Fig. 2

Optimal control amplitude (a) and phase (b), along with the corresponding time evolution of stored energy (c) and populations (\(|c_0|^2\) black, \(|c_1|^2\) red, \(|c_2|^2\) green, \(|c_3|^2\) blue) (d) for full charging in minimum time, when \(\Omega _0/\Omega _z=1\) and \(\chi =J/\Omega _z=1/4\).