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arXiv · 2506.20594

Cyclicity of interaction frame transformations

Michael C D Tayler·Mohamed Sabba
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Abstract

We identify a cyclic property of rotation sequences involving piecewise displacements $\beta$ about arbitrary axes in three dimensions. Specifically, when transformation to the toggling frame is applied successively $m$ times, for $\beta=2\pi/m$ the original sequence returns. This main result unites several families of rotation sequences used for error-tuned control across quantum technologies, from NMR and MRI to atomic clocks and atom-scale computing. For the widest class of cycle, $m=2$, we highlight sequence duality where every narrowband $\pi$-element sequence has a broadband $\pi$-element counterpart, and vice-versa. Higher cycles ($m>2$) connect to polyhedral models for error-tolerant sequence design, characterized by vertex axes with $m$-fold rotational symmetry. We derive original sequences and outline their applications to spin control and spin decoupling.

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Michael C D Tayler, Mohamed Sabba. 2025-06-25. Cyclicity of interaction frame transformations. https://doi.org/10.1103/w42p-hcwg

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