General conditions for a quantum adiabatic evolution

TL;DR

This paper establishes general criteria and bounds for quantum adiabatic evolution, clarifying when the traditional slow variation condition guarantees adiabaticity, especially highlighting limitations with oscillating Hamiltonians.

Contribution

It derives universal conditions and bounds for adiabaticity in quantum systems, extending the understanding beyond slow Hamiltonian changes to more general cases.

Findings

Slow variation condition suffices for real, non-oscillating Hamiltonians.

Provides exact bounds for state and phase during adiabatic evolution.

Highlights limitations of traditional adiabatic criteria with oscillating Hamiltonians.

Abstract

Adiabaticity occurs when, during its evolution, a physical system remains in the instantaneous eigenstate of the hamiltonian. Unfortunately, existing results, such as the quantum adiabatic theorem based on a slow down evolution (H(epsilon t), epsilon ? 0), are insufficient to describe an evolution driven by the hamiltonian H(t) itself. Here we derive general criteria and exact bounds, for the state and its phase, ensuring an adiabatic evolution for any hamiltonian H(t). As a corollary we demonstrate that the commonly used condition of a slow hamiltonian variation rate, compared to the spectral gap, is indeed sufficient to ensure adiabaticity but only when the hamiltonian is real and non oscillating (for instance containing exponential or polynomial but no sinusoidal functions).