The toroidal dipole operator in nanostructures

The Journal of Chemical Physics, 2022

The continuous search for metamaterials with special properties, and suitable for new technological applications, is presently being driven by a preceding theoretical development, which took place after the introduction of new physical entities, anapole and a family of toroidal multipoles, having a border in common with those considered in the more familiar electric and magnetic multipole expansions. The related concept of toroidisation, that is, toroidal moment per unit volume, has been advocated in analogy to electric polarisation and magnetisation operated by electromagnetic fields, and should be considered on the same footing as regards its relevance and practicality for understanding certain properties, e.g., ferrotoroidicity in condensed matter physics, and for rationalizing the behaviour of charge-current distributions that neither radiate nor interact with external fields in classical and quantum electrodynamics. Toroidisability, i.e., the ability of sustaining toroidal moments, can also be defined by analogy with electric polarisability and magnetisability. The present study shows that such a property is general and characterizes atoms and molecules, and that the optical electric field of a light beam induces an oscillating anapole moment, i.e., the superposition of toroidal moment with an electric dipole moment. However, values of anapole polarisabilities induced by monochromatic light, estimated by time-dependent perturbation theory for rare gas atoms and a few molecules, are quite small and possibly hard to detect experimentally.

Magnetochemistry

For single-molecule toroics (SMTs) based on noncollinear Ising spins, intramolecular magnetic dipole–dipole coupling favours a head-to-tail vortex arrangement of the semi-classical magnetic moments associated with a toroidal ground state. However, to what extent does this effect survive beyond the semi-classical Ising limit? Here, we theoretically investigate the role of dipolar interactions in stabilising ground-state toroidal moments in quantum Heisenberg rings with and without on-site magnetic anisotropy. For the prototypical triangular SMT with strong on-site magnetic anisotropy, we illustrate that, together with noncollinear exchange, intramolecular magnetic dipole–dipole coupling serves to preserve ground-state toroidicity. In addition, we investigate the effect on quantum tunnelling of the toroidal moment in Kramers and non-Kramers systems. In the weak anisotropy limit, we find that, within some critical ion–ion distances, intramolecular magnetic dipole–dipole interactions, d...

Journal of Physics A: Mathematical and General, 1997

After a short presentation of the toroidal moments and the necessity to introduce them in the multipole expansion of current density, the correspondent quantum operators are introduced. The toroidal momentum operator (the quantum operator corresponding to the lowestorder toroidal multipole) is analysed. A natural set of coordinates is found. Using this set of coordinates it becomes possible to find the eigenvalues and a complete orthonormal set of eigenfunctions of the projection of this operator on the Oz axis. † Permanent address:

2021

The quantum operator T̂3, corresponding to the projection of the toroidal moment on the z axis, admits several self-adjoint extensions, when defined on the whole R space. T̂3 commutes with L̂3 (the projection of the angular momentum operator on the z axis) and they have what we call a natural set of coordinates, denoted (k, u, φ), where φ is the azimuthal angle. The second set of natural coordinates is (k1, k2, u), where k1 = k cosφ, k2 = k sinφ. In both sets these coordinates, the operators get the simple forms T̂3 ≡ −i~∂/∂u and L̂3 = −i~∂/∂φ. In both sets, T̂3 = −i~∂/∂u, so any operator that is a function of k and the partial derivatives with respect to the natural variables (k, u, φ) commute with T̂3 and L̂3. Similarly, operators that are functions of k1, k2, and the partial derivatives with respect to k1, k2, and u commute with T̂3. Therefore, we introduce here the operators p̂k ≡ −i~∂/∂k, p̂ ≡ −i~∂/∂k1, and p̂ ≡ −i~∂/∂k2 and express them in the (x, y, z) coordinates. One may al...

New Journal of Physics, 2007

It is shown that a new type of metamaterial, a 3D-array of toroidal solenoids, displays a significant toroidal response that can be readily measured. This is in sharp contrast to materials that exist in nature, where the toroidal component is weak and hardly measurable. The existence of an optimal configuration, maximizing the interaction with an external electromagnetic field, is demonstrated. In addition, it is found that a characteristic feature of the magnetic toroidal response is its strong dependence on the background dielectric permittivity of the host material, which suggests possible applications. Negative refraction and backward waves exist in a composite toroidal metamaterial, consisting of an array of wires and an array of toroidal solenoids.

Physical Review X, 2015

Toroidal multipoles are the terms missing in the standard multipole expansion; they are usually overlooked due to their relatively weak coupling to the electromagnetic fields. Here we propose and theoretically study all-dielectric metamaterials of a special class that represent a simple electromagnetic system supporting toroidal dipolar excitations in the THz part of the spectrum. We show that resonant transmission and reflection of such

2006

The time dependent Schrodinger equation inclusive of curvature effects is developed for a spinless electron constrained to motion on a toroidal surface and subjected to circularly polarized and linearly polarized waves in the microwave regime. A basis set expansion is used to determine the character of the surface currents as the system is driven at a particular resonance frequency. Surface current densities and magnetic moments corresponding to those currents are calculated. It is shown that the currents can yield magnetic moments large not only along the toroidal symmetry axis, but along directions tangential and normal to the toroidal surface as well.

