Quasiperiodic Lattices Research Articles

Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry

We investigate localization-delocalization transition in one-dimensional non-Hermitian quasiperiodic lattices with exponential short-range hopping, which possess parity-time ($\mathcal{PT}$) symmetry. The localization transition induced by the non-Hermitian quasiperiodic potential is found to occur at the $\mathcal{PT}$-symmetry-breaking point. Our results also demonstrate the existence of energy dependent mobility edges, which separate the extended states from localized states and are only associated with the real part of eigen-energies. The level statistics and Loschmidt echo dynamics are also studied.

Quantum Walks in Periodic and Quasiperiodic Fibonacci Fibers

Quantum walk is a key operation in quantum computing, simulation, communication and information. Here, we report for the first time the demonstration of quantum walks and localized quantum walks in a new type of optical fibers having a ring of cores constructed with both periodic and quasiperiodic Fibonacci sequences, respectively. Good agreement between theoretical and experimental results has been achieved. The new multicore ring fibers provide a new platform for experiments of quantum effects in low-loss optical fibers which is critical for scalability of real applications with large-size problems. Furthermore, our new quasiperiodic Fibonacci multicore ring fibers provide a new class of quasiperiodic photonics lattices possessing both on- and off-diagonal deterministic disorders for realizing localized quantum walks deterministically. The proposed Fibonacci fibers are simple and straightforward to fabricate and have a rich set of properties that are of potential use for quantum applications. Our simulation and experimental results show that, in contrast with randomly disordered structures, localized quantum walks in new proposed quasiperiodic photonics lattices are highly controllable due to the deterministic disordered nature of quasiperiodic systems.

Bose–Einstein condensates in an eightfold symmetric optical lattice**Project supported by the National Natural Science Foundation of China (Grant Nos. 11434011, 11522436, and 11774425), the National Key R&D Program of China (Grants No. 2018YFA0306501), the Beijing Natural Science Foundation, China (Grant No. Z180013), and the Research Funds of Renmin University of China (Grants Nos. 16XNLQ03 and 18XNLQ15).

We investigate the properties of Bose–Einstein condensates (BECs) in a two-dimensional quasi-periodic optical lattice (OL) with eightfold rotational symmetry by numerically solving the Gross–Pitaevskii equation. In a stationary external harmonic trapping potential, we first analyze the evolution of matter-wave interference pattern from periodic to quasi-periodic as the OL is changed continuously from four-fold periodic to eight-fold quasi-periodic. We also investigate the transport properties during this evolution for different interatomic interaction and lattice depth, and find that the BEC crosses over from ballistic diffusion to localization. Finally, we focus on the case of eightfold symmetric lattice and consider a global rotation imposed by the external trapping potential. The BEC shows vortex pattern with eightfold symmetry for slow rotation, becomes unstable for intermediate rotation, and exhibits annular solitons with approximate axial symmetry for fast rotation. These results can be readily demonstrated in experiments using the same configuration as in Phys. Rev. Lett. 122 110404 (2019).

Localization Driven Superradiant Instability.

The prominent Dicke superradiant phase arises from coupling an ensemble of atoms to a cavity optical field when an external optical pumping exceeds a threshold strength. Here we report a prediction of the superradiant instability driven by Anderson localization, realized with a hybrid system of the Dicke and Aubry-André (DAA) model for bosons trapped in a one-dimensional (1D) quasiperiodic optical lattice and coupled to a cavity. Our central finding is that for bosons condensed in a localized phase given by the DAA model, the resonant superradiant scattering is induced, for which the critical optical pumping of the superradiant phase transition approaches zero, giving an instability driven by the Anderson localization. The superradiant phase for the DAA model with or without a mobility edge is investigated, showing that the localization driven superradiant instability is in sharp contrast to the superradiance as widely observed for a Bose-Einstein condensate in extended states, and should be insensitive to the temperature of the system. This study unveils a novel effect of localization on the Dicke superradiance, and is well accessible based on the current experiments.

