The Adiabatic Theorem for Quantum Systems with Spectral Degeneracy

TL;DR

This paper rigorously formulates the adiabatic theorem for quantum systems with degenerate spectra, providing necessary and sufficient conditions for its validity, crucial for quantum computation and braiding applications.

Contribution

It establishes a clear, rigorous adiabatic theorem for degenerate quantum systems, including practical tests and analysis of an exactly solvable model.

Findings

Derived necessary and sufficient conditions for adiabatic approximation validity in degenerate systems

Simplified the sufficient condition into a practical test

Analyzed an exactly solvable degenerate quantum problem

Abstract

By stating the adiabatic theorem of quantum mechanics in a clear and rigorous way, we establish a necessary condition and a sufficient condition for its validity, where the latter is obtained employing our recently developed adiabatic perturbation theory. Also, we simplify further the sufficient condition into a useful and simple practical test at the expenses of its mathematical rigor. We present results for the most general case of quantum systems, i.e., those with degenerate energy spectra. These conditions are of upmost importance to assess the validity of practical implementations of non-Abelian braiding and adiabatic quantum computation. To illustrate the degenerate adiabatic approximation, and the necessary and sufficient conditions for its validity, we analyze in depth an exactly solvable time-dependent degenerate problem.