Differences between quantum and classical adiabatic evolution

Abstract

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.

7 pages, 1 figure

References in corpus (24)

• Topological Photonics
• Topological States and Adiabatic Pumping in Quasicrystals
• Holonomic Quantum Computation
• Photonic topological pumping through the edges of a dynamical four-dimensional quantum Hall system
• Classification of topological phonons in linear mechanical metamaterials
• Quantized Nonlinear Thouless Pumping
• Consistency of the Adiabatic Theorem
• Non-Abelian braiding on photonic chips
• Observation of non-Abelian topological semimetals and their phase transitions
• Edge states and topological pumping in elastic lattices with periodically modulated coupling
• Experimental observation of temporal pumping in electro-mechanical waveguides
• Adiabaticity condition for non-Hermitian Hamiltonians
• From the adiabatic theorem of quantum mechanics to topological states of matter
• Minimal non-abelian nodal braiding in ideal metamaterials
• Adiabatic pumping via avoided crossings in stiffness modulated quasiperiodic beams
• Smart patterning for topological pumping of elastic surface waves
• Decomposition of Time-Ordered Products and Path-Ordered Exponentials
• Adiabatic edge-to-edge transformations in time-modulated elastic lattices and non-Hermitian shortcuts
• Elastic Temporal Waveguiding
• Topological braiding of non-Abelian mid-gap defects in classical meta-materials
• The role of the geometric phases in adiabatic populations tracking for non-hermitian hamiltonians
• Hannay Angle: Yet Another Symmetry Protected Topological Order Parameter in Classical Mechanics
• Introduction to {\it Quantum Matter}
• Topological adiabatic dynamics in classical mass-spring chains with clamps