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

Yoshitaka Tanimura

doi:10.1063/1.4890441

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.

Highlights

Quantum open systems have been a subject of fundamental interest for many years
Problems in this category include those of understanding how the irreversibility of time appears in system dynamics, why macroscopic systems can be treated with classical mechanics instead of quantum mechanics, how wave functions collapse as a result of measurements done with macroscopic instruments, and why and how quantum systems approach a thermal equilibrium state through interaction with their environments
The real-time hierarchal equations of motion (HEOM) were integrated from the factorized initial conditions ρ0(0,.)..,0(0) = exp[−βHA]/tr{exp[−βHA]} and ρj(n1,)...,jK (0) = 0 at t = 0, and steady states were realized between t = 100 and t = 200

Summary

Introduction

Quantum open systems have been a subject of fundamental interest for many years. Problems in this category include those of understanding how the irreversibility of time appears in system dynamics, why macroscopic systems can be treated with classical mechanics instead of quantum mechanics, how wave functions collapse as a result of measurements done with macroscopic instruments, and why and how quantum systems approach a thermal equilibrium state through interaction with their environments. Theories of quantum open systems have been used to construct models of practical interest, in particular to account for line shapes in EPR, NMR, and laser spectra, to evaluate chemical reaction rates and electron and charge transfer rates in chemical physics, and to explore the lifetimes of quantum entanglement states in quantum information theory.11The phenomena mentioned above arise from the unavoidable interaction of a system with its environment. Quantum open systems have been a subject of fundamental interest for many years Problems in this category include those of understanding how the irreversibility of time appears in system dynamics, why macroscopic systems can be treated with classical mechanics instead of quantum mechanics, how wave functions collapse as a result of measurements done with macroscopic instruments, and why and how quantum systems approach a thermal equilibrium state through interaction with their environments.. In the quantum mechanical case, dissipative systems are often modeled as main systems coupled to heat-bath degrees of freedom at finite temperature. This coupling gives rise to thermal fluctuations and dissipation that drive the systems toward the thermal equilibrium state. The heat-bath degrees of freedom are reduced using such methods as the projection operator method and the path integral method

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