Abstract
The problem of a time-dependent quantum system, which consists of coupled sub-harmonic generation with the parametric down conversion injected within a single cavity containing an atom, is considered. The problem is transformed into the interaction between the su(1, 1) quantum system and a single two-level atom. The wave function is calculated and the expectation values for the dynamical operators are obtained. It is shown that the phenomenon of collapses and revivals are sensitive to the variation of the Bargmann index k and the Perelomov parameter eta; moreover, a coherent trapping behavior is seen at the atomic coherence angle theta = pi/2. The phenomena of bunching and anti-bunching are reported where the atomic angle theta and the phase angle phi as well as eta are the most effective parameters. They are also effective for the variance squeezing, however the Bargmann index k increases the fluctuations and decreases the amount of squeezing. For the linear entropy we have observed that the index k has no effect, but the main effective parameters are theta, phi and eta in addition to the ratio between the atomic frequency and the coupling parameter.