 Solution

For the first two targets assume $N\ge 5$, so the indexed levels exist. A possible level population reward isIt selects $\hat{A}=|5⟩⟨5|$ and $t_F=10$.

For the coherent target, use its rank-one orthogonal projector, not the sum of its two level populations:The imaginary quantum coherence term fixes the desired relative phase $-i$. Maximizing level populations alone would not select this superposition.

For energy, choose a specified final energy observable $H_F$:For stored energy after the control is switched off, $H_F=H_0$ is the natural choice. If a fixed external field remains, use that known final Quantum Hamiltonian instead. If the terminal Quantum Hamiltonian itself is varied through the controls, the objective has explicit control dependence and additional terminal derivatives must be included. Each reward can be combined with the given dynamics and fluence penalty as $J_j={\mathcal{A}}_j-\mathcal{D}-\mathcal{C}$.