Showing papers on "Normal mode published in 2008"

Climbing the Jaynes–Cummings ladder and observing its nonlinearity in a cavity QED system

Abstract: The field of cavity quantum electrodynamics (QED), traditionally studied in atomic systems1,2,3, has gained new momentum by recent reports of quantum optical experiments with solid-state semiconducting4,5,6,7,8 and superconducting9,10,11 systems. In cavity QED, the observation of the vacuum Rabi mode splitting is used to investigate the nature of matter–light interaction at a quantum-mechanical level. However, this effect can, at least in principle, be explained classically as the normal mode splitting of two coupled linear oscillators12. It has been suggested that an observation of the scaling of the resonant atom–photon coupling strength in the Jaynes–Cummings energy ladder13 with the square root of photon number n is sufficient to prove that the system is quantum mechanical in nature14. Here we report a direct spectroscopic observation of this characteristic quantum nonlinearity. Measuring the photonic degree of freedom of the coupled system, our measurements provide unambiguous spectroscopic evidence for the quantum nature of the resonant atom–field interaction in cavity QED. We explore atom–photon superposition states involving up to two photons, using a spectroscopic pump and probe technique. The experiments have been performed in a circuit QED set-up15, in which very strong coupling is realized by the large dipole coupling strength and the long coherence time of a superconducting qubit embedded in a high-quality on-chip microwave cavity. Circuit QED systems also provide a natural quantum interface between flying qubits (photons) and stationary qubits for applications in quantum information processing and communication16.

Spatially and spectrally resolved imaging of modal content in large-mode-area fibers

Abstract: A new measurement technique, capable of quantifying the number and type of modes propagating in large-mode-area fibers is both proposed and demonstrated. The measurement is based on both spatially and spectrally resolving the image of the output of the fiber under test. The measurement provides high quality images of the modes that can be used to identify the mode order, while at the same time returning the power levels of the higher-order modes relative to the fundamental mode. Alternatively the data can be used to provide statistics on the level of beam pointing instability and mode shape changes due to random uncontrolled fluctuations of the phases between the coherent modes propagating in the fiber. An added advantage of the measurement is that is requires no prior detailed knowledge of the fiber properties in order to identify the modes and quantify their relative power levels. Because of the coherent nature of the measurement, it is far more sensitive to changes in beam properties due to the mode content in the beam than is the more traditional M2 measurement for characterizing beam quality. We refer to the measurement as Spatially and Spectrally resolved imaging of mode content in fibers, or more simply as S2 imaging.

Basic physics of Alfvén instabilities driven by energetic particles in toroidally confined plasmasa)

Abstract: Superthermal energetic particles (EP) often drive shear Alfven waves unstable in magnetically confined plasmas. These instabilities constitute a fascinating nonlinear system where fluid and kinetic nonlinearities can appear on an equal footing. In addition to basic science, Alfven instabilities are of practical importance, as the expulsion of energetic particles can damage the walls of a confinement device. Because of rapid dispersion, shear Alfven waves that are part of the continuous spectrum are rarely destabilized. However, because the index of refraction is periodic in toroidally confined plasmas, gaps appear in the continuous spectrum. At spatial locations where the radial group velocity vanishes, weakly damped discrete modes appear in these gaps. These eigenmodes are of two types. One type is associated with frequency crossings of counterpropagating waves; the toroidal Alfven eigenmode is a prominent example. The second type is associated with an extremum of the continuous spectrum; the reversed shear Alfven eigenmode is an example of this type. In addition to these normal modes of the background plasma, when the energetic particle pressure is very large, energetic particle modes that adopt the frequency of the energetic particle population occur. Alfven instabilities of all three types occur in every toroidal magnetic confinement device with an intense energetic particle population. The energetic particles are most conveniently described by their constants of motion. Resonances occur between the orbital frequencies of the energetic particles and the wave phase velocity. If the wave resonance with the energetic particle population occurs where the gradient with respect to a constant of motion is inverted, the particles transfer energy to the wave, promoting instability. In a tokamak, the spatial gradient drive associated with inversion of the toroidal canonical angular momentum Pζ is most important. Once a mode is driven unstable, a wide variety of nonlinear dynamics is observed, ranging from steady modes that gradually saturate, to bursting behavior reminiscent of relaxation oscillations, to rapid frequency chirping. An analogy to the classic one-dimensional problem of electrostatic plasma waves explains much of this phenomenology. EP transport can be convective, as when the wave scatters the particle across a topological boundary into a loss cone, or diffusive, which occurs when islands overlap in the orbital phase space. Despite a solid qualitative understanding of possible transport mechanisms, quantitative calculations using measured mode amplitudes currently underestimate the observed fast-ion transport. Experimentally, detailed identification of nonlinear mechanisms is in its infancy. Beyond validation of theoretical models, the future of the field lies in the development of control tools. These may exploit EP instabilities for beneficial purposes, such as favorably modifying the current profile, or use modest amounts of power to govern the nonlinear dynamics in order to avoid catastrophic bursts.

