Open Quantum Dynamics Theory for Coulomb Potentials: Hierarchical Equations of Motion for Atomic Orbitals (AO-HEOM)

TL;DR

This work develops a rotationally invariant open quantum dynamics framework for Coulomb potential systems in thermal baths, addressing the failure of standard Markovian approaches due to bathentanglement. It introduces the 3D-RISB model and derives AO-HEOM, a nonperturbative hierarchical equations of motion for atomic orbitals using three independent directional baths. Numerical demonstrations show that AO-HEOM captures temperature- and coupling-dependent spectral features, including the emergence of discrete transitions at low temperatures and strong coupling, which are inaccessible to classical or perturbative treatments. The approach enables accurate modeling of quantum dynamics in Coulombic and cavity-QED-like settings, with potential extensions to nonlinear spectra and multi-electron systems, and promises publicly available GPU-accelerated code.

Abstract

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 (3D-RISB) model to derive numerically ``exact'' hierarchical equations of motion for atomic orbitals (AO-HEOM) 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 isotropic thermal environment, with systematic variation of system-bath coupling strength and temperature.

Figures (3)

• Figure 1: Linear absorption spectra $I_{zz}(\omega)$ along the $z$ direction for three temperature regimes: (a) high ($\beta=1.0$), (b) intermediate ($\beta=2.0$), and low ($\beta=5.0$). Each panel shows results for three S–B coupling strengths: (i) weak (solid curve, $\eta=0.001$), (ii) intermediate (dashed curve, $\eta=0.005$), and (iii) strong (dotted curve, $\eta=0.01$). To provide a reference, the spectral intensity, evaluated via Fermi's Golden Rule [factors in Eq. \ref{['Goldenrule']}], was computed for each value of $\beta$ and plotted after scaling by (a) 15, (b) 30, and (c) 100, to highlight the peak positions and relative intensities of each series. Among these, the red lines correspond to the Lyman series, the blue lines to the Balmer series, the green lines to the Paschen series, and the purple lines to the Brackett series.
• Figure 2: Linear absorption spectra $I_{zz}(\omega)$ along the $z$ direction. The calculation conditions are the same as in Fig. \ref{['weak figure']} (a)-(c), except that the SB bond strength is weakened as (i) $\eta=0.0001$ (solid curve), (ii) $\eta=0.0003$ (dashed curve), and (iii) $\eta=0.005$ (dotted curve). To provide a reference, the spectral intensity, evaluated via Fermi’s Golden Rule [factors in Eq. \ref{['Goldenrule']}], was computed for each value of $\beta$ and plotted after scaling by (a) 50, (b) 80, and (c) 100. Color-coded spectral lines: red for Lyman, blue for Balmer, green for Paschen, and purple for Brackett.
• Figure 3: Linear absorption spectra $I_{zz}(\omega)$ in the $z$ direction for principal quantum numbers of the final state ranging from $n'=2$ to 5. The inverse temperature and the S-B coupling are fixed at $\beta=1.0$ and $\eta$ = 0.01. Each curve is normalized to its maximum peak intensity.