Toroidal Trap Research Articles

Spin-orbit-coupled Bose-Einstein condensates held under a toroidal trap

We study a quasispin-$1/2$ Bose-Einstein condensate with synthetically generated spin-orbit coupling in a toroidal trap, and show that the system has a rich variety of ground and metastable states. As the central hole region increases, i.e., the potential changes from harmonic-like to ring-like, the condensate exhibits a variety of structures, such as triangular stripes, flower-petal patterns, and counter-circling states. We also show that the rotating systems have exotic vortex configurations. In the limit of a quasi-one dimensional ring, the quantum many-body ground state is obtained, which is found to be the fragmented condensate.

Dimensional reduction in Bose-Einstein condensed clouds of atoms confined in tight potentials of any geometry and any interaction strength.

Motivated by numerous experiments on Bose-Einstein condensed atoms which have been performed in tight trapping potentials of various geometries [elongated and/or toroidal (annular)], we develop a general method which allows us to reduce the corresponding three-dimensional Gross-Pitaevskii equation for the order parameter into an effectively one-dimensional equation, taking into account the interactions (i.e., treating the width of the transverse profile variationally) and the curvature of the trapping potential. As an application of our model we consider atoms which rotate in a toroidal trapping potential. We evaluate the state of lowest energy for a fixed value of the angular momentum within various approximations of the effectively one-dimensional model and compare our results with the full solution of the three-dimensional problem, thus getting evidence for the accuracy of our model.

Plasmastatic model of toroidal trap “Galatea-belt”

Magnetic galatea-traps for thermonuclear plasma confinement with current carrying conductors immersed into the plasma volume, are represented by an example of the toroidal trap “The Belt” with two circular conductors. Numerical models of equilibrium plasma and field configurations are investigated in straightened into cylinder analogues of some toroidal galateas in a series of works by the authors. This paper presents a plasmastatic model of configurations in the toroidal variant of “The Belt” in terms of a boundary problem with the Grad-Shafranov equation. Distinctions of their geometry and quantitative characteristics from the cylindrical analogues and their dependence of parameters are determined in computation.

Computation of Mathieu stability plot for an arbitrary toroidal ion trap mass analyser

This paper presents a method to construct Mathieu stability plot for any arbitrary toroidal ion trap mass analyser. The proposed method is demonstrated on two toroidal ion trap mass analyser geometries. Two additional studies have also been carried out. These include prediction of ion secular frequency and ion trajectories at resonances in these traps.In this numerical study, toroidal multipole coefficients of the respective traps are first evaluated and these are used for computing Mathieu parameters, a and q. These Mathieu parameters are used for constructing the stability plot, predicting secular frequency of ion motion and evaluating nonlinear resonances.The stability regions of both the traps considered in this paper are qualitatively similar to the corresponding plot of the linear ion trap mass analyser. There are, however two significant differences. First, the stability plots for toroidal ion trap mass analysers have prominent resonances. Second the apices of the stability plots of the toroidal ion trap mass analysers are different from those of the linear ion trap mass analyser.The secular frequency obtained using the Mathieu parameters in this paper agrees well with those obtained numerically, except close to the stability boundaries.Finally, the prominent nonlinear resonances have been identified as βr=23 and βz=23 in both the traps. In one of the two traps there is an additional prominent nonlinear resonance at βr+βz=1.

Fundamental modes in waveguide pipe twisted by saturated double-well potential

We study fundamental modes trapped in a rotating ring with a saturated nonlinear double-well potential. This model, which is based on the nonlinear Schrodinger equation, can be constructed in a twisted waveguide pipe in terms of light propagation, or in a Bose–Einstein condensate (BEC) loaded into a toroidal trap under a combination of a rotating π-out-of-phase linear potential and nonlinear pseudopotential induced by means of a rotating optical field and the Feshbach resonance. Three types of fundamental modes are identified in this model, one symmetric and the other two asymmetric. The shape and stability of the modes and the transitions between different modes are investigated in the first rotational Brillouin zone. A similar model used a Kerr medium to build its nonlinear potential, but we replace it with a saturated nonlinear medium. The model exhibits not only symmetry breaking, but also symmetry recovery. A specific type of unstable asymmetric mode is also found, and the evolution of the unstable asymmetric mode features Josephson oscillation between two linear wells. By considering the model as a configuration of a BEC system, the ground state mode is identified among these three types, which characterize a specific distribution of the BEC atoms around the trap.

