Journal of Statistical Physics

, Volume 159, Issue 6, pp 1408–1423 | Cite as

Hierarchical Equations for Open System Dynamics in Fermionic and Bosonic Environments

  • D. Suess
  • W. T. Strunz
  • A. Eisfeld


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.


Non-Markovian Stochastic Schrödinger equation Master equation Quantum trajectories Fermionic Bosonic 



DS acknowledges support by the Excellence Initiative of the German Federal and State Governments (Grant ZUK 43), the ARO under contracts W911NF-14- 1-0098 and W911NF-14-1-0133 (Quantum Characterization, Verification, and Validation), and the DFG.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Max Planck Institute for Physics of Complex SystemsDresdenGermany
  2. 2.Institut für Theoretische PhysikTechnische Universität DresdenDresdenGermany
  3. 3.Institute of PhysicsUniversity of FreiburgFreiburgGermany

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