Quantum nonlinear spin switching model

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 $Ĥ=-H_z{\mathrm{}\left(t\right)S}_z+V\left(\mathbf{S}\right)$ with $H_z(\pm \infty )=\pm \infty$ and $\Psi \left(-\infty \right)=|-S〉$ to find $P\left(t\right)=(1\mathrm{}-〈S_z{〉}_t/S)/2.\mathrm{}$ The case $V\left(\mathbf{S}\right)=-H_x{S}_x$ 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, $P\left(t\right)\mathrm{}$ 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 $-{\mathrm{DS}}_z^2$ 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 $D>\mathrm{}0.\mathrm{}$ If the classical motion shows special points, as is the case for $D<\mathrm{}0\mathrm{}$ for particular values of the sweep rate, or is irregular (the biaxial-anisotropy model with field along the hard axis) the cumulant expansion fails.