Evolution-Time Dependence in Near-Adiabatic Quantum Evolutions

TL;DR

This paper investigates how the probability of remaining in the ground state during near-adiabatic quantum evolutions depends on total evolution time, revealing oscillatory behaviors influenced by spectral gaps and confirming findings with numerical simulations.

Contribution

It extends the quantum adiabatic theorem to analyze time dependence in near-adiabatic regimes, highlighting oscillatory probabilities linked to spectral gaps.

Findings

Probability oscillates with evolution time in the near-adiabatic limit.

Small spectral gaps cause complex oscillatory behavior in the final state.

Numerical simulations confirm the analytical predictions.

Abstract

We expand upon the standard quantum adiabatic theorem, examining the time-dependence of quantum evolution in the near-adiabatic limit. We examine a Hamiltonian that evolves along some fixed trajectory from $\hat{H}_0$ to $\hat{H}_1$ in a total evolution-time $\tau$, and our goal is to determine how the final state of the system depends on $\tau$. If the system is initialized in a non-degenerate ground state, the adiabatic theorem says that in the limit of large $\tau$, the system will stay in the ground state. We examine the near-adiabatic limit where the system evolves slowly enough that most but not all of the final state is in the ground state, and we find that the probability of leaving the ground state oscillates in $\tau$ with a frequency determined by the integral of the spectral gap along the trajectory of the Hamiltonian, so long as the gap is big. If the gap becomes exceedingly small, the final probability is the sum of oscillatory behavior determined by the integrals of the gap before and after the small gap. We confirm these analytic predictions with numerical evidence from barrier tunneling problems in the context of quantum adiabatic optimization.