Scattering of dark-state polaritons in optical lattices and quantum phase gate for photons

We discuss the quasi 1-D scattering of two counter-propagating, dark-state polaritons (DSP), each containing a single excitation. DSPs are formed from photons in media with electromagnetically induced transparency and are associated with ultra-slow group velocities. State-dependent elastic collisions of atoms at the same lattice site lead to a nonlinear interaction. It is shown that the scattering process in a deep optical lattice filled by cold atoms generates a large and homogeneous conditional phase shift between two individual polaritons. The latter has potential applications for a photonic phase gate. The quasi 1-D scattering problem is solved analytically and the influence of degrading processes such as dephasing due to collisions with ground-state atoms is discussed.