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  • Open Access

Cyclicity of interaction-frame transformations

Michael C. D. Tayler*

Mohamed Sabba

  • *Contact author: michael.tayler@icfo.eu
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Phys. Rev. A 112, 052432 – Published 18 November, 2025

DOI: https://doi.org/10.1103/w42p-hcwg

Abstract

We study the structure of rotation sequences composed of piecewise displacements by an angle β about arbitrary axes in three dimensions. We find that when β=2π/m, repeated toggling-frame transformations return the sequence to its original form after m applications. We develop this cyclicity property to unify several families of error-tuned control sequences used across quantum technologies, including NMR, MRI, atomic clocks, and atom-scale computing. For the m=2 case, we show a duality between narrowband and broadband π-element sequences. For m>2, we relate the cyclic structure to polyhedral symmetry, leading to original sequence designs based on m-fold rotational axes. The result may guide the construction of robust control protocols grounded in geometric and group-theoretic principles.

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Phys. Rev. A 112, 052432– Published 18 November, 2025
  • Received 27 March 2025
  • Accepted 3 October 2025
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