Four-level Hamiltonian model with an additional Kerr medium and multiphoton processes

The dynamics of the Q-function for a four-level system coupled to one mode cavity field via multiphoton processes in the presence of a Kerr-like medium are investigated. Starting the field mode in a coherent state and the atom in a coherent superposition of its ground and upper states, the atom-field wave function and the reduced density matrix of the field are obtained. The cavity field statistics are investigated through Mandel Q parameter. The influence of the detunings and the Kerr medium on the evolution of the Q parameter and the Q-function is analyzed numerically for single- and two-photon processes. It is shown that splitting, rotation and broadening phenomena have occurred in the evolution of the Q-function.