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Quasi-Bell states in a strongly coupled qubit-oscillator system and their delocalization in the phase space

We study the evolution of bipartite entangled quasi-Bell states in a strongly coupled qubit-oscillator system in the presence of a static bias, and extend it to the ultra-strong coupling regime. Using the adiabatic approximation the reduced density matrix of the qubit is obtained for the strong coupling domain in closed form that involves linear combinations of the Jacobi theta functions. The reduced density matrix of the oscillator yields the phase space Husimi Q-distribution. 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 the delocalization in the phase space of the oscillator is studied by using the Wehrl entropy and the complexity of the quantum state. For a small phase space amplitude the entangled quasi-Bell state develops, during its time evolution, squeezing property and nonclassicality of the photon statistics which are measured by the quadrature variance and the Mandel parameter, respectively.