Hierarchical Equations Of Motion Formalism Research Articles

Tensor-Train Format Hierarchical Equations of Motion Formalism: Charge Transfer in Organic Semiconductors via Dissipative Holstein Models.

Hierarchical Equations of Motion (HEOM) in the Tensor-Train (TT) representation is applied to study the charge-transfer dynamics in organic semiconductors (OSCs). The theoretical formulation as well as the basic computational aspects of HEOM-TT are discussed in detail. Charge transfer in OSCs is modeled using dissipative polaronic models that incorporate the effects of both high- and low-frequency molecular vibrations, and it is simulated in a fully quantum and nonperturbative manner, which has not been studied intensively. The capability of treating complex electron-vibrational systems is examined by analyzing and comparing the numerical behavior of the time-dependent variational approach and the time-Alternating Minimal Energy methods and by calculating the current autocorrelation function and diffusivity across various models. Our results indicate that the HEOM-TT framework offers a robust tool for the detailed analysis of complex polaronic systems, suggesting its potential for broader applications.

Nonadiabatic dynamics of molecules interacting with metal surfaces: Extending the hierarchical equations of motion and Langevin dynamics approach to position-dependent metal-molecule couplings.

Electronic friction and Langevin dynamics is a popular mixed quantum-classical method for simulating the nonadiabatic dynamics of molecules interacting with metal surfaces, as it can be computationally more efficient than fully quantum approaches. In this work, we extend the theory of electronic friction within the hierarchical equations of motion formalism to models with a position-dependent metal-molecule coupling. We show that the addition of a position-dependent metal-molecule coupling adds new contributions to the electronic friction and other forces, which are highly relevant for many physical processes. Our expressions for the electronic forces within the Langevin equation are valid both in and out of equilibrium and for molecular models containing strong interactions. We demonstrate the approach by applying it to different models of interest.

Transient simulation of the electrical hysteresis in a metal/polymer/metal nanostructure.

The time-dependent quantum transportation through a metal/polymer/metal system is theoretically investigated on the basis of a Su-Schrieffer-Heeger model combined with the hierarchical equations of motion formalism. Using a non-adiabatic dynamical method, the evolution of the electron subspace and lattice atoms with time can be obtained. It is found that the calculated transient currents vary with time and reach stable values after a response time under the bias voltages. However, the stable current as the system reaches its dynamical steady state exhibits a discrepancy between two sweep directions of the bias voltage, which results in pronounced electrical hysteresis loops in the current-voltage curve. By analyzing the evolution of instantaneous energy eigenstates, the occupation number of the instantaneous eigenstates, and the lattice of the polymer, we show that the formation of excitons and the delay of their annihilation are responsible for the hysteretic current-voltage characteristics, where electron-phonon interactions play the key factor. Furthermore, the hysteresis width and amplitude can also be modulated by the strength of the electron-phonon coupling, level-width broadening function, and temperature. We hope these results about past condition-dependent switching performance at a sweep voltage can provide further insight into some of the basic issues of interest in hysteresis processes in conducting polymers.

Current-induced forces in nanosystems: A hierarchical equations of motion approach

An approach to calculating current-induced forces in charge transport through nanosystems is introduced. Starting from the fully quantum mechanical hierarchical equations of motion formalism, a timescale separation between electronic and vibrational degrees of freedom is used to derive a classical Langevin equation of motion for the vibrational dynamics as influenced by current-induced forces, such as the electronic friction. The resulting form of the friction is shown to be equivalent to previously derived expressions. The numerical exactness of the hierarchical equations of motion approach, however, allows the investigation of transport scenarios with strong intrasystem and system-environment interactions. As a demonstration, the electronic friction of three example systems is calculated and analyzed: a single electronic level coupled to one classical vibrational mode, an Anderson impurity model coupled to one classical vibrational mode, and a single electronic level coupled to both a classical and quantum vibrational mode.

Recent advances in fermionic hierarchical equations of motion method for strongly correlated quantum impurity systems

Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.

