Energy landscape in a Penrose tiling

Abstract

Since their spectacular experimental realisation in the early 80's [1], quasicrystals [2] have been the subject of very active research, whose domains extend far beyond the scope of solid state physics. In optics, for instance, photonic quasicrystals have attracted strong interest [3] for their specific behaviour, induced by the particular spectral properties, in light transport [4][5][6], plasmonic [7] and laser action [8]. Very recently, one of the most salient spectral feature of quasicrystals, namely the gap labelling [9], has been observed for a polariton gas confined in a one dimensional quasi-periodic cavity [10]. This experimental result confirms a theory which is now very complete in dimension one [11,12]. In dimension greater than one, the theory is very far from being complete. Furthermore, some intriguing phenomena, like the existence of self-similar eigenmodes can occur [13]. All this makes two dimensional experimental realisations and numerical simulations pertinent and attractive. Here, we report on measurements and energy-scaling analysis of the gap labelling and the spatial intensity distribution of the eigenstates for a microwave Penrose-tiled quasicrystal. Far from being restricted to the microwave system under consideration, our results apply to a more general class of systems.Quasicrystals are alloys that are ordered but lack translational symmetry. In dimension two, they can be modelled with a collection of polygons (tiles) that cover the whole plane, so that each pattern (a sub-collection of tiles) appears, up to translation, with a given density but the tiling is not periodic. A typical example is given by the Penrose tiling [14]. Here, we implement a microwave realisation of a Penrose-tiled lattice using a set of coupled dielectric resonators [see Fig. 1 (a)]. The microwave setup used has shown its versatility by successfully addressing various physical situations ranging from Anderson localisation [15] to topological phase transition in graphene [16], and provided the first experimental realisation of the Dirac oscillator [17].We establish a two-dimensional tight-binding regime [18], where the electromagnetic field is mostly confined within the resonators. For an isolated resonator, only a single mode is important in a broad spectral range around the bare frequency E b 6.65 GHz. This mode spreads out evanescently, so that the coupling FIG. 1:Microwave Penrose-tiled quasicrystal. (a) Diamond-vertex Penrose-tiled quasicrystal, where the tiling is superposed to guide the eye (thin diamonds in green). The sites of the lattice are occupied by dielectric resonators (ceramic cylinders of 5 mm height and 8 mm diameter) with a high index of refraction (n = 6). The lattice is sandwiched between two aluminium plates (the upper one is not shown). The microwaves are excited by a movable loop antenna. (b) Experimentally obtained DOS as a function of frequency, the white and gray zones indicate the main frequency bands, Ei, and the gaps, ∆Ei, respectively. The bare frequency E b is indicated by the wh...

Many-body localization in a quasiperiodic Fibonacci chain

“…Many experiments conducted with phonons [90], photons [91], microwaves [92], polaritons [93] and cold atoms [94] explore the properties of non-interacting quantum particles in a quasiperiodic landscape. A cold-atomic experimental setup has been proposed [95] to realize the Fibonacci sequence considered here, which could be used to probe the many-body physics discussed in our work.…”

Measuring topological invariants from generalized edge states in polaritonic quasicrystals

“…These integers can take N distinct values, N being the number of letters in the chain [34]. Despite important advances on the topological properties of quasicrystals [19][20][21][22][23][24][28][29][30]] the topological invariants have not yet been directly measured as winding numbers.The physical origin of topological numbers in a Fibonacci quasicrystal can be related to its structural properties [35]. To understand this, let us introduce a general method to generate a Fibonacci sequence: it is based on the characteristic function…”

“…From a general viewpoint, wave or quantum systems possessing a gapped energy spectrum, such as band insulators, superconductors, or 2D conductors in a magnetic field, can be assigned topological invariants generally called Chern numbers [3]. These numbers control a variety of physical phenomena: for instance in the integer quantum Hall effect, they determine the value of the Hall conductance as a function of magnetic field [4,5] [22,[28][29][30] and exploited to implement topological pumping, a key concept of topology [22]. A paradigmatic example of quasicrystal is given by the 1D Fibonacci chain.…”

“…In particular the topological edge states [25][26][27] of quasicrystals have been recently investigated in photonic systems [22,[28][29][30] and exploited to implement topological pumping, a key concept of topology [22]. A paradigmatic example of quasicrystal is given by the 1D Fibonacci chain.…”

Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains

“…Those are known as "critical states", with a spatial profile lying between the exponential decay of modes in a randomly disordered chain and uniformly extending Bloch eigenmodes in periodic chains [13][14][15] . Such modes have recently been shown to consist of locally resonating patterns (i.e., characteristic sequences of amplitudes) which occur on repeating segments of a quasiperiodic structures and are characteristic for a given quasiband 17,19,64 . This is particularly visible for the bulk modes of the uppermost quasiband in Fig.…”