Classical and quantum chaos in a circular billiard with a straight cut

Chaos-Assisted Tunneling

The ability of particles to "tunnel" through potential energy barriers is a purely quantum phenomenon. A classical particle in a symmetric double-well potential, with energy below the potential barrier, will be trapped on one side of the potential well. A quantum particle, however, can sit on both sides, in either a symmetric state or an antisymmetric state. An analogous phenomenon occurs in conservative classical systems with two degrees of freedom and no potential barriers. If only the energy is conserved, the phase space will be a mixture of regular "islands" embedded in a sea of chaos. Classically, a particle sitting in one regular island cannot reach another symmetrically located regular island when the islands are separated by chaos. However, a quantum particle can sit on both regular islands, in symmetric and antisymmetric states, due to chaos-assisted tunneling. Here, we give an overview of the theory and recent experimental observations of this phenomenon.

Impact of Cavity Geometry on Microlaser Dynamics

We experimentally investigate spatiotemporal lasing dynamics in semiconductor microcavities with various geometries, featuring integrable or chaotic ray dynamics. The classical ray dynamics directly impacts the lasing dynamics, which is primarily determined by the local directionality of long-lived ray trajectories. The directionality of optical propagation dictates the characteristic length scales of intensity variations, which play a pivotal role in nonlinear light-matter interactions. While wavelength-scale intensity variations tend to stabilize lasing dynamics, modulation on much longer scales causes spatial filamentation and irregular pulsation. Our results will pave the way to control the lasing dynamics by engineering the cavity geometry and ray dynamical properties.

Quasibound states in a triple Gaussian potential

We derive the transmission probabilities and delay times, and identify quasibound state structures in an open quantum system consisting of three Gaussian potential energy peaks, a system whose classical scattering dynamics we show to be chaotic. Such open quantum systems can serve as models for nanoscale quantum devices and their wave dynamics are similar to electromagnetic wave dynamics in optical microcavities. We use a quantum web to determine energy regimes for which the system exhibits the quantum manifestations of chaos, and we show that the classical scattering dynamics contains a significant amount of chaos. We also derive an exact expression for the non-Hermitian Hamiltonian whose eigenvalues give quasibound state energies and lifetimes of the system.

Signatures of chaos in the Brillouin zone

When the classical dynamics of a particle in a finite two-dimensional billiard undergoes a transition to chaos, the quantum dynamics of the particle also shows manifestations of chaos in the form of scarring of wave functions and changes in energy level spacing distributions. If we "tile" an infinite plane with such billiards, we find that the Bloch states on the lattice undergo avoided crossings, energy level spacing statistics change from Poisson-like to Wigner-like, and energy sheets of the Brillouin zone begin to "mix" as the classical dynamics of the billiard changes from regular to chaotic behavior.

Chaos in the band structure of a soft Sinai lattice

We study the effect of broken spatial and dynamical symmetries on the band structure of two lattices with unit cells that are soft versions of the classic Sinai billiard. We find significant signatures of chaos in the band structure of these lattices, in energy regimes where the underlying classical unit cell undergoes a transition to chaos. Broken dynamical symmetries and the presence of chaos can diminish the feasibility of changing and controlling band structure in a wide variety of two-dimensional lattice-based devices, including two-dimensional solids, optical lattices, and photonic crystals.

Gaussian orthogonal ensemble statistics in graphene billiards with the shape of classically integrable billiards

A crucial result in quantum chaos, which has been established for a long time, is that the spectral properties of classically integrable systems generically are described by Poisson statistics, whereas those of time-reversal symmetric, classically chaotic systems coincide with those of random matrices from the Gaussian orthogonal ensemble (GOE). Does this result hold for two-dimensional Dirac material systems? To address this fundamental question, we investigate the spectral properties in a representative class of graphene billiards with shapes of classically integrable circular-sector billiards. Naively one may expect to observe Poisson statistics, which is indeed true for energies close to the band edges where the quasiparticle obeys the Schrödinger equation. However, for energies near the Dirac point, where the quasiparticles behave like massless Dirac fermions, Poisson statistics is extremely rare in the sense that it emerges only under quite strict symmetry constraints on the straight boundary parts of the sector. An arbitrarily small amount of imperfection of the boundary results in GOE statistics. This implies that, for circular-sector confinements with arbitrary angle, the spectral properties will generically be GOE. These results are corroborated by extensive numerical computation. Furthermore, we provide a physical understanding for our results.

Controlling mode competition by tailoring the spatial pump distribution in a laser: a resonance-based approach

We introduce a simplified version of the steady-state ab initio laser theory for calculating the effects of mode competition in continuous wave lasers using the passive cavity resonances. This new theory harnesses widely available numerical methods that can efficiently calculate the passive cavity resonances, with negligible additional computational overhead. Using this theory, we demonstrate that the pump profile of the laser cavity can be optimized both for highly multi-mode and single-mode emission. An open source implementation of this method has been made available.

