Time-local master equations for non-Markovian processes in quantum and classical open systems

• Applicant: Professor Dr. Heinz-Peter Breuer
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics ; Optics, Quantum Optics and Physics of Atoms, Molecules and Plasmas
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 431945604

The dynamics of open quantum systems is often governed by non-Markovian behavior and pronounced memory effects, featuring a backflow of information from the environment to the open system. Mathematically, this dynamics is commonly treated by means of master equations for the open system's density matrix, which can be derived employing suitable projection operator techniques. In the quantum regime there are basically two different variants of these techniques, namely the Nakajima-Zwanzig projection operator technique leading to a memory kernel master equation, and the time-convolutionless (TCL) projection operator technique yielding a time-local first order differential equation. As part of the second funding period of the Research Unit "Reducing complexity of nonequilibrium systems" the present project is concerned with the general properties of TCL master equations describing classical and quantum non-Markovian processes in open systems. One of the central problems is the analysis of the mathematical structure of singularities of the time-dependent TCL generator. This is a challenging problem which has not yet been studied in general terms. We will develop a classification of the singularities and establish general conditions for their occurrence in physically relevant system-environment models. A further challenge is to investigate the impact of singularities on the performance of the TCL perturbation expansion in terms of the ordered cumulants of environmental correlations functions. Another problem of great relevance is the application to a recent formulation of nonequilibrium quantum thermodynamics, which employs time-local master equations to formulate work and heat exchange in nonequilibrium processes and, hence, requires a detailed discussion of the mathematical structure of the singularities and of their physical meaning and implications. In addition to these mainly analytical works, we will also develop algorithms for the numerical determination of the time-dependent TCL generator. This is a problem of practical relevance as it allows to efficiently carry out long-time simulations. We will also compare the quality of the various approximations obtained from the time-local and from the memory-kernel master equation, and examine the performance of the TCL master equation in the classical regime of non-Markovianity. Finally, we will develop a mixed quantum-classical approach, combining the TCL master equation with a classical treatment of the slow degrees of freedom and a hopping process between energy surfaces of the effective Hamiltonian.

• DFG Programme: Research Units
• Subproject of: FOR 5099: Reducing complexity of nonequilibrium systems
• Co-Investigators: Dr. Johan Runeson; Professor Dr. Michael Thoss