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 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 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 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.
Building upon our previous implementation for the hydrogen atom [Y. Zhang and Y. Tanimura, J. Chem. Phys. 163, 184108 (2025)], we have developed source code for atomic orbital-hierarchical equations of motion (AO-HEOM), a quantum mechanical framework based on HEOM formulated within an AO basis. This method enables numerically "exact" simulations of atomic systems coupled to three independent thermal baths, under both isotropic and anisotropic conditions, while preserving rotational symmetry. AO-HEOM rigorously accounts for system-bath entanglement, which is critical for describing the quantum nature of environmental interactions. Incorporating spatial bath degrees of freedom significantly increases the computational cost in the HEOM formalism because of the proliferation of electronic states at high temperatures and the inclusion of Matsubara frequency terms at low temperatures. To address this challenge, we developed a graphics processing unit-accelerated implementation. As a demonstration, we computed the emission spectra of He I and He II atoms. The source code is broadly applicable to atomic systems and enables detailed analysis of electronic transitions in thermal environments.
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 developed a computer code for the thermodynamic quantum Fokker-Planck equations (T-QFPE), derived from a thermodynamic system-bath model. This model consists of an anharmonic subsystem coupled to multiple Ohmic baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code numerically integrates the T-QFPE and their classical expression to simulate isothermal, isentropic, thermostatic, and entropic processes in both quantum and classical cases. The accuracy of the results was verified by comparing the analytical solutions of the Brownian oscillator. Additionally, we illustrated a breakdown of the Markovian Lindblad-master equation in the pure quantum regime. As a demonstration, we simulated a thermostatic Stirling engine employed to develop non-equilibrium thermodynamics [S. Koyanagi and Y. Tanimura, J. Chem. Phys 161, 114113 (2024)] under quasi-static conditions. The quasi-static thermodynamic potentials, described as intensive and extensive variables, were depicted as work diagrams. In the classical case, the work done by the external field is independent of the system-bath coupling strength. In contrast, in the quantum case, the work decreases as the coupling strength increases due to quantum entanglement between the subsystem and bath. The codes were developed for multicore processors using Open multiprocessing (OpenMP) and for graphics processing units (GPU) using the Compute United Device Architecture (CUDA). These codes are provided as supplementary materials.
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.