Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential

Non-Hermitian Delocalization in a Two-Dimensional Photonic Quasicrystal

Theoretical and experimental studies suggest that both Hermitian and non-Hermitian quasicrystals show localization due to the fractal spectrum and to the transition to diffusive bands via exceptional points, respectively. Here, we present an experimental study of a dodecagonal photonic quasicrystal based on electromagnetically induced transparency in a Rb vapor cell. First, we observe the suppression of the wave packet expansion in the Hermitian case. We then discover a new regime, where increasing the non-Hermiticity leads to delocalization, demonstrating that the behavior in non-Hermitian quasicrystals is richer than previously thought.

Quantum criticality of generalized Aubry-André models with exact mobility edges using fidelity susceptibility

In this study, we explore the quantum critical phenomena in generalized Aubry-André models, with a particular focus on the scaling behavior at various filling states. Our approach involves using quantum fidelity susceptibility to precisely identify the mobility edges in these systems. Through a finite-size scaling analysis of the fidelity susceptibility, we are able to determine both the correlation-length critical exponent and the dynamical critical exponent at the critical point of the generalized Aubry-André model. Based on the Diophantine equation conjecture, we can determines the number of subsequences of the Fibonacci sequence and the corresponding scaling functions for a specific filling fraction, as well as the universality class. Our findings demonstrate the effectiveness of employing the generalized fidelity susceptibility for the analysis of unconventional quantum criticality and the associated universal information of quasiperiodic systems in cutting-edge quantum simulation experiments.

Physical Properties of an Aperiodic Monotile with Graphene-like Features, Chirality, and Zero Modes

The discovery of the Hat, an aperiodic monotile, has revealed novel mathematical aspects of aperiodic tilings. However, the physics of particles propagating in such a setting remains unexplored. In this work we study spectral and transport properties of a tight-binding model defined on the Hat. We find that (i) the spectral function displays striking similarities to that of graphene, including sixfold symmetry and Dirac-like features; (ii) unlike graphene, the monotile spectral function is chiral, differing for its two enantiomers; (iii) the spectrum has a macroscopic number of degenerate states at zero energy; (iv) when the magnetic flux per plaquette (ϕ) is half of the flux quantum, zero modes are found localized around the reflected "anti-hats"; and (v) its Hofstadter spectrum is periodic in ϕ, unlike for other quasicrystals. Our work serves as a basis to study wave and electron propagation in possible experimental realizations of the Hat, which we suggest.

Exact New Mobility Edges between Critical and Localized States

The disorder systems host three types of fundamental quantum states, known as the extended, localized, and critical states, of which the critical states remain being much less explored. Here we propose a class of exactly solvable models which host a novel type of exact mobility edges (MEs) separating localized states from robust critical states, and propose experimental realization. Here the robustness refers to the stability against both single-particle perturbation and interactions in the few-body regime. The exactly solvable one-dimensional models are featured by a quasiperiodic mosaic type of both hopping terms and on-site potentials. The analytic results enable us to unambiguously obtain the critical states which otherwise require arduous numerical verification including the careful finite size scalings. The critical states and new MEs are shown to be robust, illustrating a generic mechanism unveiled here that the critical states are protected by zeros of quasiperiodic hopping terms in the thermodynamic limit. Further, we propose a novel experimental scheme to realize the exactly solvable model and the new MEs in an incommensurate Rydberg Raman superarray. This Letter may pave a way to precisely explore the critical states and new ME physics with experimental feasibility.

Thermodynamic Phase Diagram of Two-Dimensional Bosons in a Quasicrystal Potential

Quantum simulation of quasicrystals in synthetic bosonic matter now paves the way for the exploration of these intriguing systems in wide parameter ranges. Yet thermal fluctuations in such systems compete with quantum coherence and significantly affect the zero-temperature quantum phases. Here we determine the thermodynamic phase diagram of interacting bosons in a two-dimensional, homogeneous quasicrystal potential. We find our results using quantum Monte Carlo simulations. Finite-size effects are carefully taken into account and the quantum phases are systematically distinguished from thermal phases. In particular, we demonstrate stabilization of a genuine Bose glass phase against the normal fluid in sizable parameter ranges. We interpret our results for strong interactions using a fermionization picture and discuss experimental relevance.

