Analysis of the quadrupole mass filter with quadrupole excitation by the envelope equation method

The method of the envelope equation has been developed to describe the stability of the motion of ions in a quadrupole mass filter in the presence of periodic excitations of the feeding voltage. Dynamic equations that describe the variations in the envelope of ion vibrations in the vicinity of the vertex of the first common stability region have been obtained and reduced to the form of the Mathieu equations. The splitting of the stability diagram of the motion of ions into stability islands due to excitation has been described. The results of the approximate theory have been confirmed by an exact analysis of the stability diagram for rational values of the relative excitation frequency. The boundaries of the applicability domain for the developed theory limited by first-order resonances have been determined.     

Orbital and epicyclic motion of charged test particles around non-rotating Einstein-Æther black holes

The study of charged test particle dynamics in the combined black hole gravitational field and magnetic field around it could provide important theoretical insight into astrophysical processes around such compact object. We have explored the orbital and epicyclic motion of charged test particles in the background of non-rotating Einstein-Æther black holes in the presence of external uniform magnetic field. We numerically integrate the equations of motion and analyze the trajectories of the charged test particles. We examined the stability of circular orbits using effective potential technique and study the characteristics of innermost stable circular orbits. We analyze the key features of quasi-harmonic oscillations of charged test particles nearby the stable circular orbits in an equatorial plane of the black hole, and investigate the radial profiles of the frequencies of latitudinal as well as radial harmonic oscillations in dependence on the strength of magnetic field, mass of the black hole and dimensionless coupling constants of the theory. We demonstrate that the magnetic field and dimensionless parameters of the theory have strong influence on charged particle motion around Einstein-Æther black holes.     

Acceptance of the quadrupole mass filter in the upper stability island under biharmonic excitation

Dynamic characteristics of the quadrupole mass filter (QMF) with the parametric resonance excitation of ion oscillations by a low additional HF voltage are studied by numerical methods. The upper stability island formed by instability bands that follow the isolines of stability parameters of the unperturbed first stability band is considered. Isolines of characteristic indices on the island’s parameter plane are calculated. Ion trajectories at characteristic points near the x and y island boundaries, which have the form of beats, are presented. Parameters of the ellipses are determined as functions of the initial phase at which the ions enter the HF field. These parameters are shown to be periodic with a period of π with respect to the phase shift between the main and additional HF voltages. As a result, the QMF acceptance and, consequently, transmittance have maxima as functions of the phase shift. Therefore, to increase the QMF transmittance, the biharmonic signal must be synchronized.     

Theoretical investigation of unstable axial mass-selective extraction of ions from a nonlinear trap

The unstable axial mass-selective extraction of ions from a three-dimensional quadrupole ion trap is studied theoretically. A method for mapping the ion coordinates over the period of the RF power-supply voltage is developed with allowance for nonlinear distortions of the quadrupole potential. Equations for the envelope of ion oscillations are derived in the form of the equation of motion of a material point in the field of effective forces. The effect of the “delayed extraction” of ions in the presence of negative even field harmonics is explained. The positive even harmonics of the distorted quadrupole potential are shown to be favorable for ion extraction. The dynamics of the extracted ions is investigated.     

Homotopy analysis method to study a quadrupole mass filter

The homotopy analysis method (HAM) is applied to study the behavior of a hyperbolic rods of quadrupole mass filter and a sinusoidal potential form V(ac) cos(Ωt). Numerical computation method of a 20th-order HAM is employed to compare the physical properties of the confined ions with fifth-order Runge-Kutta method. Also, comparison is made for the first stability region, the ion trajectories in real time, the polar plots, and the ion trajectory in x?-?y plan. The results show that the two methods are fairly similar; therefore, the HAM method has potential application to solve linear and nonlinear equations of the charge particle confinement in quadrupole field.     

Overcritical rotation of a trapped Bose-Einstein condensate

The rotational motion of an interacting Bose-Einstein condensate confined by a harmonic trap is investigated by solving the hydrodynamic equations of superfluids, with the irrotationality constraint for the velocity field. We point out the occurrence of an overcritical branch where the system can rotate with angular velocity larger than the oscillator frequencies. We show that in the case of isotropic trapping the system exhibits a bifurcation from an axisymmetric to a triaxial configuration, as a consequence of the interatomic forces. The dynamical stability of the rotational motion with respect to the dipole and quadrupole oscillations is explicitly discussed.     

Nonlinear dynamics of a spinor Bose-Einstein condensate in a double-well potential

We examine the nonlinear dynamical behavior of a spinor Bose-Einstein condensate in a double-well potential. Considering a condensate with large number of atoms, such that it can be described using the mean field theory, we separate the spinor dynamics from the spatial dynamics under the single-mode approximation. We limit ourselves to certain initial conditions under which the spatial mode is frozen so that we can focus on the spinor dynamics only. Identifying collective spin variables of our system, we derive the corresponding nonlinear equations of motion for them. Employing standard stability analysis, we find and characterize fixed points of the system. For a wide range of physical parameters such as tunneling strength and non-linear interactions, as well as for various initial preparations of the system, we identify qualitatively different dynamical regimes possible in the system. In particular, complete and incomplete oscillations of spin variables between quantum wells are found. We also show that by bringing some fixed points close to each other in the phase space of the system, it is possible to induce amplitude modulation to those otherwise regular tunneling oscillations.     

