Two entangled photons incident upon two distant interferometers can give a coincidence counting rate that depends nonlocally on the sum of the phases of the two interferometers. It has recently been shown that experiments of this kind may violate a simple inequality that must be satisfied by any classical or semi-classical field theory. The inequality provides a graphic illustration of the lack of objective realism of the electric field. The results of a recent experiment which violates this inequality and in which the optical path length between the two interferometers was greater than 100 m are briefly described

Franson, J. D.

English

NASA Technical Reports Server

N92-22048
VIOLATIONS OF A NEW INEQUALITY FOR CLASSICAL FIELDS
J.D. Franson
The Johns Hopkins University
Applied Physics Laboratory
Laurel, MD 20723
ABSTRACT
Two entangled photons incident upon two distant interferometers can give
a coincidence counting rate that depends nonlocally on the sum of the phases
of the two interferometers. It has recently been shown that experiments of
this kind may violate a simple inequality that must be satisfied by any
classical or semi-classical field theory. The inequality provides a graphic
illustration of the lack of objective realism of the electric field. The
results of a recent experiment which violates this inequality and in which the
optical path length between the two interferometers was greater than I00 m are
briefly described.
INTRODUCTION
It has been shown 1'2 that two-photon interferometer experiments can
violate Bell's inequality 3 and a number of experiments 4-7 have demonstrated
effects of that kind. Several experiments 4'6 based upon the two-photon
interferometer of Ref. I have not, however, violated Bell's inequality due to
the limited visibility (509) of the interference fringes that results when the
resolving time of the photon detectors and electronics is not sufficiently
fast.
Those experiments may 8, however, violate a surprisingly simple
inequality that must be satisfied by any classical or semi-classical field
theory. The inequality follows directly from the assumption that the classical
field has some well-defined value and thus illustrates the lack of objective
realism exhibited by the quantum-mechanical field.
This paper will briefly review the nature of two-photon interferometry
and then derive the new inequality; the derivation closely follows that of
Ref. 8. Some additional details of the derivation that are not contained in
Ref. 8 but are required for applications to actual experiments are presented
in the Appendix. The results of a recent two-photon interferometer experiment
performed over a distance of i00 meters will be briefly described. Finally,
some comments will be made with regard to the connection between uncertainty
relations and inequalities of this type.
TWO-PHOTON INTERFEROMETRY
The experiments of interest 4'6'7 are outlined in Figure i. Two
coincident photons are emitted by parametric down-conversion and travel in
different directions toward two identical interferometers. Each interferometer
contains a shorter and a longer path, and the difference AT in transit times
over the two paths is taken to be much larger than the coherence time of the
photons. Nevertheless, interference between the quantum-mechanical amplitudes
for the photons to have both traveled the shorter paths or the longer paths
produces a modulation in the coincidence counting rate R c given I by
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https://ntrs.nasa.gov/search.jsp?R=19920012805 2020-03-17T13:13:19+00:00Z
1 R ( 01 + 02 + OOAT) (i)Rc = _ coc°s2
Here Re0 is the coincidence rate with the beam splitters removed, 81 and 82 are
phase-shifts introduced into the two longer paths, and _0 is the frequency of
the pump laser. Eq. (I) violates Bell's inequality but is only valid if the
resolution of the coincidence measurements is better than AT. The maximum
visibility is 50% for time resolutions much worse than AT.
There has been some question as to whether or not the experiments with
visibilities of 50% or less are nevertheless inconsistent with any semi-
classical field theory. Ou and Mandel 9 have suggested that that is the case
but counter-examples to their argument have been given by Carmichael I° and by
Chiao and Kwiat1_ Although their semi-classical models are able to reproduce
the modulation in the coincidence rate, they are not able to represent the
fact that the photons are known from other experiments 12 to be coincident to
within a time interval much smaller than AT. That provides the physical basis
for the inequalities derived below.
BASIC INEQUALITY
The basic inequality that must be satisfied by any classical field is
based on Cauchy's inequality 13, which follows from the fact that
(a - b) 2 _ 0 (2)
where a and b are any two real numbers. Multiplying the two factors and
rearranging gives Cauchy's inequality:
2ab _ a 2 + b 2
(3)
When a and b are complex it is still the case that
labl = lal Ibl _ la12 + Ibl_
2
(4)
The modulation of the coincidence rate will be found to be proportional
to the quantity Q defined by
e _ <IE;(_) E_(t) E2(t - AT) EI(C - AT)]> (5)
Here E 1 and E 2 refer to the fields at the positions of detectors 1 and 2
(which will be assumed to be equidistant from the source) with the beam
splitters removed and <> denotes an average over a long time interval.
