On truncations of hierarchical equations of motion for finite-dimensional systems

TL;DR

This paper analyzes truncation methods for hierarchical equations of motion in finite-dimensional quantum systems, proving spectral convergence and absence of spurious modes, with applications to the spin-boson model.

Contribution

It introduces a Schur-complement based truncation scheme for HEOM, demonstrating spectral convergence and stability preservation in finite-dimensional quantum systems.

Findings

Spectral convergence of truncated HEOM to the full system as depth increases

Truncations are free of spectral pollution if the full HEOM is stable

Application demonstrated on the spin-boson model

Abstract

We study truncations of hierarchical equations of motion (HEOM) for finite-dimensional open quantum systems. We prove that for finite-dimensional approximations constructed with a Schur-complement type of terminator, the spectrum converges to that of the full HEOM as the truncation depth increases. We also prove that this approximation is free of spectral pollution: sufficiently deep truncations do not produce spurious unstable modes, provided the exact HEOM is stable. We illustrate the results for the spin-boson model.