Mode Fluctuation Distribution Research Articles

Spectral statistics of a spin- [formula omitted] particle in coupled quartic oscillator potentials

The quantum spectral statistical properties, i.e. the nearest neighbor level spacing distribution (NNSD) and the mode fluctuation distribution of a spin- 1 2 particle in three-dimensional coupled quartic oscillator potentials are numerically studied. Introducing the coupling terms, the system can be continuously transformed from an integrable to chaotic one. With use of three parameter sets, three kinds of the Gaussian ensembles can be realized. The Brody-like NNSD interpolation formula between the Poisson distribution and that of each ensemble is found to work well for the numerically obtained NNSD.

Transition of Spectral Statistics of a Spin-1/2 Coupled Quartic Oscillators from GSE to Integrable Regime

A large amount of work has been done in the past decades to understand quantum chaos. Most work has been concentrated on a spinless particle in two dimensions, which provides the simplest and the most fundamental knowledge of quantum chaotic systems (see Ref.1, and references therein). In this work, we study the system of a spin-1/2 particle in three dimensions, which is also essential for physics. By choosing parameters, our system has three kinds of chaotic limits, which correspond to the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble(GUE) and the Gaussian Symplectic Ensemble(GSE). However, the intermediate regime between the integrable and the GSE (or the GUE) class has been scarcely studied. In addition to the nearest neighbor level spacing distribution(NNSD), We also adopt the mode fluctuation distribution(MFD), which was introduced for the signature of chaoticity. Half-integer spin and three dimensions lead us to the chaotic system that is not the GOE class. In this work, spectral statistics of the GOE, the GSE, the integrable classes and their intermediate ones are studied.

Spectral Statistics of a Spin-1/2 Coupled Quartic Oscillators

The quantum spectral statistical properties, i.e. the nearest neighbor level spacing distribution (NNSD), the mode fluctuation distribution (MFD), the spectral rigidity, etc., of a spin-1/2 particle in three-dimensional coupled quartic oscillator potentials are numerically studied. Changing coupling parameters, the system is continuously transformed from an integrable one to chaotic ones. Various kinds of the Gaussian ensembles (GE’s) and the intermediate systems between the integrable class and each GE class can be achieved by selecting non-zero coupling parameters. Especially, this system gives a rare example of the Gaussian symplectic ensemble class. In order to have reliable statistics of quantum levels, it is necessary to evaluate thousands of energy levels from the ground state without missing. We compute the quantum energy levels by numerical diagonalization of the truncated matrix of the Hamiltonian in the basis of harmonic oscillators. Several types of the interpolation formulae of the Poissonian and the Wigner-like distribution of each GE for the NNSD are examined. It is found that the MFD always becomes Gaussian, if the system is chaotic, irrespective of the kinds of the GE’s. Study of statistical properties of quantum levels has started from 1950’s. The random matrix theory (RMT) was first applied to the spectra of neutron resonances in the nuclear scattering experiment. 1) The RMT can be also applicable to predict the distribution of quantum levels in chaotic regime. Nearest neighbor level spacing distribution (NNSD), spectral rigidity, mode fluctuation distribution, distribution of the values of the wave functions, level curvatures with respect to internal parameters, etc., have been shown to be useful for studying the quantum chaos. Among them the NNSD is best known as a measure of the chaoticity and the integrability. It becomes the Wigner distribution if the systems are chaotic, whereas it becomes the Poisson distribution for integrable systems. For intermediate systems between chaotic and integrable regions, we have not reached a satisfactory surmise yet. The several formulae which are based on a physical inspection have been proposed and tested. However, at least in the low energy limit the most successful one is still the Brody distribution. Most of quantum levels which are numerically and experimentally obtained stay in a relatively lower energy region. The Brody distribution is essentially an interpolation formula between the Poissonian and the Wigner surmise and has no particular physical motivation. 2) The Berry-Robnik distribution is a very precious exception with the proper physical inspection. 3) It was introduced under the assumption that the phase space can be separated into two regions, that is, the integrable region (the KAM tori area) and

Numerical analysis of level statistical properties of two- and three-dimensional coupled quartic oscillators

The spectral statistical properties, i.e. the nearest neighbor spacing distribution, the spectral rigidity and the mode fluctuation distribution (MFD) are numerically studied for two- and three-dimensional quantum coupled quartic oscillators. By changing a single parameter, the system is continuously transformed from an integrable system to a chaotic one. The diagonalization method of the truncated Hamiltonian matrix is shown to be effective for the calculation of quantum spectra. The MFD is found to be more sensitive to the integrability of the system than its chaoticity. The Poincaré surface of sections of three-dimensional systems are discussed.