Determination of multipole coefficients in toroidal ion trap mass analysers

In this paper, we present methods to determine multipole coefficients for describing the potential in toroidal ion trap mass analysers. Three different methods have been presented to compute the toroidal multipole coefficients. The first method uses at least square fit (LS) and is useful when we have ability to compute potential at a set of points in the trapping region. In the second method we use the Discrete Fourier Transform (DFT) of potentials on a circle in the trapping region. The third method uses surface charge distribution obtained from the Boundary Element Method (BEM) to compute these coefficients. Using these multipole coefficients we have presented (1) equations of ion motion in toroidal ion traps, (2) the Mathieu parameters in terms of multipole coefficients and (3) the secular frequency of ion motion in these traps. It has been shown that the secular frequency obtained from our method has a good match with that obtained from numerical trajectory simulation.

Hydraulic jumps in inhomogeneous strongly coupled toroidal dust flows

The inhomogeneous flow of strongly coupled dust particles in a toroidal particle trap with harmonic radial confinement is analyzed in the incompressible fluid limit. It is shown that the flow can spontaneously generate shock-like events, which are similar to the hydraulic jump in open channel flows. A definition of the Froude number for this model is given and the critical speed is recovered as the group velocity of surface waves. This hydraulic model is compared with molecular-dynamics simulations, which show that a sudden bifurcation of the flow lines and a localized temperature peak appear just at the point where the critical condition for the hydraulic jump is located.

Persistent Superfluid Flow Arising from the He-McKellar-Wilkens Effect in Molecular Dipolar Condensates.

We show that the He-McKellar-Wilkens effect can induce a persistent flow in a Bose-Einstein condensate of polar molecules confined in a toroidal trap, with the dipolar interaction mediated via an electric dipole moment. For Bose-Einstein condensates of atoms with a magnetic dipole moment, we show that although it is theoretically possible to induce persistent flow via the Aharonov-Casher effect, the strength of the electric field required is prohibitive. We also outline an experimental geometry tailored specifically for observing the He-McKellar-Wilkens effect in toroidally trapped condensates.

A toroidal trap for cold atoms using an rf-dressed quadrupole trap

We demonstrate the trapping of cold atoms in a toroidal geometry using a radio frequency (rf) dressed quadrupole magnetic trap formed by superposing a strong rf-field on a quadrupole trap. This rf-dressed quadrupole trap has the minimum potential away from the quadrupole trap centre on a circular path which facilitates trapping in toroidal geometry. In these experiments, the laser cooled atoms were first trapped in a quadrupole trap, then cooled evaporatively using a weak rf-field, and finally trapped in an rf-dressed quadrupole trap. The radius of the toroid could be varied by varying the frequency of the dressing rf-field. It has also been demonstrated that a single rf source and an antenna can be used for the rf-evaporative cooling as well as for the rf-dressing of atoms. The atoms trapped in the toroidal trap may have applications in the realization of an atom gyroscope as well as in studying the quantum gases in low dimensions.

Optimization of Performance of Toroidal Ion Trap with Trianglar Electrode by Theoretical Simulation

Abstract Owing to the ion storage capacity far superior to that of three-dimensional ion trap of the same volume, toriodal ion trap is deemed to be another useful candidate of miniaturization of ion trap mass analyzer in recent years. To further optimize the performance of mass spectrometry of toriodal ion trap, especially the ion detection efficiency and the mass resolving power, a new toriodal ion trap built with triangular electrodes was proposed and investigated theoretically. The new toroidal ion trap was composed of two identical ring electrodes with triangular cross section and two cylindrical electrodes of different sizes. Ions were extracted from the ion trap through resonance excitation for mass analysis. The asymmetrical triangular electrode structure was obtained through theoretical simulation and optimization of ion detection efficiency and mass resolving power. The mass resolution (M/ΔM) of ∼1500 at m/z 609 was obtained for a toriodal ion trap with an optimal electrode structure.