Mobility edge and intermediate phase in one-dimensional incommensurate lattice potentials

We study theoretically the localization properties of two distinct one-dimensional quasiperiodic lattice models with a single-particle mobility edge (SPME) separating extended and localized states in the energy spectrum. The first one is the familiar Soukoulis-Economou trichromatic potential model with two incommensurate potentials, and the second is a system consisting of two coupled 1D Aubry-Andre chains each containing one incommensurate potential. We show that as a function of the Hamiltonian model parameters, both models have a wide single-particle intermediate phase (SPIP), defined as the regime where localized and extended single-particle states coexist in the spectrum, leading to a behavior intermediate between purely extended or purely localized when the system is dynamically quenched from a generic initial state. Our results thus suggest that both systems could serve as interesting experimental platforms for studying the interplay between localized and extended states, and may provide insight into the role of the coupling of small baths to localized systems. We also calculate the Lyapunov (or localization) exponent for several incommensurate 1D models exhibiting SPME, finding that such localization critical exponents for quasiperiodic potential induced localization are nonuniversal and depend on the microscopic details of the Hamiltonian.

Transport through degenerate tori and quantum-to-classical crossover in a driven Aubry-Andre model

Dynamics of ultracold atoms in a one-dimensional incommensurate quasiperiodic optical lattice is considered. The lattice is created as superposition of a large-amplitude primary lattice and a small-amplitude secondary lattice. In the tight-binding approximation motion of atoms is described by the Aubry-Andre model. Attention is focused on atom delocalization caused by amplitude modulation of the secondary lattice. We consider the semiclassical regime that is realized if the lattice spacing of the primary lattice is small compared to that of the secondary one. It is shown that efficient delocalization occurs if frequency of the modulation corresponds to resonant influence on degenerate tori in classical phase space. Inclusion of the next-to-nearest tunneling into the Aubry-Andre model leads to shift of the corresponding resonance to the lower frequency range. As ratio of primary and secondary lattice periods increases, effect of the resonance ceases.

Emergent Periodic and Quasiperiodic Lattices on Surfaces of Synthetic Hall Tori and Synthetic Hall Cylinders.

Synthetic spaces allow physicists to bypass constraints imposed by certain physical laws in experiments. Here, we show that a synthetic torus, which consists of a ring trap in the real space and internal states of ultracold atoms cyclically coupled by Laguerre-Gaussian Raman beams, could be threaded by a net effective magnetic flux through its surface-an impossible mission in the real space. Such a synthetic Hall torus gives rise to a periodic lattice in real dimensions, in which the periodicity of the density modulation of atoms fractionalizes that of the Hamiltonian. Correspondingly, the energy spectrum is featured by multiple bands grouping into clusters with nonsymmorphic-symmetry-protected band crossings in each cluster, leading to swaps of wave packets in Bloch oscillations. Our scheme allows physicists to glue two synthetic Hall tori such that localization may emerge in a quasicrystalline lattice. If the Laguerre-Gaussian Raman beams and ring traps were replaced by linear Raman beams and ordinary traps, a synthetic Hall cylinder could be realized and deliver many of the aforementioned phenomena.

Numerical investigations and experimental measurements on the structural dynamic behaviour of quasi-periodic meta-materials

Abstract The periodic structures have various applications in vibroacoustic engineering fields since they introduce frequency band effects due to the periodic discontinuities in the geometrical or material configurations: this can lead to increased performances. This paper is focused on the analysis of quasi-periodic structures: instead of using strictly repeated patterns, a certain degree of irregularity is introduced. Quasi-periodic lattices are defined as assemblies of two different elements in two directions. The assembly follows a Thue-Morse Morphism sequence which results in asymmetry in both directions. Numerical studies and experimental measurements on two-dimensional periodic and quasi-periodic lattices are thus performed. First validations are carried out by comparing the quasi-periodic lattice modelled by using finite element model with a prototype manufactured by laser machine. The wave characteristics in quasi-periodic lattice introduce elements of novelty for designing wider frequency stop bands in low frequency ranges.