It should be emphasized from the start that the angular brackets denote
an average over time and not an ensemble average. That is what the
experiments actually measure, since the results from a single system are
simply averaged over time. In addition, no assumption of ergodicity is
required in the proof that follows; the average over an ensemble is not
considered and it therefore makes no difference whether or not the time
average is equivalent to an ensemble average. It will also be found that the
proof does not assume stationarity, either.
24
The basic inequality can be obtained by choosing
a = E_(t) E2(t - AT) (6)
b : Ei(t)E1(t - AT)
Inserting eqs. (6) and (7) into eq. (4) gives
<IEI(t) EI(t)E2(t - AT) E_(t - AT)I>
< E[ ( t ) Ei ( t - A T) E2 ( t - AT) El(t)> 2
+ <E](t) E[(t - AT) EI(t - AT)E2(t)>/2
(7)
(8)
The physical significance of the above inequality can be seen in Figure
2, in which both fields El(t) and E2(t ) correspond to narrow pulses emitted at
the same time. If E I is evaluated at time t and E2 is evaluated at time
t z AT, as illustrated by the arrows in the figure, then one or the other of
the fields must be zero and their product vanishes. The right-hand-side of
eq. (8) is then zero, which requires that the left-hand-side also vanish.
Although this inequality may seem trivial in nature, it is a consequence of
the fact that the classical fields are well-defined (complex) numbers and the
inequality is violated by quantum fields, as will be discussed below.
INEQUALITY FOR THE VISIBILITY
The inequality of eq. (8) can be used to set a limit on the amount of
modulation that can occur in a classical treatment of the two-photon
interferometer experiments. Once again, let El(t) be the classical field that
would arrive at detector I in the absence of the two beam splitters and assume
for the moment that the half-width w of the coincidence window is negligibly
small. The corresponding coincidence rate as a function of the time offset
is then
Rco(r) = _<11(t)I2(t + r)> (9)
: _<E[(t) E](t + r)E2(: + r)E1(t)>
where I 1 and 12 are the intensities of the two beams and the constant _ is
related to the detection efficiencies and w. With the insertion of the two
beam splitters, the total electric field Eft(t) at detector 1 becomes
_ 1 [El(t) + ei%E1(t _ AT)] (10)E_ 2
A similar expression exists for the total field at detector 2 and the
classical coincidence rate R c with the beam splitters inserted and z = 0 is
given by
_ _-in<l[E1(t) + eie,E1(t - AT)] [E2(t) + eie2E2(t - AT)] 12> (II)
Rc - 16
Multiplying out all the factors in eq.(ll) gives a total of sixteen terms:
25
Rc _- i_6_i<El(C) E2"(t)EI(C) E2 (t)
+ei%E[(t) El(t) El(C) E2(t - AT)
+ eie_E[(t) El(t) E_(t - AT) E_(t)
+ei[e_*"2] E_ ( t) E2 ( t) El ( t - AT) E2 ( t - AT)
+ e-ie2E[(t) S](t - AT) E,(t) E2(t)
+E_ (t) Ei (t - AT) El(t) E2 (t - AT)
+ e i[e'-ej E;(t) EI(t - AT) E1(t - AT) E2(t )
+ei"*Ei(t) EI(t - AT) E_(t - AT) E2(t - AT) (12)
+ e-ieiE_(t -aT) El(t) E_(t) %(:)
+e i[8'm81] E_ ( t - AT) E2 ( t ) % (t) E2 ( t - AT)
+ E_(t - AT) Ei(t ) E,(t - AT) E2(t )
+eie' E_ ( t - A T) E2 ( t) EI ( t - A T) E2 ( t - AT)
+ ei[-e*-eJE_(t - AT) EI(t - ar) E,(t)E2(t)
+ e-i%E_(C - aT) EI(C - aT) E,(t) E2(t - aT)
+ e-ie'E_(t - AT) E_(t - AT) El(t - AT) E2(t )
* E; ( t - A T) E2 ( t - AT) El(t - A T) E2 ( t - AT)>
As suggested by eq. (1), the experiments can be performed in such a way
as to measure the averaged coincidence rate as a function of 8r = 01 + 02 :
_(8_) - i .2. -2.2 ]o + % - (13)
The averages over O z and 0 2 were explicitly performed in one of the
experiments 6. In the remaining experiments the individual phases were not
directly measured and had essentially random values from one run to the next,
since variations in the temperature of the laboratory would have shifted the
phase of both interferometers by several fringes from one day to the next.