Dark-soliton collisions in a toroidal Bose-Einstein condensate

We study the dynamics of two gray solitons in a Bose-Einstein condensate confined by a toroidal trap with a tight confinement in the radial direction. Gross-Pitaevskii simulations show that solitons can be long living objects passing through many collisional processes. We have observed quite different behaviors depending on the soliton velocity. Very slow solitons, obtained by perturbing the stationary solitonic profile, move with a constant angular velocity until they collide elastically and move in the opposite direction without showing any sign of lowering their energy. In this case the density notches are always well separated and the fronts are sharp and straight. Faster solitons present vortices around the notches, which play a central role during the collisions. We have found that in these processes the solitons lose energy, as the outgoing velocity turns out to be larger than the incoming one. To study the dynamics, we model the gray soliton state with a free parameter that is related to the soliton velocity. We further analyze the energy, soliton velocity and turning points in terms of such a free parameter, finding that the main features are in accordance with the infinite one-dimensional system.

Coherent quantum phase slip in two-component bosonic atomtronic circuits

Coherent quantum phase slip consists in the coherent transfer of vortices in superfluids. We investigate this phenomenon in two miscible coherently coupled components of a spinor Bose gas confined in a toroidal trap. After imprinting different vortex states, i.e. states with quantized circulation, on each component, we demonstrate that during the whole dynamics the system remains in a linear superposition of two current states in spite of the nonlinearity, and can be mapped onto a linear Josephson problem. We propose this system as a good candidate for the realization of a Mooij–Harmans qubit and remark its feasibility for implementation in current experiments with 87Rb, since we have used values for the physical parameters currently available in laboratories.

Experimental observation of crystalline particle flows in toroidal dust clouds

The dust flow in a toroidal dust trap is studied experimentally. The flow is driven by the Hall component of the ion drag force in a magnetized plasma. Dust density waves are found in a torus with a large minor radius a, which allows for several wavelength, 2a>5λ, in the (mostly) radial direction of the ion flow. Beyond an intermediate state with radial sloshing oscillations, a crystalline dust flow with suppressed wave activity could be realized for 2a<2λ. The particles arrange themselves in distinct layers with hexagonal-like local order. Smooth transitions between states with different numbers of layers are found in the inhomogeneous flow.

Impurity coupled to an artificial magnetic field in a Fermi gas in a ring trap

The dynamics of a single impurity interacting with a many-particle background is one of the central problems of condensed-matter physics. Recent progress in ultracold-atom experiments makes it possible to control this dynamics by coupling an artificial gauge field specifically to the impurity. In this paper, we consider a narrow toroidal trap in which a Fermi gas is interacting with a single atom. We show that an external magnetic field coupled to the impurity is a versatile tool to probe the impurity dynamics. Using a Bethe ansatz, we calculate the eigenstates and corresponding energies exactly as a function of the flux through the trap. Adiabatic change of flux connects the ground state to excited states due to flux quantization. For repulsive interactions, the impurity disturbs the Fermi sea by dragging the fermions whose momentum matches the flux. This drag transfers momentum from the impurity to the background and increases the effective mass. The effective mass saturates to the total mass of the system for infinitely repulsive interactions. For attractive interactions, the drag again increases the effective mass which quickly saturates to twice the mass of a single particle as a dimer of the impurity and one fermion is formed. For excited states with momentum comparable to number of particles, effective mass shows a resonant behavior. We argue that standard tools in cold-atom experiments can be used to test these predictions.

