Quantum nonlinear spin switching model

Abstract

We present a method, the dynamical cumulant expansion, that allows us to calculate quantum corrections for time-dependent quantities of interacting spin systems or single spins with anisotropy. This method is applied to the quantum spin model $\ifmmode \hat{H}\else \^{H}\fi{}=\ensuremath{-}{H}_{z}{(t)S}_{z}+V(\mathbf{S})$ with ${H}_{z}(\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty})=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ and $\ensuremath{\Psi}(\ensuremath{-}\ensuremath{\infty})=|\ensuremath{-}S〉$ to find $P(t)=(1\ensuremath{-}〈{S}_{z}{〉}_{t}/S)/2.$ The case $V(\mathbf{S})=\ensuremath{-}{H}_{x}{S}_{x}$ corresponds with the standard Landau-Zener-Stueckelberg model of tunneling at avoided level crossings for $N=2S$ independent particles mapped onto a single-spin-S problem, $P(t)$ being the staying probability. Here the solution does not depend on S and it follows, e.g., from the classical Landau-Lifshitz equation. A term $\ensuremath{-}{\mathrm{DS}}_{z}^{2}$ accounts for the particle interaction and it makes the model nonlinear and essentially quantum mechanical. The $1/S$ corrections obtained with our method are in good accord with a full quantum-mechanical solution if the classical motion is regular, as for $D>0.$ If the classical motion shows special points, as is the case for $D<0$ for particular values of the sweep rate, or is irregular (the biaxial-anisotropy model with field along the hard axis) the cumulant expansion fails.