Focusing waves at arbitrary locations in a ray-chaotic enclosure using time-reversed synthetic sonas

Time-reversal methods are widely used to achieve wave focusing in acoustics and electromagnetics. Past time-reversal experiments typically require that a transmitter be initially present at the target focusing point, which limits the application of this technique. In this paper, we propose a method to focus waves at an arbitrary location inside a complex enclosure using a numerically calculated wave excitation signal. We use a semiclassical ray algorithm to calculate the signal that would be received at a transceiver port resulting from the injection of a short pulse at the desired target location. The time-reversed version of this signal is then injected into the transceiver port, and an approximate reconstruction of the short pulse is created at the target. The quality of the pulse reconstruction is quantified in three different ways, and the values of these metrics are shown to be predicted by the statistics of the scattering parameter |S_{21}|^{2} between the transceiver and target points in the enclosure over the bandwidth of the pulse. We demonstrate the method experimentally using a flat microwave billiard, and we quantify the reconstruction quality as a function of enclosure loss, port coupling, and other considerations.

Regular modes in a mixed-dynamics-based optical fiber

A multimode optical fiber with a truncated transverse cross section acts as a powerful versatile support to investigate the wave features of complex ray dynamics. In this paper, we concentrate on the case of a geometry inducing mixed dynamics. We highlight that regular modes associated with stable periodic orbits present an enhanced spatial intensity localization. We report the statistics of the inverse participation ratio whose features are analogous to those of Anderson localized modes. Our study is supported by both numerical and experimental results on the spatial localization and spectral regularity of the regular modes.

Cross-section fluctuations in open microwave billiards and quantum graphs: The counting-of-maxima method revisited

The fluctuations exhibited by the cross sections generated in a compound-nucleus reaction or, more generally, in a quantum-chaotic scattering process, when varying the excitation energy or another external parameter, are characterized by the width Γcorr of the cross-section correlation function. Brink and Stephen [Phys. Lett. 5, 77 (1963)] proposed a method for its determination by simply counting the number of maxima featured by the cross sections as a function of the parameter under consideration. They stated that the product of the average number of maxima per unit energy range and Γcorr is constant in the Ercison region of strongly overlapping resonances. We use the analogy between the scattering formalism for compound-nucleus reactions and for microwave resonators to test this method experimentally with unprecedented accuracy using large data sets and propose an analytical description for the regions of isolated and overlapping resonances.

Wave-Field Shaping in Cavities: Waves Trapped in a Box with Controllable Boundaries

Electromagnetic cavities are used in numerous domains of applied and fundamental physics, from microwave ovens and electromagnetic compatibility to masers, quantum electrodynamics (QED), and quantum chaos. The wave fields established in cavities are statically fixed by their geometry, which are usually modified by using mechanical parts like mode stirrers in reverberation chambers or screws in masers and QED. Nevertheless, thanks to integral theorems, tailoring the cavity boundaries theoretically permits us to design at will the wave fields they support. Here, we show in the microwave domain that it is achievable dynamically simply by using electronically tunable metasurfaces that locally modify the boundaries, switching them in real time from Dirichlet to Neumann conditions. We prove that at a high modal density, counterintuitively, it permits us to create wave patterns presenting hot spots of intense energy. We explain and model the physical mechanism underlying the concept, which allows us to find a criterion ensuring that modifying parts of a cavity's boundaries turn it into a completely different one. We finally prove that this approach even permits us, in the limiting case where the cavity supports only well-separated resonances, to choose the frequencies at which the latter occur.

Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging

The spatial coherence of laser sources has limited their application to parallel imaging and projection due to coherent artifacts, such as speckle. In contrast, traditional incoherent light sources, such as thermal sources or light emitting diodes (LEDs), provide relatively low power per independent spatial mode. Here, we present a chip-scale, electrically pumped semiconductor laser based on a novel design, demonstrating high power per mode with much lower spatial coherence than conventional laser sources. The laser resonator was fabricated with a chaotic, D-shaped cavity optimized to achieve highly multimode lasing. Lasing occurs simultaneously and independently in ∼1,000 modes, and hence the total emission exhibits very low spatial coherence. Speckle-free full-field imaging is demonstrated using the chaotic cavity laser as the illumination source. The power per mode of the sample illumination is several orders of magnitude higher than that of a LED or thermal light source. Such a compact, low-cost source, which combines the low spatial coherence of a LED with the high spectral radiance of a laser, could enable a wide range of high-speed, full-field imaging and projection applications.

