Semiclassical approximations for adiabatic slow-fast systems

In this letter we give a systematic derivation and justification of the semiclassical model for the slow degrees of freedom in adiabatic slow-fast systems first found by Littlejohn and Flynn (Phys. Rev. A, 44 (1991) 5239). The classical Hamiltonian obtains a correction due to the variation of the adiabatic subspaces and the symplectic form is modified by the curvature of the Berry connection. We show that this classical system can be used to approximate quantum-mechanical expectations and the time evolution of operators also in sub-leading order in the combined adiabatic and semiclassical limit. In solid-state physics the corresponding semiclassical description of Bloch electrons has led to substantial progress during the recent years, see works by Niu and coworkers. Here, as an illustration, we show how to compute the piezo-current arising from a slow deformation of a crystal in the presence of a constant magnetic field