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arXiv:2407.10744 · The Journal of Chemical Physics

Capturing non-Markovian polaron dressing with the master equation formalism

Jake Iles-Smith, Owen Diba, Ahsan Nazir

Published Oct 3, 2024• 13 citations
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Abstract

Understanding the dynamics of open quantum systems in strong coupling and non-Markovian regimes remains a formidable theoretical challenge. One popular and well-established method of approximation in these circumstances is provided by the polaron master equation (PME). In this work, we re-evaluate and extend the validity of the PME to capture the impact of non-Markovian polaron dressing, induced by non-equilibrium open system dynamics. By comparing with numerically exact techniques, we confirm that while the standard PME successfully predicts the dynamics of system observables that commute with the polaron transformation (e.g., populations in the Pauli z-basis), it can struggle to fully capture those that do not (e.g., coherences). This limitation stems from the mixing of system and environment degrees of freedom inherent to the polaron transformation, which affects the accuracy of calculated expectation values within the polaron frame. Employing the Nakajima-Zwanzig projection operator formalism, we introduce correction terms that provide an accurate description of observables that do not commute with the transformation. We demonstrate the significance of the correction terms in two cases, the canonical spin-boson model and a dissipative time-dependent Landau-Zener protocol, where they are shown to impact the system dynamics on both short and long timescales.

DOI
10.1063/5.0228779

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Understanding the dynamics of open quantum systems in strong coupling and non-Markovian regimes remains a formidable theoretical challenge. One popular and well-established method of approximation in these circumstances is provided by the polaron master equation (PME). In this work, we re-evaluate and extend the validity of the PME to capture the impact of non-Markovian polaron dressing, induced by non-equilibrium open system dynamics. By comparing with numerically exact techniques, we confirm that while the standard PME successfully predicts the dynamics of system observables that commute with the polaron transformation (e.g., populations in the Pauli z-basis), it can struggle to fully capture those that do not (e.g., coherences). This limitation stems from the mixing of system and environment degrees of freedom inherent to the polaron transformation, which affects the accuracy of calculated expectation values within the polaron frame. Employing the Nakajima-Zwanzig projection operator formalism, we introduce correction terms that provide an accurate description of observables that do not commute with the transformation. We demonstrate the significance of the correction terms in two cases, the canonical spin-boson model and a dissipative time-dependent Landau-Zener protocol, where they are shown to impact the system dynamics on both short and long timescales.
While the quantum regression theorem (QRT) is the standard tool for computing multi-time correlation functions in open quantum systems, it relies on system-bath separability and an environment that remains in equilibrium, assumptions that are violated once dynamical correlations develop. Using the projection operator formalism, we derive an extension to the QRT that explicitly incorporates these correlation-induced corrections, enabling us to capture multi-time correlation functions in non-Markovian regimes. We apply this framework to the variational polaron master equation for the spin-boson model in ohmic and super-ohmic regimes, where the polaron transformation mixes system-bath degrees of freedom to produce a non-thermal effective environment. Benchmarking against numerically exact tensor-network simulations, we demonstrate quantitative agreement for single- and two-time observables, including linear-response spectra, even at strong coupling. Our approach broadens the reach of analytic master equations to multi-time observables in strong-coupling and non-Markovian regimes, where environmental memory effects and system-bath correlations are crucial.
We investigate the effects of initial states on the dynamics of a quantum spin-boson model using an efficient variational polaron master equation approach. The transformation of a factorized system-environment initial state into the variational polaron frame introduces inhomogeneous terms in the master equation, essential for accurately describing non-Markovian dynamics beyond the weak coupling regime. To efficiently account for these effects, we formulate the dynamics in an extended auxiliary Liouville space, converting the resulting integro-differential equations into coupled ordinary differential equations. We further introduce a simplified variational polaron technique for computing the time evolution of expectation values of general observables, including those that do not commute with the polaron transformation, overcoming a major limitation of previous approaches and enabling the unified treatment of coherence and population dynamics. The effects of transformed initial states generated from arbitrary factorized initial states are systematically characterized through expectation values and trace-distance analysis, providing insight into the associated bath-relaxation dynamics over a broad range of tunneling amplitudes, temperatures, and coupling strengths. Comparison with the weak-coupling master equation reveals a delay phenomenon associated with multi-bath-quanta processes, pronounced at intermediate coupling strengths and tunneling amplitudes. We further show that coherent superposition initial states induce stronger nonequilibrium effects than localized states, with inhomogeneous terms playing a crucial role in the short-time dynamics. The proposed formalism applies to both undriven and resonantly driven spin-boson models, providing a computationally efficient framework for open quantum-system dynamics with arbitrary factorized initial states and general physical observables.
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 interactions. 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 \emph{et~al.}, Phys. Rev. Lett. \textbf{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 ultra-strong interaction regimes. We present a unified and transparent derivation of these two approaches and combine them to obtain the polar-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 \emph{slowing} down of thermalization in the strong-coupling regime of the spin-boson model.
The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath ϱB(t)\varrho_{B}(t)ϱB​(t) and the system ϱS(t)\varrho_{S}(t)ϱS​(t) are written down. A formal solution of the equation for ϱB(t)\varrho_{B}(t)ϱB​(t), having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in ϱS(t)\varrho_{S}(t)ϱS​(t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of ϱB(t)\varrho_{B}(t)ϱB​(t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.
In recent years, there has been tremendous focus on identifying whether effective descriptions of open quantum systems such as master equations can accurately describe the dynamics of open quantum systems. One particular question is whether they provide the correct steady state in the long time limit. The transient regime is also of interest. Description of evolution by various master equations---some of them being not completely positive---is benchmarked against exact solutions [see, e.g., R. Hartmann and W. T. Strunz, Phys. Rev. A 101, 012103 (2020)]. An important property of true evolution is its non-Markovian features, which are not captured by the simplest completely positive master equations. In this paper we consider a non-Markovian yet completely positive evolution (known as refined weak coupling or cumulant equation) for the spin-boson model with an overdamped Drude-Lorentz spectral density and arbitrary coupling. We benchmark it against a numerically exact solution, as well as against other master equations, for different coupling strengths and temperatures. We find the cumulant to be a better description in the weak-coupling regime where it is supposed to be valid. For the examples considered it shows superiority at moderate and strong couplings in the low-temperature regime for all examples considered. In the high-temperature regime, however, its advantage vanishes. This indicates that the cumulant equation is a good candidate for simulations at weak to moderate coupling and low temperature. Our calculations are greatly facilitated due to our concise formulation of the cumulant equation by means of representation of the density matrix in the SU(N) basis.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.