Quantum Billiards Research Articles

Intermediate spectral statistics of rational triangular quantum billiards.

Triangular billiards whose angles are rational multiples of π are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle, which permits one to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion, and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short-range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.

Systematically Constructing Mesoscopic Quantum States Relevant to Periodic Orbits in Integrable Billiards from Directionally Resolved Level Distributions

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.

Calculation method for the conductance of mesoscopic systems: assembly of Tight Binding Hamiltonians with spin effects—revisited

We discretize the Schrödinger equation in the approximation of the effective mass for the two-dimensional electron gas of GaAs, without magnetic field and on the other hand, with magnetic field. This discretization leads naturally to Tight Binding (TB) Hamiltonians in the approximation of the effective mass. An analysis of this discretization allows us to gain insight into the role of site and hopping energies, which allows us to model the TB Hamiltonian assembly with spin: Zeeman and spin–orbit coupling effects, especially the case Rashba. With this tool we can assemble Hamiltonians of quantum boxes, Aharanov–Bohm interferometers, anti-dots lattices and effects of imperfections, as well as disorder in the system. The extension to mount quantum billiards is natural. We also explain here how to adapt the recursive equations of Green’s functions for the case of spin modes, apart from transverse modes, for the calculation of conductance in these mesoscopic systems. The assembled Hamiltonians allow to identify the matrix elements (depending on the different parameters of the system) associated with splitting or spin flipping, which gives a starting point to model specific systems of interest, manipulating certain parameters. In general, the approach of this work allows us to clearly see the relationship between the wave and matrix description of quantum mechanics. We discuss here also, the extension of the method for 1D and 3D systems, for the extension apart from the first neighbors and for the inclusion of other types of interaction. The way we approach the method, has the objective of showing how specifically the site and hopping energies change in the presence of new interactions. This is very important in the case of spin interactions, because by looking at the matrix elements (site or hopping) we can directly identify the conditions that can lead to splitting, flipping or a mixture of these effects. Which is essential for the design of devices based on spintronics. Finally, we discuss spin-conductance modulation (Rashba spin precession) for the states of an open quantum dot (resonant states). Unlike the case of a quantum wire, the spin-flipping observed in the conductance is not perfectly sinusoidal, there is an envelope that modulates the sinusoidal component, which depends on the discrete-continuous coupling of the resonant states.

Enhancement factor in the regime of semi-Poisson statistics in a singular microwave cavity.

We investigated properties of a singular billiard, that is, a quantum billiard which contains a pointlike (zero-range) perturbation. A singular billiard was simulated experimentally by a rectangular microwave flat resonator coupled to microwave power via wire antennas which act as singular scatterers. The departure from regularity was quantitatively estimated by the short-range plasma model in which the parameter η takes the values 1 and 2 for the Poisson and semi-Poisson statistics, respectively. We show that in the regime of semi-Poisson statistics the experimental power spectrum and the second nearest-neighbor-spacing distribution P(2,s) are in good agreement with their theoretical predictions. Furthermore, the measurement of the two-port scattering matrix allowed us to evaluate experimentally the enhancement factor F(γ^{tot}) in the regime of the semi-Poisson statistics as a function of the total absorption factor γ^{tot}. The experimental results were compared with the analytical formula for F(γ^{tot}) evaluated in this article. The agreement between the experiment and theory is good.

Unidirectionality and Husimi functions in constant-width neutrino billiards

We investigate the spectral properties and Husimi functions of relativistic quantum billiards (QBs) consisting of a spin-1/2 particle governed by the Dirac equation and confined to a planar domain of constant-width (CW) by imposing boundary conditions (BCs) on the spinor components. We consider those of neutrino billiards (NBs) proposed in (Berry and Mondragon 1987 Proc. R. Soc. A 412 53). The classical dynamics of billiards of corresponding shape is predominantly chaotic. CW billiards attracted particular attention because they exhibit unusual properties. Their classical dynamics features unidirectionality, whereas in the corresponding nonrelativistic QB a change of the rotational direction of motion is possible via dynamical tunneling, and the spectral properties coincide with those of typical quantum systems with violated time-reversal invariance. Unidirectionality of the quantum dynamics would arise in the structure of the Husimi functions. We analyze them for two realizations of CW NBs in the ultra-relativistic, i.e. the massless case and for massive cases and come to the result, that the modes can be separated into clockwise and counterclockwise modes and dynamical tunneling is absent. This is attributed to the BCs and the unidrectionality of the local current arising from them.

