Quantum Billiards Research Articles

One cannot hear the density of a drum (and further aspects of isospectrality)

It is well known that certain pairs of planar domains have the same spectra of the Laplacian operator. We prove that these domains are still isospectral for a wider class of physical problems, including the cases of heterogeneous drums and quantum billiards in an external field. In particular we show that the isospectrality is preserved when the density or the potential is symmetric under reflections along the folding lines of the domain. These results are also confirmed numerically using the finite-difference method: We find that the pairs of numerical matrices obtained in the discretization are exactly isospectral up to machine precision.

Semiclassical wave functions for open quantum billiards

We present a semiclassical approximation to the scattering wave function Ψ(r,k) for an open quantum billiard, which is based on the reconstruction of the Feynman path integral. We demonstrate its remarkable numerical accuracy for the open rectangular billiard and show that the convergence of the semiclassical wave function to the full quantum state is controlled by the mean path length or equivalently the dwell time for a given scattering state. In the numerical implementation a cutoff length in the maximum path length or, equivalently, a maximum dwell time τ(max) included implies a finite energy resolution ΔE~τ(max)(-1). Possible applications include leaky billiards and systems with decoherence present.

Quantum billiards in multidimensional models with fields of forms

Bianchi type I cosmological model in (n+1)-dimensional gravity with several forms is considered. When the electric non-composite brane ansatz is adopted, the Wheeler-DeWitt (WDW) equation for the model, written in a conformally covariant form, is analyzed. Under certain restrictions, asymptotic solutions to the WDW equation near the singularity are found, which reduce the problem to the so-called quantum billiard on the (n-1)-dimensional Lobachevsky space H^{n-1}. Two examples of quantum billiards are considered: a 2-dimensional quantum billiard for a 4D model with three 2-forms and a 9D quantum billiard for an 11D model with 120 4-forms which mimics SM2-brane sector of D=11 supergravity. For certain solutions, vanishing of the wave function at the singularity is proved.

Statistical Properties of Eigenfunctions for Quantum Billiards with and without Positive Lyapunov Exponent

Statistical properties of eigenfunctions of quantum polygonal and dispersing billiards are numerically investigated in detail. Residual parameter and skewness, which estimate the deviation of amplitude distribution from the Gaussian distribution, are obtained for hundreds of eigenfunctions. Almost all eigenfunctions of the dispersing billiards show random pattern. For polygonal billiards, it is found that the majority shows quite complicated pattern, but some eigenfunctions are highly regular. Among them two types of highly regular pattern are conspicuous, and both eigenfunctions are apparently influenced by the vertices. The difference between the origin of irregular eigenfunctions of the dispersing billiards and that of the polygonal billiards is also discussed.

Theoretical description of cylindrical nano-structures, including pores in semiconductors

Cylindrical and prismatic nano-structures (nano-wires and pores) with circular and hexagonal cross-section are studied using mirror boundary conditions to solve the Schrödinger equation in effective mass approximation. In comparison with “quantum billiard” problem, the solution using mirror boundary conditions allows to obtain the results in a much simpler way. It is possible to use even and odd mirror boundary conditions depending on the sign of the wave-function equated in the original and the image points, respectively. The even mirror boundary conditions provides non-vanishing wave-function at the boundary, corresponding to weak confinement allowing quantum tunneling. The odd mirror boundary conditions set wave-function to zero at the boundary, corresponding to a strong confinement. We report on spatial distributions of probability density in cross-section of a cylindrical and prismatic nano-structures, presenting the formulas for the energy of the corresponding quantum states.

Correction: Corrigendum: An exactly solvable model for the integrability–chaos transition in rough quantum billiards

Corrigendum: An exactly solvable model for the integrability–chaos transition in rough quantum billiards

Semiclassical approximation for transmission through open Sinai billiards

We use a semiclassical approximation to study the transport through the weakly open chaotic Sinai quantum billiards which can be considered as the schematic of a Sinai mesoscopic device, with the diffractive scatterings at the lead openings taken into account. The conductance of the ballistic microstructure which displays universal fluctuations due to quantum interference of electrons can be calculated by Landauer formula as a function of the electron Fermi wave number, and the transmission amplitude can be expressed as the sum over all classical paths connecting the entrance and the exit leads. For the Sinai billiards, the path sum leads to an excellent numerical agreement between the peak positions of power spectrum of the transmission amplitude and the corresponding lengths of the classical trajectories, which demonstrates a good agreement between the quantum theory and the semiclassical theory.

