Counterdiabatic driving for random-gap Landau-Zener transitions

The Landau--Zener (LZ) model describes a two-level quantum system that undergoes an avoided crossing. In the adiabatic limit, the transition probability vanishes. An auxiliary control field H_CD can be reverse-engineered so that the full Hamiltonian H₀ + H_CD reproduces adiabaticity for all parameter values. Our aim is to construct a single control field H₁ that drives an ensemble of LZ-type Hamiltonians with a distribution of energy gaps. H₁ works best statistically, minimizing the average transition probability. We restrict our attention to a special class of H₁ controls, motivated by H_CD. We found a systematic trade-off between instantaneous adiabaticity and the final transition probability. Certain limiting cases with a linear sweep can be treated analytically; one of them being the LZ system with Dirac δ(t) function. Comprehensive and systematic numerical simulations support and extend the analytic results.