Collective coupling of driven multilevel atoms and its effect on four-wave mixing

Microscopic models based on multilevel atoms are central to optimizing non-linear optical responses and the coherent control of light. These models are traditionally based on single-atom effects that are parametrically extrapolated to include collective effects, such as an enhanced response or propagation within atomic media. In this work we present a systematic analysis of the cooperative effects arising in driven systems composed of multilevel atoms coupled via a common electromagnetic environment. The analysis is based on an interplay between dressed states induced by the driving field and photon exchanges, and collective decay channels. This theory is applied to the case of four-wave mixing induced by a pair of lasers acting on an atomic pair with internal levels in the diamond configuration. The effect of inter-atomic correlations and collective decay over the photons created in this nonlinear process is then explored. The dependence of single and two-photon correlations are studied in detail for each region by varying atomic orientations and laser parameters { consistent with current experiments involving atomic gases.}Photonic correlation functions are shown to exhibit a transition from a Lorentz-like dependence on the two-photon detuning -- with general features that can be obtained in an isolated atom scheme -- to a two-peaked distribution when the dipole-dipole interactions become relevant. For weak Rabi frequencies whose value is smaller than the highest collective decay rate, the atoms are trapped inside their ground state as they approach each other. It is found that the anisotropy of the dipole-dipole interaction and its wave nature are essential to understand the behavior of the photons correlations. Signatures of these processes are identified for existing experimental realizations.
Comment: 16 pages, 8 figures