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.     

Non-Markovian dynamics and entanglement in quantum Brownian motion

Dynamical aspects of quantum Brownian motion in a low temperature environment are investigated. We give a systematic calculation of quantum entanglement among two Brownian oscillators without invoking Born–Markov approximation widely used for the study of open systems. Our approach is suitable to probe short time dynamics at cold temperatures where many experiments on quantum information processing are performed.     

Optimized hierarchical equations of motion theory for Drude dissipation and efficient implementation to nonlinear spectroscopies

Hierarchical equations of motion theory for Drude dissipation is optimized, with a convenient convergence criterion proposed in advance of numerical propagations. The theoretical construction is on the basis of a Pade? spectrum decomposition that has been qualified to be the best sum-over-poles scheme for quantum distribution function. The resulting hierarchical dynamics under the a priori convergence criterion are exemplified with a benchmark spin-boson system, and also the transient absorption and related coherent two-dimensional spectroscopy of a model exciton dimer system. We combine the present theory with several advanced techniques such as the block hierarchical dynamics in mixed Heisenberg-Schro?dinger picture and the on-the-fly filtering algorithm for the efficient evaluation of third-order optical response functions.     

Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach

A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.     

Efficient calculation of open quantum system dynamics and time‐resolved spectroscopy with distributed memory HEOM (DM‐HEOM)

Time‐ and frequency‐resolved optical signals provide insights into the properties of light‐harvesting molecular complexes, including excitation energies, dipole strengths and orientations, as well as in the exciton energy flow through the complex. The hierarchical equations of motion (HEOM) provide a unifying theory, which allows one to study the combined effects of system‐environment dissipation and non‐Markovian memory without making restrictive assumptions about weak or strong couplings or separability of vibrational and electronic degrees of freedom. With increasing system size the exact solution of the open quantum system dynamics requires memory and compute resources beyond a single compute node. To overcome this barrier, we developed a scalable variant of HEOM. Our distributed memory HEOM, DM‐HEOM, is a universal tool for open quantum system dynamics. It is used to accurately compute all experimentally accessible time‐ and frequency‐resolved processes in light‐harvesting molecular complexes with arbitrary system‐environment couplings for a wide range of temperatures and complex sizes. © 2018 Wiley Periodicals, Inc.     

A local coherent-state approximation to system-bath quantum dynamics

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.     

Wigner function approach to the quantum Brownian motion of a particle in a potential

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.     

Relaxation of quantum hydrodynamic modes

In this article, we develop a series of hierarchical mode‐coupling equations for the momentum cumulants and moments of the density matrix for a mixed quantum system. Working in the Lagrange representation, we show how these can be used to compute quantum trajectories for dissipative and nondissipative systems. This approach is complementary to the de Broglie–Bohm approach in that the moments evolve along hydrodynamic/Lagrangian paths. In the limit of no dissipation, the paths are the Bohmian paths. However, the “quantum force” in our case is represented in terms of momentum fluctuations and an osmotic pressure. Representative calculations for the relaxation of a harmonic system are presented to illustrate the rapid convergence of the cumulant expansion in the presence of a dissipative environment. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Nonadiabatic quantum-classical reaction rates with quantum equilibrium structure

Time correlation function expressions for quantum reaction-rate coefficients are computed in a quantum-classical limit. This form for the correlation function retains the full quantum equilibrium structure of the system in the spectral density function but approximates the time evolution of the operator by quantum-classical Liouville dynamics. Approximate analytical expressions for the spectral density function, which incorporate quantum effects in the many-body environment and reaction coordinate, are derived. The results of numerical simulations of the reaction rate are presented for a reaction model in which a two-level system is coupled to a bistable oscillator which is, in turn, coupled to a bath of harmonic oscillators. The nonadiabatic quantum-classical dynamics is simulated in terms of an ensemble of surface-hopping trajectories and the effects of the quantum equilibrium structure on the reaction rate are discussed.     

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.     

