Time integrated functional about krotov.jl

[goerz]

Krotov's method in principle allows for state-dependent running costs, see Reich at al. J. Chem. Phys. 136, 104103 (2012) (arXiv). However, it has not yet been implemented in this package.

The exact math for Krotov would require a (time-continuous) inhomogeneous backward propagation, which currently isn't implemented in QuantumPropagators.jl. I'll probably get around to that eventually, but not anytime soon. I have a conjecture that it might work to use a "discrete inhomogeneous backward propagation", as I speculate in Quantum 6, 871 (2022), which uses a regular backward propagation and then just adds the inhomogenous term as a constant. That would be much easier to implement, basically the same as JuliaQuantumControl/GRAPE.jl#53 for GRAPE.

So if there's a good example, I might give it a shot after implementing the running costs for GRAPE, to see if it works. If it does, great, you'll be able to do the optimization with either GRAPE or Krotov. If not, you probably won't be able to solve this kind of optimization problem with Krotov.jl in the near future.