Evolution of a localized electron spin in a nuclear spin environment

TL;DR

This paper investigates the dynamics of a localized electron spin interacting with a nuclear spin environment in quantum dots, revealing undamped oscillations and slow decay of correlations due to near-integrability.

Contribution

It introduces a semi-classical model accounting for position-dependent hyperfine coupling and analyzes the resulting classical dynamics, highlighting the non-decaying oscillations in spin correlations.

Findings

Electron spin correlation functions show no decay with arbitrary initial conditions.

Correlation functions exhibit complicated undamped oscillations.

Ensemble-averaged correlations decay slowly as 1/ln(t).

Abstract

Motivated by recent interest in the role of the hyperfine interaction in quantum dots we study the dynamics of a localized electron spin coupled to many nuclei. An important feature of the model is that the coupling to an individual nuclear spin depends on its position in the quantum dot. We introduce a semi-classical description of the system valid in the limit of a large number of nuclei and analyze the resulting classical dynamics. Contrary to a natural assumption, the correlation functions of electron spin with an arbitrary initial condition show no decay in time. Rather, they exhibit complicated undamped oscillations. This may be attributed to the fact that the system has many integrals of motion and is close to an integrable one. The ensemble averaged correlation functions do exhibit a slow decay (1/ln(t)) for t -> \infty.