Non-Abelian gauge potentials driven localization transition in quasiperiodic optical lattices

Derived from quantum waves immersed in an Abelian gauge potential, the quasiperiodic Aubry-André-Harper (AAH) model is a simple yet powerful Hamiltonian to study the Anderson localization of ultracold atoms. Here, we investigate the localization properties of ultracold atoms in quasiperiodic optical lattices subject to a non-Abelian gauge potential, which are depicted by non-Abelian AAH models. We identify that the non-Abelian AAH models can bear the self-duality. We analyze the localization of such non-Abelian self-dual optical lattices, revealing a rich phase diagram driven by the non-Abelian gauge potential involved: a transition from a pure delocalization phase, then to coexistence phases, and finally to a pure localization phase. This is in stark contrast to the Abelian counterpart that does not support the coexistence phases. Our results establish the connection between localization and gauge symmetry, and thus comprise a new insight on the fundamental aspects of localization in quasiperiodic systems, from the perspective of non-Abelian gauge potential.

Consensus formation in continuous opinion dynamics on quasiperiodic lattices

We studied the Biswas–Chatterjee–Sen (BCS) consensus formation model, also known as the Kinetic Continuous Opinion Dynamics (KCOD) model on quasiperiodic lattices by using Kinetic Monte Carlo simulations and Finite Size Scaling technique. Our results are consistent with a continuous phase transition, controlled by an external noise. We obtained the order parameter [Formula: see text], defined as the averaged opinion, the fourth-order Binder cumulant [Formula: see text] and susceptibility [Formula: see text] as functions of the noise parameter. We estimated the critical noises for Penrose and Ammann–Beenker lattices. We also considered seven-fold and nine-fold quasiperiodic lattices and estimated the respective critical noises as well. Irrespective of rotational and translational long-range order of the lattice, the system falls in the same universality class of the two-dimensional Ising model. Quasiperiodic order is irrelevant and it does not change any critical exponents for BCS model.

Epidemic outbreaks on two-dimensional quasiperiodic lattices

We present a novel kinetic Monte Carlo technique to study the susceptible-infected-removed model in order to simulate epidemic outbreaks on two quasiperiodic lattices, namely, Penrose and Ammann-Beenker. Our analysis around criticality is performed by investigating the order parameter, which is defined as the probability of growing a spanning cluster formed by removed sites, evolving from an initial system configuration with a single random chosen infective site. This system is studied by means of the cluster size distribution, obtained by the Newman-Ziff algorithm. Additionally, we obtained the mean cluster size, and a cumulant ratio to estimate the epidemic threshold. In spite of the quasiperiodic order moves the transition point, compared to periodic lattices, this is not able to alter the universality class of the model, leading to the same critical exponents. In addition, our technique can be generalized to study epidemic outbreaks in networks and diffusing populations.

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 , 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 , where the states of the system change from extended to localized, precisely like in a metal-insulator phase transition.

Interplay of non-Hermitian skin effects and Anderson localization in nonreciprocal quasiperiodic lattices

Non-Hermiticity from non-reciprocal hoppings has been shown recently to demonstrate the non-Hermitian skin effect (NHSE) under open boundary conditions (OBCs). Here we study the interplay of this effect and the Anderson localization in a \textit{non-reciprocal} quasiperiodic lattice, dubbed non-reciprocal Aubry-Andr\'{e} model, and a \textit{rescaled} transition point is exactly proved. The non-reciprocity can induce not only the NHSE, but also the asymmetry in localized states with two Lyapunov exponents for both sides. Meanwhile, this transition is also topological, characterized by a winding number associated with the complex eigenenergies under periodic boundary conditions (PBCs), establishing a \textit{bulk-bulk} correspondence. This interplay can be realized by an elaborately designed electronic circuit with only linear passive RLC devices instead of elusive non-reciprocal ones, where the transport of a continuous wave undergoes a transition between insulating and amplifying. This initiative scheme can be immediately applied in experiments to other non-reciprocal models, and will definitely inspires the study of interplay of NHSEs and more other quantum/topological phenomena.

Density of states and specific heat of three dimensional quasiperiodic lattices using real space renormalization group technique

In this paper we have introduced a new method based on real space renormalization group technique to find the density of states of three dimensional quasiperiodic lattices in the thermodynamic limit. The importance of this method lies in the fact that for three dimensional quasiperiodic lattices in this limit, no other technique is available in the literature for the density of states till now. Finding the density of states, we have also studied the electronic specific heat of such lattices and revealed the nature of temperature dependence of the specific heat.