Analysis of resonant population transfer in time-dependent elliptical quantum billiards

A Fermi golden rule for population transfer between instantaneous eigenstates of elliptical quantum billiards with oscillating boundaries is derived. Thereby the occurrence of both the recently observed resonant population transfer between instantaneous eigenstates and the empirical criterion stating that these transitions occur when the driving frequency matches the mean difference of the latter [Lenz et al., New J. Phys. 13, 103019 (2011)] is explained. As a second main result a criterion judging which resonances are resolvable in a corresponding experiment of certain duration is provided. Our analysis is complemented by numerical simulations for three different driving laws. The corresponding resonance spectra are in agreement with the predictions of both criteria.

A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards

We show that the spectral zeta functions of inhomogeneous strings and drums can be calculated using Rayleigh-Schrödinger perturbation theory. The inhomogeneities that can be treated with this method are small but otherwise arbitrary and include the previously studied case of a piecewise constant density. In two dimensions the method can be used to derive the spectral zeta function of a domain obtained from the small deformation of a square. We also obtain exact sum rules that are valid for arbitrary densities and that correspond to the values taken by the spectral zeta function at integer positive values; we have tested numerically these sum rules in specific examples. We show that the Dirichlet or Neumann Casimir energies of an inhomogeneous string, evaluated to first order in perturbation theory, contain in some cases an irremovable divergence, but that the combination of the two is always free of divergences. Finally, our calculation of the Casimir energies of a string with piecewise constant density and of two perfectly conducting concentric cylinders, of similar radius, reproduce the results previously published.

Tunneling phenomena in the open elliptic quantum billiard

The study of open quantum billiards has gained popularity in the last decades, including different common and uncommon geometries such as the circular and stadium billiards. We present an extensive analysis of the elliptic quantum billiard with hyperbolic channels. We concentrate on the tunneling through an elliptic resonator-like structure. We analyze three different variations of the system: the first configuration has horizontal channels, then we study the system with vertical leads, and finally we displace both channels by the same angle to gain a more general perspective. We observed a very unusual phase distribution in the resonator cavity when there is no tunneling through the system.

Generation of large orbital angular momentum from superposed Bessel beams corresponding to resonant geometric modes

We theoretically verify that a coherent superposition of nearly degenerate Bessel beams characterizes the family of resonant geometric modes in circular billiards. With the ray-wave correspondence, we experimentally exploit a large-aperture cylindrical waveguide to generate the resonant geometric modes with large orbital angular momentum. Furthermore, the free-space propagation of the geometric modes emerging from the cylindrical waveguide is demonstrated to display the transient dynamics of the coherent states released from a circular quantum billiard.

Relativistic wavepackets in classically chaotic quantum cosmological billiards

Close to a spacelike singularity, pure gravity and supergravity in 4 to 11 spacetime dimensions admit a cosmological billiard description based on hyperbolic Kac-Moody groups. We investigate the quantum cosmological billiards of relativistic wavepackets towards the singularity, employing flat and hyperbolic space descriptions for the quantum billiards. We find that the strongly chaotic classical billiard motion of four-dimensional pure gravity corresponds to a spreading wavepacket subject to successive redshifts and tending to zero as the singularity is approached. We discuss the possible implications of these results in the context of singularity resolution and compare them with those of known semiclassical approaches. As an aside, we obtain exact solutions for the one-dimensional relativistic quantum billiards with moving walls.

An exactly solvable model for the integrability–chaos transition in rough quantum billiards

A central question of dynamics, largely open in the quantum case, is to what extent it erases a system's memory of its initial properties. Here we present a simple statistically solvable quantum model describing this memory loss across an integrability-chaos transition under a perturbation obeying no selection rules. From the perspective of quantum localization-delocalization on the lattice of quantum numbers, we are dealing with a situation where every lattice site is coupled to every other site with the same strength, on average. The model also rigorously justifies a similar set of relationships, recently proposed in the context of two short-range-interacting ultracold atoms in a harmonic waveguide. Application of our model to an ensemble of uncorrelated impurities on a rectangular lattice gives good agreement with ab initio numerics.

