We study optimal quantum control of the dynamics of trapped Bose-Einstein condensates:
The targets are to split a condensate, residing initially in a single well, into a
double well, without inducing excitation; and to excite a condensate from the ground
to the first excited state of a single well. The condensate is described in the mean-
field approximation of the Gross-Pitaevskii equation. We compare two optimization
approaches in terms of their performance and ease of use, namely gradient ascent pulse
engineering (GRAPE) and Krotov’s method. Both approaches are derived from the
variational principle but differ in the way the control is updated, additional costs
are accounted for, and second order derivative information can be included. We find
that GRAPE produces smoother control fields and works in a black-box manner, whereas
Krotov with a suitably chosen step size parameter converges faster but can produce
sharp features in the control fields.