Complete population transfer in 4-level system via Pythagorean triple coupling

We describe a relation between the requirement of complete population transfer in a four-mode system and the generating function of Pythagorean triples from number theory. We show that complete population transfer will occur if ratios between coupling coefficients exactly match one of the Pythagorean triples (a; b; c) in Z, c^{2} = a^{2} + b^{2}. For a four-level ladder system this relation takes a simple form (V12; V23; V34) ~ (c; b; a), where coefficients Vij describe the coupling between modes. We find that the structure of the evolution operator and the period of complete population transfer are determined by two distinct frequencies. A combination of these frequencies provides a generalization of the two-mode Rabi frequency for a four-mode system.