Topological modes in radiofrequency resonator arrays

Abstract

Topological properties of solid states have sparked considerable recent interest due to their importance in the physics of lattices with a non-trivial basis and their potential in the design of novel materials. Here we describe an experimental and an accompanying numerical toolbox to create and analyze topological states in coupled radiofrequency resonator arrays. The arrays are coupled harmonic oscillator systems that are very easily constructed, offer a variety of geometric configurations, and whose eigenfunctions and eigenvalues are amenable to detailed analysis. These systems offer well defined analogs to coupled oscillator systems in general in that they are characterized by resonances whose frequency spectra depend on the individual resonators, the interactions between them, and the geometric and topological symmetries and boundary conditions. In particular, we describe an experimental analog of a small one-dimensional system, with excellent agreement with theory. The numerical part of the toolbox allows for simulations of larger mesoscopic systems with a semi-continuous band structure, in which all resonators still exhibit individual signatures. Systematic parameter variation yields an astonishing richness of band structures in this simple linear system, allowing for further explorations into novel phenomena of topological modes.

Sources of quantized excitations via dichotomic topological cycles

“…This robust phenomenon was confirmed experimentally not long ago with ultra-cold fermionic and bosonic atoms [4,5], even though the prediction was made forty years ago. Since then, the phenomenon has received renewed attention from the meta-material community because of the possibility of resonant spectrum engineering [6][7][8][9][10][11], of topological wave-steering [12][13][14][15][16][17][18][19][20], as well as of dynamic edge-to-edge pumping [21][22][23][24]. The last batch of experimental achievements are highly relevant for the present work, because they demonstrated that fast dynamic re-configurations of the meta-materials are now possible, to a level where the pumped signals do not succumb to dissipation and the effect can be revealed.…”

Recovery of eigenvectors from eigenvalues in systems of coupled harmonic oscillators

“…(2) (Non-Hermitian case) A dimeric array with alternating capacitances, C 1 = C 3 = C 5 , corresponding to a frequency of f = 200 MHz, and C 2 = C 4 corresponding to f = 220 MHz. Such a dimeric array exhibits a band gap that is evident with five elements but evolves more fully in the asymptotic limit of infinite N [18].…”

“…In the experiment, the resonator arrays were composed of N = 5 rectangular LC loops that were aligned coaxially and have been described in detail previously [18]. Individual resonators were tuned by a trim capacitor to their base frequencies, which are defined as the resonance frequencies of the isolated uncoupled elements.…”

“…From the experimental perspective, it becomes increasingly difficult to resolve all the spectral peaks as N increases. Typical linewidths of the spectrum of our experimental rf array can be assessed from our previous publication of a similar experiment [18]. From the computational perspective, larger N might cause numerical problems.…”

“…The system matrix of the monomeric system considered here is Hermitian, and its eigenmodes could be straightforwardly estimated from measurements. The matrix of the dimeric system is non-Hermitian, but the dimeric system is, in fact, more interesting as it can give rise to localized edge modes [18]. This matrix has the same eigenvalue spectrum as a similar Hermitian matrix [16], and a naive application of the eigenvector-eigenvalue identity to its spectrum naturally provides the eigenvectors of the similar Hermitian matrix.…”