Exact solution and stability of postbuckling configurations of beams

Abstract: We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.

Absence of wave packet diffusion in disordered nonlinear systems.

Abstract: We study the spreading of an initially localized wave packet in two nonlinear chains (discrete nonlinear Schrodinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverges with time. We find that the participation number of a wave packet does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times we find a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wave packet, we rule out the possibility of slow energy diffusion. The dynamical state could approach a quasiperiodic solution (Kolmogorov-Arnold-Moser torus) in the long time limit.

Acoustic Oscillations and Elastic Moduli of Single Gold Nanorods

Abstract: We present the first acoustic vibration measurements of single gold nanorods with well-characterized dimensions and crystal structure The nanorods have an average size of 90 nm × 30 nm and display two vibration modes, the breathing mode and the extensional mode Correlation between the dimensions obtained from electron microscope images and the vibrational frequencies of the same particle allows us to determine the elastic moduli for each individual nanorod Contrary to previous reports on ensembles of gold nanorods, we find that the single particle elastic moduli agree well with bulk values

Calculation of vibrational transition frequencies and intensities in water dimer: comparison of different vibrational approaches.

Abstract: We have calculated frequencies and intensities of fundamental and overtone vibrational transitions in water and water dimer with use of different vibrational methods. We have compared results obtained with correlation-corrected vibrational self-consistent-field theory and vibrational second-order perturbation theory both using normal modes and finally with a harmonically coupled anharmonic oscillator local mode model including OH-stretching and HOH-bending local modes. The coupled cluster with singles, doubles, and perturbative triples ab initio method with augmented correlation-consistent triple-ζ Dunning and atomic natural orbital basis sets has been used to obtain the necessary potential energy and dipole moment surfaces. We identify the strengths and weaknesses of these different vibrational approaches and compare our results to the available experimental results.

Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory

Abstract: The effects of flow velocity on the vibration frequency and mode shape of the fluid-conveying single-walled carbon nanotube are analyzed using nonlocal elastic theory. Results show that the frequency and mode shape are significantly influenced by the nonlocal parameter e0a/L. Increasing the nonlocal parameter decreases the real component of frequency and the decrease is more obvious for a lower flow velocity and a higher-order mode. In addition, a higher mode shape is observed with increasing the value of e0a/L. When a critical flow velocity is reached, the combination of first and second modes takes place. The mode shape for the combination is large relative to mode 3 due to the coupled frequency effect, especially including negative imaginary frequency. Furthermore, the mode shape of the combination increases as the nonlocal effect increases.

Vibrational analysis of single-layered graphene sheets

Abstract: A molecular structural mechanics method has been implemented to investigate the vibrational behavior of single-layered graphene sheets. By adopting this approach, mode shapes and natural frequencies are obtained. Vibrational analysis is performed with different chirality and boundary conditions. Numerical results from the atomistic modeling are employed to develop predictive equations via a statistical nonlinear regression model. With the proposed equations, fundamental frequencies of single-layered graphene sheets with considered boundary conditions can be predicted within 3% difference with respect to the atomistic simulation.

Effect of 3-D viscoelastic structure on post-seismic relaxation from the 2004 M= 9.2 Sumatra earthquake

Abstract: SUMMARY The 2004 M = 9.2 Sumatra‐Andaman earthquake profoundly altered the state of stress in a large volume surrounding the ∼1400 km long rupture. Induced mantle flow fields and coupled surface deformation are sensitive to the 3-D rheology structure. To predict the post-seismic motions from this earthquake, relaxation of a 3-D spherical viscoelastic earth model is simulated using the theory of coupled normal modes. The quasi-static deformation basis set and solution on the 3-D model is constructed using: a spherically stratified viscoelastic earth model with a linear stress‐strain relation; an aspherical perturbation in viscoelastic structure; a ‘static’ mode basis set consisting of Earth’s spheroidal and toroidal free oscillations; a “viscoelastic” mode basis set; and interaction kernels that describe the coupling among viscoelastic and static modes. Application to the 2004 Sumatra‐Andaman earthquake illustrates the profound modification of the post-seismic flow field at depth by a slab structure and similarly large effects on the near-field post-seismic deformation field at Earth’s surface. Comparison with postseismic GPS observations illustrates the extent to which viscoelastic relaxation contributes to the regional post-seismic deformation.