Intermediate statistics in singular quarter-ellipse shaped microwave billiards* *Dedicated to the Memory of Fritz Haake.

We report on experiments with a flat, superconducting microwave billiard with the shape of a quarter ellipse simulating a singular billiard, that is, a quantum billiard containing zero-range perturbations. The pointlike scatterers were realized with long antennas. Their coupling to the microwaves inside the cavity depends on frequency. A complete sequence of 1013 eigenfrequencies was identified rendering possible the investigation of spectral properties as function of frequency. They exhibit intermediate statistics and are well described by analytical results derived by Bogomolny, Gerland, Giraud and Schmit for singular billiards with shapes that generate an integrable classical dynamics. This comparison revealed a quadratic frequency dependence of the coupling parameter. The size of the chaotic component induced by the diffractive effects of the scatterers was determined by comparison with analytical results derived by Haake and Lenz for an additive random-matrix model, which interpolates between the models applicable for quantum systems with an integrable and chaotic classical dynamics, respectively.

Quantum signatures of chaos in relativistic quantum billiards with shapes of circle- and ellipse-sectors* *Dedicated to the memory of Fritz Haake. Fritz introduced me (Barbara) into the field of quantum chaos. He supervised my diploma and doctoral thesis and since then has been a mentor, friend and helpful advisor during all the stages of my scientific career. I take his

According to the Berry–Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE-like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré–Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.

Superscarred quasimodes on flat surfaces with conical singularities

For any given momentum vector $\xi _0\in \mathbb {S}^1$ we construct a continuous family of quasimodes for the Laplace–Beltrami operator on a translation surface. We apply our result to rational polygonal quantum billiards and thus construct a continuous

Time-reversal-invariant hexagonal billiards with a point symmetry.

A biparametric family of hexagonal billiards enjoying the C_{3} point symmetry is introduced and numerically investigated. First, the relative measure r(ℓ,θ;t) in a reduced phase space was mapped onto the parameter plane ℓ×θ for discrete time t up to 10^{8} and averaged in tens of randomly chosen initial conditions in each billiard. The resulting phase diagram allowed us to identify fully ergodic systems in the set. It is then shown that the absolute value of the position autocorrelation function decays like |C_{q}(t)|∼t^{-σ}, with 0<σ⩽1 in the hexagons. Following previous examples of irrational triangles, we were able to find billiards for which σ∼1. This is further evidence that, although not chaotic (all Lyapunov exponents are zero), billiards in polygons might exhibit a near strongly mixing dynamics in the ergodic hierarchy. Quantized counterparts with distinct classical properties were also characterized. Spectral properties of singlets and doublets of the quantum billiards were investigated separately well beyond the ground state. As a rule of thumb, for both singlet and doublet sequences, we calculate the first 120000 energy eigenvalues in a given billiard and compute the nearest neighbor spacing distribution p(s), as well as the cumulative spacing function I(s)=∫_{0}^{s}p(s^{'})ds^{'}, by considering the last 20000 eigenvalues only. For billiards with σ∼1, we observe the results predicted for chaotic geometries by Leyvraz, Schmit, and Seligman, namely, a Gaussian unitary ensemble behavior in the degenerate subspectrum, in spite of the presence of time-reversal invariance, and a Gaussian orthogonal ensemble behavior in the singlets subset. For 0<σ<1, formulas for intermediate quantum statistics have been derived for the doublets following previous works by Brody, Berry and Robnik, and Bastistić and Robnik. Different regimes in a given energy spectrum have been identified through the so-called ergodic parameter α=t_{H}/t_{C}, the ratio between the Heisenberg time and the classical diffusive-like transport time, which signals the possibility of quantum dynamical localization when α<1. A good quantitative agreement is found between the appropriate formulas with parameters extracted from the classical phase space and the data from the calculated quantum spectra. A rich variety of standing wave patterns and corresponding Poincaré-Husimi representations in a reduced phase space are reported, including those associated with lattice modes, scarring, and high-frequency localization phenomena.