Local Phonon Environment as a Design Element for Long-Lived Excitonic Coherence: Dithia-anthracenophane Revisited

The purpose of this study is to investigate the role of a structured immediate phonon environment in determining the exciton dynamics and the possibility of using it as an optimal design element. Through the case study of dithia-anthracenophane, a bichromophore using the Hierarchical Equations Of Motion formalism, we show that the experimentally observed coherent exciton dynamics can be reproduced only by considering the actual structure of the phonon environment. While the slow dephasing of quantum coherence in dithia-anthracenophane can be attributed to strong vibronic coupling to high-frequency modes, vibronic quenching is the source of long oscillation periods in population transfer. This study sheds light on the crucial role of the structure of the immediate phonon environment in determining the exciton dynamics. We conclude by proposing some design principles for sustaining long-lived coherence in molecular systems.

The influence of a Hamiltonian vibration vs a bath vibration on the 2D electronic spectra of a homodimer.

We elucidate the influence of the system-bath boundary placement within an open quantum system, with emphasis on the two-dimensional electronic spectra, through the application of the hierarchical equations of motion formalism for an exciton system. We apply two different models, the Hamiltonian vibration model (HVM) and bath vibration model (BVM), to a monomer and a homodimer. In the HVM, we specifically include the vibronic states in the Hamiltonian capturing vibronic quenching, whereas in the BVM, all vibrational details are contained within the bath and described by an underdamped spectral density. The resultant spectra are analyzed in terms of energetic peak position and thermodynamic broadening precision in order to evaluate the efficacy of the two models. The HVM produces 2D spectra with accurate peak positional information, while the BVM is well suited to modeling dynamic peak broadening. For the monomer, both models produce equivalent spectra in the limit where additional damping associated with the underdamped vibration in the BVM approaches zero. This is supported by analytical results. However, for the homodimer, the BVM spectra are redshifted with respect to the HVM due to an absence of vibronic quenching in the BVM. The computational efficiency of the two models is also discussed in order to inform us of the most appropriate use of each method.

Dissipaton equation of motion for system-and-bath interference dynamics

In this review we give a comprehensive account on the dissipaton equation of motion (DEOM) approach to quantum mechanics of open systems. This approach provides a statistical quasi-particle (dissipaton) picture for the environment, as it participates in the correlated system-and-bath dynamics. The underlying dissipaton algebra is de facto established via a close comparison with the celebrated hierarchical equations of motion formalism that is rooted at the Feynman-Vernon influence functional path integral formalism. As a quasi-particle generalization, DEOM identifies unambiguously the physical meanings of all involving dynamical variables as many-dissipaton configurations. It addresses the dynamics of not only systems but also hybridizing bath degrees of freedom. We demonstrate these features of DEOM via its real-time evaluation of the Fano interference of an analytically solvable model system, with the highlight that the statistical quasi-particle picture is ubiquitous, implied even in those commonly used quantum master equations.

Hierarchical Liouville-space approach to nonequilibrium dynamical properties of quantum impurity systems

We propose a hierarchical dynamics approach for evaluation of nonequilibrium dynamic response properties of quantum impurity systems. It is based on a hierarchical equations of motion formalism, in conjunction with a linear response theory established in the hierarchical Liouville space. This provides an accurate and universal tool for characterization of a variety of response and correlation functions of local impurities, as well as transport related response properties. The practicality of our proposed approach is demonstrated via the evaluation of various dynamical properties of a single-impurity Anderson model. These include the impurity spectral density, local charge fluctuation, local magnetic susceptibility, and current-voltage admittance, in both equilibrium and nonequilibrium situations. The numerical results are considered to be quantitatively accurate, as long as they converge quantitatively with respect to the truncation of the hierarchy.

Explicit system-bath correlation calculated using the hierarchical equations of motion method

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.

