Quantum analysis of nonlinear beam splitter with second order nonlinearity.

A linear beam splitter mixes two input modes having annihilation operators $\hat {a}$ and $\hat {b}$ and generate two output modes having annihilation operators $\hat {c}$and $\hat {d}$, which are linear in $\hat {a}$ and $\hat {b}$and may be written as $\hat {c}=t\hat {a}+ir\hat {b}$ and $\hat {d}=t\hat {b}+ir\hat {a}$, where t and r are the real coefficients of transmission and reflection, respectively, with r$^{2}$ + t$^{2}$ =1. We include the second order nonlinearity and as a result we find generation of second- harmonic non-classical light. If two coherent beams are mixed at input, squeezing and sub-Poissonian photon statistics are seen to be exhibited by the second-harmonic output beam.