## Numerical simulation of quantum spins in a dissipative environment.

##### Abstract

When modelling real physical systems, one should always consider the influence of the
environment. The study of open quantum systems aims precisely at the assessment of
the role of the environment in the evolution of these systems. The computational study
of these systems is in general a formidable task. However, in many instances one can
approximate the environment by means of classical dynamics. For such cases, when the
environment is in a thermal state, one can generalize non-Hamiltonian equations of motion
to quantum-classical dynamics in order to control the thermodynamic temperature. Such a
generalization is achieved within the partial Wigner representation of quantum mechanics,
and leads to a quantum-classical evolution of the system. We have studied how to simulate
a thermal bath, by means of the least possible number of additional extended variables [1].
This has been achieved upon reformulating the Nos´e-Hoover Power (NHP) thermostat in
quantum-classical theory. When applied to the dissipative evolution of a quantum spin the
NHP thermostat, obtained numerical results in agreement with those obtained using Nos`e-
Hoover chains. However, since a fewer number of variables are needed to achieve the correct
sampling of the canonical distribution at equilibrium, the NHP thermostat promises to be
better suited for the simulation of low dimensional open quantum system on discrete grids.
Following this the quantum dynamics of a XXZ two-spin chain interacting with multiple
bosonic baths have been studied, within the mixed Wigner-Heisenberg representation of
quantum mechanics. In particular, we have simulated the dynamics of the reduced density
matrix ρs(t), along with the time evolution of the quantum entanglement of the two spins
via the concurrence C (ρ).