Quantum hierarchical Fokker–Planck equations with U(1) gauge fields [U(1)-QHFPE]: A computational framework for Aharonov–Bohm effects

Abstract

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.