The Aubry–André model as a hobbyhorse for understanding the localization phenomenon

We present a thorough pedagogical analysis of the single particle localization phenomenon in a quasiperiodic lattice in one dimension. Beginning with a detailed derivation of the Aubry–André Hamiltonian we describe the localization transition through the analysis of stationary and dynamical observables. Emphasis is placed on both the properties of the model and technical aspects of the performed calculations. In particular, the stationary properties investigated are the inverse participation ratio, the normalized participation ratio and the energy spectrum as a function of the disorder strength. Two dynamical quantities allow us to discern the localization phenomenon, being the spreading of an initially localized state and the evolution of population imbalance in even and odd sites across the lattice. The present manuscript could be useful in bringing advanced undergraduate and graduate students closer to the comprehension of localization phenomena, a topic of current interest in fields of condensed matter, ultracold atoms and complex systems.

Localization and Anderson delocalization of light in fractional dimensions with a quasi-periodic lattice.

We address the properties of wavepacket localization-delocalization transition (LDT) in fractional dimensions with a quasi-periodic lattice. The LDT point, which is generally determined by the competition between two sub-lattices comprising the quasi-periodic lattice, turns out to be inversely proportional to the Lévy index. Surprisingly, we find that, in the presence of weak structural disorder, anti-Anderson localization occurs, i.e., the introduced disorder results in an increasing of the size of the linear modes. Inclusion of a weak focusing nonlinearity is shown to improve localization. The propagation simulation achieves excellent agreement with the linear and nonlinear eigenmode analysis.

Density of states of two dimensional quasiperiodic lattices in the thermodynamic limit

A scheme based on real-space renormalization group method to find density of states of two dimensional lattices in the thermodynamic limit is introduced in this paper. The main importance of this method is its applicability in two dimensional quasiperiodic systems for which no analytic solution is possible. Previously there were many works with real-space renormalization scheme in one dimensional lattices, but its application in two dimensional systems was not successful. In the present scheme, we have mapped the two dimensional system into a number of one dimensional systems, and applying real-space renormalization method to those new one dimensional systems we have found out the density of states of the entire two dimensional system.

Properties and applications of one dimensional quasiperiodic lattices

Quasiperiodic lattices have been widely studied in cold atoms, which make one study extended-Anderson localization transition in one-dimensional (1D) or two-dimensional (2D) systems. In 2008, Inguscio's group prepared one-dimensional quasiperiodic lattice in cold atomic system and observed Anderson localization, which promoted the theoretical and experimental studies of quasiperiodic systems. Later, Bloch's group observed the many body localization in their prepared 1D and 2D quasiperiodic lattices. Recently, they also successfully observed the mobility edge and many body localization in a system with mobility edge in quasiperiodic lattices. These cold atom experiments have promoted the researches of many body localization and mobility edge. Quasiperiodic lattices have become a platform and its effects on many physical phenomena are being extensively studied, which can be expectantly observed in cold atomic experiments. This paper is based on the authors' some related work and briefly review some recent studies on the 1D quasiperiodic lattices, including some important cold atom experiments, some important properties of the quasiperiodic lattices and their effects on some physical phenomena, such as topological states.

On the information entropy of matter-waves in quasi-periodic lattice potentials

We consider ultracold Bose gases in quasi-random potentials and quantify localization of matter waves by means of Shannon information entropy. We explicitly examine the role of quasi-random potentials in producing localized states in the linear and nonlinear regimes. It is seen that the information entropic-based approach can be more useful to quantify localization of different types of states observed in Bose-Einstein condensates.

Vibrational properties of quasi-periodic beam structures

Quasi-periodic structures have been widely studied, notably in the atomic vibration domain. In this paper a beam structure based on Octagonal quasi-periodic tiling is studied. We provide a complete description of its vibrational response, including the density of its vibrational states, a detailed description of its vibration modes, and the computation of the dynamical structure factor (spectral density of energy) for transverse and for longitudinal waves. It is shown that quasi-periodic structures exhibit localized low frequency vibration modes that are due to resonant vibrations of isolated patterns in the quasi-periodic structure, but in opposite, high-frequency modes are (non-trivially) extended. Moreover, the paper shows the possible existence of band gaps in the vibrational response of periodic and quasi-periodic beam lattices as a function of the ratio between the bending and the tensile stiffness of the beams.