Jiushu SHAO
Beijing Normal University

Upon employing the Hubbard-Stratonovich transformation or Ito calculus as well as the Girsanov transformation, the quantum dynamics of a dissipative system described by a system-plus-bath model is shown to satisfy a stochastic Liouville equation driven by complex noises, which might be regarded as the quantum analogue to the traditional Langevin equation.

The stochastic formulation can be used not only as a theoretical tool for deriving master equations for specific systems or developing approximations, but also as a practical technique for simulating nonequilibrium dynamics numerically via a direct implementation or transforming to a deterministic algorithm a la hierarchical equations. It has been demonstrated that a mixed random-deterministic scheme taking both advantages of the random and deterministic treatments is effective to calculate the zero-temperature dynamics of the spin-boson model with Ohmic dissipation. It is observed that for strong dissipation the population in the localized state obeys a simple rate dynamics and time scale is proportional to the reciprocal of the cutoff frequency.