Fermionic stochastic Schrödinger equation and master equation: An open-system model

Abstract

This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath, to the models in which the system operators commute with the fermion bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schrödinger equation (SSE), called fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model and the N-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.

Hierarchical Equations for Open System Dynamics in Fermionic and Bosonic Environments

“…For certain simple models, this ansatz is exact and the Q-operator independent of the noise [15]. The latter property can be used to derive a master equation for the reduced density operator (6).…”

“…The theory of non-Markovian quantum state diffusion for fermionic environments has been derived in [15,16]. Here, we will briefly recapitulate the crucial steps in order to establish the notation used throughout the paper.…”

“…However, these can be eliminated using the Grassmannian Novikov theorem (see Ref. [15] and Appendix B). Finally, we obtain the following hierarchy of density operators…”

Maximum-entropy description of animal movement

“…(2), (11), or (B1) with any finite number of constraints, even though this model can be considered as an oscillatory generalization of the OU process in other, physical contexts [22][23][24].…”

Non-Markovian dynamics of quantum open systems embedded in a hybrid environment

“…For fermionic bath, similar tools have also been developed, including scattering theory [29], non-equilibrium Green's function approach [30], and fermionic path integral [31,32]. Notably, the fermionic quantum state diffusion equations have been developed recently [33,34,35]. Although bosonic and fermionic formalisms share many similarities, a unified description of both types of baths is still useful for the purpose of direct applications.…”