Toroidal Potential Research Articles

Three-dimensional vortex and multipole quantum droplets in a toroidal potential

We investigate the existence, stability, and evolution dynamics of three-dimensional quantum droplets (QDs) in binary Bose–Einstein condensates trapped in a toroidal potential. The interplay of competing mean-field (MF) and beyond-MF nonlinear terms (attractive and repulsive ones) with the potential enables the formation of a variety of stable QDs with complex structures, from higher-order vortices and multipoles to necklace complexes. There are two branches of the vortex and multipole QDs with opposite slopes of the norm-vs.-chemical-potential curves. The upper-branch vortex QDs are completely stable, regardless of the value of their topological charge (winding number), m. On the other hand, multipole QDs are stable only near the turning point of the curves. Although the stability region shrinks with the growth of the number of poles, necklace-shaped QDs remain stable for the number up to 16. Applying a torque perpendicular to the original QD’s angular momentum, we observe stable precession of vortex QDs. Periodic rotation is also observed when multipole QDs are set in motion by a planar phase torque.

Algorithmic Aspects of Simulation of Magnetic Field Generation by Thermal Convection in a Plane Layer of Fluid

We present an algorithm for numerical solution of the equations of magnetohydrodynamics describing the convective dynamo in a plane horizontal layer rotating about an arbitrary axis under geophysically sound boundary conditions. While in many respects we pursue the general approach that was followed by other authors, our main focus is on the accuracy of simulations, especially in the small scales. We employ the Galerkin method. We use products of linear combinations (each involving two to five terms) of Chebyshev polynomials in the vertical Cartesian space variable and Fourier harmonics in the horizontal variables for space discretisation of the toroidal and poloidal potentials of the flow (satisfying the no-slip conditions on the horizontal boundaries) and magnetic field (for which the boundary conditions mimick the presence of a dielectric over the fluid layer and an electrically conducting bottom boundary), and of the deviation of temperature from the steady-state linear profile. For the chosen coefficients in the linear combinations, the products satisfy the respective boundary conditions and constitute non-orthogonal bases in the weighted Lebesgue space. Determining coefficients in the expansion of a given function in such a basis (for instance, for computing the time derivatives of these coefficients) requires solving linear systems of equations for band matrices. Several algorithms for determining the coefficients, which are exploiting algebraically precise relations, have been developed, and their efficiency and accuracy have been numerically investigated for exponentially decaying solutions (encountered when simulating convective regimes which are spatially resolved sufficiently well). For the boundary conditions satisfied by the toroidal component of the flow, our algorithm outperforms the shuttle method, but the latter proves superior when solving the problem for the conditions characterising the poloidal component. While the conditions for the magnetic field on the horizontal boundaries are quite specific, our algorithms for the no-slip boundary conditions are general-purpose and can be applied for treating other boundary-value problems in which the zero value must be admitted on the boundary.

Effective potentials in a rotating spin-orbit-coupled spin-1 spinor condensate

We theoretically study the stationary-state vortex lattice configurations of rotating spin-orbit (SO)- and coherently-coupled spin-1 Bose–Einstein condensates (BECs) trapped in quasi-two-dimensional harmonic potentials. The combined effects of rotation, SO and coherent couplings are analyzed systematically from the single-particle perspective. Through the single-particle Hamiltonian, which is exactly solvable for one-dimensional coupling, we illustrate that a boson in these rotating SO- and coherently-coupled condensates are subjected to effective toroidal, symmetric double-well, or asymmetric double-well potentials under specific coupling and rotation strengths. In the presence of mean-field interactions, using the coupled Gross–Pitaevskii formalism at moderate to high rotation frequencies, the analytically obtained effective potential minima and the numerically obtained coarse-grained density maxima position are in excellent agreement. On rapid rotation, we further find that the spin-expectation per particle of an antiferromagnetic spin-1 BEC approaches unity indicating a similarity in the response with ferromagnetic SO-coupled condensates.