Thermal drifts during the course of an experimental run would have a similar
effect on the individual phases while leaving the modulation of the
coincidence rate unaltered.
In any event, the terms in eq. (12) with phase factors of exp(iOl),
exp(i62), exp[i(81 - 82)], etc., average to zero, heaving only those terms
with no phase dependence or a dependence on 81 + 82 . The remaining terms can
be written as
--Rc= _I_<E_ (t) Ef (t) E_ (t)E* (c)>
+ in<E[(t) E](t - AT) E2(t - AT) E_(t)> (14)
8
+ 1
l---_n[e_eT<_.1 (t) E2,(t)E_(t_" - AT)Ez(t - AT)> + c.c.]
where the average over a long time interval ensures that
<E[(t - AT) Ef(t - AT) E2(t - AT) E_(t - AT)> (is)
= <E[(t) Ef(t)E2(t)E1(t)>
and the symmetry of the two beams gives
26
< E_ ( t ) E_ ( t - A T) E2 ( t - AT) El(t)> (16)
: <E](t) E_(t - AT)EI(r - AT) E2(t)>
The assumption inherent in eq. (16) is not essential and can he avoided by
simply replacing Rc0(AT) with [Rc0(AT ) + Rc0(-AT)]/2 in what follows.
The maximum and minimum coincidence rates from eq. (14) satisfy
1
Rma x < -{n<E2 C t) E2 C t) E2 C t) E1( t) >
+ 1 <EI(t) EI(t AT) E2(: AT) E_(t)> (17)
- -
+ -{n<lEi(t) Ei(t)E2(t - AT) E_(t - AT)I>
_in _ _<E_(t) E_(t)E2(t)E_(t)>
+ 1
-_1]<E_ ( t) E_ ( t - AT) E_ ( t - AT) El(t)> (iB)
1
- -{_<[E[(t)E_(t)E2(t - AT) EI(t - AT)I>
The visibility is defined as usual by
V = Rmax - Rmin (1.9)
Using the inequality of eq. (8) and expressing the right-hand-side in terms of
R_0 (AT) gives
Rco (A T)
V < (20)
Rco(O) + Rco(AT)
Eq. (20) gives the maximum visibility that can occur in any classical field
theory and gives zero modulation for the case in which the fields correspond
to coincident pulses.
If the experiments are performed using detectors with limited time
responses and large coincidence windows, as is often the case, then the above
inequality can be generalized to
' 1 [ 3AT/2T/2Rco("C) d'r + -2JAT/2 Rc°('_) d_ (21)v N
2 ]o-Rco ( "c) dT
as is shown in the Appendix. Rc0 is again the coincidence rate that would be
obtained using detectors with a negligible time response and a negligibly-
small window.
COMPARISON WITH EXPERIMENT
Earlier experiments 12 have shown that the down-converted photons are
coincident to within a time interval much less than the value of AT in at
least three 4'_'7 of the two-photon interferometer experiments, in which case
the inequalities of eqs. (20) or (21) show that there is no classical or semi-
27
classical field theory consistent with all of the available observations.
The author has recently completed an experiment 6 in which the optical
path length between the two interferometers was larger than I00 meters The
main goal of the experiment was to investigate these effects in the limit of
large distances. Furry has suggested that the collapse of the wavefunction may
be degraded in someway when it occurs over sufficiently large distances,
leading to an eventual modification of the quantum-theory predictions. The
visibility of the interference pattern observed agreed with that expected from
the quantum theory to within the experimental uncertainty of 4%and violated
the inequality of eq. (21) by four standard deviations. This provides some
indication that the collapse of the wavefunction is unaffected even when it
occurs over relatively large optical path lengths.
CLASSICALMODELS
In the classical models suggested by Carmichael9 and by Chiao and
KwiatI°, the fields EI and E2 have well-defined frequencies that sumto the
pumplaser frequency for a time interval larger than AT or the time resolution
of the coincidence circuits. In that case the coincidence rate of eq. (14)
simplifies to
= + 1/2 (22)
2
This differs from the quantum-mechanical result by the additional factor of
1/2 and corresponds to a visibility of 50%.
Such models cannot simultaneously localize the fields into coincident
pulses whose widths are less than AT, however. Any classical model that does
would have the visibility reduced accordingly as required by the inequalities
of eqs. (20) or (21).
