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arXiv:2509.11462

Quantum hierarchical Fokker-Planck equations with U(1) gauge fields: Application to the Aharonov-Bohm ring

Yang, Hyeonseok, Shoki Koyanagi, Yoshitaka Tanimura

Published Sep 14, 2025• 0 citations
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Abstract

We investigate a three-dimensional subsystem under a time-dependent U(1) gauge field coupled to rotationally invariant environments. To capture the dynamic behavior of the subsystem under thermal excitations and dissipations, it is imperative to treat the bath in a non-Markovian and nonperturbative manner. This is because quantum noise is constrained by the uncertainty principle, which dictates the relationship between the noise correlation time and the amplitude of the energy fluctuation. To this end, we derive the hierarchical equations of motion (HEOM) incorporating the gauge field, enabling a rigorous investigation of the dynamics of the reduced subsystem. Transforming the HEOM into the Wigner representation yields quantum hierarchical Fokker-Planck equations [U(1)-QHFPE] with U(1) gauge fields. These equations incorporate vector fields into the damping operators while preserving both gauge invariance and rotational symmetry. To demonstrate the practical use of the formalism, the effects of a heat bath in the Aharonov-Bohm (AB) ring. Our investigation includes simulations of the equilibrium distribution, linear absorption spectra, and AB currents under thermal conditions. Within a rotationally invariant system-bath (RISB) model, we predict the emergence of a persistent current even in dissipative environments, provided the bath is non-Markovian and the temperature is sufficiently low. We also assessed the validity of the Caldeira-Leggett model in this context.

