Hierarchical equations of motion method based on Fano spectrum decomposition for low temperature environments

Abstract

The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.

Quantum parameter estimation in a dissipative environment

“…[62,63,77,78], the HEOM method has been extended to arbitrary spectral density function as well as finite temperature environment situation. Moreover, it has been reported that the HEOM method can be extended to simulate the dissipative dynamics of a few-level system embedded in a fermionic environment [53,65,79] or a spin environment [80,81]. It would be very interesting to extrapolate our study to these more general situations.…”

Effect of quantum resonances on local temperature in nonequilibrium open systems

“…To investigate the properties of the impurity, the hierarchical equations of motion (HEOM) approach [65][66][67][68][69][70] is employed, which takes the reduced density matrix of the system ρ and a hierarchical set of auxiliary density operators as the basic variables. The HEOM theory is, in principle, formally exact, and its numerical outcomes are guaranteed to be quantitatively accurate if the results converge with respect to the truncation tier of the hierarchy.…”

Speeding up quantum dissipative dynamics of open systems with kernel methods

“…An alternative numerical exact bench-marking approach is the hierarchical equations of motion (HEOM) [44][45][46][47][48][49][50][51][52][53] approach pioneered by Tanimura and Kubo. 54 HEOM approach captures the combined effects of system-environment dissipation, non-Markovian memory effect, and many-body correlation in a non-perturbative manner.…”