Semiclassical universality of parametric spectral correlations

We consider quantum systems with a chaotic classical limit that depend on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor $K(\tau,x)$ which depends on a scaled parameter difference $x$. For parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the small $\tau$ expansion of the form factor agrees with Random Matrix Theory for systems with and without time reversal symmetry.Comment: 18 pages, no figure

Internal and External Resonances of Dielectric Disks

Circular microresonators (microdisks) are micron sized dielectric disks embedded in a material of lower refractive index. They possess modes with complex eigenvalues (resonances) which are solutions of analytically given transcendental equations. The behavior of such eigenvalues in the small opening limit, i.e. when the refractive index of the cavity goes to infinity, is analysed. This analysis allows one to clearly distinguish between internal (Feshbach) and external (shape) resonant modes for both TM and TE polarizations. This is especially important for TE polarization for which internal and external resonances can be found in the same region of the complex wavenumber plane. It is also shown that for both polarizations, the internal as well as external resonances can be classified by well defined azimuthal and radial modal indices.Comment: 5 pages, 8 figures, pdflate

Spectral statistics in chaotic systems with a point interaction

We consider quantum systems with a chaotic classical limit that are perturbed by a point-like scatterer. The spectral form factor K(tau) for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order tau^2 and tau^3 that off-diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer.Comment: LaTeX, 21 pages, 7 figures, small corrections, new references adde

Small-scale instabilities in dynamical systems with sliding

NOTICE: this is the author’s version of a work that was accepted for publication in Physica D: Nonlinear Phenomena . Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D: Nonlinear Phenomena , Vol. 239 Issues 1-2 (2010), DOI: 10.1016/j.physd.2009.10.003We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos

Early-warning indicators for rate-induced tipping

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.A dynamical system is said to undergo rate-induced tipping when it fails to track its quasi-equilibrium state due to an above-critical-rate change of system parameters. We study a prototypical model for rate-induced tipping, the saddle-node normal form subject to time-varying equilibrium drift and noise. We find that both most commonly used early-warning indicators, increase in variance and increase in autocorrelation, occur not when the equilibrium drift is fastest but with a delay. We explain this delay by demonstrating that the most likely trajectory for tipping also crosses the tipping threshold with a delay, and therefore, the tipping itself is delayed. We find solutions of the variational problem determining the most likely tipping path using numerical continuation techniques. The result is a systematic study of the most likely tipping time in the plane of two parameters, distance from tipping threshold and noise intensity

Spectral Statistics of "Cellular" Billiards

For a bounded planar domain $\Omega^0$ whose boundary contains a number of flat pieces $\Gamma_i$ we consider a family of non-symmetric billiards $\Omega$ constructed by patching several copies of $\Omega^0$ along $\Gamma_i$'s. It is demonstrated that the length spectrum of the periodic orbits in $\Omega$ is degenerate with the multiplicities determined by a matrix group $G$. We study the energy spectrum of the corresponding quantum billiard problem in $\Omega$ and show that it can be split in a number of uncorrelated subspectra corresponding to a set of irreducible representations $\alpha$ of $G$. Assuming that the classical dynamics in $\Omega^0$ are chaotic, we derive a semiclassical trace formula for each spectral component and show that their energy level statistics are the same as in standard Random Matrix ensembles. Depending on whether ${\alpha}$ is real, pseudo-real or complex, the spectrum has either Gaussian Orthogonal, Gaussian Symplectic or Gaussian Unitary types of statistics, respectively.Comment: 18 pages, 4 figure

Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs I: construction and numerical results

In a series of two papers we investigate the universal spectral statistics of chaotic quantum systems in the ten known symmetry classes of quantum mechanics. In this first paper we focus on the construction of appropriate ensembles of star graphs in the ten symmetry classes. A generalization of the Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry classes. The conjecture is supported by numerical results that demonstrate the fidelity of the spectral statistics of star graphs to the corresponding Gaussian random-matrix theories.Comment: 15 page

On the Form Factor for the Unitary Group

We study the combinatorics of the contributions to the form factor of the group U(N) in the large $N$ limit. This relates to questions about semiclassical contributions to the form factor of quantum systems described by the unitary ensemble.Comment: 35 page

Geometrical theory of diffraction and spectral statistics

We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit $\hbar \to 0$. This result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space.Comment: 14 pages, no figure, to appear in J. Phys.

Semiclassical expansion of parametric correlation functions of the quantum time delay

We derive semiclassical periodic orbit expansions for a correlation function of the Wigner time delay. We consider the Fourier transform of the two-point correlation function, the form factor $K(\tau,x,y,M)$, that depends on the number of open channels $M$, a non-symmetry breaking parameter $x$, and a symmetry breaking parameter $y$. Several terms in the Taylor expansion about $\tau=0$, which depend on all parameters, are shown to be identical to those obtained from Random Matrix Theory.Comment: 21 pages, no figure