Quantum Theory for Generation of Nonclassical Photon Pairs by a Medium with Coherent Atomic Memory

We present a fully quantum mechanical treatment of recent experiments on creation of collective quantum memory and generation of non-classically correlated photon pairs from an atomic ensemble via the protocol of Duan et al. [Nature {\bf 414,} 413(2001)]. The temporal evolution of photon numbers, photon statistics and cross-correlation between the Stokes and anti-Stokes fields is found by solving the equation of motion for atomic spin-wave excitations. We consider a low-finesse cavity model with collectively enhanced signal-to-noise ratio, which remains still considerably large in the free-space limit. Our results describe analytically the dependence of quantum correlations on spin decoherence time and time-delay between the write and read lasers and reproduce the observed data very well including the generated pulse shapes, strong violation of Cauchy-Schwarz inequality and conditional generation of anti-Stokes single-photon pulse. The theory we developed may serve as a basic approach for quantum description of storage and retrieval of quantum information, especially when the statistical properties of non-classical pulses are studied.