Resonance- and Chaos-Assisted Tunneling

We consider dynamical tunneling between two symmetry-related regular islands that are separated in phase space by a chaotic sea. Such tunneling processes are dominantly governed by nonlinear resonances, which induce a coupling mechanism between ``regular'' quantum states within and ``chaotic'' states outside the islands. By means of a random matrix ansatz for the chaotic part of the Hamiltonian, one can show that the corresponding coupling matrix element directly determines the level splitting between the symmetric and the antisymmetric eigenstates of the pair of islands. We show in detail how this matrix element can be expressed in terms of elementary classical quantities that are associated with the resonance. The validity of this theory is demonstrated with the kicked Harper model.Comment: 25 pages, 5 figure

Gallavotti-Cohen theorem, Chaotic Hypothesis and the zero-noise limit

The Fluctuation Relation for a stationary state, kept at constant energy by a deterministic thermostat - the Gallavotti-Cohen Theorem -- relies on the ergodic properties of the system considered. We show that when perturbed by an energy-conserving random noise, the relation follows trivially for any system at finite noise amplitude. The time needed to achieve stationarity may stay finite as the noise tends to zero, or it may diverge. In the former case the Gallavotti-Cohen result is recovered, while in the latter case, the crossover time may be computed from the action of `instanton' orbits that bridge attractors and repellors. We suggest that the `Chaotic Hypothesis' of Gallavotti can thus be reformulated as a matter of stochastic stability of the measure in trajectory space. In this form this hypothesis may be directly tested

A realistic example of chaotic tunneling: The hydrogen atom in parallel static electric and magnetic fields

Statistics of tunneling rates in the presence of chaotic classical dynamics is discussed on a realistic example: a hydrogen atom placed in parallel uniform static electric and magnetic fields, where tunneling is followed by ionization along the fields direction. Depending on the magnetic quantum number, one may observe either a standard Porter-Thomas distribution of tunneling rates or, for strong scarring by a periodic orbit parallel to the external fields, strong deviations from it. For the latter case, a simple model based on random matrix theory gives the correct distribution.Comment: Submitted to Phys. Rev.

Semiclassical Trace Formulas for Noninteracting Identical Particles

We extend the Gutzwiller trace formula to systems of noninteracting identical particles. The standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur in continuous families. The identical nature of the particles also introduces discrete permutational symmetries. We exploit the formalism of Creagh and Littlejohn [Phys. Rev. A 44, 836 (1991)], who have studied semiclassical dynamics in the presence of continuous symmetries, to derive many-body trace formulas for the full and symmetry-reduced densities of states. Numerical studies of the three-particle cardioid billiard are used to explicitly illustrate and test the results of the theory.Comment: 29 pages, 11 figures, submitted to PR

Maslov indices and monodromy

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived

Spectral Statistics in the Quantized Cardioid Billiard

The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the level-spacing distribution is in good agreement with the GOE distribution of random-matrix theory. In case of the number variance and rigidity we observe agreement with the random-matrix model for short-range correlations only, whereas for long-range correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosine-modulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits instead of exclusively Gaussian peaks.Comment: 20 pages, uu-encoded ps.Z-fil

Signatures of chaotic tunnelling

Recent experiments with cold atoms provide a significant step toward a better understanding of tunnelling when irregular dynamics is present at the classical level. In this paper, we lay out numerical studies which shed light on the previous experiments, help to clarify the underlying physics and have the ambition to be guidelines for future experiments.Comment: 11 pages, 9 figures, submitted to Phys. Rev. E. Figures of better quality can be found at http://www.phys.univ-tours.fr/~mouchet

Three disks in a row: A two-dimensional scattering analog of the double-well problem

We investigate the scattering off three nonoverlapping disks equidistantly spaced along a line in the two-dimensional plane with the radii of the outer disks equal and the radius of the inner disk varied. This system is a two-dimensional scattering analog to the double-well-potential (bound state) problem in one dimension. In both systems the symmetry splittings between symmetric and antisymmetric states or resonances, respectively, have to be traced back to tunneling effects, as semiclassically the geometrical periodic orbits have no contact with the vertical symmetry axis. We construct the leading semiclassical ``creeping'' orbits that are responsible for the symmetry splitting of the resonances in this system. The collinear three-disk-system is not only one of the simplest but also one of the most effective systems for detecting creeping phenomena. While in symmetrically placed n-disk systems creeping corrections affect the subleading resonances, they here alone determine the symmetry splitting of the 3-disk resonances in the semiclassical calculation. It should therefore be considered as a paradigm for the study of creeping effects. PACS numbers: 03.65.Sq, 03.20.+i, 05.45.+bComment: replaced with published version (minor misprints corrected and references updated); 23 pages, LaTeX plus 8 Postscript figures, uses epsfig.sty, espf.sty, and epsf.te

Chaos assisted tunnelling with cold atoms

In the context of quantum chaos, both theory and numerical analysis predict large fluctuations of the tunnelling transition probabilities when irregular dynamics is present at the classical level. We consider here the non-dissipative quantum evolution of cold atoms trapped in a time-dependent modulated periodic potential generated by two laser beams. We give some precise guidelines for the observation of chaos assisted tunnelling between invariant phase space structures paired by time-reversal symmetry.Comment: submitted to Phys. Rev. E ; 16 pages, 13 figures; figures of better quality can be found at http://www.phys.univ-tours.fr/~mouchet