Post-adiabatic forces and Lagrangians with higher-order derivatives

TL;DR

This paper explores the dynamics of a classical particle coupled to a quantum system, deriving higher-order adiabatic corrections that lead to complex Lagrangians with higher derivatives and novel geometric features.

Contribution

It introduces a systematic method to derive higher-order adiabatic corrections for classical particles coupled to quantum systems, revealing new geometric and dynamical properties.

Findings

At order ε^2, motion is geodesic on a curved manifold with mixed metric signatures.

At order ε^3, the Lagrangian depends linearly on acceleration, indicating a spin tensor.

The Hamiltonian structure involves non-linear Poisson brackets and higher derivatives.

Abstract

We study a slow classical system [particle] coupled to a fast quantum system with discrete energy spectrum. We adiabatically exclude the quantum system and construct an autonomous dynamics for the classical particle in successive orders of the small ratio $\epsilon$ of the characteristic times. It is known that in the order $\epsilon^0$ the particle gets an additional [Born-Oppenheimer] potential, while in the order $\eps^1$ it feels an effective magnetic field related to the Berry phase. In the order $\epsilon^2$ the motion of the classical particle can be reduced to a free [geodesic] motion on a curved Riemannian manifold, with the metric generated by the excluded quantum system. This motion has a number of unusual features, e.g., it combines subspaces of different (Riemannian and pseudo-Riemannian) signature for the metric tensor. In the order $\epsilon^3$ the motion of the classical particle is still described by a Lagrangian, but the latter linearly depends on the particle's acceleration. This implies the existence of a spin tensor [non-orbital angular momentum] for the particle. This spin tensor is related to the momentum via an analogue of the zitterbewegung effect. The Hamiltonian structure of the system is non-trivial and is defined via non-linear Poisson brackets. The linear dependence of the effective classical Lagrangian on higher-order derivatives is seen as well in the higher orders $\epsilon^n$.