A Davydov Ansatz approach to accurate system-bath dynamics in the presence of multiple baths with distinct temperatures

TL;DR

This work develops and benchmarks a time-dependent variational approach based on the multi-Davydov Ansatz (mDA) for a two-bath spin-boson model at finite temperatures, using Thermofield Dynamics to incorporate thermal fluctuations. The authors demonstrate that mDA, with multiplicity $M$, can reproduce benchmark results from HEOM and QUAPI across weak, intermediate, and strong coupling regimes and across high, low, and very low temperatures, often with favorable scaling in the number of bath modes. A key contribution is the introduction of a deviation-vector metric $\sigma^2(t)$ to rigorously assess variational accuracy and convergence, establishing practical thresholds. The study highlights mDA’s versatility and efficiency in capturing non-Markovian, long-time dynamics and energy transport in non-equilibrium quantum systems, suggesting its potential for broader quantum thermodynamics applications and more challenging spectral densities.

Abstract

We perform benchmark simulations using the time-dependent variational approach with the multiple Davydov Ansatz (mDA) to study realtime nonequilibrium dynamics in a single qubit model coupled to two thermal baths with distinct temperatures. A broad region of the parameter space has been investigated, accompanied by a detailed analysis of the convergence behavior of the mDA method. In addition, we have compared our mD2 results to those from two widely adopted, numerically "exact" techniques: the methods of hierarchical equations of motion (HEOM) and the quasi-adiabatic path integral (QUAPI). It is found that the mDA approach in combination with thermal field dynamics yields numerically accurate, convergent results in nearly all regions of the parameter space examined, including those that pose serious challenges for QUAPI and HEOM. Our results reveal that mDA offers a highly adaptable framework capable of capturing long-time dynamics, even in challenging regimes where other methods face limitations. These findings underscore the potential of mDA as a versatile tool for exploring quantum thermodynamics, energy transfer processes, and non-equilibrium quantum systems.

Figures (10)

• Figure 1: Schematic of the two-bath spin-boson model. A central two-level system is simultaneously coupled to two independent bosonic reservoirs of temperatures $T_l$ and $T_r$, and the $k$th reservoir mode is characterized by annihilation and creation operators $\hat{b}_{\nu,k}$ and $\hat{b}_{\nu,k}^\dagger$ ($\nu=l,r$), and the bidirectional arrows denote the system-bath coupling.
• Figure 2: Time evolution of the deviation $\sigma^2(t)$ on a log–linear scale for $T=2$, $\alpha/\omega_0=0.02$, $\omega_{c}/\omega_0=1.5$. The dashed horizontal line marks the convergence threshold $\sigma^2=10^{-2}$.
• Figure 3: Parameter space of the model with zero tunneling parameter ($\Delta = 0$). The horizontal axis (in logarithmic scale) corresponds to the coupling strength, while the vertical axis (in linear scale) represents the cutoff frequency $\omega_c$. Three temperature regimes are considered: (a) $T = 0.02$ (very low temperature), (b) $T = 0.2$ (low temperature), and (c) $T = 2.0$ (high temperature). The marked points denote the specific parameter values used in our simulations.
• Figure 4: Time evolution of the nonequilibrium population difference $\sigma_z(t)$ for the weak coupling case ($\alpha=0.02$) at high temperature ($T=2.0$) with zero tunneling parameter ($\Delta=0$). The mD2 results (shown in red and orange) are presented for two scenarios: (a) a non-adiabatic (fast) bath with $\omega_c=1.5$, and (b) an adiabatic (slow) bath with $\omega_c=0.1$, with the mD2 multiplicity indicated by $M$. Triangles denote the QUAPI data (using $\Delta t=0.1/\omega_0$, $k_{\max}=7$ for (a) and $\Delta t=0.1/\omega_0$, $k_{\max}=9$ for (b)), and dashed lines mark the HEOM results obtained with $Nk=2$ and ${\rm max}_{\rm depth}=20$.
• Figure 5: Time evolution of the nonequilibrium population difference $\sigma_z(t)$ for the strong coupling case ($\alpha=1.0$) at high temperature ($T=2.0$) with zero tunneling parameter ($\Delta=0$). The mD2 results (shown in red and orange) are presented for two scenarios: (a) a non-adiabatic (fast) bath with $\omega_c=2$, and (b) an adiabatic (slow) bath with $\omega_c=0.5$, with the mD2 multiplicity indicated by $M$. Triangles denote the QUAPI data (using $\Delta t=0.1/\omega_0$, $k_{\max}=8$ for (a) and $\Delta t=0.3/\omega_0$, $k_{\max}=11$ for (b)), and dashed lines mark the HEOM results obtained with $Nk=2$ and ${\rm max}_{\rm depth}=20$.