RC-HEOM Hybrid Method for Non-Perturbative Open System Dynamics

Abstract

The Hierarchical equations of motion (HEOM) method is an important non-perturbative technique, allowing numerically exact treatment of open quantum systems with strong coupling and non-Markovian memory. However, its encoding of bath memory into auxiliary density operators often limits direct access to detailed bath information. In contrast, the reaction-coordinate (RC) mapping allows direct and transparent access to the dominant collective bath mode, but its perturbative and often Markovian treatment of the residual bath restricts its reliability. In this work, we introduce RC-HEOM, a hybrid method that unifies the strengths of both approaches by combining RC mapping with a fully non-perturbative HEOM description of the residual bath. RC-HEOM simultaneously retains exact non-Markovian memory and access to the RC mode, which enables analysis of system-RC information. Applying this method to the Anderson impurity models, we directly track the emergence of the Kondo singlet from the growth of the Kondo resonance and uncover a nontrivial RC-mediated coherence revival. These results demonstrate that RC-HEOM is a promising method for characterizing open quantum systems in regimes that are difficult to access with conventional master-equation methods.

Figures (3)

• Figure 1: Illustration of three methods used in open system dynamics calculations. (a) RC$\text{--}$HEOM. The system is coupled to the RC which couples to the ADOs of the residual bath. (b) HEOM*. System interactions with the bath is expressed as the system coupled to many ADOs. (c) RC$\text{--}$ME. The system is coupled to the RC which couples to the residual bath. System+RC interactions with the residaul bath are treated with the Lindblad master equation.
• Figure 2: The density of states $A(\omega)$ of the single impurity Anderson model at different temperatures. Lorentz bath parameters set to $W=1.25\Gamma,\mu=0$, and system parameters set to $U=3\pi\Gamma/2,\epsilon=-U/2$. Results using HEOM*, RC$\text{--}$ME, RC$\text{--}$HEOM are shown in black solid, blue dashdot, red dashed curves, respectively. Computation details in Supplemental Material sup.
• Figure 3: Coherence revival of the two-impurity Anderson model. (a) $l_1$ norm of coherence for the two impurities: $l_1^{\text{sys}}$ with respect to time $t$. Results using HEOM*, RC$\text{--}$ME, RC$\text{--}$HEOM are shown in black solid, blue dashdot, red dashed curves, respectively. Gray vertical line at $t'$ indicates the coherence revival time. (b) Individual coherence contributions $|\rho_{i,j}^{\text{sys}}|$ (that make up $l_1^{\text{sys}}$) which display revival are shown in black, blue, red, and green solid curves. Additionally, the interference factor $\mathcal{I}$ which measures the RC mediated path interference between the individual system+RC coherence that contribute to $|C_{\text{rev}}|$ (black solid curve) is shown as the magenta dashdot curve. Lorentz bath parameters are set to $\Gamma=20W, \mu=0, k_BT=5W$, and system parameters are set to$\epsilon_1=-2W, \epsilon_2=-W, U_1=U_2=10W$. Impurities are initialized in the vacuum state. RC is initialized in the thermal equilibrium state. Computation details are shown in Supplemental Material sup.