We study the stability of counterdiabatic drive by computing the joint dependence of the probability of nonadiabatic transitions $P_{\eta }\left(\delta \right)$ on the adiabatic parameter $\delta$ and on the normalized amplitude of the adiabatic gauge potential (AGP) $\eta$ in the Landau-Zener-Stückelberg-Majorana model. We show that the Dykhne-Davis-Pechukas formula cannot be readily applied since the AGP introduces a singularity in the Hamiltonian which makes the wave function multivalued in the complex-time plane. This can be understood as the non-Abelian Aharonov-Bohm phase introduced by the AGP and leads to the counterdiabatic correction of the Landau-Zener formula. In particular, it shows that, unlike the nonperturbative suppression of transitions in the adiabatic limit $\delta \to 0$, the probability is only perturbatively suppressed in the counterdiabatic limit $\eta \to 1$. We then consider the extension of our results to integrable time-dependent quantum Hamiltonians. We prove that the AGP satisfies the flatness constraint which characterizes integrability in these models, which allows us to derive simple expressions for the AGP and the probability of transitions $P_{\eta }\left(\delta \right)$ near adiabatic or counterdiabatic evolution in three- and four-state integrable examples.

• Received 24 April 2023
• Revised 16 November 2023
• Accepted 11 January 2024

DOI:https://doi.org/10.1103/PhysRevA.109.022201