Polarization phase gate with a tripod atomic system

We analyze the nonlinear optical response of a four-level atomic system driven into a tripod configuration. The large cross-Kerr nonlinearities that occurr in such a system are shown to produce nonlinear phase shift of order $\pi$. Such a substantial shift may be observed in a cold atomic gas in a magneto-optical trap where it coupl be fasibly exploited towards the realization of a polarization quantum phase gate. The experimental feasibility of such a gate is here examined in detail.