Realization of hierarchical equations of motion from stochastic perspectives

Abstract

The hierarchical equations of motion (HEOM) for a generalized quantum dissipative system is rigorously constructed in the frameworks of two different stochastic dynamical descriptions, i.e., the non-Markovian quantum state diffusion approach as well as the stochastic decoupling scheme. We demonstrated that the HEOMs obtained by these two different stochastic dynamical methods are identical. Moreover, we present some numerical examples to verify the feasibility of our formalism.

Controllable dynamics of a dissipative two-level system

“…To handle the reduced dynamics without the rotating-wave approximation, we employ a purely numerical method, the HEOM approach 39 – 43 , to obtain the exact reduced dynamics of the TLS. The HEOM can be viewed as a bridge connecting the standard Schrödinger equation, which is exact but commonly hard to solve directly, and a set of ordinary differential equations, which can be treated numerically by using the well-developed Runge–Kutta algorithm.…”

“…Without invoking the Born, weak-coupling and rotating-wave approximations, the HEOM can provide a rigorous numerical result as long as the initial state of the whole system is a system-environment separable state. To realize the traditional HEOM algorithm, it is necessary that the zero-temperature environmental correlation function can be (or at least approximately) written as a finite sum of exponentials 43 , 44 . Fortunately, one can easily demonstrate that for the Lorentzian spectral density considered in this paper.…”

Controllable Dynamics of a Dissipative Two-level System

“…Moreover, we extend the single-mode-HO-based steer scheme to a more general qauntum dissipative system, in which the counter-rotating-wave terms are included. To handle the reduced dynamics without the rotating-wave approximation, we employ a purely numerical method, the hierarchical equations of motion (HEOM) approach [29][30][31][32][33] , to obtain the exact reduced dynamics of the TLS. The HEOM can be viewed as a bridge connecting the standard Schrödinger equation, which is exact but commonly hard to solve directly, and a set of ordinary differential equations, which can be treated numerically by using the well-developed Runge-Kutta algorithm.…”

“…To obtain the reduced dynamics of the TLS without invoking any approximation, we employ the HEOM approach, which is a highly efficient and nonperturbative numerical method. To realize the traditional HEOM algorithm, it is necessary that the zero-temperature environmental correlation function C(t) = dωJ(ω)e −iωt can be (or at least approximately) written as a finite sum of exponentials 33,40 . Fortunately, one can easily demonstrate that C(t) = αe −(ω c +iε)t for the Lorentz spectral density considered in this paper.…”

“…Then, following the procedure shown in Refs. 33,40 , one can obtain the following hierarchy equations…”