Multivariate analysis of short time series in terms of ensembles of correlation matrices.

When dealing with non-stationary systems, for which many time series are available, it is common to divide time in epochs, i.e. smaller time intervals and deal with short time series in the hope to have some form of approximate stationarity on that time scale. We can then study time evolution by looking at properties as a function of the epochs. This leads to singular correlation matrices and thus poor statistics. In the present paper, we propose an ensemble technique to deal with a large set of short time series without any consideration of non-stationarity. Given a singular data matrix, we randomly select subsets of time series and thus create an ensemble of non-singular correlation matrices. As the selection possibilities are binomially large, we will obtain good statistics for eigenvalues of correlation matrices, which are typically not independent. Once we defined the ensemble, we analyze its behavior for constant and block-diagonal correlations and compare numerics with analytic results for the corresponding correlated Wishart ensembles. We discuss differences resulting from spurious correlations due to repetitive use of time-series. The usefulness of this technique should extend beyond the stationary case if, on the time scale of the epochs, we have quasi-stationarity at least for most epochs.

A generalized fidelity amplitude for open systems.

We consider a central system which is coupled via dephasing to an open system, i.e. an intermediate system which in turn is coupled to another environment. Considering the intermediate and far environment as one composite system, the coherences in the central system are given in the form of fidelity amplitudes for a certain perturbed echo dynamics in the composite environment. On the basis of the Born-Markov approximation, we derive a master equation for the reduction of that dynamics to the intermediate system alone. In distinction to an earlier paper (Moreno et al 2015 Phys. Rev. A 92, 030104. (doi:10.1103/PhysRevA.92.030104)), where we discussed the stabilizing effect of the far environment on the decoherence in the central system, we focus here on the possibility of using the measurable coherences in the central system for probing the open quantum dynamics in the intermediate system. We illustrate our results for the case of chaotic dynamics in the near environment, where we compare random matrix simulations with our analytical result.

First experimental realization of the Dirac oscillator.

We present the first experimental microwave realization of the one-dimensional Dirac oscillator, a paradigm in exactly solvable relativistic systems. The experiment relies on a relation of the Dirac oscillator to a corresponding tight-binding system. This tight-binding system is implemented as a microwave system by a chain of coupled dielectric disks, where the coupling is evanescent and can be adjusted appropriately. The resonances of the finite microwave system yield the spectrum of the one-dimensional Dirac oscillator with and without a mass term. The flexibility of the experimental setup allows the implementation of other one-dimensional Dirac-type equations.

Microwave fidelity studies by varying antenna coupling.

The fidelity decay in a microwave billiard is considered, where the coupling to an attached antenna is varied. The resulting quantity, coupling fidelity, is experimentally studied for three different terminators of the varied antenna: a hard-wall reflection, an open wall reflection, and a 50 Ω load, corresponding to a totally open channel. The model description in terms of an effective Hamiltonian with a complex coupling constant is given. Quantitative agreement is found with the theory obtained from a modified VWZ approach [J. J. M. Verbaarschot, Phys. Rep. 129, 367 (1985)].

Surprising relations between parametric level correlations and fidelity decay.

Relations among fidelity, cross-form-factor (i.e., parametric level correlations), and level velocity correlations are found both by deriving a Ward identity in a two-matrix model and by comparing exact results, using supersymmetry techniques, in the framework of random matrix theory. A power law decay near Heisenberg time, as a function of the relevant parameter, is shown to be at the root of revivals recently discovered for fidelity decay. For cross-form-factors the revivals are illustrated by a numerical study of a multiply kicked Ising spin chain.

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.

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 reanalyze 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 surprising. For the first sample, the random matrix analysis yields perturbation strengths compatible with semiclassical predictions. For the cuboid, the measured perturbation strengths are by a common factor of 5/3 too large. Apart from that, the experimental curves for the distortion are well reproduced.

Nonperiodic echoes from mushroom billiard hats.

Mushroom billiards have the remarkable property to show one or more clear cut integrable islands in one or several chaotic seas, without any fractal boundaries. The islands correspond to orbits confined to the hats of the mushrooms, which they share with the chaotic orbits. It is thus interesting to ask how long a chaotic orbit will remain in the hat before returning to the stem. This question is equivalent to the inquiry about delay times for scattering from the hat of the mushroom into an opening where the stem should be. For fixed angular momentum we find that no more than three different delay times are possible. This induces striking nonperiodic structures in the delay times that may be of importance for mesoscopic devices and should be accessible to microwave experiments.

