Truncated Wigner approximation for the bosonic model of large spin baths

TL;DR

This paper evaluates the effectiveness of the truncated Wigner approximation in modeling large spin baths represented as harmonic oscillators, comparing it with other methods through numerical analysis of chain length and bath size effects.

Contribution

It introduces a chain representation for bosonic degrees of freedom in the central spin model and assesses the truncated Wigner approximation's accuracy against other approaches.

Findings

Truncated Wigner approximation performs well for certain chain lengths.

The accuracy depends on the number of bath spins and truncation level.

Numerical results highlight the method's applicability and limitations.

Abstract

The central spin model has a wide applicability, it is ideally suited to describe a small quantum system, for instance a quantum bit, in contact to a bath of spins, e.g., nuclear spins, or other small quantum systems in general. According to previous work~[R\"ohrig \textit{et al.}, Phys. Rev. B {\bf 97}, 165431 (2018)], a large bath of quantum spins can be described as a bath of quantum harmonic oscillators. But the resulting quantum model is still far from being straightforward solvable. Hence we consider a chain representation for the bosonic degrees of freedom to study how well a truncated Wigner approximation of the effective model of harmonic oscillators works in comparison with other approximate and exact methods. Numerically, we examine the effect of the number of bath spins and of the truncation level, i.e., the chain length.