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

Didier Robert

Abstract

We consider the time dependent Schrödinger equation with a coupling spin-orbit in the semi-classical regime $\hbar\searrow 0$ and large spin number $\spin\rightarrow +\infty$ such that $\hbar^δ\spin=c$ where $c>0$ and $δ>0$ are constant. The initial state $Ψ(0)$ 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 $δ<1$, at the leading order in $\hbar$, the time evolution $Ψ(t)$ of $ Ψ(0)$ is well approximated by the product of an orbital and a spin coherent state. Nevertheless for $1/2<δ<1$ the quantum orbital leaves the classical orbital. For $δ=1$ 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 $Ψ(t)$ is a mixed state in $L^2(\R)$ for small $t>0$.

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

Abstract

We consider the time dependent Schrödinger equation with a coupling spin-orbit in the semi-classical regime and large spin number such that where and are constant. The initial state is a product of an orbital coherent state in and a spin coherent state in a spin irreducible representation space . For , at the leading order in , the time evolution of is well approximated by the product of an orbital and a spin coherent state. Nevertheless for the quantum orbital leaves the classical orbital. For 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 is a mixed state in for small .
Paper Structure (14 sections, 16 theorems, 110 equations)

This paper contains 14 sections, 16 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction to the interaction propagator
  4. A realization of spin-$\sf{s}$ representation
  5. The classical spin space
  6. The Schrödinger coherent states
  7. The Schrödinger equation for the spin-orbit interaction
  8. The regime $\sf{s} = c\hbar^{-\delta}$, $0<\delta < 1$
  9. The regime $\hbar\sf{s}=\kappa$=constant
  10. More on the Dicke model
  11. Preliminary computations and reductions
  12. Proof of Theorem \ref{['Dmod']}
  13. Classical perturbations of Hamiltonians
  14. Proof of Proposition \ref{['tind']}

Key Result

Theorem 1.1

BG For the initial state $\Psi_{z_0,{\bf n}_0} =\varphi^{\Gamma_0}_{z_0}\otimes\psi^{(\sf{s})}_{{\bf n}_0}$ we have where $z_0\mapsto z(t)$ is the classical flow for the Hamiltonian for $H_0(t)$, $S(t,t_0)$ is the classical action, the covariance matrix $\Gamma(t)$ is computed from the dynamics generated by the linearized flow of $H_0(t)$ and $\alpha(t)$ is a real phase computed from the spin mot

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 18 more