Experimental verification of fidelity decay: from perturbative to fermi golden rule regime.

The scattering matrix was measured for a flat microwave cavity with classically chaotic dynamics. The system can be perturbed by small changes of the geometry. We define the "scattering fidelity" in terms of parametric correlation functions of scattering matrix elements. In chaotic systems and for weak coupling, the scattering fidelity approaches the fidelity of the closed system. Without free parameters, the experimental results agree with random matrix theory in a wide range of perturbation strengths, reaching from the perturbative to the Fermi golden rule regime.

Slow phase relaxation as a route to quantum computing beyond the quantum chaos border.

We reveal that phase memory can be much longer than energy relaxation in systems with exponentially large dimensions of Hilbert space; this finding is documented by 50 years of nuclear experiments, though the information is somewhat hidden. For quantum computers Hilbert spaces of dimension 2(100) or larger will be typical and therefore this effect may contribute significantly to reduce the problems of scaling of quantum computers to a useful number of qubits.

Symmetry breaking: a heuristic approach to chaotic scattering in many dimensions.

As the theory of chaotic scattering in high-dimensional systems is poorly developed, it is very difficult to determine initial conditions for which interesting scattering events, such as long delay times, occur. We propose to use symmetry breaking as a way to gain the insight necessary to determine low-dimensional subspaces of initial conditions in which we can find such events easily. We study numerically the planar scattering off a disk moving on an elliptic Kepler orbit, as a simplified model of the elliptic restricted three-body problem. When the motion of the disk is circular, the system has an integral of motion, the Jacobi integral, which is no longer conserved for nonvanishing eccentricity. In the latter case, the system has an effective five-dimensional phase space and is therefore not amenable for study with the usual methods. Using the symmetric problem as a starting point we define an appropriate two-dimensional subspace of initial conditions by fixing some coordinates. This subspace proves to be useful to define scattering experiments where the rich and nontrivial dynamics of the problem is illustrated. We consider in particular trajectories which take very long before escaping or are trapped by consecutive collisions with the disk.

Symmetry properties of periodic orbits extracted from scattering data.

Discrete symmetries of a system are reflected in the properties of the shortest periodic orbits. By applying a recent method to extract these from the scaling of the fractal structure in scattering functions, we show how the symmetries can be extracted from scattering data simultaneously with the periods and the Lyapunov exponents. We pay particular attention to the change of scattering data under a small symmetry breaking.

First experimental evidence for quantum echoes in scattering systems.

A self-pulsing effect termed quantum echoes has been observed in experiments with an open superconducting and a normal conducting microwave billiard whose geometry provides soft chaos, i.e., a mixed phase space portrait with a large stable island. For such systems a periodic response to an incoming pulse has been predicted. Its period has been associated with the degree of development of a horseshoe describing the topology of the classical dynamics. The experiments confirm this picture and reveal the topological information.

Wigner-Dyson statistics for a class of integrable models.

We construct an ensemble of second-quantized Hamiltonians with two bosonic degrees of freedom, whose members display with probability one Gaussian orthogonal ensemble (GOE) or Gaussian unitary ensemble (GUE) statistics. Nevertheless, these Hamiltonians have a second integral of motion, namely, the boson number, and thus are integrable. To construct this ensemble we use some "reverse engineering" starting from the fact that n bosons in a two-level system with random interactions have an integrable classical limit by the old Heisenberg association of boson operators to actions and angles. By choosing an n-body random interaction and degenerate levels we end up with GOE or GUE Hamiltonians. Ergodicity of these ensembles completes the example.

Phase shift experiments identifying Kramers doublets in a chaotic superconducting microwave billiard of threefold symmetry.

The spectral properties of a two-dimensional microwave billiard showing threefold symmetry have been studied with a new experimental technique. This method is based on the behavior of the eigenmodes under variation of a phase shift between two input channels, which strongly depends on the symmetries of the eigenfunctions. Thereby a complete set of 108 Kramers doublets has been identified by a simple and purely experimental method. This set clearly shows Gaussian unitary ensemble statistics, although the system is time-reversal invariant.

Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a microwave billiard with threefold symmetry.

