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 $H_\text{CD}$ can be reverse-engineered so that the full Hamiltonian $H_0 + H_\text{CD}$ reproduces adiabaticity for all parameter values. Our aim is to construct a single control field $H_1$ that drives an ensemble of LZ-type Hamiltonians with a distribution of energy gaps. $H_1$ works best statistically, minimizing the average transition probability. We restrict our attention to a special class of $H_1$ controls, motivated by $H_\text{CD}$. 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 $\delta(t)$ function. Comprehensive and systematic numerical simulations support and extend the analytic results.