Electronic Quantum Materials Simulated with Artificial Model Lattices

The band structure and electronic properties of a material are defined by the sort of elements, the atomic registry in the crystal, the dimensions, the presence of spin–orbit coupling, and the electronic interactions. In natural crystals, the interplay of these factors is difficult to unravel, since it is usually not possible to vary one of these factors in an independent way, keeping the others constant. In other words, a complete understanding of complex electronic materials remains challenging to date. The geometry of two- and one-dimensional crystals can be mimicked in artificial lattices. Moreover, geometries that do not exist in nature can be created for the sake of further insight. Such engineered artificial lattices can be better controlled and fine-tuned than natural crystals. This makes it easier to vary the lattice geometry, dimensions, spin–orbit coupling, and interactions independently from each other. Thus, engineering and characterization of artificial lattices can provide unique insights. In this Review, we focus on artificial lattices that are built atom-by-atom on atomically flat metals, using atomic manipulation in a scanning tunneling microscope. Cryogenic scanning tunneling microscopy allows for consecutive creation, microscopic characterization, and band-structure analysis by tunneling spectroscopy, amounting in the analogue quantum simulation of a given lattice type. We first review the physical elements of this method. We then discuss the creation and characterization of artificial atoms and molecules. For the lattices, we review works on honeycomb and Lieb lattices and lattices that result in crystalline topological insulators, such as the Kekulé and “breathing” kagome lattice. Geometric but nonperiodic structures such as electronic quasi-crystals and fractals are discussed as well. Finally, we consider the option to transfer the knowledge gained back to real materials, engineered by geometric patterning of semiconductor quantum wells.

Strongly Interacting Bosons in a Two-Dimensional Quasicrystal Lattice

Quasicrystals exhibit exotic properties inherited from the self-similarity of their long-range ordered, yet aperiodic, structure. The recent realization of optical quasicrystal lattices paves the way to the study of correlated Bose fluids in such structures, but the regime of strong interactions remains largely unexplored, both theoretically and experimentally. Here, we determine the quantum phase diagram of two-dimensional correlated bosons in an eightfold quasicrystal potential. Using large-scale quantum Monte Carlo calculations, we demonstrate a superfluid-to-Bose glass transition and determine the critical line. Moreover, we show that strong interactions stabilize Mott insulator phases, some of which have spontaneously broken eightfold symmetry. Our results are directly relevant to current generation experiments and, in particular, drive prospects to the observation of the still elusive Bose glass phase in two dimensions and exotic Mott phases.

Electrical circuits involving fractal time

In this paper, we develop fractal calculus by defining improper fractal integrals and their convergence and divergence conditions with related tests and by providing examples. Using fractal calculus that provides a new mathematical model, we investigate the effect of fractal time on the evolution of the physical system, for example, electrical circuits. In these physical models, we change the dimension of the fractal time; as a result, the order of the fractal derivative changes; therefore, the corresponding solutions also change. We obtain several analytical solutions that are non-differentiable in the sense of ordinary calculus by means of the local fractal Laplace transformation. In addition, we perform a comparative analysis by solving the governing fractal equations in the electrical circuits and using their smooth solutions, and we also show that when α=1, we get the same results as in the standard version.

Quantum Criticality in the 2D Quasiperiodic Potts Model

Quantum critical points in quasiperiodic magnets can realize new universality classes, with critical properties distinct from those of clean or disordered systems. Here, we study quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q-state Potts model in 2+1D. Using a controlled real-space renormalization group approach, we find that the critical behavior is largely independent of q, and is controlled by an infinite-quasiperiodicity fixed point. The correlation length exponent is found to be ν=1, saturating a modified version of the Harris-Luck criterion.

Lieb-Liniger Bosons in a Shallow Quasiperiodic Potential: Bose Glass Phase and Fractal Mott Lobes

The emergence of a compressible insulator phase, known as the Bose glass, is characteristic of the interplay of interactions and disorder in correlated Bose fluids. While widely studied in tight-binding models, its observation remains elusive owing to stringent temperature effects. Here we show that this issue may be overcome by using Lieb-Liniger bosons in shallow quasiperiodic potentials. A Bose glass, surrounded by superfluid and Mott phases, is found above a critical potential and for finite interactions. At finite temperature, we show that the melting of the Mott lobes is characteristic of a fractal structure and find that the Bose glass is robust against thermal fluctuations up to temperatures accessible in quantum gases. Our results raise questions about the universality of the Bose glass transition in such shallow quasiperiodic potentials.

Multi-orbital tight binding model for cavity-polariton lattices

In this work we present a tight-binding model that allows to describe with a minimal amount of parameters the band structure of exciton-polariton lattices. This model based on s and p non-orthogonal photonic orbitals faithfully reproduces experimental results reported for polariton graphene ribbons. We analyze in particular the influence of the non-orthogonality, the inter-orbitals interaction and the photonic spin-orbit coupling on the polarization and dispersion of bulk bands and edge states.

