Differences between quantum and classical adiabatic evolution

Adiabatic evolution is an emergent design principle for time modulated metamaterials, often inspired by insights from topological quantum computing such as braiding operations. However, the pursuit of classical adiabatic metamaterials is rooted in the assumption that classical and quantum adiabatic evolution are equivalent. We show that this is only true in the limit where the frequencies of all the bands are at infinite distance from $0$; and some instances of quantum adiabatic evolution, such as those containing zero modes, cannot be reproduced in classical systems. This is because mode coupling is fundamentally different in classical mechanics. We derive classical conditions to ensure adiabaticity and demonstrate that only under these conditions - which are different from quantum adiabatic conditions -, the single band Berry phase and Wilczek-Zee matrix for everywhere degenerate bands emerge as meaningful quantities encoding the geometry of classical adiabatic evolution. Finally, for general multiband systems we uncover a correction term in the non-Abelian gauge potential for classical systems.