  In this paper a formalism for studying the dynamics of quantum systems
coupled to classical spin environments is reviewed. The theory is based on
generalized antisymmetric brackets and naturally predicts open-path
off-diagonal geometric phases in the evolution of the density matrix. It is
shown that such geometric phases must also be considered in the
quantum-classical Liouville equation for a classical bath with canonical phase
space coordinates; this occurs whenever the adiabatics basis is complex (as in
the case of a magnetic field coupled to the quantum subsystem). When the
quantum subsystem is weakly coupled to the spin environment, non-adiabatic
transitions can be neglected and one can construct an effective non-Markovian
computer simulation scheme for open quantum system dynamics in classical spin
environments. In order to tackle this case, integration algorithms based on the
symmetric Trotter factorization of the classical-like spin propagator are
derived. Such algorithms are applied to a model comprising a quantum two-level
system coupled to a single classical spin in an external magnetic field.
Starting from an excited state, the population difference and the coherences of
this two-state model are simulated in time while the dynamics of the classical
spin is monitored in detail. It is the author's opinion that the numerical
evidence provided in this paper is a first step toward developing the
simulation of quantum dynamics in classical spin environments into an effective
tool. In turn, the ability to simulate such a dynamics can have a positive
impact on various fields, among which, for example, nano-science.
