The intermediate level statistics in dynamically localized chaotic eigenstates

The quantum localization is one of the remarkable phenomena in the studies of quantum chaos and plays an important role in various contexts. Thus, an understanding of the properties of quantum localization is essential. In spite of much effort dedicated to investigating the manifestations of localization in the time-dependent systems, the features of localization in time-independent systems are still less explored, particularly in quantum systems which correspond to the classical systems with smooth Hamiltonian. In this work, we present such a study for a quantum many-body system, namely, the Dicke model. The classical counterpart of the Dicke model is given by a smooth Hamiltonian with two degrees of freedom. We examine the signatures of localization in its chaotic eigenstates. We show that the entropy localization measure, which is defined in terms of the information entropy of Husimi distribution, behaves linearly with the participation number, a measure of the degree of localization of a quantum state. We further demonstrate that the localization measure probability distribution is well described by the β distribution. We also find that the averaged localization measure is linearly related to the level repulsion exponent, a widely used quantity to characterize the localization in chaotic eigenstates. Our findings extend the previous results in billiards to the quantum many-body system with classical counterpart described by a smooth Hamiltonian, and they indicate that the properties of localized chaotic eigenstates are universal.

The localization measures A (based on the information entropy) of localized chaotic eigenstates in the Poincaré-Husimi representation have a distribution on a compact interval [0;A0], which is well approximated by the beta distribution, based on our extensive numerical calculations. The system under study is the Bunimovich' stadium billiard, which is a classically ergodic system, also fully chaotic (positive Lyapunov exponent), but in the regime of a slightly distorted circle billiard (small shape parameter ") the diffusion in the momentum space is very slow. The parameter α = tH/tT , where tH and tT are the Heisenberg time and the classical transport time (diffusion time), respectively, is the important control parameter of the system, as in all quantum systems with the discrete energy spectrum. The measures A and their distributions have been calculated for a large number of ε and eigenenergies. The dependence of the standard deviation σ on α is analyzed, as well as on the spectral parameter β (level repulsion exponent of the relevant Brody level spacing distribution). The paper is a continuation of our recent paper (B. Batistić, Č. Lozej and M. Robnik, Nonlinear Phenomena in Complex Systems 21, 225 (2018)), where the spectral statistics and validity of the Brody level spacing distribution has been studied for the same system, namely the dependence of β and of the mean value < A > on α.

We propose localization measures in phase space of the ground state of bilayer quantum Hall systems at fractional filling factors , to characterize the three quantum phases (shortly denoted by spin, canted and ppin) for arbitrary -isospin λ. We use a coherent state (Bargmann) representation of quantum states, as holomorphic functions in the 8-dimensional Grassmannian phase-space (a higher-dimensional generalization of the Haldane’s 2-dimensional sphere ). We quantify the localization (inverse volume) of the ground state wave function in phase-space throughout the phase diagram (i.e. as a function of Zeeman, tunneling, layer distance, etc, control parameters) with the Husimi function second moment, a kind of inverse participation ratio that behaves as an order parameter. Then we visualize the different ground state structure in phase space of the three quantum phases, the canted phase displaying a much higher delocalization (a Schrödinger cat structure) than the spin and ppin phases, where the ground state is highly coherent. We find a good agreement between analytic (variational) and numeric diagonalization results.

The entropy of an electronic system is defined in terms of the Husimi function, a nonnegative distribution function in phase space. The Husimi function is calculated by maximizing the entropy subject to the constraints that the Husimi function give a Gaussian convolution of the desity when integrating over the momentum coordinates and that its second moment with respect to momentum give a sum of Gaussian convolutions of the density and the kinetic energy density. The result is compared with the Wigner function. Equations are given for calculating the density matrix from the Husimi function. The resulting equation for the exchange energy requires a difficult numerical integration. An alternate method is used to obtain the density matrix from an approximate partially collapsed Husimi matrix that gives the maximum entropy Husimi function as its diagonal. The results are exact for the harmonic oscillator ground state. Exchange energies calculated for H and the He isoelectronic series through C+4 show slight improvements over those calculated using a maximum entropy Wigner function.