Ground-state phases and spin textures of spin–orbit-coupled dipolar Bose–Einstein condensates in a rotating toroidal trap* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11475144 and 11047033), the Natural Science Foundation of Hebei Province, China (Grant Nos. A2019203049 and A2015203037), the University Science and Technology Foundation of Hebei Provincial Department of Education, China (Grant No. Z2017056), Science

We investigate the ground-state phases and spin textures of spin–orbit-coupled dipolar pseudo-spin-1/2 Bose–Einstein condensates in a rotating two-dimensional toroidal potential. The combined effects of dipole–dipole interaction (DDI), spin–orbit coupling (SOC), rotation, and interatomic interactions on the ground-state structures and topological defects of the system are analyzed systematically. For fixed SOC strength and rotation frequency, we provide a set of phase diagrams as a function of the DDI strength and the ratio between inter- and intra-species interactions. The system can show rich quantum phases including a half-quantum vortex, symmetrical (asymmetrical) phase with quantum droplets (QDs), asymmetrical segregated phase with hidden vortices (ASH phase), annular condensates with giant vortices, triangular (square) vortex lattice with QDs, and criss-cross vortex string lattice, depending on the competition between DDI and contact interaction. For given DDI strength and rotation frequency, the increase of the SOC strength leads to a structural phase transition from an ASH phase to a tetragonal vortex lattice then to a pentagonal vortex lattice and finally to a vortex necklace, which is also demonstrated by the momentum distributions. Without rotation, the interplay of DDI and SOC may result in the formation of a unique trumpet-shaped Bloch domain wall. In addition, the rotation effect is discussed. Furthermore, the system supports exotic topological excitations, such as a half-skyrmion (meron) string, triangular skyrmion lattice, skyrmion–half-skyrmion lattice, skyrmion–meron cluster, skyrmion–meron layered necklace, skyrmion–giant-skyrmion necklace lattice, and half-skyrmion–half-antiskyrmion necklace.

Event Studies of High‐Latitude FACs With Inverse and Assimilative Analysis of AMPERE Magnetometer Data

AbstractWe present examples of high‐latitude field‐aligned current (FAC) and toroidal magnetic potential patterns in both hemispheres reconstructed at a 2‐min cadence using an updated optimal interpolation (OI) method that ingests magnetic perturbation data provided by the Active Magnetosphere and Planetary Electrodynamics Response Experiment (AMPERE) program. A solstice and an equinoctial event are studied to demonstrate the reconstructed patterns and to provide scientific insights into FAC response to different solar wind drivers. For the 14 June 2011 high‐speed stream event with mostly northward Bz driving, we found persistently stronger FACs in the Northern Hemisphere. Extreme interhemispheric asymmetry is associated with the interplanetary magnetic field (IMF) direction and large dipole tilt, consistent with earlier studies. FAC asymmetries seen during an isolated substorm can be attributed to dipole tilt. During relatively low geomagnetic activity, the FAC response to IMF Bx changes is identified. For the 17–18 March 2013 period, we provide global snapshots of rapid FAC changes related to an interplanetary shock passage. We further present comparisons between instantaneous and mean behaviors of FAC for the solar wind sheath passage and interplanetary coronal mass ejection southward Bz interval and northward Bz intervals. We show that (1) sheath passage results in strong FAC and high variation in the dayside polar cap region and pre‐midnight region, different from the typical R1/R2 currents during prolonged southward Bz; (2) four‐cell reverse patterns appear during northward Bz but are not stable; and (3) persistent dawn‐dusk asymmetry is seen throughout the storm, especially during an extreme substorm, likely associated with a dawnside current wedge.

Analysis of the rotative potential of two-frequency oscillation of water molecule

The rotational oscillations of a water molecule are considered using the model of a two-frequency physical pendulum and it is shown that the type of its potential is correctly described as toroidal. It is shown that the ellipticity of the toroidal potential in a non-uniform force field decreases with an increase in the exponent n than for an ellipsoidal one, however, both potentials become close. A decrease in the ellipticity of the toroidal potential in this field can lead to the expansion of the region of existence of elliptical-like oscillations of the two-frequency pendulum towards its lower speeds

Poloidal-Toroidal Decomposition of Solenoidal Vector Fields in the Ball

Under study is the polynomial orthogonal basis system of vector fields in the ball which corresponds to the Helmholtz decomposition and is divided into the three parts: potential, harmonic, and solenoidal. It is shown that the decomposition of a solenoidal vector field with respect to this basis is a poloidal-toroidal decomposition (the Mie representation). In this case, the toroidal potentials are Zernike polynomials, whereas the poloidal potentials are generalized Zernike polynomials. The polynomial system of toroidal and poloidal vector fields in a ball can be used for solving practical problems, in particular, to represent the geomagnetic field in the Earth’s core.

