Entanglement dynamics via semiclassical propagators in systems of two spins

TL;DR

This paper derives a semiclassical formula to analyze how entanglement evolves over time in two-spin systems, linking quantum entanglement dynamics to classical trajectory stability.

Contribution

It introduces a semiclassical expression for entanglement dynamics in two-spin systems, connecting quantum properties to classical stability analysis.

Findings

Semiclassical formula accurately predicts short-time entanglement evolution.

Entanglement dynamics depend on classical trajectory stability.

Formula reproduces quantum properties in large spin limit.

Abstract

We analyze the dynamical generation of entanglement in systems of two interacting spins initially prepared in a product of spin coherent states. For arbitrary time-independent Hamiltonians, we derive a semiclassical expression for the purity of the reduced density matrix as function of time. The final formula, subsidiary to the linear entropy, shows that the short-time dynamics of entanglement depends exclusively on the stability of trajectories governed by the underlying classical Hamiltonian. Also, this semiclassical measure is shown to reproduce the general properties of its quantum counterpart and give the expected result in the large spin limit. The accuracy of the semiclassical formula is further illustrated in a problem of phase exchange for two particles of spin $j$.