Hierarchical Dynamics of Correlated System−Environment Coherence and Optical Spectroscopy

We propose an efficient construction of the hierarchical equations of motion formalism of quantum dissipation on the basis of the Padé spectrum decomposition of the Bose function and the multiple brownian oscillators decomposition of the environment spectral density. The related hierarchical Liouville space algebra for quantum dissipative mechanics is outlined in relation to the evaluation of nonlinear optical response functions. With the simulated transient pump-probe spectroscopy of model exciton systems, we demonstrate the correlated system-environment coherence by the present nonperturbative and non-markovian quantum dissipation theory.

Electron transfer dynamics: Zusman equation versus exact theory

The Zusman equation has been widely used to study the effect of solvent dynamics on electron transfer reactions. However, application of this equation is limited by the classical treatment of the nuclear degrees of freedom. In this paper, we revisit the Zusman equation in the framework of the exact hierarchical equations of motion formalism, and show that a high temperature approximation of the hierarchical theory is equivalent to the Zusman equation in describing electron transfer dynamics. Thus the exact hierarchical formalism naturally extends the Zusman equation to include quantum nuclear dynamics at low temperatures. This new finding has also inspired us to rescale the original hierarchical equations and incorporate a filtering algorithm to efficiently propagate the hierarchical equations. Numerical exact results are also presented for the electron transfer reaction dynamics and rate constant calculations.

Dynamic electronic response of a quantum dot driven by time-dependent voltage

We present a comprehensive theoretical investigation on the dynamic electronic response of a noninteracting quantum dot system to various forms of time-dependent voltage applied to the single contact lead. Numerical simulations are carried out by implementing a recently developed hierarchical equations of motion formalism [J. S. Jin et al., J. Chem. Phys. 128, 234703 (2008)], which is formally exact for a fermionic system interacting with grand canonical fermionic reservoirs, in the presence of arbitrary time-dependent applied chemical potentials. The dynamical characteristics of the transient transport current evaluated in both linear and nonlinear-response regimes are analyzed, and the equivalent classic circuit corresponding to the coupled dot-lead system is also discussed.

Dynamic Coulomb blockade in single-lead quantum dots

We investigate the transient dynamic response of an Anderson impurity quantum dot (QD) to a family of ramp-up driving voltages applied to the single coupling lead. Transient current is calculated based on a hierarchical equations of motion formalism for open dissipative systems (Jin et al 2008 J. Chem. Phys. 128 234703). In the nonlinear response and nonadiabatic charging regime, characteristic resonance features of transient response current reveal distinctly and faithfully the energetic configuration of the QD. We also discuss and comment on both the physical and numerical aspects of the theoretical formalism used in this work.

Dynamics of quantum dissipation systems interacting with fermion and boson grand canonical bath ensembles: Hierarchical equations of motion approach

A hierarchical equations of motion formalism for a quantum dissipation system in a grand canonical bath ensemble surrounding is constructed on the basis of the calculus-on-path-integral algorithm, together with the parametrization of arbitrary non-Markovian bath that satisfies fluctuation-dissipation theorem. The influence functionals for both the fermion or boson bath interaction are found to be of the same path integral expression as the canonical bath, assuming they all satisfy the Gaussian statistics. However, the equation of motion formalism is different due to the fluctuation-dissipation theories that are distinct and used explicitly. The implications of the present work to quantum transport through molecular wires and electron transfer in complex molecular systems are discussed.

Dynamics of quantum dissipation systems interacting with bosonic canonical bath: Hierarchical equations of motion approach

A nonperturbative theory is developed, aiming at an exact and efficient evaluation of a general quantum system interacting with arbitrary bath environment at any temperature and in the presence of arbitrary time-dependent external fields. An exact hierarchical equations of motion formalism is constructed on the basis of a calculus-on-path-integral algorithm, via the auxiliary influence generating functionals related to the interaction bath correlation functions in a parametrization expansion form. The corresponding continued-fraction Green's functions formalism for quantum dissipation is also presented. Proposed further is the principle of residue correction, not just for truncating the infinite hierarchy, but also for incorporating the small residue dissipation that may arise from the practical difference between the true and parametrized bath correlation functions. The final residue-corrected hierarchical equations of motion can therefore be used practically for the evaluation of arbitrary dissipative quantum systems.