## Quantum dynamics in the partial Wigner picture.

##### Abstract

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.