Local master equations bypass the secular approximation

Stefano Scali1, Janet Anders1,2, and Luis A. Correa1

1Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom
2Institut für Physik und Astronomie, University of Potsdam, 14476 Potsdam, Germany.

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Master equations are a vital tool to model heat flow through nanoscale thermodynamic systems. Most practical devices are made up of interacting sub-system, and are often modelled using either $\textit{local}$ master equations (LMEs) or $\textit{global}$ master equations (GMEs). While the limiting cases in which either the LME or the GME breaks down are well understood, there exists a 'grey area' in which both equations capture steady-state heat currents reliably, but predict very different $\textit{transient}$ heat flows. In such cases, which one should we trust? Here, we show that, when it comes to dynamics, the local approach can be more reliable than the global one for weakly interacting open quantum systems. This is due to the fact that the $\textit{secular approximation}$, which underpins the GME, can destroy key dynamical features. To illustrate this, we consider a minimal transport setup and show that its LME displays $\textit{exceptional points}$ (EPs). These singularities have been observed in a superconducting-circuit realisation of the model [1]. However, in stark contrast to experimental evidence, no EPs appear within the global approach. We then show that the EPs are a feature built into the Redfield equation, which is more accurate than the LME and the GME. Finally, we show that the local approach emerges as the weak-interaction limit of the Redfield equation, and that it entirely avoids the secular approximation.

What do non-Hermitian degeneracies i.e., exceptional points, mean for open-quantum systems? In this work, we show what the exceptional points represent for the system, how to find them, and how to use them to benchmark master equations. To do so, we consider an example model, we build its local and global master equations and compare them to the Redfield equation. An unexpected link between local and partial Redfield equation emerges at the end.

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[1] Adam Hewgill, Gabriele De Chiara, and Alberto Imparato, "Quantum thermodynamically consistent local master equations", Physical Review Research 3 1, 013165 (2021).

[2] Shishir Khandelwal, Nicolas Brunner, and Géraldine Haack, "Signatures of exceptional points in a quantum thermal machine", arXiv:2101.11553.

[3] Roie Dann and Ronnie Kosloff, "Open system dynamics from thermodynamic compatibility", Physical Review Research 3 2, 023006 (2021).

[4] Feng Tian, Jian Zou, Lei Li, Hai Li, and Bin Shao, "Effect of Inter-System Coupling on Heat Transport in a Microscopic Collision Model", Entropy 23 4, 471 (2021).

[5] Anton Trushechkin, "Unified GKLS quantum master equation beyond the secular approximation", arXiv:2103.12042.

[6] Gerard McCaul, Kurt Jacobs, and Denys I. Bondar, "Fast computation of dissipative quantum systems with ensemble rank truncation", Physical Review Research 3 1, 013017 (2021).

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