Entanglement dynamics of a nano-mechanical resonator coupled to a central qubit

We introduce a unitary operator which may be constructed conveniently by exploiting the properties of the Glauber displacement and parity operators. We show that it can be considered as a constant of motion of a quantum system including pure dephasing interaction of a central qubit with a nano-mechanical resonator. Using the eigenvectors of the Glauber displacement operator, we paves a way for an extensive study of the dynamics of resonator-qubit states. In addition, we show that the establishment of such a constant of motion, which includes the parity operator, provides a way to introduce a capable mechanical framework to generate some desired mechanical state as superposition of the Glauber coherent states. We investigate nonclassical properties of the generated mechanical states, by evaluating mechanical-squeezing, Wigner function, position-momentum entropic squeezing, and entanglement between spin-mechanical modes. Furthermore, the pairwise classical and quantum correlations are derived based on a necessary and sufficient condition for the zero-discord state. Finally, in order to study the mechanical state's behavior in an environment, we consider that the output state is subject to amplitude (phase)-damping channels and their dissipative properties are analyzed.