Three-Dimensional Quartic Oscillators for the Study of Mode Fluctuation Distribution

The mode fluctuation distribution (MFD) of three-dimensional quartic oscillators coupled by quartic perturbations is numerically studied. The system is continuously transferred from the integrable to the chaotic regime by changing a single coupling strength. The MFD of the three-dimensional systems is found to be a sensitive measure of integrability as in two-dimensional systems. If the system is exactly integrable, the MFD has the characteristic skew distribution. On the other hand, even in the region that the chaotic and integrable volumes are comparable in the phase space, the MFD already becomes almost the Gaussian distribution as in the completely chaotic regime. We also find that the Brody parameter increases gradually without a plateau by increasing the coupling constant. It shows a clear contrast to the two-dimensional system.

Mode Fluctuation Distribution of Coupled Quartic Oscillators Using Diagonalization of Truncated Sparse Matrix of Hamiltonian

Mode Fluctuation Distribution of Coupled Quartic Oscillators Using Diagonalization of Truncated Sparse Matrix of Hamiltonian

Mode fluctuation distribution of coupled quartic oscillators

The mode fluctuation distribution (MFD) of a system of two quartic oscillators coupled by a quartic perturbation is numerically studied. The coupling strength serves as a control parameter to simulate the transition from integrable to chaotic regimes. It is demonstrated that even in this potential system the MFD turns out to be a very sensitive measure, as it manifests the same transformation seen in billiard systems. It is characterized by Gaussian and skewed distributions in the regions, known to be chaotic and integrable, respectively, from classical Poincar\'e sections and quantum-mechanical level spacing statistics. In the intermediate regions where the Kol'mogorov-Arnol'd-Moser tori survive, the MFD has various distorted forms.

Mode fluctuation distribution for spectra of superconducting microwave billiards

High-resolution eigenvalue spectra of several two- and three-dimensional superconducting microwave cavities have been measured in the frequency range below 20 GHz and analyzed using a statistical measure that is given by the distribution of the normalized mode fluctuations. For chaotic systems the limit distribution is conjectured to show a universal Gaussian, whereas integrable systems should exhibit a non-Gaussian limit distribution. For the investigated Bunimovich stadium and the three-dimensional Sinai billiard we find that the distribution is in good agreement with this prediction. We study members of the family of lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on billiards, having mixed dynamics. It turns out that in this case the number of approximately 1000 eigenvalues for each billiard does not allow us to observe significant deviations from a Gaussian, whereas an also measured circular billiard with regular dynamics shows the expected difference from a Gaussian.

Mode Fluctuations as Fingerprints of Chaotic and Non-Chaotic Systems

The mode-fluctuation distribution P(W) is studied for chaotic as well as for non-chaotic quantum billiards. This statistic is discussed in the broader framework of the E(k,L) functions being the probability of finding k energy levels in a randomly chosen interval of length L, and the distribution of n(L), where n(L) is the number of levels in such an interval, and their cumulants ck(L). It is demonstrated that the cumulants provide a possible measure for the distinction between chaotic and non-chaotic systems. The vanishing of the normalized cumulants Ck, k ≥ 3, implies a Gaussian behaviour of P(W), which is realized in the case of chaotic systems, whereas non-chaotic systems display non-vanishing values for these cumulants leading to a non-Gaussian behaviour of P(W). For some integrable systems there exist rigorous proofs of the non-Gaussian behaviour which are also discussed. Our numerical results and the rigorous results for integrable systems suggest that a clear fingerprint of chaotic systems is provided by a Gaussian distribution of the mode-fluctuation distribution P(W).