Quantum circuits for amplification of Kerr nonlinearity via quadrature squeezing

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.