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arXiv:physics/9910035 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Analyzing symmetry breaking within a chaotic quantum system via Bayesian inference

C. I. Barbosa, Hanns Ludwig Harney

Published Aug 1, 2000• 4 citations
arXivHTMLPDF

Abstract

Bayesian inference is applied to the level fluctuations of two coupled microwave billiards in order to extract the coupling strength. The coupled resonators provide a model of a chaotic quantum system containing two coupled symmetry classes of levels. The number variance is used to quantify the level fluctuations as a function of the coupling and to construct the conditional probability distribution of the data. The prior distribution of the coupling parameter is obtained from an invariance argument on the entropy of the posterior distribution.

DOI
10.1103/physreve.62.1897

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Bayesian inference is applied to the level fluctuations of two coupled microwave billiards in order to extract the coupling strength. The coupled resonators provide a model of a chaotic quantum system containing two coupled symmetry classes of levels. The number variance is used to quantify the level fluctuations as a function of the coupling and to construct the conditional probability distribution of the data. The prior distribution of the coupling parameter is obtained from an invariance argument on the entropy of the posterior distribution.
Flat microwave cavities can be used to experimentally simulate quantum mechanical systems. By coupling two such cavities, we study the equivalent to symmetry breaking in quantum mechanics. As the coupling is tunable, we can measure resonance strength distributions as a function of the symmetry breaking. We analyze the data by employing a qualitative model based on random matrix theory and show that the results derived from the strength distribution are consistent with those previously obtained from spectral statistics.
Flat microwave cavities can be used to experimentally simulate quantum mechanical systems. By coupling two such cavities, we study the equivalent to the symmetry breaking in quantum mechanics. As the coupling is tunable, we can measure resonance strength distributions as a function of the symmetry breaking. We analyze the data employing a qualitative model based on Random Matrix Theory (RMT) and show that the results derived from the strength distribution are consistent with those previously obtained from spectral statistics.
We report on the experimental study of the spectral properties of quantum systems consisting of two quantum billiards (QBs), one with chaotic, the other one with integrable classical dynamics, that are coupled to each other via an opening in a common wall. They are compared to those of a special case of the Rosenzweig-Porter model with random matrices composed of two diagonal blocks modeling the spectral properties of the QBs, that are coupled with a tunable parameter. We demonstrate that this model is suitable for the description of the experimental data and thus may be employed to determine the strength of the coupling. It results from the increasing overlap of eigenmodes in the QBs penetrating through the opening into the other one, leading to a mixing of their eigenstates, and the breaking of the symmetry present in the QB with integrable dynamics. This implicates deviations of the spectral properties from those of typical quantum systems with integrable and chaotic dynamics, respectively, and approaches those of a fully chaotic system for sufficiently large coupling strength. In contrast in previous studies the transition from integrable to chaotic dynamics was induced by introducing a random potential of increasing strength into such a QB and applicability of a variant of the Rosenzweig-Porter model was tested.
We investigate the average eigenstate entanglement in a bipartite many-body system which exhibits a breaking of two local conservation laws into a global conserved quantity. Such a setting is realized by the particle number conservation in Bose-Hubbard systems. We devise a corresponding random matrix model which captures the universal features of this symmetry breaking and allows for applying powerful random matrix methods. By combining the concept of symmetry resolved entanglement with perturbation theory for quantum chaotic systems we obtain a universal entanglement transition depending on a single fundamental parameter. Furthermore, the symmetry resolved entanglement allows for separating the genuine entanglement from the part which originates from the conserved quantity. This latter contribution is quantified by the number entropy. For this it is shown that the symmetry breaking generates a localization of the eigenstates due to a banded structure of the time-evolution operator. By extrapolating the results beyond the perturbative regime we obtain an analytic description of the full transition.
Gibbs states are familiar from statistical mechanics, yet their use is not limited to that domain. For instance, they also feature in the maximum entropy reconstruction of quantum states from incomplete measurement data. Outside the macroscopic realm, however, estimating a Gibbs state is a nontrivial inference task, due to two complicating factors: the proper set of relevant observables might not be evident a priori; and whenever data are gathered from a small sample only, the best estimate for the Lagrange parameters is invariably affected by the experimenter's prior bias. I show how the two issues can be tackled with the help of Bayesian model selection and Bayesian interpolation, respectively, and illustrate the use of these Bayesian techniques with a number of simple examples.