DOI
10.48550/arxiv.2509.11462

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We investigate a three-dimensional subsystem under a time-dependent U(1) gauge field coupled to rotationally invariant environments. To capture the dynamic behavior of the subsystem under thermal excitations and dissipations, it is imperative to treat the bath in a non-Markovian and nonperturbative manner. This is because quantum noise is constrained by the uncertainty principle, which dictates the relationship between the noise correlation time and the amplitude of the energy fluctuation. To this end, we derive the hierarchical equations of motion (HEOM) incorporating the gauge field, enabling a rigorous investigation of the dynamics of the reduced subsystem. Transforming the HEOM into the Wigner representation yields quantum hierarchical Fokker-Planck equations [U(1)-QHFPE] with U(1) gauge fields. These equations incorporate vector fields into the damping operators while preserving both gauge invariance and rotational symmetry. To demonstrate the practical use of the formalism, the effects of a heat bath in the Aharonov-Bohm (AB) ring. Our investigation includes simulations of the equilibrium distribution, linear absorption spectra, and AB currents under thermal conditions. Within a rotationally invariant system-bath (RISB) model, we predict the emergence of a persistent current even in dissipative environments, provided the bath is non-Markovian and the temperature is sufficiently low. We also assessed the validity of the Caldeira-Leggett model in this context.
We investigate a three-dimensional subsystem under a time-dependent U(1) gauge field coupled to rotationally invariant environments. To capture the dynamic behavior of the subsystem under thermal excitations and dissipations, it is imperative to treat the bath in a non-Markovian and nonperturbative manner. This is because quantum noise is constrained by the uncertainty principle, which dictates the relationship between the noise correlation time and the amplitude of the energy fluctuation. To this end, we derive the hierarchical equations of motion (HEOM) incorporating the gauge field, enabling a rigorous investigation of the dynamics of the reduced subsystem. Transforming the HEOM into the Wigner representation yields quantum hierarchical Fokker-Planck equations [U(1)-QHFPE] with U(1) gauge fields. These equations incorporate vector fields into the damping operators while preserving both gauge invariance and rotational symmetry. To demonstrate the pratical use of the formalism, we will use it to study the effects of a heat bath on the Aharonov-Bohm (AB) ring. Our investigation includes simulations of the equilibrium distribution, linear absorption spectra, and AB currents under thermal conditions. Within a rotationally invariant system-bath model, we predict the emergence of a persistent current even in dissipative environments, provided the bath is non-Markovian and the temperature is sufficiently low. We also assessed the validity of the Caldeira-Leggett model in this context.
We present a software package that solves the quantum Fokker-Planck equation with gauge fields, formulated within the hierarchical equations of motion framework [U(1)-QHFPE]. The framework rigorously preserves gauge invariance and rotational symmetry under non-Markovian and non-perturbative system-bath (S-B) interactions, enabling accurate simulations of transport phenomena such as the Aharonov-Bohm effect under thermal environments. In a strong S-B coupling regime, quantum S-B entanglement emerges naturally. The demonstration programs perform calculations of response functions in Aharonov-Bohm ring geometries with a mechanical potential that may induce quantum tunneling, thereby illustrating the software's capability to resolve topological quantum interference in dissipative open systems. Written in C++, the code includes a central processing unit version with highly readable OpenMP directives and a graphics processing unit version tailored for high-performance computing.
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 possesses 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. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
We investigate the quantum dynamics of Coulomb potential systems in thermal baths. We study these systems within the framework of open quantum dynamics theory, focusing on preserving the rotational symmetry of the entire system, including the baths. Thus, we employ a three-dimensional rotationally invariant system-bath model to derive numerically "exact" hierarchical equations of motion for atomic orbitals that enable a non-perturbative and non-Markovian treatment of system-bath interactions at finite temperatures. To assess the formalism, we calculated the linear absorption spectrum of an atomic system under an isotropic thermal environment, with systematic variation of system-bath coupling strength and temperature.
We develop a quantum-classical hierarchical equations of motion (QC-HEOM) approach for simulating non-Markovian open quantum systems. The method combines the ensemble-averaged classical path reference of the quantum-classical path integral formalism with a hierarchy of auxiliary quantum influence functionals. By incorporating thermal fluctuations through an ensemble average over reference trajectories, the hierarchy is required to represent only the residual quantum memory associated with the imaginary part of the bath response function. Consequently, unlike conventional hierarchical equations of motion, QC-HEOM does not require Matsubara or Padé expansions of the thermal kernel and exhibits only weak temperature dependence of the hierarchy size. Furthermore, because thermal fluctuations are supplied through reference classical trajectories, the framework naturally extends beyond harmonic baths and enables the incorporation of anharmonic and molecular environments through externally generated trajectories. We derive the formalism and demonstrate its exactness for a harmonic bath. Applications to an asymmetric spin-boson model and the seven-site Fenna--Matthews--Olson complex illustrate the accuracy of QC-HEOM. It reproduces benchmark quasi-adiabatic path integral and hierarchical equations of motion results while requiring substantially fewer auxiliary objects, particularly at low temperatures. These results establish QC-HEOM as an efficient framework for treating residual quantum memory in quantum-classical descriptions of open-system dynamics. The separation of thermal fluctuations from residual quantum memory through the use of Wigner trajectories provides an approximate route toward hierarchical treatments of complex anharmonic environments that are inaccessible to conventional HEOM approaches.
The theory of hierarchical equations of motion (HEOM) is one of the standard methods to give exact evaluations of the dynamics as coupled to harmonic oscillator environments. However, the theory is numerically demanding due to its hierarchy, which is the set of auxiliary elements introduced to capture the non-Markovian and non-perturbative effects of environments. When system-bath coupling becomes relatively strong, the required computational resources and precision move beyond the regime that can be currently handled. This article presents a new representation of HEOM theory in which the hierarchy is mapped into a continuous space of a collective bath coordinate and several auxiliary coordinates as the form of the quantum Fokker-Planck equation. This representation gives a rigorous time evolution of the bath coordinate distribution and is more stable and efficient than the original HEOM theory, particularly when there is a strong system-bath coupling. We demonstrate the suitability of this approach to treat vibronic system models coupled to environments.