Cyclic ladder operators and hidden Weyl-Heisenberg structure in a Floquet system

Ladder operators, found in the quantum harmonic oscillator and other quantized systems, provide an elegant approach to solving or understanding otherwise intricate physics problems. In this Letter, we discuss cyclic ladder operators in both Hermitian and non-Hermitian systems with a finite Hilbert space, with the highest (lowest) level directly descending (ascending) to the lowest (highest) level via a single raising (lowering) operation. We show that an equally spaced energy ladder emerges when these systems have an underlying Weyl-Heisenberg commutation relation, with the cyclic ladder operators and the temporal evolution operator behaving as the generators of the Weyl-Heisenberg group. We further illustrate such a system using a one-dimensional Floquet lattice, where the cyclic ladder operators become diagonal and the temporal evolution simplifies to a permutation matrix after a Floquet period. Our findings reveal a hidden relation between non-trivial dynamics and algebraic principles in Floquet systems, which may exist for other quantum numbers as well besides the energy levels.