Dark-State Polaritons in Single- and Double-$\mathrm{\Lambda }$ Media

We derive the properties of polaritons in single-$\mathrm{\Lambda }$ and double-$\mathrm{\Lambda }$ media using a microscopic equation-of-motion technique. In each case, the polaritonic dispersion relation and composition arise from a matrix eigenvalue problem for arbitrary control field strengths. We show that the double-$\mathrm{\Lambda }$ medium can be used to up- or down-convert single photons while preserving quantum coherence. The existence of a dark-state polariton protects this single-photon four-wave mixing effect against incoherent decay of the excited atomic states. The efficiency of this conversion is limited mainly by the sample size and the lifetime of the metastable state.