Quantum dynamics in the partial Wigner picture.
The Wigner formalism can be used to provide a representation of quantum dynamics in a classical-like phase space. However, there are many cases, such as when dealing with spin systems in a dissipative environment, in which one can more conveniently resort to a partial Wigner representation. The quantum propagator in the partial Wigner representation is, in general, a very complicated mathematical object. However, using a linear approximation, the propagator can be taken as a basis for describing the dynamics of hybrid quantum-classical systems. Such a hybrid system is composed of a quantum subsystem interacting with a coupled environment subsystem which evolves under classical-like dynamics represented in the Wigner phase space. In studying these hybrid dynamics it becomes apparent that, for a general environment system, there exists a series of quantum correction terms that restore the hybrid equation to exact quantum dynamics. Thus it is these correction terms that influence the existence of quantum effects in the dynamics of the environment subsystem and could therefore provide unique dynamical signatures indicating the existence of quantum effects. With the above motivation, we have derived an analytical expression for the quantum propagator, including correction terms, in the case of position-dependent couplings and polynomial-potential environment systems, and we have studied, numerically, the resulting quantum dynamics in a few relevant cases through comparison of quantum-classical and quantum-corrected evolutions. The type of system chosen for numerical study consisted of a two-level, or pseudo-spin, quantum system coupled to an environment represented by a quartic potential. It was found that the Rabi oscillations of the pseudo-spin are sensitive to the quantum corrections in a certain range of parameter values, either exhibiting stronger damping or stronger oscillations, depending on the tunnelling behaviour introduced by the corrections. If one were to interpret the pseudo-spin as a Cooper-pair box and the polynomial potential as representing the oscillatory behaviour of a buckled nano-rod, then this works suggests that one might be able to witness the transition of a non-linear nano-oscillator from the realm of classical dynamics to the quantum regime by observation of the pseudo-spin Rabi oscillations.