Experimental Observation of Spatial Antibunching of Photons

Abstract

We report an interference experiment that shows transverse spatial antibunching of photons. Using collinear parametric down-conversion in a Young-type fourth-order interference setup we show interference patterns that violate classical Schwarz inequality and should not exist at all in a classical description.Photon antibunching in a stationary field is recognized as a signature of nonclassical behavior, for its description is not possible in terms of a nonsingular positive GlauberSudarshan P distribution [1]. It is well known that any state of the electromagnetic field that has a classical analog can be described by means of a positive P distribution which has the properties of a classical probability functional over an ensemble of coherent states.The classical intensity correlation function for stationary fields must obey the following inequality [1]:All field states described in terms of a positive nonsingular P distribution must obey the standard quantum mechanical counterpart of (1), where products of intensities are replaced by ordered products of photon density operators [1], that is,where T : : stands for time and normal ordering. Photon density operators are defined aŝwhereV (r, t) = k,σâ k,σ ǫ k,σ e i(k·r−ωt) , a k,σ is the annihilation operator for the mode with wave vector k and polarization σ, ǫ k,σ is the unit polarization vector, and ω = ck. Inequality (2) means that for such class of fields, photons are detected either bunched or randomly distributed in time. Photon antibunching in time, characterized by the violation of (2) Let us now turn to space domain and consider that the transverse field profile of a given stationary light beam propagating along z direction is described by a complex stochastic vector amplitude V(ρ, t) with an associated probability functional P(V). Here, ρ lies in a plane transverse to the propagation direction. The average intensity at a point ρ isand the two-point intensity correlation functionIts time dependence is restricted to the difference τ = t 1 − t 2 , since the field is assumed to be stationary. In the space domain, the concept analogous to stationarity is homogeneity. For a homogeneous field, the expectation value of any quantity that is a function of position is invariant under translation of the origin [1]. In particular, Γ (2,2) (ρ 1 , ρ 2 , τ ) = Γ (2,2) (δ, τ )andwhere δ = ρ 1 − ρ 2 and N = 1, 2, . . . Applying Schwarz inequality, I(ρ, t)I(ρ + δ, t + τ ) 2 ≤ I 2 (ρ, t) I 2 (ρ + δ, t + τ ) .

Spatial correlations in parametric down-conversion

“…In the same spirit, the anti-bunching of photons was later demonstrated in the transverse spatial domain in Ref. [18], and was probably the first formal demonstration of the non-classical behavior of spatial correlations. In 2004, Howell et al [21] and D'Angelo et al [22] demonstrated that the pairs of down-converted photons are indeed entangled by experimentally violating classical separability criteria.…”

“…A more comprehensive study of these far field correlations were made by Grayson and Barbosa [10]. Experiments exploring interference at a double slit experiment using twin photons and coincidence counting came in 1994 [13] and 1995 [14] and was further investigated in later work [50,34,19,17,51,52,53,18,54,55]. In this section, we review some of the key experiments exploring aspects of interference of photon pairs using a double slit.…”

“…The π/2 phase factors are due to the delay introduced by the zero-order quarter-wave plates. The coincidence probability is [18,19] …”

“…(18) by a classical amplitude. In order to simplify expression (18), it is convenient to make the following approximations:…”

“…Monken et al [16] demonstrated that the spatial correlations could be controlled through the shaping of the angular spectrum of the pump beam. This control allowed the use of the spatial correlations in a series of applications, ranging from fundamental aspects of Quantum Mechanics such as the measurement of the photonic de Broglie wavelength of a two-photon wave packet [17] and the observation of the spatial anti-bunching of photons [18,19], to applications to quantum information such as, for instance, the production of spatial qudits [20]. Howell et al [21] and D'Angelo et al [22] formally demonstrated entanglement between the spatial properties of the twin photons.…”