Transverse effects in paired-photon generation via an electromagnetically induced transparency medium. II. Beyond perturbation theory

Abstract

Generation of counterpropagating paired Stokes and anti-Stokes photons in a double-$\ensuremath{\Lambda}$- (or $M$-) type electromagnetically induced transparency (EIT) system is examined in quantum theory in the three-dimensional space. We use the Maxwell-Bloch equations to study the transverse effects of generated entangled photon pairs via such an EIT medium. We reexamine correlation measurements for both amplitude and intensity correlations. In the low parametric gain region $[\sqrt{{v}_{g}∕8{\ensuremath{\gamma}}_{31}L}\ensuremath{\gg}\sqrt{{k}_{s}∕{k}_{as}}(\ensuremath{\mid}{\ensuremath{\Omega}}_{p}\ensuremath{\mid}∕\ensuremath{\mid}{\ensuremath{\Gamma}}_{p}\ensuremath{\mid})]$, when the paired Stokes and anti-Stokes are at the single-photon level, we obtain the results shown in Wen and Rubin, Phys. Rev. A 74, 023807 (2006). If $\sqrt{{k}_{s}∕{k}_{as}}(\ensuremath{\mid}{\ensuremath{\Omega}}_{p}\ensuremath{\mid}∕\ensuremath{\mid}{\ensuremath{\Gamma}}_{p}\ensuremath{\mid})$ is comparable to or greater than $\sqrt{{v}_{g}∕8{\ensuremath{\gamma}}_{31}L}$, i.e., in the high parametric gain regime, the system may not reach the single-photon level, and accidental coincidences occur which cause loss of visibility in the interference experiments.