Recently it has been shown that time-reversal invariant systems with discrete symmetries may display, in certain irreducible subspaces, spectral statistics corresponding to the Gaussian-unitary ensemble (GUE) rather than to the expected orthogonal one (GOE). A Kramers-type degeneracy is predicted in such situations. We present results for a microwave billiard with a threefold rotational symmetry and with the option to display or break a reflection symmetry. This allows us to observe the change from GOE to GUE statistics for one subset of levels. Since it was not possible to separate the three subspectra reliably, the number variances for the superimposed spectra were studied. The experimental results are compared with a theoretical and numerical study considering the effects of level splitting and level loss.

Signatures of the correlation hole in total and partial cross sections.

In a complex scattering system with few open channels, say a quantum dot with leads, the correlation properties of the poles of the scattering matrix are most directly related to the internal dynamics of the system. We may ask how to extract these properties from an analysis of cross sections. In general this is very difficult, if we leave the domain of isolated resonances. We propose to consider the cross correlation function of two different elastic or total cross sections. For these we can show numerically and to some extent also analytically a significant dependence on the correlations between the scattering poles. The difference between uncorrelated and strongly correlated poles is clearly visible, even for strongly overlapping resonances.

Spectral statistics of the two-body random ensemble revisited.

Using longer spectra we reanalyze spectral properties of the two-body random ensemble studied 30 years ago. At the center of the spectra the old results are largely confirmed, and we show that the nonergodicity is essentially due to the variance of the lowest moments of the spectra. The longer spectra allow us to test and reach the limits of validity of French's correction for the number variance. At the edge of the spectra we discuss the problems of unfolding in more detail. With a Gaussian unfolding of each spectrum the nearest-neighbor spacing distribution between ground state and first exited state is shown to be stable. Using such an unfolding the distribution tends toward a semi-Poisson distribution for longer spectra. For comparison with the nuclear table ensemble we could use such unfolding obtaining similar results as in the early papers, but an ensemble with realistic splitting gives reasonable results if we just normalize the spacings in accordance with the procedure used for the data.

Semi-Poisson statistics and beyond.

Semi-Poisson statistics are shown to be obtained by removing every other number from a random sequence. Retaining every (r+1)th level we obtain a family of sequences, which we call daisy models. Their statistical properties coincide with those of Bogomolny's nearest-neighbor interaction Coulomb gas if the inverse temperature coincides with the integer r. In particular, the case r=2 reproduces closely the statistics of quasioptimal solutions of the traveling salesman problem.

A pilot study employing Dehydroepiandrosterone (DHEA) in the treatment of chronic fatigue syndrome.

Patients with chronic fatigue syndrome (CFS) frequently associate the disease onset with a period of high physical and/or emotional stress. Alterations in hypothalamic-pituitary adrenal axis (HPA) function have been demonstrated. Although Cortisol production in patients with CFS has proven to be low, Dehydroepiandrosterone (DHEA) production has not been measured. DHEA output may be altered in this population. The purpose of this uncontrolled, prospective, 6 month study of 23 white women, ages 35-55 was to identify CFS patients with suboptimal serum levels of DHEA-sulphate (DHEA-S), defined as DHEA-S <2.0 microg/mL, and to treat those patients with oral DHEA. DHEA-S levels were re-measured after 4-6 weeks of oral DHEA therapy (25 mg). If DHEA-S remained <2.0 microg/ mL, or if no clinical response was achieved after 4-6 weeks of therapy, then an increased dose of DHEA was given. Physical and psychological impairment and disability status were measured by the MHAQII before DHEA intervention and at 3-month intervals. Of initially screened patients with CFS, 76% (116 of 153) were ages 35-55, and 89% (103 of 116) had suboptimal (<2.0 microg/mL) production of DHEA-S.Supplementation with DHEA to CFS patients lead to a significant reduction in the symptoms of CFS: pain (improved by 18%, p = 0.035), fatigue (decreased by 21%, p = 0.009)), activities of daily living (improved by 8.5%, p = 0.058), helplessness (decreased by 11%, p = 0.015), anxiety (decreased by 35%, p < 0.01), thinking (improved by 26%, p < 0.01), memory (improved by 17%, p < 0.05), and sexual problems (improved by 22%, p = 0.06) over the period of the trial. Further study is necessary to determine the safety and efficacy of supplementation of DHEA to this population in a controlled setting.