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arXiv:2306.05714 · Physical Review A

Thermalization of open quantum systems using the multiple-Davydov- D2 variational approach

Mantas Jakučionis, Darius Abramavičius

Published Jun 8, 2023• 5 citations
arXivHTMLPDF

Abstract

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-${\text{D}}_{2}$ 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-${\text{D}}_{2}$ trial wave function and local heating effects in molecular complexes.

DOI
10.1103/physreva.107.062205

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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 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.
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.
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.
This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states.
In non-perturbative non-Markovian open quantum systems, reaching either low temperatures with the hierarchical equations of motion (HEOM) or high temperatures with the Thermalized Time Evolving Density Operator with Orthogonal Polynomials Algorithm (T-TEDOPA) formalism in Hilbert space remains challenging. We compare different ways of modeling the environment. Sampling the Fourier transform of the bath correlation function, also called temperature dependent spectral density, proves to be very effective. T-TEDOPA [Tamascelli et al., Phys. Rev. Lett. 123, 090402 (2019)] uses a linear chain of oscillators with positive and negative frequencies, while HEOM is based on the complex poles of an optimized rational decomposition of the temperature dependent spectral density [Xu et al., Phys. Rev. Lett. 129, 230601 (2022)]. Resorting to the poles of the temperature independent spectral density and of the Bose function separately is an alternative when the problem due to the huge number of Bose poles at low temperatures is circumvented. Two examples illustrate the effectiveness of the HEOM and T-TEDOPA approaches: a benchmark pure dephasing case and a two-bath model simulating the dynamics of excited electronic states coupled through a conical intersection. We show the efficiency of T-TEDOPA to simulate dynamics at a finite temperature by using either continuous spectral densities or only all the intramolecular oscillators of a linear vibronic model calibrated from ab initio data of a phenylene ethynylene dimer.
Accelerating quantum dynamical simulations with quantum computing has received considerable attention but remains a significant challenge. In variational quantum algorithms for quantum dynamics, designing an expressive and shallow-depth parametrized quantum circuit (PQC) is a key difficulty. Here, we propose a multiset variational quantum dynamics algorithm (MS-VQD) tailored for nonadiabatic dynamics involving multiple electronic states. The MS-VQD employs multiple PQCs to represent the electronic-nuclear coupled wave function, with each circuit adapting to the motion of the nuclear wavepacket on a specific potential energy surface. By simulating excitation energy transfer dynamics in molecular aggregates described by the Frenkel-Holstein model, we demonstrate that the MS-VQD achieves the same accuracy as the traditional VQD while requiring significantly shallower PQCs. Notably, its advantage increases with the number of electronic states, making it suitable for simulating nonadiabatic quantum dynamics in complex molecular systems.