Quantum Nonlinear Switching Model

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We present a method, the dynamical cumulant expansion, that allows 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 H = -Hz(t)Sz + V(S) with Hz(±∞) = ±∞ and Ψ(-∞)=|-S> we study the quantity P(t)=(1-<Sz>ₜ/S)/2. The case V(S)=-Hₓ Sₓ corresponds to the standard Landau-Zener-Stueckelberg model of tunneling at avoided-level crossing 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 follows, e.g., from the classical Landau-Lifshitz equation. A term -DSz² accounts for particles' interaction and it makes the model nonlinear and essentially quantum mechanical. The 1/S corrections obtained with our method are in a good accord with a full quantum-mechanical solution if the classical motion is regular, as for D>0.