Regime Of Small Oscillations Research Articles

Strongly interacting trapped one-dimensional quantum gases: Exact solution

Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. This Review collects the predictions coming from a family of exact solutions which allows us to obtain the many-body wavefunction of strongly correlated quantum fluids confined by a tight waveguide and subjected to any form of longitudinal confinement. It directly describes the experiments with trapped ultracold atoms where the strongly correlated regime in one dimension has been achieved. The exact solution applies to bosons, fermions, and mixtures. It allows us to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach. It also predicts the exact quantum dynamics at all the times, including the small oscillation regime yielding the collective modes of the system and the large quench regime where the system parameters are changed considerably. The solution can be extended to describe finite-temperature conditions, spin, and magnetization effects. The Review illustrates the idea of the solution, presents the key theoretical achievements, and the main experiments on strongly correlated one-dimensional quantum gases.

Behavior of solitary waves of coupled nonlinear Schrödinger equations subjected to complex external periodic potentials with odd- symmetry

We discuss the response of both moving and trapped solitary wave solutions of a two-component nonlinear Schrödinger system in 1 + 1 dimensions to an odd- external periodic complex potential. The dynamical behavior of perturbed solitary waves is explored by conducting numerical simulations of the nonlinear system and using a collective coordinate variational approximation. We present case examples corresponding to choices of parameter values and initial conditions involved therein. The results of the collective coordinate approximation are compared against numerical simulations where we observe qualitatively good agreement between the two. Unlike the case for a single-component solitary wave in a complex periodic -symmetric potential, the collective coordinate equations do not have a small oscillation regime, and initially the height of the two components changes in opposite directions often causing instability. We find that the dynamic stability criteria we have used in the one-component case are a good indicator for the onset of dynamic instabilities in the present setup.

Invariant-Based Inverse Engineering for Fast and Robust Load Transport in a Double Pendulum Bridge Crane.

We set a shortcut-to-adiabaticity strategy to design the trolley motion in a double-pendulum bridge crane. The trajectories found guarantee payload transport without residual excitation regardless of the initial conditions within the small oscillations regime. The results are compared with exact dynamics to set the working domain of the approach. The method is free from instabilities due to boundary effects or to resonances with the two natural frequencies.

Robust load transport by an overhead crane with respect to cable length uncertainties

We exploit a Fourier method to inverse engineer fast load transport protocols for an overhead crane, which is robust with respect to the deviation of cable length, corresponding to an ideal value. In the small oscillations regime, the protocols guarantee final adiabatic energies, i.e. null excitation with respect to the initial energy, regardless of the initial conditions. We demonstrate with calculations for the exact dynamics the stability of this result.

Interferometer with a driven trapped ion

We propose an interferometric measurement of weak forces using a single ion subjected to designed time-dependent spin-dependent forces. Explicit expressions of the relation between the unknown force and the final populations are found considering different scenarios, such as the weak force acting on two branches, on one branch, or having errors in the driving force. The flexibility to design the trap trajectories is used to minimize errors due to anharmonicities in the trap. The advantages of the approach are the use of geometrical phases, which provides stability, the possibility to design faster-than-adiabatic processes with sensitivity control, and the independence of the results on the motional states for the small-oscillations regime in which the effective potentials are purely harmonic.

Invariant-Based Inverse Engineering of Crane Control Parameters

By applying invariant-based inverse engineering in the small-oscillations regime, we design the time dependence of the control parameters of an overhead crane (trolley displacement and rope length), to transport a load between two positions at different heights with minimal final energy excitation for a microcanonical ensemble of initial conditions. The analogies between ion transport in multisegmented traps or neutral atom transport in moving optical lattices and load manipulation by cranes opens a route for a useful transfer of techniques among very different fields.

Analyzing chaos in higher order disordered quartic-sextic Klein-Gordon lattices using q-statistics

In the study of subdiffusive wave-packet spreading in disordered Klein–Gordon (KG) nonlinear lattices, a central open question is whether the motion continues to be chaotic despite decreasing densities, or tends to become quasi-periodic as nonlinear terms become negligible. In a recent study of such KG particle chains with quartic (4th order) anharmonicity in the on-site potential it was shown that q−Gaussian probability distribution functions of sums of position observables with q > 1 always approach pure Gaussians (q=1) in the long time limit and hence the motion of the full system is ultimately “strongly chaotic”. In the present paper, we show that these results continue to hold even when a sextic (6th order) term is gradually added to the potential and ultimately prevails over the 4th order anharmonicity, despite expectations that the dynamics is more “regular”, at least in the regime of small oscillations. Analyzing this system in the subdiffusive energy domain using q-statistics, we demonstrate that groups of oscillators centered around the initially excited one (as well as the full chain) possess strongly chaotic dynamics and are thus far from any quasi-periodic torus, for times as long as t=109.

Rattleback dynamics and its reversal time of rotation.

