Quantum analysis of a Nonlinear Beam splitter: third-order nonlinearity.

A linear beam splitter mixes two input modes having annihilation operators, say $\hat {a}$ and $\hat {b}$, to generate two output modes having annihilation operators, say, $\hat {c}$ and $\hat {d}$. It is common to write $\hat {c}=t\hat {a}+ir\hat {b}$, $\hat {d}=t\hat {b}+ir\hat {a}$, where the real constants t and r denote coefficients of transmission and reflection, respectively, and t$^{2}$ + r$^{2}$ = 1. We generalize the linear beam splitter input-output relations to include third-order nonlinearity and show that the nonlinear terms can give non-classical outputs with classical inputs. We study generation of squeezing and sub-Poissonian statistics for coherent light inputs.