Electric-circuit realization of a hyperbolic drum

Abstract

The Laplace operator encodes the behaviour of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space. Here we demonstrate that the spectral ordering of Laplacian eigenstates for hyperbolic (negative curvature) and flat (zero curvature) two-dimensional spaces has a universally different structure. We use a lattice representation of hyperbolic space in an electric-circuit network to measure the eigenstates of a 'hyperbolic drum', and to analyze signal propagation along the curved geodesics. Our experiments showcase a versatile platform to emulate hyperbolic lattices in tabletop experiments, which can be utilized to explore propagation dynamics as well as to realize topological hyperbolic matter.

Designing non-Hermitian real spectra through electrostatics

“…Our work opens up a new paradigm for engineering non-Hermitian spectra, particularly real spectra, in various settings, such as cold atoms [44,45], photonics [19,22], metamaterials [50,51], mechanical and acoustic systems [47]. While real spectra are important for state stability in the majority of experiments, we point out that non-real spectra present further possibilities in terms of topological sophistication [34], and are just as physically relevant in the form of the Laplacian spectra of steady-state networks such as electrical circuits [28,48,52,53].…”

“…whose real spectrum possess eigenstate localization profiles closely obeying V (x), despite hoppings with no apparent symmetry whatsoever that even suggests of the possibility of a real spectrum. Containing only up to next-nearest neighbor hoppings, it is simple enough to feasibly realize in photonic, mechanical, electrical or ultracold atomic systems [28,[44][45][46][47][48].…”

Bose-Einstein condensation on hyperbolic spaces

“…More recently, experimentalists have been able to physically construct hyperbolic structures in the laboratory by confining particles to discrete hyperbolic lattices in circuit QED [Hu+19; KFH19; Kol+20; Alt+21; SMR21] and by using topoelectric circuits [Len+21]. These setups can serve, among other things, as tabletop simulators of quantum gravity.…”

Hyperbolic band theory through Higgs bundles

“…To date, such circuits, albeit photonic rather than electronic in nature, have been artificially engineered [43]. A subsequent experiment realizes a hyperbolic drum as an electric circuit [44], culminating in the experimental detection of negative curvature via the detection of the eigenmodes of the Laplace-Beltrami operator and of signal propagation along hyperbolic geodesics. Further theoretical works emerging recently in hyperbolic matter and band theory include [37,1,8,64,4,6].…”