ANNALS OF OPTICS - XXV ENFMC - 2002 246 Quantum interference of nondiffracting beams Itamar Vidal, Solange B. Cavalcanti, Jandir M. Hickmann and Eduardo J. S. Fonseca Departamento de Física, Universidade Federal de Alagoas Cidade Universitária, 57072-970, Maceió, AL, Brazil Itamar@loqnl.ufal.br Abstract We report a quantum interference of nondiffracting beams by using photon pairs generated by spontaneous parametric down-conversion. The photon pairs transmitted by two double annular aperture produce fourth-order interference pattern along the transverse plane of detection as well as along the propagation distance. The latter presents a typical behaviour of the so-called Talbot effect . Introduction During the last few years, the properties of nondiffracting beams [1], also known as Bessel beams, have received attention by various authors. Recently, the superluminal behavior of these beams has been under investigation from a theoretical [2] and an experimental [3] point of view. An interesting self-imaging effect phenomenon, resulting from the superposition of two Bessel beams propagating in free space, has been predicted and experimentally demonstrated at a classical level [4]. On the other hand, the correlated photon pairs generated by spontaneous parametric down-conversion produce light fields possessing strong quantum features. Parametric down conversion is a versatile source of two-photon entangled states. Recently, there has been an increase of interest in the use of spatially entangled photon pairs in quantum imaging [5] and quantum lithography [6]. In addition, a two-photon quantum interference, generated by Young’s double-slits, has produced surprising new experimental results [7] allowing the measurement of the de Broglie wavelength of a biphoton (two-photon wave packets) as well as the production of conditional interference fringes, among other effects. In the present work, we investigate the superimposed Bessel beams self-imaging effect using photon pairs produced by a spontaneous parametric down-conversion source, instead of classical light. Our results show that the photon pairs transmitted by the two annular slits form an interference pattern analogous to the ones produced by the Talbot effect, which can be explained using a theory based on the angular spectrum transfers from the pump field to the photon pairs [8]. Theory In the process of parametric down-conversion, one photon of pump laser with frequency p w and wave vector p k  is converted into a photon pair conventionally called signal ( s w , s k  ) and idler ( i w , i k  ). In this process the energy ) ( i s p w w w    + = and the momentum ) ( i s p k k k       + = of the photons are conserved. A double annular slit is placed at a distance A z z = after the crystal (Fig. 1). The number of transmitted two-photon packets is proportional to the fourth-order correlation function, ) , ( ) , ( ) , ( ) , ( ) , ( ) ( ) ( ) ( ) ( i i i s s s i i i s s s s i z E z E z E z E r r C ρ ρ ρ ρ       + + - - = , (1) where ) , ( ) ( s s s z E ρ  + and ) , ( ) ( i i i z E ρ  + are the idler and the signal transmitted electric field operators, respectively and ρ  is the transversal component of . r  After being scattered by an aperture ) ( ρ  A placed on the plane A z z = , the far field in the paraxial approximation is given by               ′ - - - ⋅ ′ - ′ ′ = + A A z k q z z k q q i q q T q a q d q d const z E 2 ) ( 2 exp ) ( ) ( ˆ . ) , ( 2 2 ) (         ρ ρ , (2)