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

TL;DR

This paper establishes a novel link between Pythagorean triples and complete population transfer in a four-level quantum system, revealing conditions for perfect transfer based on number theory and analyzing the system's dynamics.

Contribution

It introduces a new theoretical framework connecting Pythagorean triples to population transfer conditions in four-level systems, expanding understanding of quantum control.

Findings

Complete population transfer occurs when coupling ratios match Pythagorean triples.

The evolution operator's structure and transfer period depend on two distinct frequencies.

A generalized Rabi frequency for four-mode systems is derived.

Abstract

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.