Relativistic Quantum Billiards with Threefold Rotational Symmetry: Exact, Symmetry-Projected Solutions for Equilateral Neutrino Billiard

Relativistic Quantum Billiards with Threefold Rotational Symmetry: Exact, Symmetry-Projected Solutions for Equilateral Neutrino Billiard

Indicators of wavefunction (de)localisation for avoided crossing in a quadrupole quantum billiard

The relationship between wavefunction (de)localisation and avoided crossing in a quadrupole billiard is analysed. The following three-types of measures are employed for wavefunction (de)localisation: inverse participation ratio, inverse of Rényi entropy, and root-mean-square (RMS) image contrast. All these measures exhibit minimal values at the centre of the avoided crossing, where the wavefunction is maximally delocalised. Our results indicate that these quantities can be sufficient for the indication of wavefunction (de)localisation.

Exploiting Modern Chladni Plates to Analogously Manifest the Point Interaction

A truncated basis is employed to analyze the influence of the point interaction on the eigenvalues and eigenfunctions in quantum billiards. The point interaction is numerically confirmed to cause the shift in the eigenvalue that leads the original eigenfunctions to be superposed to form the new eigenfunction. The amplitude and nodal-line patterns of the eigenfunctions are found to vary significantly with the coupling strength. It is further verified that the point-driven Chladni plates can be employed to analogously manifest the noticeable dependence of the nodal-line pattern on the coupling strength. More importantly, the dispersion relation between the frequency and the wave number for the flexural wave can be precisely determined in the process of utilizing the point interaction to model the modern Chladni plates.

Bound states in the continuum in asymmetrical quantum-mechanical and electromagnetic waveguides

We study transport properties and the formation of bound states in the continuum (BIC) in asymmetric quantum mechanical and electromagnetic waveguides. An analytical model for an arbitrary asymmetric two-terminal quantum mechanical waveguide is proposed, and conditions of BIC formation are formulated. We show that the Friedrich-Wintgen mechanism of BIC formation in a system coupled to two continua takes place regardless of the symmetry of the system as long as the proportionate coupling condition is fulfilled. This result is illustrated by numerical simulation of two-dimensional quantum billiard and optical waveguide with a cavity. Due to the universal wave nature of BIC, the proposed BIC formation mechanism allows one to obtain BICs in the broader class of quantum mechanical, electromagnetic, acoustic, and other types of structures.

Microwave photonic crystals, graphene, and honeycomb-kagome billiards with threefold symmetry: Comparison with nonrelativistic and relativistic quantum billiards

We present results for properties related to the band structure of a microwave photonic crystal, that is, a flat resonator containing metallic cylinders arranged on a triangular grid, referred to as the Dirac (microwave) billiard, with a threefold-symmetric shape. Such systems have been used to investigate finite-size graphene sheets. It was shown recently that the eigenmodes of rectangular Dirac billiards are well described by the tight-binding model of a finite-size honeycomb-kagome lattice of corresponding shape. We compare properties of the eigenstates of the Dirac billiard with those of the associated graphene and honeycomb-kagome billiard and relativistic quantum billiard. We outline how the eigenstates of threefold-symmetric systems can be separated according to their transformation properties under rotation by $\frac{2\ensuremath{\pi}}{3}$ into three subspaces, namely singlets, that are rotationally invariant, and doublets that are noninvariant. We reveal for the doublets in graphene and honeycomb-kagome billiards in quasimomentum space a selective excitation of the valley states associated with the two inequivalent Dirac points. For the understanding of symmetry-related features, we extend known results for nonrelativistic quantum billiards and the associated semiclassical approach to relativistic neutrino billiards.