Photoelectron spectra of the niobium-molybdenum diatomic anion, obtained at 488 and 514 nm, display vibrationally resolved transitions from the ground state and one excited electronic state of the anion to the ground state and one excited electronic state of the neutral molecule. The electron affinity of NbMo is measured to be 1.130 ± 0.005 eV. Its 2Δ3/2 spin-orbit component is observed to lie 870 ± 20 cm-1 above its previously identified 2Δ5/2 ground state. For 93Nb98Mo, vibrational energies measured for levels up to v = 4 for the 2Δ5/2 and 2Δ3/2 states give harmonic frequency and anharmonicity constant values of ωe = 492 ± 12 cm-1 and ωexe = 8.0 ± 3.2 cm-1, the former value corresponding to a force constant of 6.80 ± 0.35 mdyn/Å. These two vibrational parameters suggest a bond dissociation energy that is too low by at least a factor of 3, indicating that the ground state potential energy curve of NbMo deviates markedly from a Morse potential at higher energies. An excited electronic state of NbMo, assigned as a 2Σ+ state, is observed at 2900 ± 25 cm-1 (T0). Vibrational energies up to v = 8 in this excited state give values of ωe = 544 ± 8 cm-1 and ωexe = 1.9 ± 1.2 cm-1 for 93Nb98Mo. The former value corresponds to a high vibrational force constant of 8.30 ± 0.25 mdyn/Å. Both doublet states of the neutral molecule are accessed from the anion ground state, which is assigned as 1Σ+. For the 93Nb98Mo- anion, the fundamental vibrational frequency (ΔG1/2) is 484 ± 15 cm-1. Electron affinity data indicate that the bond dissociation energy of NbMo- is 0.213 ± 0.005 eV greater than that of neutral NbMo, whose previously reported value then gives D0 = 4.85 ± 0.27 eV for the anion. An excited state of the anion lying 3050 ± 25 cm-1 (T0) above its ground state is assigned as 3Δ, and the energies of its spin-orbit components above the 3Δ3 lowest energy level are measured to be 450 ± 20 cm-1 (3Δ2) and 1100 ± 20 cm-1 (3Δ1). Their uneven spacing suggests that the energy of the 3Δ2 level is lowered by interaction with a higher energy Ω = 2 anion state. The vibrational frequency (ΔG1/2) for the 3Δ1 and 3Δ2 states is measured to be 433 ± 20 cm-1. Bond length differences among the observed states are estimated from Franck-Condon fits to vibrational band intensity profiles. When combined with the previously reported NbMo bond length, these provide bond length estimates for the ground state of the anion (1.940 ± 0.025 Å) and for the observed excited states. These species offer extreme examples of multiple metal-metal bonding, with formal bond orders of 51/2 for the 2Δ ground and 2Σ+ excited doublet states of NbMo, 6 for the singlet ground state of the anion, and 5 for its low-lying triplet state. The relationships among their bonding properties and those of related multiply bonded transition metal dimers are discussed.

The recommendations for electrical therapies described in this section are designed to improve survival from SCA and life-threatening arrhythmias. Whenever defibrillation is attempted, rescuers must coordinate high-quality CPR with defibrillation to minimize interruptions in chest compressions and to ensure immediate resumption of chest compressions after shock delivery. The high first-shock efficacy of newer biphasic defibrillators led to the recommendation of single shocks plus immediate CPR instead of 3-shock sequences that were recommended prior to 2005 to treat VF. Further data are needed to refine recommendations for energy levels for defibrillation and cardioversion using biphasic waveforms.

Femtosecond chemical activation of reactions at very high thermal energies, much above the bond energy, is developed in this report. We address the concept of nonstatistical dynamics at such high energies. The approach offers a new direction for probing the dynamics of reactions in their ground state with unique activation, collision-free, and temporally and spatially defined.

We investigate the occurrence of large degeneracies and their dependence on the size and shape of the confining potential. For a three-dimensional anisotropic harmonic oscillator and a particle in a cuboid, large degeneracies, which grow with size and energy, appear when aspect ratios are commensurable. A crossover from commensurable to incommensurable behaviour can be induced by changing either the size or shape (degree of rationality of aspect ratios). We study this crossover using a combination of analytical and numerical techniques. We find that shape-size scaling is valid, which allows for a quantitative characterization of degeneracies in terms of a single parameter involving both size and shape. The distribution of energy level spacings and scaled degeneracies in the commensurable regime are also found.

