Hierarchical Equations for Open System Dynamics in Fermionic and Bosonic Environments

We present two 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 (Suess et al. in Phys. Rev. Lett. 113, 150403, 2014). 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 for density matrices. A similar reformulation of the bosonic hierarchy of pure states to a master equation hierarchy and its relation to the hierarchical equations of motion of Tanimura and Kubo is also presented.