The use of stability bands to improve the performance of quadrupole mass filters

The possibility of increasing the resolving power of quadrupole mass filters has been discussed. It has been shown that the limitations associated with the finite time of flight imposed by Von Zahn’s rule are modified while using the islands of stability that appear when quadrupole is excited by the additional signals. By calculation of the exponential increment of growth of the oscillation amplitude the effect of the acceleration of mass separation and improvement of the peak shape, when the islands of stability are used for ion filtering, is explained. The case of the excitation by two signals at different frequencies has been studied theoretically. The conditions under which suppression of the first order resonance for one of the directions of motion is obtained. The direct modeling of the peak shape of the mass filter shows the possibility of obtaining a resolution of 10,000 with a time of flight of ions through the quadrupole of 100 cycles of the main RF supply, and low sensitivity of the new operating mode to the nonlinear field distortions in the quadrupoles with rods of circular cross sections.     

Stability of two-ion crystals in the Paul trap: A comparison of exact and pseudopotential calculations

The stability of two-ion crystals in a Paul trap with a dc component in the quadrupole potential has been studied with the use of the monodromy matrix. The pseudopotential model predicts crystals with the ions at rest either along the trap axis or in the radial plane. The solutions of the full equations of motion disagree with the predictions of the pseudopotential model when the radial and axial secular frequencies are nearly degenerate: the crystal is either unstable (as first noted by Emmertet al.) or exists in a previously unanticipated configuration in which the ions lie at an angle to the trap axes. A bifurcation diagram near the edge of the crystalline stability range does not support a frequency-doubling route to chaos.Dedicated to H. Walther on the occasion of his 60th birthday     

Ultrashort Optical Pulses in Carbon Nanotubes and Heavy-Ion Absorption

Propagation of electromagnetic waves in a medium with zigzag carbon nanotubes is studied. Based on the Maxwell equations, an effective equation for the electromagnetic field vector potential is derived, which takes into account ion oscillations in the dielectric medium with nanotubes. It is found that the pulse shape depends on the character of oscillations of heavy ions, which may absorb the electromagnetic field, and on other parameters of the problem.     

Acceptance and transmittance of a quadrupole mass filter with amplitude modulation of RF voltage with allowance for an edge field

A method for calculating the acceptance of a quadrupole mass filter with amplitude modulation of rf voltage is developed. The key factor in determining the transformation of the phase ellipse by edge fields is taking into account the phase shift of the rf field during motion of ions in the input edge field. The variation of the combined acceptance depending on the axial energy of ions (or the transit time of ions through the edge field) is in conformity with the rated transmittance, which is an indirect indication of the correctness of the proposed approach.     

An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field

We study the limiting behavior of a singularly perturbed Schr?dinger-Poisson system describing a 3-dimensional electron gas strongly confined in the vicinity of a plane (x, y) and subject to a strong uniform magnetic field in the plane of the gas. The coupled effects of the confinement and of the magnetic field induce fast oscillations in time that need to be averaged out. We obtain at the limit a system of 2-dimensional Schr?dinger equations in the plane (x, y), coupled through an effective selfconsistent electrical potential. In the direction perpendicular to the magnetic field, the electron mass is modified by the field, as the result of an averaging of the cyclotron motion. The main tools of the analysis are the adaptation of the second order long-time averaging theory of ODEs to our PDEs context, and the use of a Sobolev scale adapted to the confinement operator.     

Extended bodies with quadrupole moment interacting with gravitational monopoles: reciprocity relations

An exact solution of Einstein’s equations representing the static gravitational field of a quasi-spherical source endowed with both mass and mass quadrupole moment is considered. It belongs to the Weyl class of solutions and reduces to the Schwarzschild solution when the quadrupole moment vanishes. The geometric properties of timelike circular orbits (including geodesics) in this spacetime are investigated. Moreover, a comparison between geodesic motion in the spacetime of a quasi-spherical source and non-geodesic motion of an extended body also endowed with both mass and mass quadrupole moment as described by Dixon’s model in the gravitational field of a Schwarzschild black hole is discussed. Certain “reciprocity relations” between the source and the particle parameters are obtained, providing a further argument in favor of the acceptability of Dixon’s model for extended bodies in general relativity.     

Dirac's extended electron model

Dirac's extended electron model is elaborated here both on the classical and quantum level. The classical equations of motion are deduced from Dirac's action principle. It is shown that the model is free of the troublesome runaway solutions in the classical theory. The quantum theory of the radial oscillations is worked out in detail and the spectrum is discussed. The stability of the model is studied and it is found that Dirac's extended electron is unstable against quadrupole deformations.     

