# Superintegrability of the Fock-Darwin system

## TL;DR

This paper demonstrates that the Fock-Darwin system exhibits superintegrability in both quantum and classical regimes for rational frequency ratios, revealing higher-order symmetries, degeneracies, and closed classical trajectories.

## Contribution

It establishes the superintegrability of the Fock-Darwin system for rational frequency ratios and characterizes its quantum and classical symmetries and trajectories.

## Key findings

- Quantum superintegrability for rational frequency ratios.
- Higher-order polynomial algebra of symmetries.
- Classical closed trajectories and constants of motion.

## Abstract

The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.