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arXiv:2103.02586 · Physical Review A

Temperature-controlled open-quantum-system dynamics using a time-dependent variational method

Mantas Jakučionis, Darius Abramavičius

Published Mar 1, 2021• 5 citations
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Abstract

The Dirac-Frenkel variational method with Davydov ${\text{D}}_{2}$ trial wavefunction is extended by introducing a thermalization algorithm and applied to simulate dynamics of a general open quantum system. The algorithm allows to control temperature variations of a harmonic finite-size bath when in contact with the quantum system. Thermalization of the bath vibrational modes is realized via stochastic scatterings, implemented as a discrete-time Bernoulli process with Poisson statistics. It controls bath temperature by steering vibrational modes' evolution towards their canonical thermal equilibrium. Numerical analysis of the exciton relaxation dynamics in a small molecular cluster reveals that thermalization additionally provides significant calculation speedup due to the reduced number of vibrational modes needed to obtain the convergence.

DOI
10.1103/physreva.103.032202

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The Dirac-Frenkel variational method with Davydov D2{\text{D}}_{2}D2​ trial wavefunction is extended by introducing a thermalization algorithm and applied to simulate dynamics of a general open quantum system. The algorithm allows to control temperature variations of a harmonic finite-size bath when in contact with the quantum system. Thermalization of the bath vibrational modes is realized via stochastic scatterings, implemented as a discrete-time Bernoulli process with Poisson statistics. It controls bath temperature by steering vibrational modes' evolution towards their canonical thermal equilibrium. Numerical analysis of the exciton relaxation dynamics in a small molecular cluster reveals that thermalization additionally provides significant calculation speedup due to the reduced number of vibrational modes needed to obtain the convergence.
Numerical implementation of an explicit phonon bath requires a large number of oscillator modes in order to maintain oscillators at the initial temperature when modeling energy relaxation processes. An additional thermalization algorithm may be useful in controlling the local temperature. In this paper we extend our previously proposed thermalization algorithm [M. Jaku\ifmmode \check{c}\else \v{c}\fi{}ionis and D. Abramavi\ifmmode \check{c}\else \v{c}\fi{}ius, Phys. Rev. A 103, 032202 (2021) ] to be used with the numerically exact multiple-Davydov-D2{\text{D}}_{2}D2​ trial wave function for simulation of relaxation dynamics and spectroscopic signals of open quantum systems using the time-dependent Dirac-Frenkel variational principle. By applying it to the molecular aggregate model, we demonstrate how the thermalization approach significantly reduces the numerical cost of simulations by decreasing the number of oscillators needed to explicitly simulate the aggregate's environment fluctuations while maintaining correspondence to the exact population relaxation dynamics. Additionally, we show how the thermalization can be used to find the equilibrium state of the excited molecular aggregate, which is necessary for simulation of the fluorescence and other spectroscopic signals. The thermalization algorithm we present offers the possibility to investigate larger system-bath models than was previously possible using the multiple-Davydov-D2{\text{D}}_{2}D2​ trial wave function and local heating effects in molecular complexes.
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.
A variational approach for the quantum dynamics of statistical mixtures is developed, which is based upon the representation of the natural states of the mixture in terms of hybrid Gaussian-based Multiconfiguration Time-Dependent Hartree (G-MCTDH) wavefunctions. The method, termed ρG-MCTDH, is combined with a treatment of dissipation and decoherence based on the nonstochastic open-system Schrödinger equations. The performance and the convergence properties of the approach are illustrated for a two-dimensional tunneling system, where the primary tunneling coordinate, represented by flexible single-particle functions, is resonantly coupled to a second harmonic mode, represented by Gaussian wave packets. The harmonic coordinate is coupled to the environment and two different processes are studied: (i) vibrational relaxation at zero temperature described by a master equation in the Lindblad form and (ii) thermalization induced by the Caldeira-Leggett master equation. In the second case, the evolution from a quantum tunneling regime to a quasistationary classical-limit distribution, driven by the heat bath, is visualized using a flux analysis.
We present an application of the extended stochastic Liouville equation [G. M. G. McCaul et al., Phys. Rev. B 95, 125124 (2017)], which gives an exact solution for the reduced density matrix of an open system surrounded by a harmonic heat bath. This method considers the extended system (the open system and the bath) being thermally equilibrated prior to the action of a time-dependent perturbation, as opposed to the usual assumption that the system and the bath are initially partitioned. Our method presents an exact technique capable of accounting for arbitrary parameter regimes of the model. Here, we present our first numerical implementation of the method in the simplest case of a Caldeira-Leggett representation of the bath Hamiltonian, and apply it to a spin-boson system driven from coupled equilibrium. We observe significant behaviors in both the transient dynamics and asymptotic states of the reduced density matrix not present in the usual approximation.
We perform benchmark simulations using the time-dependent variational approach with the multiple Davydov Ansatz (mDA) to study real-time nonequilibrium dynamics in a single qubit model coupled to two thermal baths with distinct temperatures. A broad region of the parameter space has been investigated, accompanied by a detailed analysis of the convergence behavior of the mDA method. In addition, we have compared our mD2 results to those from two widely adopted, numerically "exact" techniques: the methods of hierarchical equations of motion (HEOM) and the quasi-adiabatic path integral (QUAPI). It is found that the mDA approach in combination with thermal field dynamics yields numerically accurate, convergent results in nearly all regions of the parameter space examined, including those that pose serious challenges for QUAPI and HEOM. Our results reveal that mDA offers a highly adaptable framework capable of capturing long-time dynamics, even in challenging regimes where other methods face limitations. These findings underscore the potential of mDA as a versatile tool for exploring quantum thermodynamics, energy transfer processes, and non-equilibrium quantum systems.
A novel quantum method to deal with typical system-bath dynamical problems is introduced. Subsystem discrete variable representation and bath coherent-state sets are used to write down a multiconfigurational expansion of the wave function of the whole system. With the help of the Dirac-Frenkel variational principle, simple equations of motion--a kind of Schrodinger-Langevin equation for the subsystem coupled to (pseudo) classical equations for the bath--are derived. True dissipative dynamics at all times is obtained by coupling the bath to a secondary, classical Ohmic bath, which is modeled by adding a friction coefficient in the derived pseudoclassical bath equations. The resulting equations are then solved for a number of model problems, ranging from tunneling to vibrational relaxation dynamics. Comparison of the results with those of exact, multiconfiguration time-dependent Hartree calculations in systems with up to 80 bath oscillators shows that the proposed method can be very accurate and might be of help in studying realistic problems with very large baths. To this end, its linear scaling behavior with respect to the number of bath degrees of freedom is shown in practice with model calculations using tens of thousands of bath oscillators.