Exterior and interior metrics with quadrupole moment

We present the Ernst potential and the line element of an exact solution of Einstein’s vacuum field equations that contains as arbitrary parameters the total mass, the angular momentum, and the quadrupole moment of a rotating mass distribution. We show that in the limiting case of slowly rotating and slightly deformed configuration, there exists a coordinate transformation that relates the exact solution with the approximate Hartle solution. It is shown that this approximate solution can be smoothly matched with an interior perfect fluid solution with physically reasonable properties. This opens the possibility of considering the quadrupole moment as an additional physical degree of freedom that could be used to search for a realistic exact solution, representing both the interior and exterior gravitational field generated by a self-gravitating axisymmetric distribution of mass of perfect fluid in stationary rotation.     

Nonlinear coherent dynamics of an atom in an optical lattice

We consider a simple model of the lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. The internal dynamics of the atom is governed by the laser field, which is treated as classical with a large number of photons. The center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degrees of freedom. The resulting Hamilton-Schrö dinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion, and the total atomic energy and the Bloch vector length are conserved during the interaction. In our previous papers, the motion of the atom has been shown to be regular or chaotic (in the sense of exponential sensitivity to small variations of the initial conditions and/or the system’s control parameters) depending on the values of the control parameters, atom-field detuning, and recoil frequency. At the exact atom-field resonance, the exact solutions for both the external and internal atomic degrees of freedom can be derived. The center-of-mass motion does not depend in this case on the internal variables, whereas the Rabi oscillations of the atomic inversion is a frequency-modulated signal with the frequency defined by the atomic position in the optical lattice. We study analytically the correlations between the Rabi oscillations and the center-of-mass motion in two limiting cases of a regular motion out of the resonance: (1) far-detuned atoms and (2) rapidly moving atoms. This paper is concentrated on chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that an atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential, or fly ballistically with weak chaotic oscillations of its momentum, or wander in the optical lattice, changing the direction of motion in a chaotic way. In the regime of chaotic wandering, the atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion-Poincaré mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. The Poincaré mappings are constructed using the translational invariance of the standing laser wave. We find common features with typical nonhyperbolic Hamiltonian systems-chains of resonant islands of different sizes imbedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of the atomic trajectories sticking to boundaries of regular islands, which should have a great influence on atomic transport in optical lattices, is found and demonstrated numerically.     

Stability and fluctuations in the external source problem of QCD2 with massless quarks

We discuss the stability of the static field and polarization charge produced by an abelian static external source in two-dimensional QCD with massless quarks. This is done in a previously proposed scheme allowing an approximate treatment of polarization effects. The discussion is based on linear fluctuation equations obtained from the equations of motion written in an appropriate moving gauge frame. We establish that the static configuration is stable in the sense that there are no initially small fluctuations which grow in time. However, in contrast with QED₂, there are fluctuations which are localized near the external source and whose amplitudes do not decay in time. These modes are studied in detail in the case of a single external point source.     

Dynamics of extended bodies with spin-induced quadrupole in Kerr spacetime: generic orbits

We discuss motions of extended bodies in Kerr spacetime by using Mathisson–Papapetrou–Dixon equations. We firstly solve the conditions for circular orbits, and calculate the orbital frequency shift due to the mass quadrupoles. The results show that we need not consider the spin-induced quadrupoles in extreme-mass-ratio inspirals for space-based gravitational wave detectors. We quantitatively investigate the temporal variation of rotational velocity of the extended body due to the coupling of quadrupole and background gravitational field. For generic orbits, we numerically integrate the Mathisson–Papapetrou–Dixon equations for evolving the motion of an extended body orbiting a Kerr black hole. By comparing with the monopole–dipole approximation, we reveal the influences of quadrupole moments of extended bodies on the orbital motion and chaotic dynamics of extreme-mass-ratio systems. We do not find any chaotic orbits for the extended bodies with physical spins and spin-induced quadrupoles. Possible implications for gravitational wave detection and pulsar timing observation are outlined.     

Spectrum of charged particle oscillations in an RF quadrupole field

Variations in the spectral composition of ion oscillations within several stability regions of a quadrupole mass filter were studied. The frequency spectrum was shown to consist of two line systems. Side lines ω_n=nω₀±βω₀/2 were observed in the oscillation spectrum near harmonics nω₀ (n=0, 1, 2,...), where ω₀ is the circular frequency of an RF field and β is the stability parameter. Near the boundaries of the stability regions, the oscillations took the form of beatings. For even values of the stability parameter, β=2k (k=1, 2,...), the beat frequency coincides with the fundamental frequency ω₀ and, for β=2k−1, the main beat frequencies are ω₀/2 and 3ω₀/2.     

Quadrupole test particle as a detector of gravitational waves

In this paper it is shown that in general relativity the theory of motion of quadrupole test particles (QTP's) can be used to describe the energy and angular momentum absorption by detectors of gravitational waves. By specifying the form of the quadrupole moment tensor Taub's [7] equations of motion of QTP's are simplified. In these equations the terms describing the change of the mass and of the angular momentum of a QTP due to external gravitational waves are found to occur. The limiting case of the flat space-time is also briefly discussed.