Tc.
Similar
statements
can
also
be made
about
the
light
intensities
Ii(t)
=
1V
1
(t)l
2
and
0740-3224/90/102127-05$02.00
©
1990
Optical
Society
of
America
Z.
Y.
Ou
and
L.
Mandel