  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.
