  Phase shifts induced by the Kerr effect are usually very small at the
single-photon level. We propose two circuits for enhancing the cross-Kerr phase
shift by applying one- and two-mode quadrature squeezing operators. Our results
are based on the vector coherent state theory and can be implemented by
physical operations satisfying the commutation relations for generators of the
generalized special unitary group SU(1,1). While the proposed methods could be
useful for the realization of quantum optical entangling gates based on Kerr
nonlinear media at the single-photon level, they also indicate a general
alternative approach to enhance higher-order nonlinearities by applying
lower-order nonlinear effects.