Localization, confinement, and field-controlled propagation of spin waves in Ni 80 Fe 20 antidot lattices

Abstract: Spin-wave modes in Ni80Fe20 thin-film antidot lattices are investigated using micromagnetic simulations and a semianalytical theoretical approach. The simulations reveal a rich eigenmode spectrum consisting of edge and center modes. We find both spatially localized and spin waves extending over many unit cells. To classify the different types of modes and to analyze the microscopic properties, we adapt a semianalytical approach. We show how to reduce the two-dimensional problem of the antidot lattice to a one-dimensional problem if certain high-symmetry axes are considered. For lattices of unit-cell lengths ranging from 200 to 1100 nm, we find that the characteristic mode eigenfrequencies can be correlated with both local inhomogeneities of the demagnetization field and specific wave vectors caused by geometry-imposed mode quantization conditions. We compare our results with recently published experimental data and discuss the crossover from dipolar to exchange-dominated spin waves. Moreover, we simulate propagation of spin waves and find a preferred axis of propagation perpendicular to the external magnetic field.

Eigenmode analysis of geodesic acoustic modes

Abstract: Geodesic acoustic modes (GAMs) are studied as plasma eigenmodes when an electrostatic potential nearly constant around a magnetic surface is applied to collisionless toroidal plasmas. Besides the standard GAM, a branch of low frequency mode and an infinite series of ion sound wavelike modes are identified. Eigenfrequencies of these modes are obtained analytically and numerically from a linear gyrokinetic model. The finite gyroradius effect is found to enhance the collisionless damping of the standard GAM, while this enhancement is not monotonic as the safety factor varies. Moreover, additional damping due to higher-harmonic resonances becomes important when the safety factor increases. The mode structure of the GAM is also discussed.

Laplacian spectra as a diagnostic tool for network structure and dynamics.

Abstract: We examine numerically the three-way relationships among structure, Laplacian spectra, and frequency synchronization dynamics on complex networks. We study the effects of clustering, degree distribution, and a particular type of coupling asymmetry (input normalization), all of which are known to have effects on the synchronizability of oscillator networks. We find that these topological factors produce marked signatures in the Laplacian eigenvalue distribution and in the localization properties of individual eigenvectors. Using a set of coordinates based on the Laplacian eigenvectors as a diagnostic tool for synchronization dynamics, we find that the process of frequency synchronization can be visualized as a series of quasi-independent transitions involving different normal modes. Particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes. For example, there are important partially synchronized states in which a set of low-lying modes remain unlocked while those in the main spectral peak are locked. We find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue.

Modal Properties of Planetary Gears With an Elastic Continuum Ring Gear

Abstract: The distinctive modal properties of equally spaced planetary gears with elastic ring gears are studied through perturbation and a candidate mode method. All eigenfunctions fall into one of four mode types whose structured properties are derived analytically. Two perturbations are used to obtain closed-form expressions of all the eigenfunctions. In the discrete planetary perturbation, the unperturbed system is a discrete planetary gear with a rigid ring. The stiffness of the ring is perturbed from infinite to a finite number. In the elastic ring perturbation, the unperturbed system is an elastic ring supported by the ring-planet mesh springs; the sun, planet and carrier motions are treated as small perturbations. A subsequent candidate mode method analysis proves the perturbation results and removes any reliance on perturbation parameters being small. All vibration modes are classified into rotational, translational, planet, and purely ring modes. The well defined properties of each type of mode are analytically determined. All modal properties are verified numerically.

