Polaron Transformed Canonically Consistent Quantum Master Equation

Abstract

A central challenge in the theory of open quantum systems is the development of theoretical frameworks capable of accurately describing large, strongly interacting quantum many-body systems in the regime of strong system-bath interaction. In this work, we take a step toward this goal by formulating a polaron-transformed version of the canonically consistent quantum master equation (CCQME) [T. Becker et al., Phys. Rev. Lett. 129, 200403 (2022)]. The CCQME extends beyond standard weak-coupling approaches while retaining the same numerical complexity as conventional quantum master equations, thereby enabling the treatment of large quantum systems. The polaron transformation further enhances the accessible system-bath interaction strengths, allowing us to move from moderate to strong interaction regimes. We present a unified and transparent derivation of these two approaches and combine them to obtain the polaron-transformed CCQME (PT-CCQME). Applying our method to the paradigmatic spin-boson model, we find excellent agreement with numerically exact time-evolving matrix product operator (TEMPO) simulations. Finally, we predict an initial-state-independent slowing down of thermalization in the strong-coupling regime of the spin-boson model.

Figures (3)

• Figure 1: Minimum eigenvalue of the reduced density matrix over a large time interval [$t\delta E_{21} \in (0,28)$] for the spin-boson model, plotted as a function of the system-bath coupling strength $\gamma$ and inverse temperature $\beta$. All parameters are expressed in units of the two-level system energy gap $\delta E_{21}$. The system is initialized in the state $\tilde{\rho}_S(0) = (\sigma_z + \mathbb{I})/2$. Negative eigenvalues (red regions) indicate positivity violations. The polaron-transformed equations [PT-Redfield, panel ( a); and PT-CCQME, panel ( c)] significantly outperform the original-frame Redfield ( b) and CCQME ( d) approaches, which consistently violate the positivity of the reduced density matrix. While the PT-Redfield method ( a) fails at intermediate couplings and low temperatures, the PT-CCQME ( c) only breaks down at much stronger couplings and lower temperatures, demonstrating its ability to preserve positivity over a substantially broader parameter regime. The system parameter $h=\varepsilon$ with a bath cutoff frequency $\omega_c = \varepsilon$. Black lines in panels ( a) and ( c) denote the boundaries of positivity violation.
• Figure 2: Comparison of the population dynamics $\langle \sigma_z \rangle$ calculated via different master equation approaches against the numerically exact TEMPO method across various system-bath coupling strengths $\gamma$ and inverse temperatures $\beta$. The coupling strength $\gamma$ increases from the weak-coupling (first column) to the strong-coupling (third column) regime, while the inverse temperature $\beta$ decreases from the low-temperature (first row) to the high-temperature (third row) regime. Red solid triangles represent the analytical equilibrium results obtained using the standard Gibbs state [Eq. \ref{['eq:weakcouplingrho']}], and green solid squares denote the analytical results derived from the ultra-strong coupling mean-force Gibbs (MFG) state [Eq. \ref{['eq:strongcouplingrho']}]. The most notable differences between the PT-CCQME and PT-Redfield approaches occur at moderate coupling strengths and moderate-to-low temperatures [panels ( b) and ( e)], where the PT-Redfield method breaks down (see Fig. \ref{['fig:Positivity']}). Panels ( a), ( d), and ( g) additionally show results for the CCQME (Ref. [Becker2022]) and the Redfield equation derived in the original frame. Beyond the weak-coupling regime, both the original-frame CCQME and Redfield equations fail, yielding unphysical values ($|\langle \sigma_z\rangle| > 1$). Details of the TEMPO parameters are given in Appendix \ref{['append:tempo']}. The TEMPO results are converged by varying the singular value decomposition truncation threshold, time step $dt$, and memory truncation time $t_{\rm mem}$. All other parameters are the same as in Fig. \ref{['fig:Positivity']}.
• Figure 3: The Liouvillian gap $g$, defined in Eq. \ref{['eq:gap']}, as a function of the system-bath coupling strength $\gamma$ for different inverse temperatures: $\beta \delta E_{21} = 1.4$ (high-temperature regime, red lines), $\beta \delta E_{21} = 2.8$ (moderate-temperature regime, orange lines), and $\beta \delta E_{21} = 5.6$ (deep-quantum regime, blue lines). Solid lines represent the PT-CCQME, while dashed lines correspond to the PT-Redfield equation. We also plot predictions from the original-frame Redfield equation, which deviate significantly from the polaron-transformed approaches (dotted lines). In both the weak-coupling ($\gamma \delta E_{21} \ll 1$) and ultra-strong coupling ($\gamma \delta E_{21} \approx 1$) regimes, the PT-Redfield and PT-CCQME results match, further confirming that the primary discrepancy between these two approaches lies in the intermediate-coupling regime (see discussion surrounding Fig. \ref{['fig:Positivity']}). All other parameters are identical to those in Fig. \ref{['fig:Positivity']}.