Phononic topological states in 1D quasicrystals

In this work, we address the study of phonons propagating on a one-dimensional quasiperiodic lattice, where the atoms are considered bounded by springs whose strength are modulated by equivalent Aubry-André hoppings. As an example, from the equations of motion, we obtained the equivalent phonon spectrum of the well known Hofstadter butterfly. We have also obtained extended, critical, and localized regimes in this spectrum. By introducing the equivalent Aubry-André model through the variation of the initial phase [Formula: see text], we have shown that border states for phonons are allowed to exist. These states can be classified as topologically protected states (topological states). By calculating the inverse participation rate, we describe the localization of phonons and verify a phase transition, characterized by the critical value of [Formula: see text], where the states of the system change from extended to localized, precisely like in a metal-insulator phase transition.

Critical Behavior and Fractality in Shallow One-Dimensional Quasiperiodic Potentials

Quasiperiodic systems offer an appealing intermediate between long-range ordered and genuine disordered systems, with unusual critical properties. One-dimensional models that break the so-called self-dual symmetry usually display a mobility edge, similarly as truly disordered systems in a dimension strictly higher than two. Here, we determine the critical localization properties of single particles in shallow, one-dimensional, quasiperiodic models and relate them to the fractal character of the energy spectrum. On the one hand, we determine the mobility edge and show that it separates the localized and extended phases, with no intermediate phase. On the other hand, we determine the critical potential amplitude and find the universal critical exponent ν≃1/3. We also study the spectral Hausdorff dimension and show that it is nonuniversal but always smaller than unity, hence showing that the spectrum is nowhere dense. Finally, applications to ongoing studies of Anderson localization, Bose-glass physics, and many-body localization in ultracold atoms are discussed.

Evolution of orbital angular momentum in a soft quasi-periodic structure with topological defects

A continuously deformative space possesses trivial or nontrivial topological characteristics depending on the associated homotopy groups associated with spaces describing the physical processes. Moreover, the interaction of spatial warping and structural symmetry always presents fantastic phenomena, especially in the systems with unique symmetrical properties such as quasicrystals. Here, we propose a quasi-periodic structure (QPS) with topological defects. The analytical expression of the corresponding Fourier spectrum is derived, which reflects the combined effects of topological structure and quasi-translational symmetry. Light-matter interaction therein brings unusual diffraction characteristics with exotic evolution of orbital angular momentum (OAM). Long-range correlation of QPS resulted in multi-fractal and pairwise distribution of optical singularities. A general conservation law of OAM is revealed. A liquid crystal photopatterned QPS is fabricated to demonstrate the above characteristics. Dynamic reconfigurable manipulation of optical singularities is achieved. Our approach offers the opportunity to manipulate OAM with multiple degrees of freedom, which has promising applications in multi-channel quantum information processing and high-dimensional quantum state generation.

Statistical Mechanics Involving Fractal Temperature

In this paper, the Schrödinger equation involving a fractal time derivative is solved and corresponding eigenvalues and eigenfunctions are given. A partition function for fractal eigenvalues is defined. For generalizing thermodynamics, fractal temperature is considered, and adapted equations are defined. As an application, we present fractal Dulong-Petit, Debye, and Einstein solid models and corresponding fractal heat capacity. Furthermore, the density of states for fractal spaces with fractional dimension is obtained. Graphs and examples are given to show details.

Topological bands for ultracold atoms

There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. This review summarizes these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. Descriptions are given of the different methods that have been used to generate these novel band structures for cold atoms and of the physical observables that have allowed their characterization. The focus is on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. Also described is how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, descriptions are given of the forms of interparticle interactions that emerge when atoms are subjected to these energy bands and of some of the many-body phases that may be sought in future experiments.

Revealing the Topology of Quasicrystals with a Diffraction Experiment

Topological properties of crystals and quasicrystals is a subject of recent and growing interest. This Letter reports an experiment where, for certain quasicrystals, these properties can be directly retrieved from diffraction. We directly observe, using an interferometric approach, all of the topological invariants of finite-length Fibonacci chains in their diffraction pattern. We also quantitatively demonstrate the stability of these topological invariants with respect to structural disorder.

Regularized Laplacian determinants of self-similar fractals

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.

Probing a Dissipative Phase Transition via Dynamical Optical Hysteresis

We experimentally explore the dynamical optical hysteresis of a semiconductor microcavity as a function of the sweep time. The hysteresis area exhibits a double power law decay due to the influence of fluctuations, which trigger switching between metastable states. Upon increasing the average photon number and approaching the thermodynamic limit, the double power law evolves into a single power law. This algebraic behavior characterizes a dissipative phase transition. Our findings are in good agreement with theoretical predictions for a single mode resonator influenced by quantum fluctuations, and the present experimental approach is promising for exploring critical phenomena in photonic lattices.

