Quantum theory of the micromaser with ultra-cold Λ-type three-level atoms

We present the general quantum theory for the micromaser with ultra-cold Λ-type three-level atoms. We first derive the dressed states for the interaction of a Λ-type three-level atom with a quantum cavity field, then treat the interaction between the injection atom with the cavity field as a scattering process, find the reflection and transmission coefficients in this process, and calculate the atomic emission probability.