Exotic vortex structures of the dipolar Bose–Einstein condensates trapped in harmonic-like and toroidal potential

Based on the tunable intensity and waist of Gaussian laser, harmonic-like and toroidal potentials can be achieved and the ground-state properties of the dipolar Bose–Einstein condensate (BEC) trapped in such potentials are investigated. It is found that, in the harmonic-like potential, the singly and doubly quantized vortices can exist in the scale condensate and translate respectively into vortex pairs and triangular vortex lattice with increasing dipole–dipole interaction (DDI). Especially, the sandwich-like structure can be observed in the ground-state density profiles by tuning the direction and strength of DDI for some rotating frequency. In the toroidal potential, the competition between the inter-component interaction and DDI can induce the transition between immiscible and miscible states, and results in the structures of a doubly quantized vortex surrounded by a vortex ring. It is worth emphasizing that, with the increasing of DDI, the doubly quantized vortex in the harmonic-like potential becomes two singly quantized vortices, while in the toroidal potential it is no happen due to the presence of Gaussian barrier.

Shells in a toroidal nucleus in the intermediate-mass region

Attention is fixed on shells in toroidal nuclei in the intermediate mass region using a toroidal single-particle potential. We find that there are toroidal shells in the intermediate mass region with large single-particle energy gaps at various nucleon numbers located at different toroidal deformations characterized by the aspect ratios of toroidal major to minor radius. These toroidal shells provide extra stability at various toroidal deformations. Relative to a toroidal core, Bohr-Mottelson spin-aligning particle-hole excitations may be constructed to occupy the lowest single-particle Routhian energies to lead to toroidal high-spin isomers with different spins. Furthermore, as a nucleon in a toroidal nucleus possesses a vorticity quantum number, toroidal vortex nuclei may be constructed by making particle-hole excitations in which nucleons of one type of vorticity are promoted to populate un-occupied single-particle orbitals of the opposite vorticity. Methods for producing toroidal high-spin isomers and toroidal vortex nuclei are discussed.

Exotic ground states of a spin–orbit-coupled spinor Bose–Einstein condensate trapped by a toroidal potential

The ground-state properties of a spin–orbit-coupled spin-1 Bose–Einstein condensate (BEC) in a toroidal trap are investigated. The exotic ground states, such as the necklace state, the persistent flow and even the vortex state, for the three components in the system are induced by adjustable external parameters. It is found that the petal number of the necklace states is increased with increasing the intensity of the spin–orbit coupling (SOC), and the anisotropy of the SOC can be used to control the structure of the ground states. The rotation of the condensate induced by the external field makes the densities of the three components asymmetric and tends to transform the necklace state to the persistent flow. The radius of the toroidal potential is another degree of freedom for manipulating the necklace state. In addition, the transition between the necklace state and the vortex, intermediated by the persistent flow, can be controlled by the ratio of the density-density and spin-exchange interactions. All of above results enrich our knowledge about the properties of the ground states of spin-1 BEC trapped in the toroidal trap.

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.

Smooth Archimedean-spiral ring waveguide for cold atomic gyroscope

We propose a robust scheme that creates a toroidal magnetic potential on a single-layer atom chip. The wire layout consists of two interleaved Archimedean spirals, which avoids the trapping perturbation caused by the input and output ports. By using a rotation bias field, the minimum of the time-averaged orbiting potential is lifted from zero, and then a relatively smooth and harmonic ring trap is formed. The location of the waveguide is immune to the magnetic variations, as it is only determined by the wire layout. The ring waveguide offers an ideal solution to developing a compact and portable atomic gyroscope.

Inverse procedure for high‐latitude ionospheric electrodynamics: Analysis of satellite‐borne magnetometer data

AbstractThis paper presents an analysis of data from the magnetometers on board the Defense Meteorological Satellite Program (DMSP) F‐15, F‐16, F‐17, and F‐18 satellites and the Iridium satellite constellation, using an inverse procedure for high‐latitude ionospheric electrodynamics, during the period of 29–30 May 2010. The Iridium magnetometer data are made available through the Active Magnetosphere and Planetary Electrodynamics Response Experiment (AMPERE) program. The method presented here is built upon the assimilative mapping of ionospheric electrodynamics procedure but with a more complete treatment of the prior model uncertainty to facilitate an optimal inference of complete polar maps of electrodynamic variables from irregularly distributed observational data. The procedure can provide an objective measure of uncertainty associated with the analysis. The cross‐validation analysis, in which the DMSP data are used as independent validation data sets, suggests that the procedure yields the spatial prediction of DMSP perturbation magnetic fields from AMPERE data alone with a median discrepancy of 30–50 nT. Discrepancies larger than 100 nT are seen in about 20% of total samples, whose location and magnitude are generally consistent with the previously identified discrepancy between DMSP and AMPERE data sets. Resulting field‐aligned current (FAC) patterns exhibit more distinct spatial patterns without spurious high‐frequency oscillatory features in comparison to the FAC products provided by AMPERE. Maps of the toroidal magnetic potential and FAC estimated from both AMPERE and DMSP data under four distinctive interplanetary magnetic field (IMF) conditions during a magnetic cloud event demonstrate the IMF control of high‐latitude electrodynamics and the opportunity for future scientific investigation.

