Analytic solutions for degenerate Raman-coupled model

The Raman-coupled interaction between an atom and a single mode of a cavity field is studied. For the cases in which a light field is initially in a coherent state and in a thermal state separately, we have derived the analytic expressions for the time evolutions of atomic population difference W, modulus B of the Bloch vector, and entropy E. We find that the time evolutions of these quantities are periodic with a period of \pi . The maxima of W and B appear at the scaled interaction time points \tau = k\pi (k = 0,1,2,\ldots). At these time points, E = 0, which shows that the atom and the field are not entangled. Between these time points, E \ne 0, which means that the atom and the field are entangled. When the field is initially in a coherent state, near the maxima, the envelope of W is a Gaussian function with a variance of 1 /\left( 4\bar n \right) (\bar n is the mean number of photons). Under the envelope, W oscillates at a frequency of \bar n / \pi . When the field is initially in a thermal state, near the maxima, W is a Lorentz function with a width of 1/ \bar n.