Pythagorean coupling: Complete population transfer in a four-state system

Complete population transfer in a four-coupled-modes system is analyzed from a geometrical point of view. An analytical solution of the dynamics is written by the use of two distinct frequencies, the generalization of the single Rabi frequency of the two-state dynamics. We also present its visualization on two separate Bloch spheres with two independent torque equations. With this scheme we analytically derive the requirements for complete population transfer in a four-state quantum system. Interestingly, the solutions are found to be linked to fundamental number theory, whereas complete population transfer occurs only if the ratios between coupling coefficients exactly match a set of Pythagorean triples.