SOLUTION OF THE TIME-DEPENDENT LIOUVILLE-VONNEUMANN EQUATION - DISSIPATIVE EVOLUTION

A mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions. The representation of the density operator is based on the Fourier pseudospectral method which allows a description both in configuration as well as in momentum space. A new propagation scheme which treats the complex eigenvalue structure of the dissipative Liouville superoperator has been developed. The framework has been designed to incorporate modern computer architecture such as parallelism and vectorization. Comparing the results to closed-form solutions exponentially fast convergence characteristics in phase space as well as in the time propagation is demonstrated. As an example of its usefulness, the new method has been successfully applied to dissipation under the constraint of selection rules. More specifically, a harmonic oscillator which relaxes to equilibrium under the constraint of second-order coupling to the bath was studied. The results of the calculation were compared to a mean field approximation developed for this problem. It has been found that this approximation does not capture the essence of the relaxation process. In conclusion, the new method presented is a conceptual tool to model multi-dimensional quantum physical systems which exhibit both relaxation as well as oscillation in an efficient, accurate and convenient manner.