Three-dimensional theory for light-matter interaction

We present a full quantum mechanical three-dimensional theory describing an electromagnetic field interacting with an ensemble of identical atoms. The theory is constructed such that it describes recent experiments on light-matter quantum interfaces, where the quantum fluctuations of light are mapped onto the atoms and back onto light. We show that the interaction of the light with the atoms may be separated into a mean effect of the ensemble and a deviation from the mean. The mean effect of the interaction effectively gives rise to an index of refraction of the gas. We formally change to a dressed state picture, where the light modes are solutions to the diffraction problem, and develop a perturbative expansion in the fluctuations. The fluctuations are due to quantum fluctuations as well as the random positions of the atoms. In this perturbative expansion we show how the quantum fluctuations are mapped between atoms and light while the random positioning of the atoms give rise to decay due to spontaneous emission. Furthermore we identify limits, where the full three-dimensional theory reduces to the one-dimensional theory typically used to describe the interaction.

Figure 1

Example of an atomic level structure. The atoms have a single ground level with spin $F$ and one or more exited levels. The fields have a large detuning $\Delta$ so that the exited states may be adiabatically eliminated and we obtain an effective ground-state Hamiltonian equation (2.3).

Figure 2

(Color online) Schematic setup. We assume that away from the ensemble, the light-mode resembles a plane wave with some transverse profile. A set of lenses focus the beam down into the ensemble.