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arXiv:2610.08953

An efficient variational polaron master equation for non-Markovian spin-boson dynamics: Transformed initial states and general observables

Chien-Chang Chen, Hsi-Sheng Goan

quant-phcond-mat.mes-hallcond-mat.stat-mechSubmitted Oct 6, 2026• Announced Oct 8, 2026• 0 citations
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Abstract

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.

Comments
20 pages, 6 figures

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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.
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.
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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.
We apply a full-polaron master equation and a weak-coupling non-Markovian master equation to describe the steady-state time-averaged properties of a driven two-level system, an electron coherently tunneling between double quantum dots (DQDs), interacting with a bosonic phonon bath. Comparing the results obtained using these two master equations with those from a recent DQD experiment and its corresponding weak-coupling theoretical method, we find that the original parameter set used in the experiment and theoretical method is not in the weak-coupling parameter regime. By using the full-polaron master equation with a slight adjustment on only the value of the interdot separation in the original experimental parameter set, we find that a reasonable fit to the experimentally measured time-averaged steady-state population data can be achieved. The adjusted interdot separation is within the possible values allowed by the geometry of the surface gates that define the DQD in the experiment. Our full-polaron equation approach does not require the special renormalization scheme employed in their weak-coupling theoretical method, and can still describe the experimental results of driving-induced phonon-enhanced steplike shoulder behaviors in the experiment. This demonstrates that the full-polaron master equation approach is a correct and efficient tool to describe the steady-state properties of a driven spin-boson model in the case of strong system-environment coupling.
We propose a ground-state ansatz for the Ohmic spin-boson model that improves upon the variational treatment of Silbey and Harris for biased systems in the scaling limit. In particular, it correctly captures the smooth crossover behavior expected for the ground-state magnetization when moving between the delocalized and localized regimes of the model, a feature that the variational treatment is unable to properly reproduce, while it also provides a lower ground-state energy estimate in the crossover region. We further demonstrate the validity of our intuitive ground-state by showing that it leads to predictions in excellent agreement with those derived from a nonperturbative Bethe-ansatz technique. Finally, recasting our ansatz in the form of a generalized polaron transformation, we are able to explore the dissipative two-state dynamics beyond weak system-environment coupling within an efficient time-local master equation formalism.
We advance the compact dressed Hamiltonian (CDH) framework to simulate the dynamics of open quantum systems strongly coupled to structured bosonic environments. The CDH approach combines reaction-coordinate mapping with a polaron transformation, incorporating strong system-environment hybridization into a dressed representation before truncating the collective bosonic mode. As a result of this truncation, dynamical features are captured with substantially fewer states than in the original representation. The residual environment is treated using a weak-coupling Markovian quantum master equation. We benchmark the method for spin-boson-type models including a single spin, coupled spins, and spin ensembles, considering both resonant and off-resonant regimes and weak to strong system-bath coupling. Comparisons with hierarchical equations of motion and converged reaction-coordinate calculations demonstrate that the CDH representation systematically approaches the reference dynamics with a compact Hilbert space, accurately reproducing oscillation frequencies, relaxation behavior, and long-time populations over a broad parameter regime. The method further captures cooperative phenomena, including bath-mediated interactions, synchronization between spins, and signatures for collective emission. Beyond efficient time propagation, the CDH framework provides direct access to the spectrum and eigenmodes of the Liouvillian. We demonstrate that slow dynamical modes converge rapidly in the CDH representation, enabling identification of relaxation timescales without fitting time-dependent observables. As a whole, the CDH framework provides an efficient and physical description of strongly coupled open quantum-system dynamics.