Quantum Particle on Dual Weight Lattice in Weyl Alcove

Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and Neumann walls on the borders of the Weyl alcoves. The amplitudes of the particle’s propagation to neighbouring positions are determined by a complex-valued dual-weight hopping function of finite support. The discrete dual-weight Hamiltonians are obtained as the sum of specifically constructed dual-weight hopping operators. By utilising the generalised dual-weight Fourier–Weyl transforms, the solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are exactly resolved. The matrix Hamiltonians, stationary states and eigenenergies of the discrete models are exemplified for the rank two cases C2 and G2.

High-order finite difference method for the Schrödinger equation on deforming domains

A high-order finite difference discretisation of the Schrödinger equation on domains that deform in time is presented. For the domain deformation, a time-dependent coordinate transformation is used. The utilisation of summation-by-parts finite difference operators, combined with boundary conditions imposed weakly by a penalty technique, leads to a provably stable scheme. The convergence properties of the scheme are verified by numerical computations. The capabilities of the numerical scheme are demonstrated by simulations of Berry phases in deforming quantum billiards.

Quantum billiards with correlated electrons confined in triangular transition metal dichalcogenide monolayer nanostructures

Forcing systems through fast non-equilibrium phase transitions offers the opportunity to study new states of quantum matter that self-assemble in their wake. Here we study the quantum interference effects of correlated electrons confined in monolayer quantum nanostructures, created by femtosecond laser-induced quench through a first-order polytype structural transition in a layered transition-metal dichalcogenide material. Scanning tunnelling microscopy of the electrons confined within equilateral triangles, whose dimensions are a few crystal unit cells on the side, reveals that the trajectories are strongly modified from free-electron states both by electronic correlations and confinement. Comparison of experiments with theoretical predictions of strongly correlated electron behaviour reveals that the confining geometry destabilizes the Wigner/Mott crystal ground state, resulting in mixed itinerant and correlation-localized states intertwined on a length scale of 1 nm. The work opens the path toward understanding the quantum transport of electrons confined in atomic-scale monolayer structures based on correlated-electron-materials.

Morphology of three-body quantum states from machine learning

The relative motion of three impenetrable particles on a ring, in our case two identical fermions and one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio κ of the impurity and fermion masses, the billiards can be integrable or non-integrable (also referred to in the main text as chaotic). To set the stage, we first investigate the energy level distributions of the billiards as a function of 1/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limiting values 1/κ = 1 and 1/κ = 0. Then, we use machine learning tools to analyze properties of probability distributions of individual quantum states. We find that convolutional neural networks can correctly classify integrable and non-integrable states. The decisive features of the wave functions are the normalization and a large number of zero elements, corresponding to the existence of a nodal line. The network achieves typical accuracies of 97%, suggesting that machine learning tools can be used to analyze and classify the morphology of probability densities obtained in theory or experiment.

Morphology of three-body quantum states from machine learning

The relative motion of three impenetrable particles on a ring, in our case two identical fermions and one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio κ of the impurity and fermion masses, the billiards can be integrable or non-integrable (also referred to in the main text as chaotic). To set the stage, we first investigate the energy level distributions of the billiards as a function of 1/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limiting values 1/κ = 1 and 1/κ = 0. Then, we use machine learning tools to analyze properties of probability distributions of individual quantum states. We find that convolutional neural networks can correctly classify integrable and non-integrable states. The decisive features of thewave functions are the normalization and a large number of zero elements, corresponding to the existence of a nodal line. The network achieves typical accuracies of 97%, suggesting that machine learning tools can be used to analyze and classify the morphology of probability densities obtained in theory or experiment.

A standard energy eigenvalue problem for directly solving the stationary states of quantum billiards via boundary integral analysis

By using the series expansions of the Bessel functions for the real and imaginary parts of the free particle Green’s function for the two-dimensional stationary states of quantum billiards, it can be shown that some components of the Green’s function are redundant which can be eliminated to make the boundary integral equation for the wave function of free particles inside quantum billiards independent of the wave numbers. This development leads to a much faster search for the energy eigenvalues of quantum billiards by scanning their wave numbers or the formulation of the standard eigenvalue problem which can directly be solved for the energy eigenvalues. Some numerical examples were used to demonstrate that the proposed technique is accurate, computationally effective and straightforward to be applied in practice.