In this paper, the effect of an off-center donor impurity on electronic structure of nanowire superlattices (NWSLs) is numerically studied using finite difference (FD) approximation. Energy levels and binding energies of ground and first excited states (1s and 2p) are computed for two different NWSL with circular cross-section which involved an array of spherical or cylindrical quantum dots (QDs) along their axis which are called Sph and Cyl NWSLs, respectively. The numerical results show that oscillatory behaviors appear in electronic energy levels and binding energies as impurity shifts away from the center of QDs. Maximum values of ground state binding energies occur when impurity is located at the center of QDs and these values shift towards higher energies as number of unit cells of NWSL decreases. An opposite behavior is observed for first excited state as impurity shifts away from the center of QDs. Also, as the QDs' volumes increase binding energies (i) decrease, (ii) increase, (iii) decrease, reach minimum values and then increase, or (iv) increase, reach maximum values and then decrease; which strongly depend on the type of NWSL, position of impurity and number of unit cells involved in NWSLs. Additionally, it is found that binding energies of Sph NWSL are larger than those of Cyl NWSL.

Two statistical properties, namely, the energy level spacing distribution and the Dyson-Mehta statistics, are reported for the energy band structure of a confined, surface superlattice in perpendicularly applied magnetic fields. Time-reversal (T) symmetry is broken in the system. However, the system is invariant under the anti-unitary combination of symmetric operations which includes T, leading to what is called false time-reversal violation. For the wave vector k not in close vicinity of the symmetrical points in k-space, the statistical properties of the band structure at sufficiently strong magnetic fields are found to be described by Gaussian orthogonal ensemble (GOE) statistics. This result is a clear manifestation of quantum chaos in the system and is in agreement with the prediction that the false time-reversal violation suffices to give the energy spectra the properties of the GOE, instead of the Gaussian unitary ensemble. The spectra are found to deviate from the GOE statistics when the wave vector k is moved towards the symmetrical points in k-space and/or the magnetic field towards B = 0. This is because in these limit cases, the system is invariant under at least one geometric, symmetrical operation and hence spectral degeneracy becomes possible. The implications of this work for experiments are also discussed.

Laser spectroscopy is a versatile tool to unveil fundamental atomic properties of an element and the ground state information of the atomic nucleus. The heaviest elements are of particular interest as the ordering of their shell electrons is strongly influenced by electron-electron correlations, quantum electrodynamics and relativistic effects leading to distinct chemical behaviour. The elements beyond fermium (Z > 100) are accessible in fusion evaporation reactions at minute quantities and at high energies, hampering their optical spectroscopy. Recently, the RAdiation Detected Resonance Ionization Spectroscopy (RADRIS) technique was employed to explore the electronic structure of the element nobelium (No, Z = 102). The 1S0->1P1 ground state transition of this element was identified. In this work, the pioneering experiment on laser spectroscopy of nobelium was extended to the isotopes of nobelium (252-254 No). These were produced in fusion-evaporation reactions by bombarding lead targets (206-208 Pb) with 48Ca projectiles. After separation from the primary beam by the velocity filter SHIP (Separator for Heavy Ion reaction Products), at GSI, the fusion products were stopped in 95 mbar high-purity argon gas and collected onto a thin tantalum filament. After a sufficient collection time, which depended on the half-life (T1/2) of the isotope under consideration, the primary beam was blocked in order to have a background free signal. During the beam-off period, the collected nobelium ions were re-evaporated as neutral atoms from the filament and were probed by two laser beams for ionization. The created photo-ions were detected by their characteristic alpha decay. With this technique the isotope shift of the transition was measured for the isotopes 252-254 No. A hyperfine splitting of the 1P1 level was resolved in 253No. These measurements in combination with state-ofthe-art atomic calculations provided a deep insight into the evolution of nuclear deformation of the investigated nobelium isotopes in the vicinity of the deformed shell closure at neutron number N = 152 along with an assesment of the magnetic moment, µ, and the spectroscopic quadrupole moment, Qs, for 253No. Moreover, several high-lying Rydberg states were measured for the first time in 254No. These Rydberg states, populated in different ways enabled establishing different Rydberg series and the extraction of the first ionization potential of the element plus an additional low-lying atomic state in 254No that is optically inaccessible from the ground state.