Spin squeezing by tensor twisting and Lipkin-Meshkov-Glick dynamics in a toroidal Bose-Einstein condensate with spatially modulated nonlinearity

We propose a scheme for spin-squeezing in the orbital motion of a Bose-Einstein condensate (BEC) in a toroidal trap. A circular lattice couples two counter-rotating modes and squeezing is generated by the nonlinear interaction spatially modulated at half the lattice period. By varying the amplitude and phase of the modulation, various cases of the twisting tensor can be directly realized, leading to different squeezing regimes. These include one-axis twisting and the two-axis counter-twisting which are often discussed as the most important paradigms for spin squeezing. Our scheme naturally realizes the Lipkin-Meshkov-Glick model with the freedom to vary all its parameters simultaneously.

Sagnac interferometry using bright matter-wave solitons.

We use an effective one-dimensional Gross-Pitaevskii equation to study bright matter-wave solitons held in a tightly confining toroidal trapping potential, in a rotating frame of reference, as they are split and recombined on narrow barrier potentials. In particular, we present an analytical and numerical analysis of the phase evolution of the solitons and delimit a velocity regime in which soliton Sagnac interferometry is possible, taking account of the effect of quantum uncertainty.

Dynamics in asymmetric double-well condensates

The dynamics of Bose-Einstein condensates in asymmetric double-wells is studied. We construct a two-mode model and analyze the properties of the corresponding phase-space diagram, showing in particular that the minimum of the phase-space portrait becomes shifted from the origin as a consequence of the nonvanishing overlap between the ground and excited states from which the localized states are derived. We further incorporate effective interaction corrections in the set of two-mode model parameters. Such a formalism is applied to a recent experimentally explored system, which is confined by a toroidal trap with radial barriers forming an arbitrary angle between them. We confront the model results with Gross-Pitaevskii simulations for various angle values finding a very good agreement. We also analyze the accuracy of a previously employed simple model for moving barriers, exploring a possible improvement that could cover a wider range of trap asymmetries.

Decay of a superfluid current of ultracold atoms in a toroidal trap

Using a numerical implementation of the truncated Wigner approximation, we simulate the experiment reported by Ramanathan et al. in Phys. Rev. Lett. 106, 130401 (2011), in which a Bose-Einstein condensate is created in a toroidal trap and set into rotation via a phase imprinting technique. A potential barrier is then placed in the trap to study the decay of the superflow. We find that the current decays via thermally activated phase slips, which can also be visualized as vortices crossing the barrier region in the radial direction. Adopting the notion of critical velocity used in the experiment, we determine it to be lower than the local speed of sound at the barrier, in contradiction to the predictions of the zero-temperature Gross-Pitaevskii equation. We map out the superfluid decay rate and critical velocity as a function of temperature and observe a strong dependence. Thermal fluctuations offer a partial explanation of the experimentally observed reduction of the critical velocity from the phonon velocity.

Plasmastatic models of galathea traps with magnetically transparent boundaries

Mathematical models and results of calculation of plasma equilibrium in a circular cylinder with three helical or straight imbedded current-carrying conductors (i.e., in a straightened analog of a toroidal Galathea trap) are presented. The equilibrium is described in the framework of two-dimensional boundary value problems with plane and helical analogs of the Grad-Shafranov equation for the scalar magnetic flux function. Problems with first-kind boundary conditions corresponding to a magnetically transparent boundary of the cylinder and problems with second-kind boundary conditions and a given value of the electric current flowing in plasma (in addition to those flowing in the conductors) are considered. Deformations of magnetoplasma configurations in the cylinder for different formulations of the above-specified problems are investigated numerically.

Persistent currents in two-component condensates in a toroidal trap

The stability of persistent currents in a two-component Bose-Einstein condensate in a toroidal trap is studied in both the miscible and the immiscible regimes. In the miscible regime we show that superflow decay is related to linear instabilities of the spin-density Bogoliubov mode. We find a region of partial stability, where the flow is stable in the majority component while it decays in the minority component. We also characterize the dynamical instability appearing for a large relative velocity between the two components. In the immiscible regime the stability criterion is modified and depends on the specific density distribution of the two components. The effect of a coherent coupling between the two components is also discussed.