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Physica Scripta

The Davydov D1.5 Ansatz for the quantum Rabi model*

Michael Werther, Frank Großmann

Published May 25, 2018• 18 citations
Publisher

Abstract

Abstract We propose a new version of the Davydov Ansatz for the approximate solution of the time-dependent Schrödinger equation (TDSE) and apply it to a spin 1/2 system coupled to a single bosonic mode. In complexity the new Ansatz is between the well-studied D1 and D2 Ansätze and is therefore termed D1.5 Ansatz. It is tailored to specific but still rather general initial conditions and is free of numerical inconsistencies that plague the D1 Ansatz. Finally, a generalization to nontrivial multiplicity in the oscillator state space is given that leads, in principle, to an exact solution of the TDSE for arbitrary system parameters. By employing Bargmann coherent states, the equations of motion could be made explicit, which is a big advantage, computationally.

DOI
10.1088/1402-4896/aac7f9

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Abstract We propose a new version of the Davydov Ansatz for the approximate solution of the time-dependent Schrödinger equation (TDSE) and apply it to a spin 1/2 system coupled to a single bosonic mode. In complexity the new Ansatz is between the well-studied D1 and D2 Ansätze and is therefore termed D1.5 Ansatz. It is tailored to specific but still rather general initial conditions and is free of numerical inconsistencies that plague the D1 Ansatz. Finally, a generalization to nontrivial multiplicity in the oscillator state space is given that leads, in principle, to an exact solution of the TDSE for arbitrary system parameters. By employing Bargmann coherent states, the equations of motion could be made explicit, which is a big advantage, computationally.
The dynamics of a coupled electron–boson system is investigated by employing a multitude of the Davydov D 1 trial states, also known as the multi-D 1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz ). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D 1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron–nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.
Dynamics of the two-spin spin-boson model in the presence of Ohmic and sub-Ohmic baths is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Its accuracy in dynamics simulations of the two-spin SBM is improved significantly over the single D1 Ansatz, especially in the weak to moderately strong coupling regime. Validity of the multi-D1 Ansatz for various coupling strengths is also systematically examined by making use of the deviation vector which quantifies how faithfully the trial state obeys the Schrödinger equation. The time evolution of population difference and entanglement has been studied for various initial conditions and coupling strengths. Careful comparisons are carried out between our approach and three other methods, i.e., the time-dependent numerical renormalization group (TD-NRG) approach, the Bloch-Redfield theory, and a method based on a variational master equation. For strong coupling, the multi-D1 trial state yields consistent results as the TD-NRG approach in the Ohmic regime while the two disagree in the sub-Ohmic regime, where the multi-D1 trial state is shown to be more accurate. For weak coupling, the multi-D1 trial state agrees with the two master-equation methods in the presence of both Ohmic and sub-Ohmic baths, but shows considerable differences with the TD-NRG approach in the presence of a sub-Ohmic bath, calling into question the validity of the TD-NRG results at long times in the literature.
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.
The pseudomode framework provides an efficient description of open quantum systems in non-Markovian and nonperturbative coupling regimes by utilizing simple Markovian Lindblad master equations that incorporate a set of damped bosonic pseudomodes. Developing an efficient numerical scheme to solve these equations accurately is highly desirable. In this work, we propose a modified multiple Davydov D1 density operator ansatz and combine it with the Dirac-Frenkel time-dependent variational principle to simulate the dynamics governed by these master equations. We apply this variational method to the spin-boson model, both with and without a periodic external driving field, and compute both single- and two-time expectation values in non-Markovian, strong-coupling regimes. The accuracy of our variational approach is validated by comparing its predictions with results obtained from the small matrix path integral method. We demonstrate that the variational and path-integral approaches are in excellent agreement for both single- and two-time expectations across nondriven and driven cases, spanning both low- and high-temperature regimes. Our results establish the reliability and broad applicability of the pseudomode-based variational approach for calculating two-time correlation functions in open quantum systems.
Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Accuracy in dynamics simulations is improved significantly over the single D1 Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D1 Ansatz for various coupling strengths and initial conditions is also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-D1 Ansatz. The phase diagram is obtained with a critical point sc = 0.4. For sc < s < 1, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For s < sc, only the coherent phase exists.
Developed originally for the Holstein polaron, the Davydov D1 ansatz is an efficient, yet extremely accurate trial state for time-dependent variation of the spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. In this work, the Dirac-Frenkel time-dependent variational procedure utilizing the Davydov D1 ansatz is implemented to study entanglement dynamics of two qubits under the influence of two independent baths. The Ohmic spectral density is used without the Born-Markov approximation or the rotating-wave approximation. In the strong coupling regime finite-time disentanglement is always found to exist, while at the intermediate coupling regime, the entanglement dynamics calculated by Davydov D1 ansatz displays oscillatory behavior in addition to entanglement disappearance and revival.