Spontaneous parametric down-conversion in a three-level system

We have studied the space-time entangled Stokes-anti-Stokes photons generated from a three-level system. In the presence of counterpropagating pump and control lasers, paired Stokes and anti-Stokes fields are spontaneously emitted into opposite directions. The two-photon wave packet is generally a convolution of the phase matching and the third-order nonlinear susceptibility and the feature of the joint temporal correlation is determined by either of them. When the phase matching plays a major role in determining the spectral width of biphotons, the two-photon interference exhibits as conventional type-II spontaneous parametric down-conversion (SPDC). However, if the spectral width of paired photons is determined by the third-order nonlinear susceptibility, two types of four-wave mixings (FWMs) destructively interfere and contribute to the two-photon coincidences. In such a case, the coincidence counting rate appears as a damped Rabi oscillation and exhibits the photon antibunching. The oscillation frequency is equal to the effective Rabi frequency. We have also discussed polarization-entanglement generation in a three-state case. It is found that if the phase matching is crucial in determining the properties of paired Stokes and anti-Stokes photons, the generated state is the same as the degenerate type-II SPDC. However, if the spectral width is mainly controlled by the resonant linewidths in the third-order nonlinearity, one can select one FWM to produce polarization-entangled biphotons, but paired photons from the other FWM process is nonpolarization entangled.