Hyperentanglement in Parametric Down-Conversion

A theory of spontaneous parametric down-conversion, which gives rise to a quantum state that is simultaneously entangled in three-dimensional wavevector and polarization, allows us to understand the unusual characteristics of fourth-order quantum interference in many experiments, including ultrafast parametric down-conversion, the speci c example illustrated in this paper. The comprehensive approach provided here permits the engineering of quantum states suitable for quantum information schemes and new quantum technologies. c 2001 Optical Society of America OCIS codes: 42.50.Dv, 42.65.Re, 42.65.Ky, 03.67a Entanglement [1] is, undoubtedly, one of the most fascinating features of quantum mechanics. Spontaneous parametric down-conversion (SPDC) [2], a nonlinear optical phenomenon, has been one of the most widely used sources of entangled quantum states. In this process, pairs of photons are generated in a state that can be entangled in frequency, momentum, and polarization when a laser beam illuminates a nonlinear optical crystal. The experimental arrangement for producing entangled photon pairs is simple both in conception and in execution. Ironically, a signi cant number of experimental e orts designed to verify the nonseparability of entangled states are carried out in the context of models that fail to access the overall relevant Hilbert space, but rather are restricted to only a single kind of entanglement, such as entanglement in energy [3], momentum [4], or polarization [5]. Inconsistencies in the analysis of downconversion quantum-interference experiments can emerge under such circumstances, as highlighted by the failure of the conventional theory [6] of ultrafast parametric down-conversion to characterize quantum-interference experiments [7]. In this paper we present a complete quantum-mechanical analysis of entangled-photon state generation via SPDC, considering simultaneous entanglement in three-dimensional wavevector (see Fig. 1) and polarization at the generation, propagation, and detection stages (see Fig. 2). As one speci c example of the applicability of this approach, we use it to describe both new and previously obtained [7] results of SPDC experiments with a femtosecond pump. Our analysis con rms that the inconsistencies between existing theoretical models and the observed data in femtosecond down-conversion experiments can indeed be attributed to a failure of considering the full Hilbert space spanned by the simultaneously entangled quantum variables. Femtosecond SPDC models have heretofore ignored transverse wavevector components and have thereby not accounted for the previously demonstrated angular spread [8] of the down-converted light. The approach presented here is suitable for Type-I, as well as Type-II, spontaneous parametric down-conversion. Our study leads to a deeper physical understanding of hyperentangled photon states and, concomitantly, provides a route for engineering these states for speci c applications, including quantum information processing.