Optimal control of multiple two-photon transitions

Optimal strategies to maximize the two-photon transition amplitude have been well studied. However, for a system with multiple intermediate states linking the same initial and final states, the question that how to achieve an optimal population transfer remains nontrivial. In this work, we propose a systematic block scheme to maximize the transition amplitude by explicitly considering the interferences among different transition pathways. The scheme can probably provide a quasi-optimal solution, even considering the uncertainties and noises in experiments. Cases with infinitesimal and finite spectral resolution are both investigated. A special example with all first transition frequencies being larger than half of the two-photon transition frequencies is employed to demonstrate our scheme. The analysis provides valuable insights on how to manipulate the interferences in control of quantum systems.