Polarization entanglement in a crystal with threefold symmetry

Summary form only given. The vast majority of experimental studies of entanglement is based on the use of a polarization entangled photon pair or multiples thereof. Typically, such pairs are produced by the process of spontaneous parametric down conversion in a crystal that exhibits a nonzero second-order nonlinear susceptibility and is cut for type II phase matching. These photons are orthogonally polarized along directions imposed by the symmetry, cut and orientation of the nonlinear crystal. Polarization-entangled pairs are then selected by spatial, and possibly spectral, filtering of the down-converted light. When the crystal is pumped with linearly polarized light a rather intuitive picture arises. If one selects the polarization of the signal photon to be linear, the idler photon will also be linearly polarized in such a way that /spl alpha//sub p/ = -(/spl alpha//sub s/ + /spl alpha//sub i/). Here /spl alpha//sub p,s,i/ refer to the polarization directions of pump, signal and idler, respectively. Their polarization state can be characterized by points on the equator of the Poincare sphere. If, for linear polarized pump, the signal is selected to be elliptically polarized, the idler will also be elliptically polarized. On the Poincare sphere signal and idler have equal latitude (on the same hemisphere), while their longitude is determined by the same relation as above. In all cases signal and idler are maximally entangled.