Quantum Hierarchical Fokker-Planck Equations with U(1) Gauge Fields (U(1)-QHFPE): A Computational Framework for Aharonov-Bohm Effects

TL;DR

The paper addresses numerically exact simulation of Aharonov–Bohm ring transport in dissipative open quantum systems with gauge fields, preserving gauge invariance and rotational symmetry in non-Markovian, non-perturbative regimes. It introduces U(1)-QHFPE, a gauge-invariant HEOM–Fokker–Planck framework implemented via a discretized Wigner transform under periodic boundary conditions ($DWT$-$PBC$) and a Drude spectral density $J^\alpha(\omega)$, enabling accurate treatment of strong system–bath coupling. Key contributions include the derivation of the U(1)-HQFPE equations, a CPU/GPU-accelerated software implementation with adaptive time stepping, and demonstration through equilibrium distributions and linear-response analyses that reveal AB phase oscillations and Byers–Yang periodicity. The framework provides a numerically exact tool for quantum transport in AB rings under realistic thermal environments, supports anisotropic baths, and lays groundwork for extensions to leads and 2D geometries, making it relevant for gauge-field–assisted quantum devices and topological interference phenomena.

Abstract

We introduce U(1)-QHFPE, a non-Markovian and non-perturbative open quantum dynamics software package for solving quantum Fokker-Planck equations incorporating gauge fields within the Hierarchical Equations of Motion (HEOM) formalism. The framework rigorously preserves gauge invariance and rotational symmetry, enabling accurate simulations of transport phenomena such as the Aharonov-Bohm effect under strong system-bath coupling. In this regime, quantum entanglement between the system and bath emerges naturally. Demonstration programs include calculations of symmetric and antisymmetric correlation functions in Aharonov-Bohm ring geometries, showcasing the code's ability to resolve topological quantum interference in dissipative open systems.

Figures (2)

• Figure 1: Equilibrium PDFs are presented for (a) the high-temperature case with $\beta = 1.0$ and (b) the low-temperature case with $\beta = 2.5$. The S-B coupling strength in the $x$-direction was set to be twice that in the $y$-direction. At each temperature, three regimes of anisotropic S-B coupling are illustrated: weak coupling (blue curves, $\eta_x = 0.02$, $\eta_y = 0.01$), intermediate coupling (green curves, $\eta_x = 0.2$, $\eta_y = 0.1$), and strong coupling (red curves, $\eta_x = 1.0$, $\eta_y = 0.5$). In panel (b), the influence of a magnetic flux $\bar{\Phi} = 0.5$ is shown as dashed curves. At high temperature, however, the equilibrium distribution exhibits negligible dependence on the magnetic flux. Consequently, results for $\bar{\Phi} = 0.5$ are omitted from panel (a).
• Figure 2: Linear response spectra of the dipole moment for the AB ring, calculated under isotropic and anisotropic environments. The S-B coupling strengths are set as follows: (red) $\eta_x = 1.0$, $\eta_y = 0.1$; (green) $\eta_x = 1.0$, $\eta_y = 1.0$; (blue) $\eta_x = 0.1$, $\eta_y = 1.0$. The magnetic flux values for each panel are: (a) $\bar{\Phi} = 0.0$, (b) $\bar{\Phi} = 0.1$, (c) $\bar{\Phi} = 0.2$, (d) $\bar{\Phi} = 0.3$, (e) $\bar{\Phi} = 0.4$, and (f) $\bar{\Phi} = 0.5$. The black dashed line marks the peak position predicted by the system Hamiltonian without the baths.