Dynamics of entanglement in coherent states, entangled SchrÖdinger cat state and distribution function

We exploit the dynamics of entangled system between a harmonic oscillator and a single mode electromagnetic field as two harmonic oscillators, to examine quantum entanglement in coherent states. Through the Schrödinger cat state, we compute analytically its distribution function. Then, we check graphically the evolution of entanglement and the distribution function for two particular cases: the first is defined by two harmonic oscillators in quantum motion in a Paul trap; here whatever the quantum state, the system is at most entangled. The second case is visualised by two harmonic oscillators with a perturbative force, in that case, by varying the displacement numbers of the coherent states, the quantum entanglement and the distribution function present a similar evolution and they reach very large values. Comparing the two models: in the first, the dynamics of quantum entanglement exhibits a multi-frequency lines behaviour while in the second model, it shows an exponential behaviour.