Theoretical study of the vibrational edge modes in graphene nanoribbons

Abstract: We investigate the phonon normal modes in hydrogen-terminated graphene nanoribbons (GNRs) using the second-generation reactive empirical bond order (REBOII) potential and density-functional theory calculations. We show that specific modes, absent in pristine graphene and localized at the GNR edges, are intrinsic signatures of the vibrational density of states of the GNRs. Three particular modes are described in details: a transverse phonon mode related to armchair GNRs, a hydrogen out-of-plane mode present in both armchair and zigzag GNRs, and the Raman radial-breathing-like mode. The good agreement between the frequencies of selected edge modes obtained using REBOII and first-principles methods shows the reliability of this empirical potential for the calculation and the assignment of phonon modes in carbon nanostructures where carbon atoms present a $s{p}^{2}$ hybridization.

Free vibration of nanorings/arches based on nonlocal elasticity

Abstract: This paper deals with the free vibration problem of nanorings/arches. The problem is formulated on the basis of Eringen’s nonlocal theory of elasticity in order to allow for the small length scale effect. Exact vibration frequencies are derived for the nanorings/arches and the effects of small length scale, defects, and elastic boundary conditions are investigated. The small length scale effect lowers the vibration frequencies. The defects and the use of elastic boundary conditions (instead of fixed restraints) also significantly reduce the frequencies and alter the vibration mode shapes of circular rings/arches. The results presented should be useful to engineers who are designing nanorings/arches for microelectromechanical and nanoelectromechanical devices.

Ab initio simulation of the IR spectra of pyrope, grossular, and andradite.

Abstract: IR spectra of pyrope Mg(3)Al(2)Si(3)O(12), grossular Ca(3)Al(2)Si(3)O(12) and andradite Ca(3)Fe(2)Si(3)O(12) garnets were simulated with the periodic ab initio CRYSTAL code by adopting an all-electron Gaussian-type basis set and the B3LYP Hamiltonian. Two sets of 17 F(1u) Transverse Optical (TO) and Longitudinal Optical (LO) frequencies were generated, together with their intensities. Because the generation of LO modes requires knowledge of the high frequency dielectric constant epsilon(infinity) and Born effective charges, they were preliminary evaluated by using a finite field saw-tooth model and well localized Wannier functions, respectively. As a by-product, the static dielectric constant epsilon(0) was also obtained. The agreement of the present calculated wavenumbers (i.e. peak positions) with the available experimental data is excellent, in that the mean absolute difference for the full set of data smaller than 8 cm(-1). Missing peaks in experimental spectra were found to correspond to modes with low calculated intensities. Correspondence between TO and LO modes was established on the basis of the overlap between the eigenvectors of the two sets and similarity of isotopic shifts; as result, the so called LO-TO splitting could be determined. Animation of the normal modes was employed to support the proposed pairing.

Regular patterns in the acoustic spectrum of rapidly rotating stars

Abstract: Context: Rapid rotation modifies the structure of the frequency spectrum of pulsating stars, thus making mode identification difficult. Aims: We look for new forms of organisation for the frequency spectrum that can provide a basis for mode identification at high rotation rates. Methods: Acoustic modes in uniformly rotating polytropic models of stars are computed using a numerical code that fully takes the effects of rotation (centrifugal distortion and Coriolis acceleration) into account. All low-degree modes, ? = 0 to 3, with radial orders n = 1-10 and 21-25 for N = 3 polytropic models and n = 1-10 for N = 1.5 polytropic models are followed from a zero rotation rate up to 59% of the break-up velocity. Results: We find an empirical formula that gives a good description of the high-frequency range of the computed acoustic spectrum for high rotation rates. Differences between this formula and complete eigenmode calculations are shown to be substantially smaller than those obtained with a 3rd order perturbative method valid at low rotation rates.

Coexistence of Equatorial Coupled Modes of ENSO

Abstract: To study the regimes of leading ocean–atmosphere coupled modes of relevance to the El Nino–Southern Oscillation (ENSO) phenomenon, a comprehensive eigenmode analysis of an intermediate coupled model linearized with respect to an array of basic states is performed. Different kinds of leading modes are found to coexist and become unstable under wide ranges of basic states and parameter conditions. In particular, the two most important modes have periods of around 4 and 2 yr. They are referred to as the quasi-quadrennial (QQ) and the quasi-biennial (QB) modes, respectively. The positive coupled feedback destabilizes and quantizes the near-continuous spectrum for the low-frequency modes of the upper-ocean dynamics, giving rise to these leading modes with distinct periodicities. The primary mechanism for the phase transition of the QQ mode is due to the slow oceanic dynamic adjustment of equatorial heat content, which is consistent with the simple conceptual recharge oscillator, whereas anomalous adve...