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Evolution of a hybrid micro-macro entangled state of the qubit-oscillator system via the generalized rotating wave approximation

We study the evolution of the hybrid entangled states in a bipartite (ultra) strongly coupled qubit-oscillator system. Using the generalized rotating wave approximation the reduced density matrices of the qubit and the oscillator are obtained. The reduced density matrix of the oscillator yields the phase space quasi probability distributions such as the diagonal P-representation, the Wigner W-distribution and the Husimi Q-function. In the strong coupling regime the Q-function evolves to uniformly separated macroscopically distinct Gaussian peaks representing 'kitten' states at certain specified times that depend on multiple time scales present in the interacting system. For the ultra-strong coupling realm a large number of interaction-generated modes arise with a complete randomization of their phases. A stochastic averaging of the dynamical quantities sets in while leading to the decoherence of the system. The delocalization in the phase space of the oscillator is studied by using the Wehrl entropy. The negativity of the W-distribution, while registering its departure from the classical states, allows us to compare the information-theoretic measures such as the Wehrl entropy with the Wigner entropy. Other features of nonclassicality such as the existence of the squeezed states and appearance of negative values of the Mandel parameter are realized during the course of evolution of the bipartite system. In the parametric regime studied here these properties do not survive after a time-averaging process.