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Register a new userFidelity and level correlations in the transition from regularity to chaos
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Mean fidelity amplitude and parametric energy--energy correlations are calculated exactly for a regular system, which is subject to a chaotic random perturbation. It turns out that in this particular case under the average both quantities are identical. The result is compared with the susceptibility of chaotic systems against random perturbations. Regular systems are more susceptible against random perturbations than chaotic ones.
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The interaction of a quantized electromagnetic field in a cavity with a set of two-level atoms inside can be described with algebraic Hamiltonians of increasing complexity, from the Rabi to the Dicke models. Their algebraic character allows, through the use of coherente states, a semiclassical description in phase space, where the non-integrable Dicke model has regions associated with regular and chaotic motion. The appearance of classical chaos can be quantified calculating the largest Lyapunov exponent in the whole available phase space for a given energy. In the quantum regime, employing efficient diagonalization techniques, we are able to perform a detailed quantitative study of the regular and chaotic regions, where the quantum Participation Ratio (PR) of coherent states on the eigenenergy basis plays a role equivalent to the Lyapunov exponent. It is noted that, in the thermodynamic limit, dividing the Participation Ratio by the number of atoms leads to a positive value in chaotic regions, while it tends to zero in the regular ones.
We derive fidelity decay and parametric energy correlations for random matrix ensembles where time--reversal invariance of the original Hamiltonian is broken by the perturbation. Like in the case of a symmetry conserving perturbation a simple relation between both quantities can be established.
Recent years have seen an increasing interest in quantum chaos and related aspects of spatially extended systems, such as spin chains. However, the results are strongly system dependent, generic approaches suggest the presence of many-body localization while analytical calculations for certain system classes, here referred to as the ``self-dual case, prove adherence to universal (chaotic) spectral behavior. We address these issues studying the level statistics in the vicinity of the latter case, thereby revealing transitions to many-body localization as well as the appearance of several non-standard random-matrix universality classes.
The operator fidelity is a measure of the information-theoretic distinguishability between perturbed and unperturbed evolutions. The response of this measure to the perturbation may be formulated in terms of the operator fidelity susceptibility (OFS), a quantity which has been used to investigate the parameter spaces of quantum systems in order to discriminate their regular and chaotic regimes. In this work we numerically study the OFS for a pair of non-linearly coupled two-dimensional harmonic oscillators, a model which is equivalent to that of a hydrogen atom in a uniform external magnetic field. We show how the two terms of the OFS, being linked to the main properties that differentiate regular from chaotic behavior, allow for the detection of this models transition between the two regimes. In addition, we find that the parameter interval where perturbation theory applies is delimited from above by a local minimum of one of the analyzed terms.
High resolution experiments have recently lead to a complete identification (energy, spin, and parity) of 151 nuclear levels up to an excitation Energy of Ex= 6.20 MeV in 208Pb. We present a thorough study of the fluctuation properties in the energy spectra of the unprecedented set of nuclear bound states. In a first approach we grouped states with the same spin and parity into 14 subspectra, analyzed standard statistical measures for short- and long-range correlations and then computed their ensemble average. Their comparison with a random matrix ensemble which interpolates between Poisson statistics expected for regular systems and the Gaussian Orthogonal Ensemble (GOE) predicted for chaotic systems shows that the data are well described by the GOE. In a second approach, following an idea of Rosenzweig and Porter we considered the complete spectrum composed of the independent subspectra. We analyzed their fluctuation properties using the method of Bayesian inference involving a quantitative measure, called the chaoticity parameter f, which also interpolates between Poisson (f=0) and GOE statistics (f=1). It turns out to be f~0.9. This is so far the closest agreement with GOE observed in spectra of bound states in a nucleus. The same analysis has also been performed with spectra computed on the basis of shell model calculations with different interactions (SDI, KB, M3Y). While the simple SDI exhibits features typical for nuclear many-body systems with regular dynamics, the other, more realistic interactions yield chaoticity parameters f close to the experimental values.
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