Optimal molecule production from Bose condensed atoms using non-linear magnetic field sweeps through a Feshbach resonance

In most experiments involving conversion of ultracold atomic gases into molecules via a Feshbach resonance, a magnetic field, $B(t) $, is linearly swept across the resonance. In this case, Landau- Zener (LZ) theory predicts a high conversion efficiency if $\delta_{LZ}=\Omega_R^2/4|\Delta\mu\partial B/\partial t|>1$, where $\Delta\mu$ is the difference between the atomic and molecular magnetic moments and $\Omega_R$ is the coupling between the atoms and molecules. $\delta_{LZ}>1$ corresponds to adiabatic evolution for which the fraction of atoms converted into molecules is independent of the functional form of the sweep. For very fast linear sweeps such that $\delta_{LZ}\ll 1$, LZ theory predicts that almost no atoms are converted to molecules. Here we employ a genetic algorithm to determine the time dependence of the magnetic field that produces the maximum number of molecules when the duration of the sweep, $T$, is small enough for the evolution to be non-adiabatic, $\Omega_R^2< 4|\Delta\mu(B_{initial}-B_{final})|/T$. The optimal sweep through resonance shows that more than $95\%$ of the atoms can be converted into molecules for sweep times as short as $4\pi/\Omega_R$ while the linear sweep results in a conversion of $<10\%$. The qualitative form of the non-linear optimal sweep is independent of the strength of the two-body interactions and the width of the resonance.