Ergodicity of the generalized lemon billiards

In this paper, we study a two-parameter family of convex billiard tables, by taking the intersection of two round disks (with different radii) in the plane. These tables give a generalization of the one-parameter family of lemon-shaped billiards. Initially, there is only one ergodic table among all lemon tables. In our generalized family, we observe numerically the prevalence of ergodicity among the some perturbations of that table. Moreover, numerical estimates of the mixing rate of the billiard dynamics on some ergodic tables are also provided.

In situ broadband cryogenic calibration for two-port superconducting microwave resonators

We introduce an improved microwave calibration method for use in a cryogenic environment, based on a traditional three-standard calibration, the Thru-Reflect-Line (TRL) calibration. The modified calibration method takes advantage of additional information from multiple measurements of an ensemble of realizations of a superconducting resonator, as a new pseudo-Open standard, to correct errors in the TRL calibration. We also demonstrate an experimental realization of this in situ broadband cryogenic calibration system utilizing cryogenic switches. All calibration measurements are done in the same thermal cycle as the measurement of the resonator (requiring only an additional 20 min), thus avoiding 4 additional thermal cycles for traditional TRL calibration (which would require an additional 12 days). The experimental measurements on a wave-chaotic microwave billiard verify that the new method significantly improves the measured scattering matrix of a high-quality-factor superconducting resonator.

High-Q modes in defected microcircular resonator confined by metal layer for unidirectional emission

Defected circular resonators laterally confined by a metal layer with a flat side as an emitting window are numerically investigated based on the boundary element method for realizing unidirectional emission microlasers. The results indicate that Fabry-Pérot (FP) modes become high Q confined modes in the defected circular resonator with a metallic layer. The mode coupling between the FP mode and chaotic-like mode can result in high Q confined mode for unidirectional emission with a narrow far field pattern.

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.

Experimental phase-space-based optical amplification of scar modes

Wave billiards which are chaotic in the geometrical limit are known to support nongeneric spatially localized modes called scar modes. The interaction of the scar modes with gain has been recently investigated in optics in microcavity lasers and vertical-cavity surface-emitting lasers. Exploiting the localization properties of scar modes in their wave-analogous phase-space representation, we report experimental results of scar mode selection by gain in a doped D-shaped optical fiber.

Quasimodes of a chaotic elastic cavity with increasing local losses

We report noninvasive measurements of the complex field of elastic quasimodes of a silicon wafer with chaotic shape. The amplitude and phase spatial distribution of the flexural modes are directly obtained by Fourier transform of time measurements. We investigate the crossover from real mode to complex-valued quasimode, when absorption is progressively increased on one edge of the wafer. The complexness parameter, which characterizes the degree to which a resonance state is complex valued, is measured for nonoverlapping resonances, and is found to be proportional to the nonhomogeneous contribution to the line broadening of the resonance. A simple two-level model based on the effective Hamiltonian formalism supports our experimental results.

Quantum chaos in the nuclear collective model. II. Peres lattices

This is a continuation of our paper [Phys. Rev. E 79, 046202 (2009)] devoted to signatures of quantum chaos in the geometric collective model of atomic nuclei. We apply the method by Peres to study ordered and disordered patterns in quantum spectra drawn as lattices in the plane of energy vs average of a chosen observable. Good qualitative agreement with standard measures of chaos is manifested. The method provides an efficient tool for studying structural changes in eigenstates across quantum spectra of general systems.

Leaking billiards

Billiards are idealizations for systems where particles or waves are confined to cavities, or to other homogeneous regions. In billiard systems a point particle moves freely except for specular reflections from rigid walls. However, billiard walls are not always completely reflective and measurements inside can also open the billiard. Since boundary openings have been studied extensively in the literature, we rather model leakages inside the billiard. In particular, we investigate the classical dynamics of a leakage for a continuous family of billiard systems, that is, the stadium-lemon-billiard family. With a single parameter the geometry of the billiard can be tuned from stadium (being fully hyperbolic) over circle (integrable) to the lemon-shaped billiard (mixed chaotic). For the stadium billiard we found an algebraically decaying mean escape time with the linear size of the leakage n(esc) approximately epsilon-1 together with an exponential decay of the survival probability distribution. The finding is nearly independent of the position and size of the leakage, as long as the leakage is much smaller than the system size, and it is in good agreement with a stochastic map approximation of the dynamics. Due to the mixed phase space for lemon billiards, the mean escape time depends both on the position and geometry of the leakage. For systems where quasiregular motion dominates, we found a linear dependence of the mean escape time, n(esc) approximately 1-epsilon, which we refer to as flooding law. Our findings are helpful in understanding dynamics of leaking Hamiltonian systems.

Spectral statistics in an open parametric billiard system

We present experimental results on the eigenfrequency statistics of a superconducting, chaotic microwave billiard containing a rotatable obstacle. Deviations of the spectral fluctuations from predictions based on Gaussian orthogonal ensembles of random matrices are found. They are explained by treating the billiard as an open scattering system in which microwave power is coupled in and out via antennas. To study the interaction of the quantum (or wave) system with its environment, a highly sensitive parametric correlator is used.

