Spin-orbit interaction with large spin in the semi-classical regime

Abstract: We consider the time dependent Schr\"odinger equation with a coupling spin-orbit in the semi-classical regime $\hbar\searrow 0$ and large spin number $\spin\rightarrow +\infty$ such that $\hbar^\delta\spin=c$ where $c>0$ and $\delta>0$ are constant. The initial state $\Psi(0)$ is a product of an orbital coherent state in $L^2(\R^d)$ and a spin coherent state in a spin irreducible representation space ${\mathcal H}_{2\spin +1}$. For $\delta <1$, at the leading order in $\hbar$, the time evolution $\Psi(t)$ of $ \Psi(0)$ is well approximated by the product of an orbital and a spin coherent state. Nevertheless for $1/2<\delta<1$ the quantum orbital leaves the classical orbital. For $\delta=1$ we prove that this last claim is no more true when the interaction depends on the orbital variables. For the Dicke model, we prove that the orbital partial trace of the projector on $\Psi(t)$ is a mixed state in $L^2(\R)$ for small $t>0$.