Non-Markovian quantum state diffusion (NMQSD) provides a powerful approach to the dynamics of an open quantum system in bosonic environments. Here we develop a 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 review we discuss recent progress in the theory of open quantum systems based on non-MarkovianNon-Markovian quantum state diffusionQuantum state diffusion equation and master equationsMaster equation. In particular, we show that an exact master equation for an open quantum system consisting of a few qubits can be explicitly constructed by using the corresponding non-Markovian quantum state diffusion equation. The exact master equation arises naturally from the quantum decoherence dynamics of qubit systems collectively interacting with a colored noise. We illustrate our general theoretical formalism by the explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This exact qubit master equation accurately characterizes the time evolution of the qubit system in various parameter domains, and paves the way for investigation of the memory effect of an open quantum system in a non-Markovian regime without any approximation.