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 120 000 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 20 000 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 Bastistic 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.

Classical billiards and quantum fluids.

The dynamics of a particle confined in the elliptical stadium billiard with rectangular thickness 2t, major axis 2a, and minor axis 2b=2 is numerically investigated in a reduced phase space with discrete time n. Both relative measure r(n), with asymptotic value r(n→∞)=r(∞) and Shannon entropy s, are calculated in the vicinity of a particular line in the a×t parameter space, namely t(c)=t(0)(a)=√a(2)-1, with a∈(1,√4/3). If t

Extreme events in chaotic lasers with modulated parameter.

We study a theoretical model describing a laser with a modulated parameter, concentrating on the appearance of extreme events, also called optical rogue pulses. It is shown that two conditions are required for the appearance of such events in this type of nonlinear system: the existence of generalized multi-stability and the collisions of chaotic attractors with unstable orbits in external crisis, expanding the attractor to visit new regions in phase space.

Ergodicity and quantum correlations in irrational triangular billiards.

Pseudochaotic properties are systematically investigated in a one-parameter family of irrational triangular billiards (all angles irrational with π). The absolute value of the position correlation function C(x)(t) decays like ~t(-α). Fast (α≈1) and slow (0<α<1) decays are observed, thus indicating that the irrational triangles do not share a unique ergodic dynamics, which, instead, may vary smoothly between the opposite limits of strong mixing (α=1) and regular behaviors (α=0). Upgrading previous data, spectral statistical properties of the quantized counterparts are computed from 150000 energy eigenvalues numerically calculated for each billiard. Gaussian orthogonal ensemble spectral fluctuations are observed when α≈1 and intermediate statistics are found otherwise. Our irrational billiards have zero Kolmogorov-Sinai entropy and essentially infinity genus. Thus, differently from previous works on rational (pseudointegrable) enclosures, our results provide a missing classical-quantum correspondence regarding the ergodic hierarchy for a set of nonchaotic systems that might enjoy the strong mixing property.

Quantum properties of irrational triangular billiards.

Triangles with sides given by consecutive integers (N , N+1 , N+2) are fully irrational (all angles irrational with pi) if 3120 and eventually the occurrence of gaps in the distribution for N ~ 180 define the onset of a long crossover towards the sequence observed in the integrable limit of the equilateral triangle (N-->infinity) .

Numerical experiments on quantum chaotic billiards.

A recently proposed numerical technique for generation of high-quality unstructured meshes is combined with a finite-element method to solve the Helmholtz equation that describes the quantum mechanics of a particle confined in two-dimensional cavities. Different shapes are treated on equal footing, including Sinai, stadium, annular, threefold symmetric, mushroom, cardioid, triangle, and coupled billiards. The results are shown to be in excellent agreement with available measurements in flat microwave resonator counterparts with nonintegrable geometries.

Mode locking of spin waves excited by direct currents in microwave nano-oscillators.

A spin-wave theory is presented which explains the frequency pulling and mode locking observed when two closely spaced spin-transfer nanometer-scale oscillators with slightly different frequencies are separately driven in the same magnetic thin film by spin-polarized carriers at high direct-current densities. The theory confirms recent experimental evidence that the origin of the phenomena lies in the nonlinear interaction between two overlapping spin waves excited in the magnetic nanostructure.

Spin-wave theory for the dynamics induced by direct currents in magnetic multilayers.

A spin-wave theory is presented for the magnetization dynamics in a ferromagnetic film that is traversed by spin-polarized carriers at high direct-current densities. It is shown that nonlinear effects due to four-magnon interactions arising from dipolar and surface anisotropy energies limit the growth of the driven spin wave and produce shifts in the microwave frequency oscillations. The theory explains quantitatively recent experimental results in nanometric point contacts onto magnetic multilayers showing downward frequency shifts (redshifts) with increasing current, if the external field is on the film plane, and upward shifts (blueshifts), if the field is perpendicular to the film.