  We study the three-dimensional nature of the quantum interface between an
ensemble of cold, trapped atomic spins and a paraxial laser beam, coupled
through a dispersive interaction. To achieve strong entanglement between the
collective atomic spin and the photons, one must match the spatial mode of the
collective radiation of the ensemble with the mode of the laser beam while
minimizing the effects of decoherence due to optical pumping. For ensembles
coupling to a probe field that varies over the extent of the cloud, the set of
atoms that indistinguishably radiates into a desired mode of the field defines
an inhomogeneous spin wave. Strong coupling of a spin wave to the probe mode is
not characterized by a single parameter, the optical density, but by a
collection of different effective atom numbers that characterize the coherence
and decoherence of the system. To model the dynamics of the system, we develop
a full stochastic master equation, including coherent collective scattering
into paraxial modes, decoherence by local inhomogeneous diffuse scattering, and
backaction due to continuous measurement of the light entangled with the spin
waves. This formalism is used to study the squeezing of a spin wave via
continuous quantum nondemolition (QND) measurement. We find that the greatest
squeezing occurs in parameter regimes where spatial inhomogeneities are
significant, far from the limit in which the interface is well approximated by
a one-dimensional, homogeneous model.
