Real-Time and Imaginary-Time Quantum Hierarchal Fokker-Planck Equations

Abstract

We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time, but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possess the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state, and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy.

References in corpus (11)

• Artificial Brownian motors: Controlling transport on the nanoscale
• Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials
• Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes
• Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities
• High-performance solution of hierarchical equations of motions for studying energy-transfer in light-harvesting complexes
• Universal Markovian reduction of Brownian particle dynamics
• Modified-scaled hierarchical equation of motion approach for the study of quantum coherence in photosynthetic complexes
• Real-Time and Imaginary-Time Quantum Hierarchal Fokker-Planck Equations
• Analysis of 2D THz-Raman spectroscopy using a non-Markovian Brownian oscillator model with nonlinear system-bath interactions
• An approach to quantum transport based on reduced hierarchy equations of motion: Application to a resonant tunneling diode
• Quantum behavior of the dc SQUID phase qubit

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