Non-Markovian quantum state diffusion for an open quantum system in fermionic environments
Non-Markovian quantum state diffusion (NMQSD) provides a powerful approach to the dynamics of an open quantum system in bosonic environments. Here we develop an NMQSD method to study the open quantum system in fermionic environments. This problem involves anticommutative noise functions (i.e., Grassmann variables) that are intrinsically different from the noise functions of bosonic baths. We obtain the NMQSD equation for quantum states of the system and the non-Markovian master equation. Moreover, we apply this NMQSD method to single and double quantum-dot systems.
In this paper, a linear-quadratic leader-follower (LQLF) differential game is considered, where the game system is governed by a mean-field stochastic differential equation (MF-SDE). By stochastic maximum principle, the optimal solution to the LF stochastic differential game is expressed as a feedback form of the state and its mean with the aid of two systems of Riccati equations.