Chaotic dynamics and orbit stability in the parabolic oval billiard

Chaotic properties of the one-parameter family of oval billiards with parabolic boundaries are investigated. Classical dynamics of such billiard is mixed and depends sensitively on the value of the shape parameter. Deviation matrices of some low period orbits are analyzed. Special attention is paid to the stability of orbits bouncing at the singular joining points of the parabolic arcs, where the boundary curvature is discontinuous. The existence of such orbits is connected with the segmentation of the phase space into two or more chaotic components. The obtained results are illustrated by numerical calculations of the Poincaré sections and compared with the properties of the elliptical stadium billiards.

Fractal analysis of chaotic classical scattering in a cut-circle billiard with two openings

We investigate the fractal behavior of the transmission of a classical particle through a circular billiard with a straight cut and two openings. As the size of the cut varies, the phase space of the closed billiard shows a full range of dynamical behavior, including integrable behavior, soft chaos (mixed phase space), and hard chaos (ergodic and mixing). For an open billiard, we numerically find the exit opening as a function of the incident angle. When the billiard is chaotic, the result shows self-similarity and infinite complexity. We calculate the fractal dimension of this structure using a box-counting method when two parameters, the size of the cut and the size of the openings, are varied.

Speckle statistics in a chaotic multimode fiber

Wave chaos is devoted to the study of wave motion when the geometrical limit of rays is chaotic. Imprints of ray chaos may be found either in spectral and spatial properties of modes or in spatio-temporal evolution of wave packets. In this paper, we present a thorough experimental and theoretical analysis of field statistics for light propagating in a multimode fiber with a noncircular cross section. This optical fiber serves as a powerful tool to image waves in a system where light rays exhibit a chaotic dynamics. We show that, in the speckle regime, the experimentally measured statistical properties of intensity patterns are well accounted for by a "random Gaussian" hypothesis. A comparison is also made in the case of regular ray motion by using a circular optical fiber. Possible extensions and applications of the tools and concepts of wave chaos are mentioned in modern communication technology.

Light scarring in an optical fiber

We report the first experimental study of wave scarring in an optical fiber with a noncircular cross section. This optical multimode fiber serves as a powerful tool to image waves in a system where light rays exhibit a chaotic dynamics. Far-field intensity measurements are used to provide a better identification of scars in the Fourier domain. This first experimental characterization of scarring effect in optics demonstrates the relevance of such an optical waveguide for novel experiments in wave chaos.

Optimized absorption in a chaotic double-clad fiber amplifier

Double-clad fibers with a doped single-mode core and a noncylindrical multimode chaotic cladding are shown to provide optimal pump-power absorption in power amplifiers. Based on the chaotic dynamics of rays in such fibers, we propose a quantitative theory for the pump-absorption ratio and favorably compare the predictions of the theory with numerical results obtained through an adapted beam-propagation scheme.

Quantum-classical correspondences of the Berry-Robnik parameter through bifurcations in lemon billiard systems

The quantum level statistics affected by bifurcations in classical dynamics is studied by using a one-parameter family of lemon billiard systems. The classical phase space of our system consists of regular and irregular regions. We determine an analytic solution of the phase volume for these regions as a function of the system parameter and show that the function reveals a cusp singularity at the bifurcation point. The function is compared with its quantum mechanical counterpart, the Berry-Robnik parameter. By estimating the semiclassical regime from the effective Planck constant that validates the quantum-classical correspondence of the Berry-Robnik parameter, we determine a region of the system parameter where the cusp can be reproduced by the statistical properties of the eigenenergy levels.

Integrability and action operators in quantum Hamiltonian systems

For a (classically) integrable quantum-mechanical system with two degrees of freedom, the functional dependence H=HQ(J1,J2) of the Hamiltonian operator on the action operators is analyzed and compared with the corresponding functional relationship H(p1,q1;p2,q2)=HC(J1,J2) in the classical limit of that system. The former converges toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, nonuniform. The existence of the function H=HQ(J1,J2) in the integrable regime of a parametric quantum system explains empirical results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The analysis is carried out for an integrable one-parameter two-spin model. Additional results presented for the (integrable) circular billiard model illuminate the same conclusions from a different angle.

Scattering properties of a cut-circle billiard waveguide with two conical leads

We examine a two-dimensional electron waveguide with a cut-circle cavity and conical leads. By considering Wigner delay times and the Landauer-Büttiker conductance for this system, we probe the effects of the closed billiard energy spectrum on scattering properties in the limit of weakly coupled leads. We investigate how lead placement and cavity shape affect these conductance and time delay spectra of the waveguide.