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Physical Review A

Polarization entanglement in a crystal with threefold symmetry

Jorrit Visser, E. R. Eliel, Gerard Nienhuis

Published Sep 20, 2002• 13 citations
Publisher

Abstract

Polarization entanglement of twin photons created in the process of parametric down-conversion is fully determined by the pump polarization when the pump, signal, and idler beams are collinear with the symmetry axis of the nonlinear crystal. We point out that in this situation a threefold rotational symmetry is needed for the process to occur. We describe the polarization entanglement of the twin photons in terms of correlations on the Poincar\'e sphere. The inherent nonconservation of the intrinsic angular momentum of light in this process is discussed.

DOI
10.1103/physreva.66.033814

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Polarization entanglement in a crystal with threefold symmetry

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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.

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Polarization entanglement of twin photons created in the process of parametric down-conversion is fully determined by the pump polarization when the pump, signal, and idler beams are collinear with the symmetry axis of the nonlinear crystal. We point out that in this situation a threefold rotational symmetry is needed for the process to occur. We describe the polarization entanglement of the twin photons in terms of correlations on the Poincar'e sphere. The inherent nonconservation of the intrinsic angular momentum of light in this process is discussed.
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.
A recent work [H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett. 85, 286 (2000)] has shown eigenstates describing pairs of photons from spontaneous parametric down-conversion, which are eigenstates of orbital, intrinsic, and total momentum. Experimental demonstration of transfer of angular momentum to parametric down-converted photons has also been achieved [A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001)]. This work shows conditions for phase matching and examples are given for a Laguerre-mode pump beam. Polar and azimuthal angles are connected by phase matching giving a peculiar divergence pattern for the down-converted light. Full entanglement of polarization and angular-momentum states of photons is discussed.
A theory is presented for the quantum state of photon pairs generated from spontaneous parametric down-conversion nonlinear process. In this theory, the influence of the final sizes of nonlinear optical crystals on optical eigenmodes is explicitly taken into consideration. It was found that these photon pairs are not in entangled quantum states. Polarization correlations between the signal beam and the idler beam are explained. It is also shown that the two photons generated from SPDC are not spatially separated, therefore the polarization correlation between the signal and idler beams is not an evidence for quantum non-locality.
The twin photons produced by a type‐II spontaneous parametric down conversion are well know as a potential source of photons for quantum teleportation due to the strong entanglement in polarization. This strong entanglement in polarization, however, depends on the spectral composition of the pump photon and the nature of optical isotropy of the crystal. By exact numerical calculation of the concurrence, we have shown that how pump photons spectral width and the birefringence nature of the crystal directly affect the degree of polarization entanglement of the twin photons.
Optical spin–orbit interaction (SOI), which can be used to simultaneously control the spin and orbital angular momentum of light, is important for both classical and quantum information applications. In linear and nonlinear optics, the SOI of light has been extensively explored in both artificial structures and conventional optical crystals. However, optical SOI in quantum nonlinear optical processes, such as spontaneous parametric downconversion (SPDC), has not been studied before. Here, we experimentally demonstrate that optical SOI in the SPDC process can be realized through a nonlinear crystal with threefold rotational symmetry. Two-photon quantum states with controlled angular momentum can be generated through the symmetry selection rules in nonlinear optics and the SOI of the pump wave. The proposed methodology may facilitate the generation and control of spin and orbital angular momentum of entangled photons.