Dissipative and fluctuation phenomena in quantum mechanics with applications

Classical dissipative and fluctuation phenomena have a long history beginning with the experimental work on Brownian motion in 1827 and its theoretical explanation by Einstein in 1905, followed by Langevin's elegant explanation in 1908 when he introduced the idea of a stochastic differential equation. While we briefly review this history, our emphasis is on stochastic phenomena requiring quantum mechanics. We discuss many problems in a variety of areas and point out the merits of a quantum generalized Langevin equation in providing a universal framework for the solution of such problems. As an example, we analyze in detail the problem of radiation reaction in quantum electrodynamics. © 1996 John Wiley & Sons, Inc.     

Trajectory approach to dissipative quantum phase space dynamics: Application to barrier scattering

The Caldeira-Leggett master equation, expressed in Lindblad form, has been used in the numerical study of the effect of a thermal environment on the dynamics of the scattering of a wave packet from a repulsive Eckart barrier. The dynamics are studied in terms of phase space trajectories associated with the distribution function, W(q,p,t). The equations of motion for the trajectories include quantum terms that introduce nonlocality into the motion, which imply that an ensemble of correlated trajectories needs to be propagated. However, use of the derivative propagation method (DPM) allows each trajectory to be propagated individually. This is achieved by deriving equations of motion for the partial derivatives of W(q,p,t) that appear in the master equation. The effects of dissipation on the trajectories are studied and results are shown for the transmission probability. On short time scales, decoherence is demonstrated by a swelling of trajectories into momentum space. For a nondissipative system, a comparison is made of the DPM with the "exact" transmission probability calculated from a fixed grid calculation.     

Proton transfer reactions in model condensed-phase environments: Accurate quantum dynamics using the multilayer multiconfiguration time-dependent Hartree approach

The recently proposed multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) approach to evaluating reactive quantum dynamics is applied to two model condensed-phase proton transfer reactions. The models consist of a one-dimensional double-well "system" that is bilinearly coupled to a "bath" of harmonic oscillators parameterized to represent a condensed-phase environment. Numerically exact quantum-mechanical flux correlation functions and thermal rate constants are obtained for a broad range of temperatures and system-bath coupling strengths, thus demonstrating the efficacy of the ML-MCTDH approach. Particular attention is focused on the regime where low temperatures are combined with weak system-bath coupling. Under such conditions it is found that long propagation times are often required and that quantum coherence effects may prevent a rigorous determination of the rate constant.     

Exact quantum master equation via the calculus on path integrals

An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.     

Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory

Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.     

Correlated driving-and-dissipation equation for non-Condon spectroscopy with the Herzberg–Teller vibronic coupling

Correlated driving-and-dissipation equation (CODDE) is an optimized complete second-order quantum dissipation approach, which is originally concerned with the reduced system dynamics only. However, one can actually extract the hybridized bath dynamics from CODDE with the aid of dissipaton-equation-of-motion theory, a statistical quasi-particle quantum dissipation formalism. Treated as a one–dissipaton theory, CODDE is successfully extended to deal with the Herzberg–Teller vibronic couplings in dipole–field interactions. Demonstrations will be carried out on the non-Condon spectroscopies of a model dimer system.     

Non-Markovian theory of open systems in classical limit

A fully classical limit of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys. 119, 2488 (2003)] is obtained and analyzed. The resulting kinetic equations are capable of describing the evolution of an open system on the entire time axis, including the short-time non-Markovian stage, and are valid beyond linear response regime. We have shown, that proceeding to the classical mechanics limit we restrict the class of allowed correlations between an open system and a canonical bath, so that the initial conditions and the relaxation operator has to be appropriately modified (projected). Disregard of the projection may lead to unphysical behavior, since mechanism of the decay of some correlations is essentially of quantum-mechanical nature, and is not correctly described by classical mechanics. The projection (quantum correction to the kinetics) is particularly important for the non-Markovian regime of relaxation towards canonical equilibrium. The conformity of the developed method to the conventional approaches is demonstrated using a model of Brownian motion (heavy particle in the bath of light ones), for which the obtained non-Markovian equations are reduced to the standard Fokker-Planck equation in phase space.