Three-dimensional light-matter interface for collective spin squeezing in atomic ensembles

Abstract

We study the three-dimensional nature of the quantum interface between an\nensemble of cold, trapped atomic spins and a paraxial laser beam, coupled\nthrough a dispersive interaction. To achieve strong entanglement between the\ncollective atomic spin and the photons, one must match the spatial mode of the\ncollective radiation of the ensemble with the mode of the laser beam while\nminimizing the effects of decoherence due to optical pumping. For ensembles\ncoupling to a probe field that varies over the extent of the cloud, the set of\natoms that indistinguishably radiates into a desired mode of the field defines\nan inhomogeneous spin wave. Strong coupling of a spin wave to the probe mode is\nnot characterized by a single parameter, the optical density, but by a\ncollection of different effective atom numbers that characterize the coherence\nand decoherence of the system. To model the dynamics of the system, we develop\na full stochastic master equation, including coherent collective scattering\ninto paraxial modes, decoherence by local inhomogeneous diffuse scattering, and\nbackaction due to continuous measurement of the light entangled with the spin\nwaves. This formalism is used to study the squeezing of a spin wave via\ncontinuous quantum nondemolition (QND) measurement. We find that the greatest\nsqueezing occurs in parameter regimes where spatial inhomogeneities are\nsignificant, far from the limit in which the interface is well approximated by\na one-dimensional, homogeneous model.\n