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The Journal of Chemical Physics

Multiple Davydov Ansätze as solutions to Lindblad master equations

Yiying Yan, Yang Zhao

Published Dec 15, 2025• 6 citations
Publisher

Abstract

Lindblad master equations describing driven quantum systems coupled to multiple bosonic modes are central to modeling cavity quantum electrodynamics and pseudomode models. In this work, we employ the density-operator-based Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D2Ansatz to provide optimal solutions for Lindblad master equations in both multimode and driven scenarios. Our approach is benchmarked against numerically "exact" methods in two representative models. The first describes a driven qubit subject to spontaneous emission and coupled to a lossy cavity, while the second corresponds to the pseudomode Lindblad master equation derived from the driven spin-boson model, incorporating seven discrete pseudomodes. The proposed variational approach achieves excellent agreement with numerically "exact" results. Furthermore, we analyze the accuracy of the variational solutions using an error metric based on the Frobenius norm, confirming their reliability. Overall, the present method offers an accurate and computationally efficient framework for simulating Lindblad master equations in complex open quantum systems.

DOI
10.1063/5.0306898

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The Dirac-Frenkel time-dependent variational approach with Davydov Ansätze is a sophisticated, yet efficient technique to obtain an accurate solution to many-body Schrödinger equations for energy and charge transfer dynamics in molecular aggregates and light-harvesting complexes. We extend this variational approach to finite temperature dynamics of the spin-boson model by adopting a Monte Carlo importance sampling method. In order to demonstrate the applicability of this approach, we compare calculated real-time quantum dynamics of the spin-boson model with that from numerically exact iterative quasiadiabatic propagator path integral (QUAPI) technique. The comparison shows that our variational approach with the single Davydov Ansätze is in excellent agreement with the QUAPI method at high temperatures, while the two differ at low temperatures. Accuracy in dynamics calculations employing a multitude of Davydov trial states is found to improve substantially over the single Davydov Ansatz, especially at low temperatures. At a moderate computational cost, our variational approach with the multiple Davydov Ansatz is shown to provide accurate spin-boson dynamics over a wide range of temperatures and bath spectral densities.
The pseudomode model effectively captures the nonperturbative dynamics of open quantum systems with significantly reduced degrees of freedom. However, it is limited by the exponential growth of the Hilbert space dimension. To overcome the computational challenges, we propose a novel method that combines the multiple Davydov Ansatz with the Choi-Jamiolkowski isomorphism. Within this framework, the Lindblad equation is transformed into the non-Hermitian Schrödinger equation in a double Hilbert space, with its dynamics determined using the time-dependent variational principle. Three cases are calculated to demonstrate the effectiveness of the proposed method. We first discuss how the Davydov Ansatz works for the model with a single pseudomode. Extending the method to multiple pseudomodes, we show that the Ansatz effectively circumvents the exponential growth of the Hilbert space. Additionally, the method is also capable of addressing potential intersections that emerge in multibath scenarios. This approach offers potential applicability to various types of pseudomode models and other dissipative systems, providing a promising tool for the studies of open quantum dynamics.
We employ the Dirac-Frenkel variational principle and the multiple Davydov ansatz to study time-dependent fluorescence spectra of a driven qubit in the weak to strong qubit-reservoir coupling regimes, where both the Rabi frequency and the spontaneous decay rate are comparable to the transition frequency of the qubit. Our method agrees well with the time-local master-equation approach in the weak coupling regime, and offers a flexible way to compute the spectra from the bosonic dynamics instead of two-time correlation functions. While the perturbative master equation breaks down in the strong coupling regime, our method actually becomes more accurate due to the use of bosonic coherent states under certain conditions. We show that the counter-rotating coupling between the qubit and the reservoir has considerable contributions to the photon number dynamics and the spectra under strong driving conditions even when the coupling is moderately weak. The time-dependent spectra are found to be generally asymmetric, a feature that is derived from photon number dynamics. In addition, it is shown that the spectral profiles can be dramatically different from the Mollow triplet due to strong dissipation and/or multiphoton processes associated with the strong driving. Our formalism provides a unique perspective to interpret time-dependent spectra.
We present a method to perform a variational analysis of the quantum master equation for driven-disspative bosonic fields with arbitrary large occupation numbers. Our approach combines the PPP representation of the density matrix and the variational principle for open quantum system. We benchmark the method by comparing it to wave-function Monte Carlo simulations and the solution of the Maxwell-Bloch equation for the Jaynes-Cummings model. Furthermore, we study a model describing Rydberg polaritons in a cavity field and introduce an additional set of variational parameters to describe correlations between different modes.
The time-resolved fluorescence spectrum of a driven two-level system is investigated using a time-dependent variational method. We first establish the transient build-up of the Mollow triplet via the Lindblad master equation under the rotating-wave approximation, providing a complete visualization of the spectral evolution from initial turn-on to steady state. To go beyond the perturbative regime, we employ the multiple Davydov D2 ansatz (multi-D2), which uses the σz eigenstates as the basis and naturally accommodates arbitrary system–bath coupling types and spectral densities. The multi-D2 method converges with M = 4 multiplicities in studied cases, outperforming the multi-D1 ansatz (M = 8) for the σx coupling benchmark. For pure σz (dephasing) coupling under resonant driving, we find that the time-resolved spectrum reveals a distinct fluorescence peak at the Rabi frequency—a signature of dressed-state transitions induced by the dephasing channel that remains hidden in population dynamics. Under mixed σx–σz coupling, the spectrum exhibits combined features of both Mollow triplet and σz-mediated emission. The effects of sub-Ohmic, Ohmic, and super-Ohmic spectral densities are systematically compared. While the resonant spectral weight J(ω0) governs the overall dissipation rate, a controlled comparison at fixed J(ω0) reveals that the super-Ohmic regime exhibits an intrinsic shape-dependent suppression of sideband emission under resonant driving, highlighting an asymmetric role of the spectral density exponent in engineering transient fluorescence. Our work establishes the multi-D2 ansatz as a versatile tool for simulating time-resolved fluorescence in complex bosonic environments.