- Open Access
Unveiling the quantum toroidal dipole
Phys. Rev. Research 8, 023081 – Published 28 April, 2026
DOI: https://doi.org/10.1103/zgh6-4tfh
Abstract
The electromagnetic response of matter is governed by three fundamental multipole families: electric, magnetic, and toroidal. While the electric and magnetic are cornerstones of physics, the toroidal dipole (TD) has eluded direct, quantitative measurement for over 60 yr. Its far-field signature is masked by the electric dipole, and its behavior in the quantum regime remains largely unexplored. We address this long-standing problem by presenting a complete quantum-mechanical formalism for the TD in a nanostructure and proposing a spectroscopic method for its direct measurement. Focusing on a particle confined to a toroidal manifold—a geometry physically realized in systems such as carbon nanotube tori and topological insulator nanorings—we analyze the spectral response to an external toroidal current. We demonstrate that the resulting Aharonov-Bohm-like energy shifts are directly proportional to the expectation value of the TD operator. The transition energies exhibit a linear dependence on this current, with a quantized slope that directly reveals the change in the TD quantum number between eigenstates. This provides a clear experimental blueprint to unveil, measure, and characterize this elusive third multipole moment and its quantum nature, opening avenues in quantum metamaterials, nanoscience, and the study of fundamental symmetries.
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