Conditional Talbot effect using a quantum two-photon state
I. Vidal, S. B. Cavalcanti, E. J. S. Fonseca, and J. M. Hickmann
Optics and Materials Group—Optma, Instituto de Física, Universidade Federal de Alagoas, Caixa Postal 2051,
57061-970 Maceió, Alagoas, Brazil
Received 8 May 2008; published 22 September 2008
We study the interference patterns obtained from the superposition of two Bessel beams originated from a
quantum two-photon state produced by a spontaneous parametric down-conversion process. Due to the non-
diffracting character of the light beam, a self-imaging effect is found along the propagation direction for
second-order interference patterns with periodicity proportional to / 2, characterizing a two-photon Talbot
effect. Conditional interference patterns in the longitudinal plane are theoretically shown as well.
DOI: 10.1103/PhysRevA.78.033829 PACS numbers: 42.50.Ar, 42.50.Dv, 42.65.Lm
I. INTRODUCTION
Bessel beams 1,2 have been the subject of intense inves-
tigations because of their important nondiffracting proper-
ties, which allow them to propagate without appreciable di-
vergence for several Rayleigh lengths. These beams have
been explored in a number of scenarios 3, for example,
optical tweezers 4, atom trapping 5, and for determining
the orbital angular momentum of a photon 6. Some inter-
esting investigations of these beams include the interference
patterns obtained from the superposition of two coherent
Bessel beams, which have been obtained theoretically 7
and experimentally 8. These patterns revealed interesting
and useful features, like a self-imaging effect in the propa-
gation direction and the possibility of controlling the rotation
of microparticles 9.
In the context of periodic structures, the self-imaging ef-
fect 10 appears in the near-field diffraction pattern of light
waves. Actually, this effect was first noted by Talbot in 1836,
who observed repeated patterns at regular distances of a dif-
fraction grating 11. This regular distance is the so-called
Talbot length l
T
= d
2
/ , where d represents the spatial period
of the pattern and the light wavelength, and the repeated
images are the so-called Talbot images or self-images.
Nowadays, the Talbot effect has been demonstrated in many
other areas of physics, such as atomic waves 12, macro-
scopic coherent Bose-Einstein condensates 13, and in the
interferometry of large C
70
fullerene molecules 14. More
recently, a discrete Talbot effect in waveguide arrays has
been reported 15.
Talbot images in the pattern of interfering Bessel beams
have only been obtained within a classical description of the
optical field. The measurements of these interference patterns
are equivalent to a first-order correlation, because two field
amplitudes are involved.
On the other hand, second-order correlation of light gen-
erated by spontaneous parametric down-conversion SPDC
has provided a wide variety of novel phenomena, including
many with nonclassical behavior. The transverse properties
of down-converted photons have been explored in quantum
imaging formation 16, quantum lithography 17, and,
more recently, in the generation of entangled qudits 18–20,
to name just a few examples. So far, only transverse spatial
properties of these photons have been exploited. In this pa-
per, we present a theoretical analysis of the behavior of spa-
tial interference patterns in the propagation direction of pho-
tons generated by the SPDC process.
Inspired by the classical Talbot effect with Bessel beams,
we will investigate their quantum counterpart using en-
tangled photons as the light source, and show that the
second-order correlation function exhibits conditional inter-
ference patterns in the propagation direction. Furthermore,
just as the de Broglie two-photon wavelength of an entangled
photon pair is half the conventional wavelength, we find the
same signature of biphoton behavior 21: the periodicity of
the interference pattern is / 2 not only in the transverse
plane but also along the propagation direction, thus charac-
terizing a two-photon Talbot effect.
II. BASIC THEORY
The setup for the theoretical model is depicted in Fig. 1.
Down-converted photons are generated in a noncollinear
phase-matching condition. The photon pairs go through
double annular slits A
s
s
and A
i
i
located at a propagation
distance z
A
s
= z
A
i
= z
A
after the crystal. Here
s
i
denotes the
transverse coordinate of the slit through which passes the
FIG. 1. Setup. A nonlinear crystal generates the down-converted
photons in a noncollinear phase-matching condition. A screen with
a double annular slit see inset is placed in each photon path at a
distance z
A
from the crystal. L
1
and L
2
are lenses. D
s
and D
i
are
single-photon detectors operating in coincidence mode.
PHYSICAL REVIEW A 78, 033829 2008
1050-2947/2008/783/0338294 ©2008 The American Physical Society 033829-1