Representations of classical Lie groups and quantized free convolution

We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.Comment: 43 pages, 4 figures. v3: relation to the Markov-Krein correspondence is updated and correcte

Single qubit decoherence under a separable coupling to a random matrix environment

This paper describes the dynamics of a quantum two-level system (qubit) under the influence of an environment modeled by an ensemble of random matrices. In distinction to earlier work, we consider here separable couplings and focus on a regime where the decoherence time is of the same order of magnitude than the environmental Heisenberg time. We derive an analytical expression in the linear response approximation, and study its accuracy by comparison with numerical simulations. We discuss a series of unusual properties, such as purity oscillations, strong signatures of spectral correlations (in the environment Hamiltonian), memory effects and symmetry breaking equilibrium states.Comment: 13 pages, 7 figure

A trivial observation on time reversal in random matrix theory

It is commonly thought that a state-dependent quantity, after being averaged over a classical ensemble of random Hamiltonians, will always become independent of the state. We point out that this is in general incorrect: if the ensemble of Hamiltonians is time reversal invariant, and the quantity involves the state in higher than bilinear order, then we show that the quantity is only a constant over the orbits of the invariance group on the Hilbert space. Examples include fidelity and decoherence in appropriate models.Comment: 7 pages 3 figure

A random matrix approach to decoherence

In order to analyze the effect of chaos or order on the rate of decoherence in a subsystem, we aim to distinguish effects of the two types of dynamics by choosing initial states as random product states from two factor spaces representing two subsystems. We introduce a random matrix model that permits to vary the coupling strength between the subsystems. The case of strong coupling is analyzed in detail, and we find no significant differences except for very low-dimensional spaces.Comment: 11 pages, 5 eps-figure

Anomalous slow fidelity decay for symmetry breaking perturbations

Symmetries as well as other special conditions can cause anomalous slowing down of fidelity decay. These situations will be characterized, and a family of random matrix models to emulate them generically presented. An analytic solution based on exponentiated linear response will be given. For one representative case the exact solution is obtained from a supersymmetric calculation. The results agree well with dynamical calculations for a kicked top.Comment: 4 pages, 2 figure

The multilevel trigger system of the DIRAC experiment

The multilevel trigger system of the DIRAC experiment at CERN is presented. It includes a fast first level trigger as well as various trigger processors to select events with a pair of pions having a low relative momentum typical of the physical process under study. One of these processors employs the drift chamber data, another one is based on a neural network algorithm and the others use various hit-map detector correlations. Two versions of the trigger system used at different stages of the experiment are described. The complete system reduces the event rate by a factor of 1000, with efficiency $\geq$95% of detecting the events in the relative momentum range of interest.Comment: 21 pages, 11 figure

Scattering fidelity in elastodynamics

The recent introduction of the concept of scattering fidelity, causes us to revisit the experiment by Lobkis and Weaver [Phys. Rev. Lett. 90, 254302 (2003)]. There, the ``distortion'' of the coda of an acoustic signal is measured under temperature changes. This quantity is in fact the negative logarithm of scattering fidelity. We re-analyse their experimental data for two samples, and we find good agreement with random matrix predictions for the standard fidelity. Usually, one may expect such an agreement for chaotic systems only. While the first sample, may indeed be assumed chaotic, for the second sample, a perfect cuboid, such an agreement is more surprising. For the first sample, the random matrix analysis yields a perturbation strength compatible with semiclassical predictions. For the cuboid the measured perturbation strength is much larger than expected, but with the fitted values for this strength, the experimental data are well reproduced.Comment: 4 page

Fidelity amplitude of the scattering matrix in microwave cavities

The concept of fidelity decay is discussed from the point of view of the scattering matrix, and the scattering fidelity is introduced as the parametric cross-correlation of a given S-matrix element, taken in the time domain, normalized by the corresponding autocorrelation function. We show that for chaotic systems, this quantity represents the usual fidelity amplitude, if appropriate ensemble and/or energy averages are taken. We present a microwave experiment where the scattering fidelity is measured for an ensemble of chaotic systems. The results are in excellent agreement with random matrix theory for the standard fidelity amplitude. The only parameter, namely the perturbation strength could be determined independently from level dynamics of the system, thus providing a parameter free agreement between theory and experiment

Random matrix description of decaying quantum systems

This contribution describes a statistical model for decaying quantum systems (e.g. photo-dissociation or -ionization). It takes the interference between direct and indirect decay processes explicitely into account. The resulting expressions for the partial decay amplitudes and the corresponding cross sections may be considered a many-channel many-resonance generalization of Fano's original work on resonance lineshapes [Phys. Rev 124, 1866 (1961)]. A statistical (random matrix) model is then introduced. It allows to describe chaotic scattering systems with tunable couplings to the decay channels. We focus on the autocorrelation function of the total (photo) cross section, and we find that it depends on the same combination of parameters, as the Fano-parameter distribution. These combinations are statistical variants of the one-channel Fano parameter. It is thus possible to study Fano interference (i.e. the interference between direct and indirect decay paths) on the basis of the autocorrelation function, and thereby in the regime of overlapping resonances. It allows us, to study the Fano interference in the limit of strongly overlapping resonances, where we find a persisting effect on the level of the weak localization correction.Comment: 16 pages, 2 figure