This paper considers the collisional excitation of 0('D) modeled by the crossing of two valence 1 IIg curves dissociating to 0{'P)+0{'P)[ V»{R)] and 0('P)+0('D) [ V22{R}]which in turn are further crossed by the C 'Ils Rydberg curve dissociating to 0('P )+0{ 'S ) [ V33(R ) ]. The role of structure in the potential curves and coupling matrix elements is quantitatively probed by the first-order functionalsensitivity densities 5lncr»(E)/51nVJ(R) of the excitation cross section o.l2(E) obtained from closecoupling calculations. The results reveal that, in spite of the well-separated nature of the crossing between the two valence curves from their crossings with the Rydberg potential curve, the excitation cross section 0. » displays considerable sensitivity to the Rydberg curve V33(R) at all energies in the range 3.0— 9.0 eV. For relative collisional energies corresponding to the higher closely spaced vibrational energy levels of the Rydberg state, the excitation cross section is found to be much more sensitive to the Rydberg state than to the two valence states themselves. At all energies, the sensitivity of the excitation cross section o. » to the coupling V»(R) between the valence states is much larger than the sensitivity to the couplings V»(R) or V»(R) with the Rydberg state. At higher energies, the large increase in the sensitivity of the cross section to the Rydberg potential is mirrored by a similar increase in sensitivity to its coupling V»(R) with the upper valence state. Due to the weak coupling between the three curves, a qualitative similarity exists between the sensitivity profiles and those predicted by the Landau-ZenerStueckelberg (LZS) theory. Quantitative departures witnessed in earlier work are, however, more pronounced for the multilevel curve crossings investigated here. Implications of the results for attempts to extend the LZS-type treatment to multilevel curve crossings and for functional-sensitivity-based algorithms for the inversion of cross-section data are discussed.

An improved set of electron-collision cross sections is derived for SF6 and used to calculate transport, ionization, attachment, and dissociation coefficients for pure SF6 and mixtures of SF6 with N2, O2, and Ne. The SF6 cross sections differ from previously published sets primarily at very low and high electron energies. At energies below 0.03 eV the attachment cross section is adjusted to fit recent electron swarm experiments, while the elastic momentum transfer cross section is increased to the theoretical limit. At high energies an allowance is made for the excitation of highly excited levels as observed in electron beam experiments. The cross-section sets used for the admixed gases have previously been published. Electron kinetic energy distributions computed from numerical solutions of the electron-transport (Boltzmann) equation using the two-term, spherical harmonic expansion approximation were used to obtain electron-transport and reaction coefficients as functions of E/N and the fractional concentration of SF6. Here E is the electric field strength and N is the gas number density. Attachment rate data for low concentrations of SF6 in N2 are used to test the attachment cross sections. Particular attention is given to the calculation of transport and reaction coefficients at the critical E/N=(E/N)c at which the ionization and attachment rates are equal.

The effect of pressure and rock type on optimal dual energy levels and minimalisation of saturation uncertainties for three-phase core experiments was investigated. X-ray scanning of a composite core comprising of two chalk plugs and one Berea sandstone core was done. Reference scans using CO2, natural seawater, n-Ci10 and a CO2/n-Ci10 mixture at four pressure stages and five different pre-defined dual energy levels were acquired. Two unique typical given saturations were input together with data from the reference scans in a routine for back-calculating three-phase intensity data. A detailed uncertainty analysis using these synthetic three-phase data and combining low and high energies was performed. The resulting combination of low and high energy level giving the lowest uncertainty was selected as a given setting for all dynamic scans. Saturated with live oil, the core was waterflooded while dynamic X-ray scans were acquired. After the waterflood, the core was depleted below the bubble point to atmospheric pressure while scans were acquired continuously. The results show that the rock type largely effects the selection of energy levels for both two- and three-phase X-ray scans. To minimise the uncertainty, the chalks generally required a higher low energy level independent of pressure than the sandstone. Also, at high pressures and for all rocks the optimal high energy level was found in the same range. At lower pressures, a lower high energy level was favourable.

We investigate the Josephson effect between two coupled superconductors, coupled by the tunneling of pairs of electrons, in the regime that their energy level spacing is comparable to the bulk superconducting gap, but neglecting any charging effects. In this regime, BCS theory is not valid, and the notion of a superconducting order parameter with a well-defined phase is inapplicable. Using the density matrix renormalization group, we calculate the ground state of the two coupled superconductors and extract the Josephson energy. The Josephson energy is found to display a reentrant behavior (decrease followed by increase) as a function of increasing level spacing. For weak Josephson coupling, a tight-binding approximation is introduced, which illustrates the physical mechanism underlying this reentrance in a transparent way. The DMRG method is also applied to two strongly coupled superconductors and allows a detailed examination of the limits of validity of the tight-binding model.