Deterministic classical limit of the optimal control problem of quantum particles with spin
Omar Morandi
TL;DR
This work addresses optimal control of a quantum gas of spinful particles with Rashba spin-orbit coupling under an external magnetic field, formulated in the Wigner phase-space framework. The authors derive the quantum OC problem, establish well-posedness of the Wigner dynamics, and show that in the classical limit $\hbar\to0$ the quantum problem concentrates on a single classical trajectory with a rotating spin, reducing to a Liouville-Rashba-Zeeman transport model. They prove that $f^\hbar$ converges to a Dirac-type measure $f^0=\delta(x-x(t))\delta(p-p(t))\,d(t)$ and that the adjoint and optimality conditions converge to a deterministic OC system (Cl-OS) for the classical dynamics, unifying the quantum and classical descriptions. This provides a rigorous justification for using simplified deterministic models in spintronics control problems and suggests significant computational savings for control design in quantum transport settings.
Abstract
We study the optimal control problem applied to a gas of particles with spin confined in a material with Rashba spin-orbit coupling effect, in the presence of an external magnetic field. The evolution of the particle gas is described in the Wigner formalism. We investigate the classical limit of the optimal control problem, and we prove the convergence of the solution of the quantum problem toward the solution of a simplified optimal control model, based on an ODE description of the particle gas in terms of a single spin vector traveling along a classical trajectory in the phase-space.