A rattleback is a rigid, semielliptic toy which exhibits unintuitive behavior; when it is spun in one direction, it soon begins pitching and stops spinning, then it starts to spin in the opposite direction, but in the other direction, it seems to spin just steadily. This puzzling behavior results from the slight misalignment between the principal axes for the inertia and those for the curvature; the misalignment couples the spinning with the pitching and the rolling oscillations. It has been shown that under the no-slip condition and without dissipation the spin can reverse in both directions, and Garcia and Hubbard obtained the formula for the time required for the spin reversal t_{r} [Proc. R. Soc. Lond. A 418, 165 (1988)1364-502110.1098/rspa.1988.0078]. In this work, we reformulate the rattleback dynamics in a physically transparent way and reduce it to a three-variable dynamics for spinning, pitching, and rolling. We obtain an expression of the Garcia-Hubbard formula for t_{r} by a simple product of four factors: (1) the misalignment angle, (2) the difference in the inverses of inertia moment for the two oscillations, (3) that in the radii for the two principal curvatures, and (4) the squared frequency of the oscillation. We perform extensive numerical simulations to examine validity and limitation of the formula, and find that (1) the Garcia-Hubbard formula is good for both spinning directions in the small spin and small oscillation regime, but (2) in the fast spin regime especially for the steady direction, the rattleback may not reverse and shows a rich variety of dynamics including steady spinning, spin wobbling, and chaotic behavior reminiscent of chaos in a dissipative system.

Asymptotic solutions in f(R)-gravity

We study cosmological solutions in R + βRN-gravity for an isotropic Universe filled with ordinary matter with the equation of state parameter γ. Using the Bogolyubov–Krylov–Mitropol’skii averaging method we find asymptotic oscillatory solutions in terms of new functions, which have been specially introduced by us for this problem and appeared as a natural generalization of the usual sine and cosine. It is shown that the late-time behaviour of the Universe in the model under investigation is determined by the sign of the difference γ − γcrit where γcrit = 2N/(3N − 2). If γ < γcrit, the Universe reaches the regime of small oscillations near values of Hubble parameter and matter density, corresponding to general relativity solution. Otherwise higher-curvature corrections become important at late times. We also study numerically basins of attraction for the oscillatory and phantom solutions, which are present in the theory for N > 2. Some important differences between N = 2 and N > 2 cases are discussed.

Rolling in the Higgs model and elliptic functions

Asymptotic methods in nonlinear dynamics are used to improve perturbation theory results in the oscillations regime. However, for some problems of nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and the Friedmann cosmological equations, not only small oscillations regime is of interest but also the regime of rolling (climbing), more precisely the rolling from a top (climbing to a top). In the Friedman cosmology, where the slow rolling regime is often used, the rolling from a top (not necessary slow) is of interest too. In the present work a method for approximate solution to the Higgs equation in the rolling regime is presented. It is shown that in order to improve perturbation theory in the rolling regime turns out to be effective not to use an expansion in trigonometric functions as it is done in case of small oscillations but use expansions in hyperbolic functions instead. This regime is investigated using the representation of the solution in terms of elliptic functions. An accuracy of the corresponding approximation is estimated.

Asymptotic expansion of solutions in a rolling problem

Asymptotic methods in the theory of differential equations and in nonlinear mechanics are commonly used to improve perturbation theory in the small oscillation regime. However, in some problems of nonlinear dynamics, in particular for the Higgs equation in field theory, it is important to consider not only small oscillations but also the rolling regime. In this article we consider the Higgs equation and develop a hyperbolic analogue of the averaging method. We represent the solution in terms of elliptic functions and, using an expansion in hyperbolic functions, construct an approximate solution in the rolling regime. An estimate of accuracy of the asymptotic expansion in an arbitrary order is presented.

Optimal control of vortex-core polarity by resonant microwave pulses

In a vortex-state magnetic nano-disk, the static magnetization is curling in the plane, except in the core region where it is pointing out-of-plane, either up or down leading to two possible stable states of opposite core polarity p. Dynamical reversal of p by large amplitude motion of the vortex core has recently been demonstrated experimentally,raising fundamental interest for potential application in magnetic storage devices. Here we demonstrate coherent control of p by single and double microwave pulse sequences, taking advantage of the resonant vortex dynamics in a perpendicular bias magnetic field. Optimization of the microwave pulse duration required to switch p allows to experimentally infer the characteristic decay time of the vortex core in the large oscillation regime. It is found to be more than twice shorter than in the small oscillation regime, raising the fundamental question of the non-linear behaviour of magnetic dissipation.

