Abstract:
Multi-time correlation functions play a central role in the characterization of dynamical, spectral, and thermal properties of open quantum systems. While the theory of reduced density matrix dynamics is well developed, the consistent computation of multi-time correlation functions beyond the standard weak-coupling and Markovian assumptions remains an open and challenging problem. This thesis addresses this gap by developing and analyzing analytical frameworks for multi-time correlation functions across weak, strong, and ultra-strong system–reservoir coupling regimes. We first revisit the formulation of multi-time correlation functions from a Heisenbergpicture perspective and derive the Quantum Regression Theorem (QRT) in the Markovian limit. We generalize the regression framework to time-ordered multi-time correlation functions with general time arguments and extend it to out-of-time-ordered correlators (OTOCs), highlighting the limitations of standard regression-based approaches. We further investigate the impact of non-Markovian dynamics, where the QRT acquires nontrivial correction terms and higher-order reservoir correlations become relevant. A central theme of this thesis is the consistency of correlation functions with thermalization conditions. We demonstrate that the standard QRT fails to preserve the Kubo–Martin–Schwinger (KMS) condition at finite order in the system–reservoir coupling. To address this issue, we introduce a modified Quantum Regression Theorem based on a weaker notion of Markovianity and show that it restores thermal consistency in the long-time limit. Going beyond the weak-coupling regime, we develop a self-consistent Green’s function framework based on the Schwinger–Keldysh formalism, demonstrating that steady-state two-point correlation functions satisfy the KMS condition for generic systems. Finally, we investigate two-time correlation functions in the ultra-strong coupling regime using the Nakajima–Zwanzig projection operator formalism. We show that the emergence of regression behavior depends on multiple dynamical timescales and the choice of system operators, and we benchmark our analytical results against numerically exact simulations.
