  Quantum calculations are developed on the dynamical system consisting of the
cosmic axions, photons and Rydberg atoms which are interacting in the resonant
microwave cavity. The time evolution is determined for the number of Rydberg
atoms in the upper state which are excited by absorbing the axion-converted and
thermal background photons. The calculations are made, in particular, by taking
into account the actual experimental situation such as the motion and uniform
distribution of the Rydberg atoms in the incident beam and also the spatial
variation of the electric field in the cavity. Some essential aspects on the
axion-photon-atom interaction in the resonant cavity are clarified by these
detailed calculations. Then, by using these results the detection sensitivity
of the Rydberg atom cavity detector is estimated properly. This systematic
quantum analysis enables us to provide the optimum experimental setup for the
dark matter axion search with Rydberg atom cavity detector.
