Thermalization in Open Quantum Systems

Abstract

The thermalization of a system when interacting with a thermal bath poses an interesting problem. If a system eventually reaches a thermal state in the long-time limit, it is expected that its density matrix would resemble the mean-force Gibbs state. Moreover, the correlation function must satisfy the Kubo-Martin-Schwinger (KMS) condition or equivalently the Fluctuation-Dissipation Relation (FDR). In this work, we derive a formal expression for the non-Markovian two-point function within the context of the weak coupling limit. Using this expression, we explicitly compute the two-point function for specific models, demonstrating their adherence to the KMS condition. Additionally, we have formulated a non-perturbative approach in the form of a self-consistent approximation that includes partial resummation of perturbation theory. This approach can capture strong coupling phenomena while still relying on simple equations. Notably, we verify that the two-point function obtained through this method also satisfies the KMS condition. Another important idea discussed in this thesis is how perturbing around the thermal equilibrium helps us learn about dynamics far from equilibrium. Specifically, we demonstrate derived constraints on the master equation for open quantum systems from the positivity of the production of relative von Neumann entropy for small perturbations around the thermal equilibrium. We further show that these constraints are equivalent to the thermalization and stability constraints. Drawing motivation from recent work on hydrodynamics, we illustrate how the poles of the Green’s function for the open system capture the spectrum of the Liouvillian governing open system dynamics.