Journal of Chemical Physics, 2022

The continuous search for metamaterials with special properties, and suitable for new technological applications, is presently being driven by a preceding theoretical development, which took place after the introduction of new physical entities, anapole and a family of toroidal multipoles, having a border in common with those considered in the more familiar electric and magnetic multipole expansions. The related concept of toroidisation, that is, toroidal moment per unit volume, has been advocated in analogy to electric polarisation and magnetisation operated by electromagnetic fields, and should be considered on the same footing as regards its relevance and practicality for understanding certain properties, e.g., ferrotoroidicity in condensed matter physics, and for rationalizing the behaviour of charge-current distributions that neither radiate nor interact with external fields in classical and quantum electrodynamics. Toroidisability, i.e., the ability of sustaining toroidal moments, can also be defined by analogy with electric polarisability and magnetisability. The present study shows that such a property is general and characterizes atoms and molecules, and that the optical electric field of a light beam induces an oscillating anapole moment, i.e., the superposition of toroidal moment with an electric dipole moment. However, values of anapole polarisabilities induced by monochromatic light, estimated by time-dependent perturbation theory for rare gas atoms and a few molecules, are quite small and possibly hard to detect experimentally.

2011

The Hamiltonian for a particle constrained to motion near a toroidal helix with loops of arbitrary eccentricity is developed. The resulting three dimensional Schr\"odinger equation is reduced to a one dimensional effective equation inclusive of curvature effects. A basis set is employed to find low-lying eigenfunctions of the helix. Toroidal moments corresponding to the individual eigenfunctions are calculated. The dependence

Advanced Optical Materials, 2018

The study of toroidal dipole modes has attracted a growing attention due to the specific properties of the toroidal electromagnetic response which differs from more familiar electric and magnetic dipole modes. Herein, toroidal dipole modes generated by metasurfaces composed of trimer clusters of high‐index dielectric particles are observed. Both far‐field transmission measurements and direct near‐field mapping of the electromagnetic fields are performed in microwave experiments, and two distinct types of the toroidal dipole modes are observed in a single geometry of the metasurface design, where the toroidal modes are generated either inside of the three‐particle clusters (the so‐called intra‐cluster toroidal modes) or between the neighboring particles in the clusters (inter‐cluster toroidal modes). A transient response of the toroidal dipole modes excited by a pulse is studied in detail. Since the metasurface is composed of simple dielectric disks without the use of any metallic co...

arXiv (Cornell University), 2016

The excitation of toroidal multipoles in metamaterials was investigated for high-Q response at a subwavelength scale. In this study, we explored the optimization of toroidal excitations in a planar metamaterial comprised of asymmetric split ring resonators (ASRRs). It was found that the scattering power of toroidal dipole can be remarkably strengthened by adjusting the characteristic parameter of ASRRs: asymmetric factor. Interestingly, the improvement in toroidal excitation accompanies increment on the Q-factor of the toroidal metamaterial; it is shown that both the scattering power of toroidal dipole and the Q-factor were increased more than one order by changing the asymmetric factor of ASRRs. The optimization in excitation of toroidal multipole provide opportunity to further increase the Qfactor of metamaterial and boost light-matter interactions at the subwavelength scale for potential applications in low-power nonlinear processing, and sensitive photonic applications.

A semiclassical model of the electron is proposed. This model is based on the Ring Electron Model of Parson and the Zitterbewegung Electron Model of Hestenes. This " Solenoid Helical Electron Model " is described in [1] and [2]. This model necessarily implies a toroidal moment for the electron. This toroidal moment is a direct consequence of this model and it is not predicted by Quantum Mechanics. This prediction can serve as experimental evidence to validate or discard the proposed model.

Journal of Physics D: Applied Physics, 2021

Herein, we discuss the conditions for excitation of symmetry-protected toroidal dipole modes in all-dielectric metasurfaces composed of trimer or twin-trimer clusters of dielectric disks. Such metasurfaces permit enhanced light–matter interaction due to spatially confined light in resonant systems with a high-quality factor. To describe characteristics of toroidal modes existing in the clusters, we use the magnetic dipole moments approximation, group-theoretical methods, group representation theory, symmetry-adapted linear combination method, and circuit theory. To validate the obtained theoretical results, we fulfill both full-wave numerical simulations and microwave experiments. In particular, we have shown that the toroidal dipole mode appears as a quasi-dark state of the trimer. It can be excited in the metasurface by the field of a linearly polarized wave, providing the symmetry of the trimer is properly reduced. In the metasurface, the properties of the toroidal dipole mode ar...

Physical Review B, 2016

We show how the elusive toroidal dipole moment appears as a radiative excitation eigenmode in a metamolecule resonator that is formed by pairs of plasmonic nanorods. We analyze one such nanorod configuration-a toroidal metamolecule. We find that the radiative interactions in the toroidal metamolecule can be qualitatively represented by a theoretical model based on an electric point dipole arrangement. Both a finite-size rod model and the point dipole approximation demonstrate how the toroidal dipole moment is subradiant and difficult to excite by incident light. By means of breaking the geometric symmetry of the metamolecule, the toroidal mode can be excited by linearly polarized light and appears as a Fano resonance dip in the forward scattered light. We provide simple optimization protocols for maximizing the toroidal dipole mode excitation. This opens up possibilities for simplified control and driving of metamaterial arrays consisting of toroidal dipole unit-cell resonators.

Scientific reports, 2015

The requirements of quantum computations impose high demands on the level of qubit protection from perturbations; in particular, from those produced by the environment. Here we propose a superconducting flux qubit design that is naturally protected from ambient noise. This decoupling is due to the qubit interacting with the electromagnetic field only through its toroidal moment, which provides an unusual qubit-field interaction, which is suppressed at low frequencies.