Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities

TL;DR

This paper develops reduced hierarchical equations of motion in both real and imaginary time to accurately describe the dynamics and thermodynamics of strongly coupled quantum systems with non-Markovian baths, incorporating initial system-bath coherence.

Contribution

It introduces a unified HEOM framework in real and imaginary time that accounts for initial system-bath coherence and enables calculation of thermodynamic quantities.

Findings

Real-time HEOM with initial coherence match factorized initial state results.

Imaginary-time HEOM can compute thermodynamic variables like free energy and entropy.

Steady state hierarchy elements relate real-time and imaginary-time HEOM results.

Abstract

For a system strongly coupled to a heat bath, the quantum coherence of the system and the heat bath plays an important role in the system dynamics. This is particularly true in the case of non-Markovian noise. We rigorously investigate the influence of system-bath coherence by deriving the reduced hierarchal equations of motion (HEOM), not only in real time, but also in imaginary time, which represents an inverse temperature. It is shown that the HEOM in real time obtained when we include the system-bath coherence of the initial thermal equilibrium state possess the same form as those obtained from a factorized initial state. We find that the difference in behavior of systems treated in these two manners results from the difference in initial conditions of the HEOM elements, which are defined in path integral form. We also derive HEOM along the imaginary time path to obtain the thermal equilibrium state of a system strongly coupled to a non-Markovian bath. Then, we show that the steady state hierarchy elements calculated from the real-time HEOM can be expressed in terms of the hierarchy elements calculated from the imaginary-time HEOM. Moreover, we find that the imaginary-time HEOM allow us to evaluate a number of thermodynamic variables, including the free energy, entropy, internal energy, heat capacity, and susceptibility. The expectation values of the system energy and system-bath interaction energy in the thermal equilibrium state are also evaluated.