Stationary states of a motionless atom in the quantized field of counterpropagating resonant waves

We investigate possible stationary states of a system consisting of a motionless atom in the resonant field of quantized isofrequency counterpropagating waves. Spontaneous relaxation is ignored. In the initial state (in the absence of an interaction), the photon subsystem has definite numbers of photons in each wave. We obtain analytic expressions for the energy spectrum and wave functions of this system. In particular, we show that the system can be in a stationary state only when the average numbers of photons in the counterpropagating waves are equal. We estimate the time it takes a system with an arbitrary initial number of photons in the counterpropagating waves to make a transition to a stationary state with equal numbers of photons in the waves.