7 research outputs found
Universality in the flooding of regular islands by chaotic states
We investigate the structure of eigenstates in systems with a mixed phase
space in terms of their projection onto individual regular tori. Depending on
dynamical tunneling rates and the Heisenberg time, regular states disappear and
chaotic states flood the regular tori. For a quantitative understanding we
introduce a random matrix model. The resulting statistical properties of
eigenstates as a function of an effective coupling strength are in very good
agreement with numerical results for a kicked system. We discuss the
implications of these results for the applicability of the semiclassical
eigenfunction hypothesis.Comment: 11 pages, 12 figure
Avoided intersections of nodal lines
We consider real eigen-functions of the Schr\"odinger operator in 2-d. The
nodal lines of separable systems form a regular grid, and the number of nodal
crossings equals the number of nodal domains. In contrast, for wave functions
of non integrable systems nodal intersections are rare, and for random waves,
the expected number of intersections in any finite area vanishes. However,
nodal lines display characteristic avoided crossings which we study in the
present work. We define a measure for the avoidance range and compute its
distribution for the random waves ensemble. We show that the avoidance range
distribution of wave functions of chaotic systems follow the expected random
wave distributions, whereas for wave functions of classically integrable but
quantum non-separable wave functions, the distribution is quite different.
Thus, the study of the avoidance distribution provides more support to the
conjecture that nodal structures of chaotic systems are reproduced by the
predictions of the random waves ensemble.Comment: 12 pages, 4 figure
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.Comment: 13 pages, 10 figure
Wavepacket Dynamics in Nonlinear Schrödinger Equations
International audienceCoherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schrödinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: the centre follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satis?fies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape, or of the confi?ning potential. A rich variety of e?ects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the condensate is transferred towards uncoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped towards the bottom of the well