Realization of discrete quantum billiards in a two-dimensional optical lattice

We propose the method for optical visualization of Bose-Hubbard model with two interacting bosons in the form of two-dimensional (2D) optical lattices consisting of optical waveguides, where the waveguides at the diagonal are characterized by different refractive index than others elsewhere, modeling the boson-boson interaction. We study the light intensity distribution function averaged over direction of propagation for both ordered and disordered cases, exploring sensitivity of the averaged picture with respect to the beam injection position. For our finite systems the resulting patterns reminiscent the ones set in billiards and therefore we introduce a definition of discrete quantum billiard discussing the possible relevance to its well established continuous counterpart.

Magnetic field-induced control of transport in multiterminal focusing quantum billiards

By exploring the four-terminal transmission of a semi-elliptic open quantum billiard in dependence of its geometry and an applied magnetic field, it is shown that a controllable switching of currents between the four terminals can be obtained. Depending on the eccentricity of the semi-ellipse and the width and placement of the leads, high transmittivity at zero magnetic field is reached either through states guided along the curved boundary or focused onto the straight boundary of the billiard. For small eccentricity, attachment of leads at the ellipse foci can yield optimized corresponding transmission, while departures from this behavior demonstrate the inapplicability of classical considerations in the deep quantum regime. The geometrically determined transmission is altered by the phase-modulating and deflecting effect of the magnetic field, which switches the pairs of leads connected by high transmittivity. It is shown that the elliptic boundary is responsible for these very special transport properties. At higher field strengths edge states form and the multiterminal transmission coefficients are determined by the topology of the billiard. The combination of magnetotransport with geometrically optimized transmission behavior leads to an efficient control of the current through the multiterminal structure.

Non-paraxial contributions to the far-field pattern of surface-emitting lasers: a manifestationof the momentum-space wavefunctions of quantum billiards

We investigated experimentally non-paraxial contributions to the high-order far-fieldpattern of large-area vertical-cavity surface-emitting lasers in order to explore by analogythe momentum-space wave distributions of quantum billiards. Our results revealthat non-paraxial contributions significantly influence the morphology of thehigh-order far-field pattern. A fast reliable method is developed for transformingthe experimental far-field patterns to the correct Fourier transform of thecorresponding near-field lasing modes. In this way we visualize the momentum-space (p–q) wavefunctions of quantum billiards.

Casimir force between integrable and chaotic pistons

We have computed numerically the Casimir force between two identical pistons inside a very long cylinder, considering different shapes for the pistons. The pistons can be considered as quantum billiards, whose spectrum determines the vacuum force. The smooth part of the spectrum fixes the force at short distances, and depends only on geometric quantities like the area or perimeter of the piston. However, correcting terms to the force, coming from the oscillating part of the spectrum which is related to the classical dynamics of the billiard, are qualitatively different for classically integrable or chaotic systems. We have performed a detailed numerical analysis of the corresponding Casimir force for pistons with regular and chaotic classical dynamics. For a family of stadium billiards, we have found that the correcting part of the Casimir force presents a sudden change in the transition from regular to chaotic geometries.

Quantum Billiards Under the Influence of External Magnetic Field

Quantum Billiards Under the Influence of External Magnetic Field

Extracting photon periodic orbits from spontaneous emission spectra in laterally confined vertically emitted cavities

We report our observation of the signature of photon periodic orbits in the spontaneous emission spectra of large-aperture vertical-cavity surface-emitting lasers (VCSELs). The high-resolution measurement clearly demonstrates that over a thousand cavity modes with a narrow linewidth can be perfectly exhibited in the spontaneous emission spectrum just below the lasing threshold. The Fourier-transformed spectrum is analyzed to confirm that the spontaneous emission spectra of large-aperture VCSELs can be exploited to analogously investigate the energy spectra of the 2D quantum billiards.

Modeling of open quantum dots and wave billiards using imaginary potentials for the source and the sink

A heuristic model for particle states and current flow in open ballistic two-dimensional (2D) quantum dots/wave billiards is proposed. The model makes use of complex potentials first introduced in phenomenological nuclear inelastic scattering theory (the optical model). Here we assume that external input and output leads connecting the system to the source and the drain regions may be represented by complex potentials. In this way, a current may be set up between the two ‘pseudo-leads’. Probability densities and current flows for an open quantum dot are analyzed here numerically and the results are compared with the microwave measurements used to emulate the system. The model is of conceptual as well as practical interest. In addition to quantum billiards, it may be used as a tool per se to analyze transport in classical wave analogues, such as microwave resonators, acoustic resonators, effects of leakage on such systems, etc.