Vortex motion induced by lattice defects in two-dimensional easy-plane magnets

In this work we investigate the dynamics of planar vortices in the presence of nonmagnetic impurities in a two-dimensional magnet with XY symmetry. We use analytical as well spin dynamics simulation methods. Our results indicate that vortices are attracted by defects in the lattice. Trapped by the nonmagnetic impurities they execute nonharmonic oscillations. In the regime of small oscillations our analytical results predict that the frequency is proportional to the inverse of the linear size of the system, in quite good agreement with numerical calculations. Although such oscillations barely can be observed in thermodynamic limit, it should be easy to be detected in nanoscale systems. We also consider the vortex motion near a line of nonmagnetic impurities. In such case, the vortex oscillates back and forth along the impurity line.

Stationary cantilever vibrations in oscillating-cantilever-driven adiabatic reversals: Magnetic-resonance-force-microscopy technique

We consider theoretically the novel technique in magnetic resonance force microscopy which is called ``oscillating cantilever-driven adiabatic reversals''. We present analytical and numerical analysis for the stationary cantilever vibrations in this technique. For reasonable values of parameters we estimate the resonant frequency shift as 6Hz per the Bohr magneton. We analyze also the regime of small oscillations of the paramagnetic moment near the transversal plane and the frequency shift of the damped cantilever vibrations.

Time-dependent functional Schrödinger picture approach to many-particle systems

A time-dependent many-body theory for many-particle systems is formulated using the functional Schr?dinger picture. Taking a Gaussian trial wave functional and using the variational principle, the equations of motion are obtained. The equations of motion are solved in the small oscillation regime to yield the generalized RPA results. The electron?hole excitation, the collective plasma oscillation for the paramagnetic state, the Stoner excitation and the spin wave for the ferromagnetic state are derived and discussed. As a further test on the versatility of the present method, the formalism is applied to the bound states of the exciton problem and shown to yield the governing equation in a straightforward manner.

Small oscillations of a chiral Gross-Neveu system

We study the small oscillations regime (RPA approximation) of the time-dependent mean-field equations, obtained in a previous work, which describe the time evolution of one-body dynamical variables of a uniform Chiral Gross-Neveu system. In this approximation we obtain an analytical solution for the time evolution of the one-body dynamical variables. The two-fermion physics can be explored through this solution. The condition for the existence of bound states is examined.

Spin dynamics in sodium clusters

We investigate the dynamics of the electron cloud in a metal cluster from the regime of small oscillations up to high excitations in the multi-plasmon regime. Particular attention is paid to the effect of spin modes. Test cases are a spin saturated cluster, Na9+, and a strongly spin polarized cluster, the spherical isomer of Na6+. The spectral distributions are dominated by the Mie plasmon resonance and a strong collective spin mode in all cases considered. Nonlinear effects as cross talk between multi-plasmon states or chaotic pattern cannot be observed in the investigated energy range.

Microscopic theory of thermodynamic properties of finite nuclei

We develope a method to evaluate large amplitude time-dependent Hartree-Fock (TDHF) dynamics in a single mode and in the presence of a thermal barh. We first apply the method with realistic interactions in the small oscillation regime to isothermal and isentropic breathing mode oscillations. We present the coefficients for the T = 0 incompressibilities and breathing mode frequencoes for 16O and 40Ca. We obtain a significant effect on the collective mass due to the non-locality of the realistic effective interaction. The consequence is that the interaction produces a remarkably low value for the incompressibility while providing an adequate description of the breathing mode energy. We conclude that nuclear matter may have an incompressibility significantly lower than 200 MeV. We present initial results for large amplitude static compression of 40Ca at T = 0 which include the excitation of nucleons to deltas.

Thermodynamic coefficients of hot nuclei.

We present a general method to evaluate large amplitude time-dependent Hartree Fock dynamics in a single mode and in the presence of a thermal bath. We apply the method in the small oscillation regime to isothermal and isentropic breathing mode oscillations. We employ realistic microscopic effective Hamiltonians with phenomenological extensions to large model spaces and obtain thermodynamic coefficients for $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$. Coefficients for the incompressibilities, specific heats, and thermal expansion are obtained along with breathing mode frequencies as a function of temperature. A significant effect due to the nonlocality of the interaction on the collective mass is obtained. The interaction produces a remarkably low value for the incompressibility while providing an adequate description of the breathing mode energy.

Asymmetry in theLCresonance characteristic in dc superconducting quantum interference devices: New method of observing pair-quasiparticle interference in superconducting tunnel junctions

The $\mathrm{LC}$ (inductance-capacitance) resonance phenomenon in the dc SQUID (superconducting quantum interference device) creates a current peak in the $I\ensuremath{-}V$ characteristics at a voltage determined by the resonance frequency. These $I\ensuremath{-}V$ characteristics have been analyzed with the resistively shunted Josephson-junction model in the low-damping and small-oscillation regime. The resonance $I\ensuremath{-}V$ characteristics contain sufficient information to determine all circuit parameters for the dc SQUID's. Specifically, we can apply this analysis to symmetric, tunnel-junction dc SQUID's to deduce the pair-quasiparticle interference conductance ($cos\ensuremath{\theta}$ term).