Boundary analysis in block schemes for control of some multi-level quantum systems

Two-photon absorption (TPA) is an important quantum process in the light-matter interaction. When more than one TPA pathway exists in a quantum system, it is necessary to utilize the interference of these pathways to increase the sum TPA amplitude. A simple block scheme (Phys. Rev. A 2013, 88, 053427) is proposed to maximize the constructive interference of two TPA pathways in a four-level diamond-configuration quantum system. It can also be expanded to control multiple TPA pathways (Chin. J. Chem. Phys. 2015, 28, 426). The spectral boundaries in the scheme is related to the energy structure of the system. This work tries to give an analysis of the boundaries for general four-level and five-level quantum systems, which have two and three TPA pathways, respectively. How the boundaries and thus the number of blocks change with the energy structure will be explored.