Unveiling the Quantum Toroidal Dipole

TL;DR

This work provides a complete quantum-mechanical formalism for the toroidal dipole (TD) and links TD observables to a direct spectroscopic measurement by coupling a charged particle on a torus to an external current $I$. The authors derive a detailed Hamiltonian framework, define the TD operator with Hermitian components, and demonstrate Aharonov-Bohm–type energy shifts that are proportional to the TD expectation value. They show that TD is quantized in this nanoscale system and that transition energies depend linearly on $I$ with slopes proportional to the TD change between states, offering a clear experimental signature. The analysis highlights AB periodicity, symmetry-based eigen-solutions, and a path to measuring TD in quantum nanostructures, with implications for quantum metamaterials and fundamental symmetry studies.

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 years. 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 the first spectroscopic method for its direct measurement. We analyze a particle confined to a toroidal manifold subjected to an external current. We demonstrate that the resulting Aharonov-Bohm-like energy shifts in the system's spectrum 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 new avenues in quantum metamaterials, nanoscience, and the study of fundamental symmetries.

Figures (10)

• Figure 1: The system consists of a particle of charge $q$ and mass $m_\text{p}$ on a thin layer between between two tori. Both these tori have the same center $O$, rotation axis $z$, and major radii $R$, but they have different minor radii, $r$ and $r+\xi_{max}$, such that $r+\xi_{max} < R$ and $\xi_{max}\ll r$. The system is described in the orthogonal curvilinear coordinates $\theta, \phi, \xi$, where $\phi \in [0,2\pi)$ is the azimuthal angle, $\theta \in [0,2\pi)$ is the angle between the minor and the major radii, and $\xi \in [0,\xi_{\max}]$ is the coordinate that runs along the minor radius. The unit vectors of the local coordinates system are $\hat{\hbox{\boldmath $\theta$}}, \hat{\hbox{\boldmath $\phi$}}, \hat{\hbox{\boldmath $\xi$}}$. A straight filiform current $I$ of length $L$ runs along the $z$ axis, symmetrically with respect to the $(xy)$ plane. We assume that $L \gg R$ and the circuit is closed in such a way that only the straight portion $L$ significantly influences the particle.
• Figure 2: (a) The function $\tilde{I}_n^{(\ln^2,-)}(1)$ vs. $n$ and (b) $\tilde{I}_n^{(\ln^2,-)}(a)$ vs $a$, for the values $n = -4, \ldots, 4$, as indicated in the legend. We observe that the asymptotic values are very rapidly approached as $a$ increases. The asymptotic expressions for $n = -4, \ldots, 4$ are given in Eq. (\ref{['In_ln2_p']}) and for larger values of $|n|$ are given in (\ref{['asympt_exp_tIn_ln2']}).
• Figure 3: The first eleven eigenvalues of $\tilde{H}^{(2D)}$, $\tilde{E}^{(2D)}_{l,m,1}$ (with $l=0,1,\ldots,10$), scaled by $a^2$, are shown as functions of $I/I_{per}$, with $a = 1.2, 2, 3$ and $m = 0, 1, 2$, as indicated in each plot. The legend, shown in (a), is the same for every plot: 0 represents the ground state, 1 the first excited state, and so on.
• Figure 4: The expectation values $\left\langle \vcenter{\hbox{$\tilde{T}^{(\theta)}_3$}} \right\rangle^{\vcenter{\hbox{\scriptsize (2D)}}}_{\vcenter{\hbox{\scriptsize l,m,1}}}(I)$, scaled by $\Delta\tilde{T}_3 = 2.5$, vs. $I/I_{per}$ for the first eleven eigenstates, $a = 1.2, 2, 3$, and $m = 0, 1, 2$, as indicated in each plot. The legend, shown in (a), is the same in every plot: 0 represents the ground state, 1 the first excited state, and so on.
• Figure 5: Excitation energies, scaled by $a^2$, $\Delta\tilde{E}_{l,0,1}(I)/a^2$ (for $m=0$ and $k=1$), as a function of scaled current $I/I_\text{per}$, for various values of $a$ (indicated on each panel). Left panels: levels $l = 1, 2, \dots, 20$. Right panels: levels $l = 1, 2, \dots, 100$. Apparent crossings of higher energy levels occur near $I/I_\text{per} = p/2$ (where $p$ is an integer)--the number of crossings increase with $l$. A legend identifying $l$ is shown in the upper-left panel for all left panels; a legend for the right panels is omitted due to lack of space.