Nonlocal de Broglie wavelength of a two-particle system
E. J. S. Fonseca,
1
Zoltan Paulini,
1
P. Nussenzveig,
2
C. H. Monken,
1
and S. Pa
´
dua
1,3,
*
1
Departamento de Fı ´sica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, MG 30123-970, Brazil
2
Instituto de Fı ´sica, Universidade de Sa ˜ o Paulo, Caixa Postal 66318, Sa ˜ o Paulo, SP 05315-970, Brazil
3
Dipartimento di Fisica, Universita ` degli Studi di Roma ‘‘La Sapienza,’’ Roma, 00185, Italy
Received 10 July 2000; Revised manuscript received 27 October 2000; published 21 March 2001
We show that it is possible to associate a de Broglie wavelength with a composite system even when the
constituent particles are separated spatially. The nonlocal de Broglie wavelength ( /2) of a two-photon system
separated spatially is measured with an appropriate detection system. The two-photon system is prepared in an
entangled state in space-momentum variables. We propose that the same result can be obtained for a system of
massive particles separated spatially in an entangled state in space-momentum variables.
DOI: 10.1103/PhysRevA.63.043819 PACS numbers: 42.50.Dv, 03.65.Ud, 03.65.Ta
I. INTRODUCTION
It is well known that a de Broglie wavelength can be
associated not only with single particles, but also with a mul-
tiparticle system. For a system of N identical particles, the
resulting wavelength is given by
dB
=
i
/ N , where
i
is the
de Broglie wavelength associated with the individual con-
stituent particles 1. Normally, these particles are held to-
gether by some kind of binding force as in the experiments
done with molecules by Borde
´
et al. 2 and by Chapman
et al. 3.
In a recent paper, Fonseca, Monken, and Pa
´
dua 4, adapt-
ing an original proposal by Jacobson et al. 5, measured the
de Broglie wavelength of a two-photon wave packet for
which the role of binding is played by entanglement. A
Young interference pattern of two-photon wave packets,
which behaved like single entities with twice the energy of
each constituent photon, was detected. The corresponding
wavelength was, of course, half the de Broglie wavelength of
a single photon. Although in all measurements there were
two photons in the same wave packet, it was shown in 4
that the measured de Broglie wavelength or /2 depends
on the two-photon state. Two or more photons in the same
wave packet do not necessarily interfere as a ‘‘bound sys-
tem.’’ The multiparticle de Broglie wavelength is measured
only if the composite system as a whole interferes with itself
for the formation of the interference pattern. This will de-
pend on its state at the entrance of the interferometer, on the
interferometer one uses, and on the detection system.
At this point an interesting question arises: is it possible to
associate and to measure a de Broglie wavelength with a
system of macroscopically separated free particles? In this
article we show that, at least for photons, the answer is yes.
The experimental results suggest that entanglement is a suf-
ficient condition to define a de Broglie wavelength for a
multiparticle system. We further propose that the same is
true for entangled massive particles separated spatially. As
long as they are in an entangled state in space-momentum
variables, the de Broglie wavelength of the system can be
measured with an interferometer analogous to the one de-
scribed here.
An entangled two-photon field can be generated by spon-
taneous parametric down-conversion SPDC. In the process
of SPDC, a pump photon incident upon a nonlinear crystal
splits into a pair of photons, usually called the signal and
idler 6. There have been numerous interesting two-photon
interference experiments see, for example, 4,7–19. In this
paper we focus on an interesting ‘‘nonlocal’’ aspect of en-
tanglement. By modifying the transverse field profile of the
pump laser beam in the SPDC process and manipulating the
detection system we obtained a fourth-order interference pat-
tern of the down-converted photons with periodicity /2
when the photon pairs are transmitted by two Young double
slits. Therefore, we are able to measure the de Broglie wave-
length of the two-photon system even when the constituent
photons are separated spatially. Another interesting result is
that, depending on the detection system, no fourth-order in-
terference pattern is observed. All these results are predicted
by a quantum multimode calculation 16,20,21.
In Ref. 12 it was shown that the angular spectrum of the
pump beam is transferred to the two-photon state generated
by SPDC. As a consequence, the probability distribution for
two-photon detection P
2
( x
s
, x
i
) reproduces the transverse
pump intensity profile W( x ), in the following way:
P
2
x
s
, x
i
W
x
s
s
+
x
i
i
, 1
where
s
=k
p
/ k
s
,
i
=k
p
/ k
i
, and k
p
, k
s
, k
i
are the wave
numbers of the pump, signal, and idler fields, respectively.
Regarding the photons as particles traveling with the same
velocity, the above expression means that it is possible to
control the transverse coordinates of their ‘‘center of mass’’
via the pump beam profile.
Consider that signal and idler photons are incident on two
double slits as shown in Fig. 1. Let us discuss only the cases
in which both photons are transmitted by the slits. By focus-
ing the pump beam on x =0, it is possible to force the pair to
go through either slits A
s
a
, A
i
b
or A
s
b
, A
i
a
Fig. 1a. On the
other hand, by creating a pump beam profile peaked at both
x =+d and x =-d see below, it is possible to force the *Corresponding author. Email address: spadua@fisica.ufmg.br
PHYSICAL REVIEW A, VOLUME 63, 043819
1050-2947/2001/634/0438195/$20.00 ©2001 The American Physical Society 63 043819-1