Fractional Transport of Photons in Deterministic Aperiodic Structures

The propagation of optical pulses through primary types of deterministic aperiodic structures is numerically studied in time domain using the rigorous transfer matrix method in combination with analytical fractional transport models. We demonstrate tunable anomalous photon transport, including the elusive logarithmic Sinai sub-diffusion in photonic systems for the first time. Our results are in excellent agreement with the scaling theory of transport in aperiodic media with fractal spectra, and additionally demonstrate logarithmic sub-diffusion in the presence of multifractality. Moreover, we establish a fruitful connection between tunable photon diffusion and fractional dynamics, which provides analytical insights into the asymptotic transport regime of optical media with deterministic aperiodic order. The demonstration of tunable sub-diffusion and logarithmic photon transport in deterministic aperiodic structures can open novel and fascinating scenarios for the engineering of wave propagation and light-matter interaction phenomena beyond the conventional diffusive regime.

Exciton-polariton trapping and potential landscape engineering

Exciton-polaritons in semiconductor microcavities have become a model system for the studies of dynamical Bose-Einstein condensation, macroscopic coherence, many-body effects, nonclassical states of light and matter, and possibly quantum phase transitions in a solid state. These low-mass bosonic quasiparticles can condense at comparatively high temperatures up to 300 K, and preserve the fundamental properties of the condensate, such as coherence in space and time domain, even when they are out of equilibrium with the environment. Although the presence of a confining potential is not strictly necessary in order to observe Bose-Einstein condensation, engineering of the polariton confinement is a key to controlling, shaping, and directing the flow of polaritons. Prototype polariton-based optoelectronic devices rely on ultrafast photon-like velocities and strong nonlinearities exhibited by polaritons, as well as on their tailored confinement. Nanotechnology provides several pathways to achieving polariton confinement, and the specific features and advantages of different methods are discussed in this review. Being hybrid exciton-photon quasiparticles, polaritons can be trapped via their excitonic as well as photonic component, which leads to a wide choice of highly complementary trapping techniques. Here, we highlight the almost free choice of the confinement strengths and trapping geometries that provide powerful means for control and manipulation of the polariton systems both in the semi-classical and quantum regimes. Furthermore, the possibilities to observe effects of the polariton blockade, Mott insulator physics, and population of higher-order energy bands in sophisticated lattice potentials are discussed. Observation of such effects could lead to realization of novel polaritonic non-classical light sources and quantum simulators.

Multimode photon-exciton coupling in an organic-dye-attached photonic quasicrystal

In this Letter, we present hybrid strong coupling between multiple photonic modes and excitons in an organic-dye-attached photonic quasicrystal. The excitons effectively interact with the photonic modes offered by the photonic quasicrystal, and multiple hybrid polariton bands are demonstrated in both experiments and calculations. Furthermore, by retrieving the measured dispersion map, we get the mixing fractions of photonic modes and excitons, and show that the polariton bands inherit not only the energy dispersion features, but also the damping behaviors from both the photonic modes and the excitons. Our investigation may inspire related studies on multimode light-matter interactions and achieve some potential applications for multimode sensors.

The road towards polaritonic devices

Polaritons are quasiparticles that form in semiconductors when an elementary excitation such as an exciton or a phonon interacts sufficiently strongly with light. In particular, exciton-polaritons have attracted tremendous attention for their unique properties, spanning from an ability to undergo ultra-efficient four-wave mixing to superfluidity in the condensed state. These quasiparticles possess strong intrinsic nonlinearities, while keeping most characteristics of the underlying photons. Here we review the most important features of exciton-polaritons in microcavities, with a particular emphasis on the emerging technological applications, the use of new materials for room-temperature operation, and the possibility of exploiting polaritons for quantum computation and simulation.

Anderson attractors in active arrays

In dissipationless linear media, spatial disorder induces Anderson localization of matter, light, and sound waves. The addition of nonlinearity causes interaction between the eigenmodes, which results in a slow wave diffusion. We go beyond the dissipationless limit of Anderson arrays and consider nonlinear disordered systems that are subjected to the dissipative losses and energy pumping. We show that the Anderson modes of the disordered Ginsburg-Landau lattice possess specific excitation thresholds with respect to the pumping strength. When pumping is increased above the threshold for the band-edge modes, the lattice dynamics yields an attractor in the form of a stable multi-peak pattern. The Anderson attractor is the result of a joint action by the pumping-induced mode excitation, nonlinearity-induced mode interactions, and dissipative stabilization. The regimes of Anderson attractors can be potentially realized with polariton condensates lattices, active waveguide or cavity-QED arrays.

Stochastic resonance in collective exciton-polariton excitations inside a GaAs microcavity

We report the first observation of stochastic resonance in confined exciton polaritons. We evidence this phenomena by tracking the polaritons behavior through two stochastic resonance quantifiers namely the spectral magnification factor and the signal-to-noise ratio. The evolution of the stochastic resonance in the function of the modulation amplitude of the periodic excitation signal is studied. Our experimental observations are well reproduced by numerical simulations performed in the framework of the Gross-Pitaevskii equation under stochastic perturbation.