VIOLATION OF THE INEQUALITY IN QUANTUM OPTICS
The intensity operator is given by f(t) = E-(t) E+(t) , where E + and E-
are the positive and negative-frequency components of the electric field
operator 13. As a result, the quantum-mechanical equivalent of eq. (8) is
I< E[ ( t ) Ef ( g ) E_ ( t - A T) E[ ( t - AT)>J
<Ef ( t) Ef ( t - _T) S; ( t - _T) E[ ( t) >/2 (23)
+ <E$(t) Ef(t - _T) E_(t - _T) Ef(t)>/2
It has already been noted I that in experiments of this kind the coincidence of
the photons requires
E_(t)E_(t ± AT) = 0
while conservation of energy in the parametric down-conversion process
requires that
(24)
E[(t - AT)E_(t - AT) = e i(_I +o2)arE[(t)E_(t) (25)
where the sum of the two photon frequencies °I and °2 is equal to _0. (Eq.
(24) is only valid when AT is small compared to the pump laser coherence
time.) Inserting eq. (24) into the right-hand-side of eq. (23) gives zero,
whereas inserting eq. (25) into the left-hand-side gives
<E((t) E_(t)E_(t)E[(t)>, which is the product of the individual beam
28
intensities and a nonzero quantity. Thus the inequality is violated in
quantumoptics.
The quantum-mechanical situation is shownin Figure 3. The field
corresponds to an entangled state in which there is a superposition of times
at which the pair of photons mayhave been emitted, as indicated by the
existence of both the solid and dotted curves. Although the product of El and E$
at two different times is zero, that does not imply that the left-hand-side of
eq. (22) must vanish. Equations (24) and (25) would be logically
inconsistent if the fields were well-defined complex numbers and the violation
of this inequality provides a graphic demonstration of the lack of objective
realism of the electric field.
CONNECTION WITH UNCERTAINTY RELATIONS
The main topics of this conference are squeezing and uncertainty
relations. It may thus be useful to make some general comments about the
connection between the inequality derived above and the uncertainty relations
associated with the quantized field.
The inequality derived above is a result of the fact that the fields are
not just complex numbers and thus have no well-defined value. In particular,
the field operators are non-commuting and satisfy
(26)
[A.(X),A_(x/)] = -ich6.vD(x - x/)
A variety of uncertainty relations can be derived from this commutation
relation, which illustrates the fact that the quantized field has no well-
defined value. As a result, there is an unavoidable uncertainty in the left-
hand-side of the classical inequality and this uncertainty is evidently large
enough that the left-hand-side can exceed the right-hand-side. Thus it seems
apparent that the violations of these classical inequalities in quantum optics
are related to the uncertainty relations for the quantized fields. More
detailed uncertainty relations for the actual quantities involved in the
classical inequality could be derived, if desired.
SUMMARY
Two-photon interferometer experiments with a sufficiently large
visibility will violate Bell's inequality and are thus inconsistent with any
local hidden-variable theory. Those experiments with smaller visibilities may
nevertheless violate an inequality for classical fields if the degree of
coincidence of the photon counts is taken into account. A recent two-photon
interferometer experiment with a large optical path length between the two
interferometers gave a visibility in good agreement with the quantum theory
and also violated the classical inequality, indicating that the effects
observed were quantum-mechanical in nature.
APPENDIX
When finite coincidence windows are used, Eq. (9) must be replaced by
Rc'0: + el> (27)
The modulation in the coincidence rate then involves
29
Q/ = 111/[w d
-{-J-w _<IE_(t) E_(t+_)E2(t - AT + _)E_(t - AT)I > (28)
The range of the integral can be divided into two regions depending on the
value of ITI. For ITI < AT 2, a and b can be chosen as
a = E_(t)E2(t - AT + r)
b = E](t + z)E1(t - AT)
(29)
and the analysis proceeds as in the text. For I_I a AT 2, a and b are chosen
instead as
a = E[(t)E](t + _) (30)
b = E2(t - AT * r)El(t - AT>
The inequality of eq. (21) then results from the use of eq. (8) in the limit
that w is much larger than AT.
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31
82 81d
?2 _ 7_\ /
Figure i Two-photon interferometer.
Field
amplitudes
E_ andE 2
Figure 2
I I+,_T
Time
A pair of classical coincident pulses.
Field
amplitudes
E; and E_
t t+_,T
Time
z / \,,.
Figure 3 Quantum-mechanical field corresponding to an entan$1ed
pair of coincident photons, with a superposition ot
times at which the pair may have been emitted.
32
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Violations of a new inequality for classical fields

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