Hierarchical equations for open system dynamics in fermionic and bosonic environments

TL;DR

This paper introduces new hierarchical equations to model the non-Markovian dynamics of open quantum systems interacting with fermionic or bosonic environments, including stochastic and density matrix approaches.

Contribution

It develops novel hierarchical equations for open system dynamics with fermionic and bosonic baths, including a noise-free density matrix hierarchy for fermions.

Findings

Derived a stochastic hierarchy with Grassmannian noise for fermions

Established a noise-free density matrix hierarchy for fermionic environments

Extended the bosonic hierarchy to a master equation form

Abstract

We present novel approaches to the dynamics of an open quantum system coupled linearly to a non-Markovian fermionic or bosonic environment. In the first approach, we obtain a hierarchy of stochastic evolution equations of the diffusion type. For the bosonic case such a hierarchy has been derived and proven suitable for efficient numerical simulations recently [arXiv:1402.4647]. The stochastic fermionic hierarchy derived here contains Grassmannian noise, which makes it difficult to simulate numerically due to its anti-commutative multiplication. Therefore, in our second approach we eliminate the noise by deriving a related hierarchy of density matrices. A similar reformulation of the bosonic hierarchy of pure states to a master equation hierarchy is also presented.