Semiclassical dynamics and long time asymptotics of the central-spin problem in a quantum dot

TL;DR

This paper models the electron spin dynamics in a quantum dot under hyperfine interaction, showing classical behavior in the large nuclear spin limit, analyzing long-term decay, and highlighting the influence of wavefunction shape.

Contribution

It introduces a semiclassical approach to the central spin problem, providing systematic methods to analyze classical spin dynamics and long-term decay behavior.

Findings

Electron spin evolution is well described by classical dynamics in the large nuclear spin limit.

Long time decay of electron spin polarization is insensitive to integrability and depends on weakly coupled spins.

Decay form is sensitive to the electronic wavefunction shape.

Abstract

The spin of an electron trapped in a quantum dot is a promising candidate implementation of a qubit for quantum information processing. We study the central spin problem of the effect of the hyperfine interaction between such an electron and a large number of nuclear moments. Using a spin coherent path integral, we show that in this limit the electron spin evolution is well described by classical dynamics of both the nuclear and electron spins. We then introduce approximate yet systematic methods to analyze aspects of the classical dynamics, and discuss the importance of the exact integrability of the central spin Hamiltonian. This is compared with numerical simulation. Finally, we obtain the asymptotic long time decay of the electron spin polarization. We show that this is insensitive to integrability, and determined instead by the transfer of angular momentum to very weakly coupled spins far from the center of the quantum dot. The specific form of the decay is shown to depend sensitively on the form of the electronic wavefunction.