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The Journal of Chemical Physics

Reduced quantum dynamics with arbitrary bath spectral densities: Hierarchical equations of motion based on several different bath decomposition schemes

Hao Liu, Lili Zhu, Shuming Bai, Qiang Shi

Published Apr 4, 2014• 151 citations
Publisher

Abstract

We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.

DOI
10.1063/1.4870035

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We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.
The time evolution in open quantum systems, such as a molecular aggregate in contact with a thermal bath, still poses a complex and challenging problem. The influence of the thermal noise can be treated using a plethora of schemes, several of which decompose the corresponding correlation functions in terms of weighted sums of exponential functions. One such scheme is based on the hierarchical equations of motion (HEOM), which is built using only certain forms of bath correlation functions. In the case where the environment is described by a complex spectral density or is at a very low temperature, approaches utilizing the exponential decomposition become very inefficient. Here, we utilize an alternative decomposition scheme for the bath correlation function based on Chebyshev polynomials and Bessel functions to derive a HEOM approach up to an arbitrary order in the environmental coupling. These hierarchical equations are similar in structure to the popular exponential HEOM scheme, but are formulated using the derivatives of the Bessel functions. The proposed scheme is tested up to the fourth order in perturbation theory for a two-level system and compared to benchmark calculations for the case of zero-temperature quantum Ohmic and super-Ohmic noise. Furthermore, the benefits and shortcomings of the present Chebyshev-based hierarchical equations are discussed.
The hierarchical equations of motion (HEOM) method has recently been widely applied to many problems of quantum dynamics in condensed phase. It is now well known that the auxiliary density operators (ADOs) in the HEOM formalism contain system-bath correlations that are important in calculating various dynamical properties, yet quantitative relations to explicitly calculate such correlations from the ADOs are still scarce. This paper extends a previous study [Q. Shi et al., J. Chem. Phys. 130, 164518 (2009)] in investigating the physical meaning of ADOs to general spectral densities and lower temperature cases. Using the path-integral technique, we derive exact relations between the expectation values of the collective bath coordinate and the ADOs, which could be very useful in investigating the correlated system-bath dynamics directly with the HEOM formalism. Numerical examples concerning the evolution of the expectation values of the collective bath coordinate are also presented.
This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.
The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent effects of the bath, the reservoir correlation functions are usually decomposed into summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. This method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultralow temperature and general band structures. We demonstrate the efficiency, accuracy, and long-time stability of this decomposition scheme by applying it to the Anderson impurity model in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.
We present a simple and effective approach for extracting the bath information in open-quantum-system dynamics under the framework of the numerically exact hierarchical equations-of-motion method, for which it is usually not straightforward to adjust the boundary between the system and bath. The key component of our approach is an additional bath mode that weakly interacts with the system through the same coupling channel as the mode of interest, which we call the probe mode. By combining the probe mode with the system and calculating its expectation value, we can deduce the corresponding expectation value of the mode of interest via simple relations. The effectiveness of our approach is demonstrated by exact decomposition of dissipation by a continuous spectral density.
We investigate the quantum dynamics of a spin coupling to a bath of independent spins via the dissipaton equation of motion (DEOM) approach. The bath, characterized by a continuous spectral density function, is composed of spins that are independent level systems described by the su(2) Lie algebra, representing an environment with a large magnitude of anharmonicity. Based on the previous work by Suarez and Silbey [J. Chem. Phys. 95, 9115 (1991)] and by Makri [J. Chem. Phys. 111, 6164 (1999)] that the spin bath can be mapped to a Gaussian environment under its linear response limit, we use the time-domain Prony fitting decomposition scheme to the bare-bath time correlation function (TCF) given by the bosonic fluctuation-dissipation theorem to generate the exponential decay basis (or pseudo modes) for DEOM construction. The accuracy and efficiency of this strategy have been explored by a variety of numerical results. We envision that this work provides new insights into extending the hierarchical equations of motion and DEOM approach to certain types of anharmonic environments with arbitrary TCF or spectral density.