Exact dynamics in the inhomogeneous central-spin model

TL;DR

This paper derives an exact formula for the dynamics of a central spin coupled to an inhomogeneous spin bath, revealing persistent oscillations without decay, which advances understanding of decoherence in quantum bits.

Contribution

It provides an exact analytical solution for the central-spin model with inhomogeneous couplings using Bethe ansatz, highlighting non-decaying oscillatory behavior.

Findings

Oscillations of the central spin are persistent with no decay.

Oscillation frequency scales with the number of bath spins.

Amplitude of oscillations decreases as 1/N_b for large bath sizes.

Abstract

We study the dynamics of a single spin-1/2 coupled to a bath of spins-1/2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. This formula is evaluated explicitly for initial conditions such that the bath spins are completely polarized at the beginning. For this case we find, after an initial decay, a persistent oscillatory behaviour of the central spin. For a large number of bath spins $N_b$, the oscillation frequency is proportional to $N_b$, whereas the amplitude behaves like $1/N_b$, to leading order. No asymptotic decay due to the non-uniform couplings is observed, in contrast to some recent studies.