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.

Hysteresis and metastability of Bose-Einstein-condensed clouds of atoms confined in ring potentials

We consider a Bose-Einstein-condensed cloud of atoms which rotate in a toroidal or annular potential. Assuming one-dimensional motion, we evaluate the critical frequencies associated with the effect of hysteresis and the critical coupling for stability of the persistent currents. We perform these calculations using both the mean-field approximation and the method of numerical diagonalization of the many-body Hamiltonian which includes corrections due to the finiteness of the atom number.

Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity.

Toroidal modes in the form of so-called Hopfions, with two independent winding numbers, a hidden one (twist s), which characterizes a circular vortex thread embedded into a three-dimensional soliton, and the vorticity around the vertical axis (m), appear in many fields, including field theory, ferromagnetics, and semi- and superconductors. Such topological states are normally generated in multicomponent systems, or as trapped quasilinear modes in toroidal potentials. We uncover that stable solitons with this structure can be created, without any linear potential, in the single-component setting with the strength of repulsive nonlinearity growing fast enough from the center to the periphery, for both steep and smooth modulation profiles. Toroidal modes with s=1 and vorticity m=0, 1, 2 are produced. They are stable for m≤1, and do not exist for s>1. An approximate analytical solution is obtained for the twisted ring with s=1, m=0. Under the application of an external torque, it rotates like a solid ring. The setting can be implemented in a Bose-Einstein condensate (BEC) by means of the Feshbach resonance controlled by inhomogeneous magnetic fields.

A time-splitting pseudospectral method for the solution of the Gross-Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstei...

Probing the circulation of ring-shaped Bose-Einstein condensates

This paper reports the results of a theoretical and experimental study of how the initial circulation of ring-shaped Bose-Einstein condensates (BECs) can be probed by time-of-flight (TOF) images. We have studied theoretically the dynamics of a BEC after release from a toroidal trap potential by solving the 3D Gross-Pitaevskii (GP) equation. The trap and condensate characteristics matched those of a recent experiment. The circulation, experimentally imparted to the condensate by stirring, was simulated theoretically by imprinting a linear azimuthal phase on the initial condensate wave function. The theoretical TOF images were in good agreement with the experimental data. We find that upon release the dynamics of the ring--shaped condensate proceeds in two distinct phases. First, the condensate expands rapidly inward, filling in the initial hole until it reaches a minimum radius that depends on the initial circulation. In the second phase, the density at the inner radius increases to a maximum after which the hole radius begins slowly to expand. During this second phase a series of concentric rings appears due to the interference of ingoing and outgoing matter waves from the inner radius. The results of the GP equation predict that the hole area is a quadratic function of the initial circulation when the condensate is released directly from the trap in which it was stirred and is a linear function of the circulation if the trap is relaxed before release. These scalings matched the data. Thus, hole size after TOF can be used as a reliable probe of initial condensate circulation. This connection between circulation and hole size after TOF will facilitate future studies of atomtronic systems that are implemented in ultracold quantum gases.

A time-splitting pseudospectral method for the solution of the Gross–Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

We present a method for numerically solving a Gross–Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose–Einstein condensates. The method we develop maintains spectral accuracy by employing Fourier or spherical harmonics in the angular coordinates combined with generalised-Laguerre basis functions in the radial direction. Using an error analysis, we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time-splitting method for integrating the equations forward in time. In contrast to a number of previous studies, no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2D and 3D time-dependent simulations. This is accomplished by developing an efficient algorithm that accurately performs the generalised-Laguerre transforms of rotating Bose–Einstein condensates for different orders of the Laguerre polynomials. Using this spatial discretisation together with a second order Strang time-splitting method, we illustrate the scheme on a number of 2D and 3D computations of the ground state of a non-rotating and rotating condensate. Comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems. The method is further applied to simulate a number of time-dependent problems including the Kelvin–Helmholtz instability in a two-component rotating condensate and the motion of quantised vortices in a 3D condensate.

Kibble–Zurek mechanism in a trapped ferromagnetic Bose–Einstein condensate

Spontaneous spin vortex formation in a magnetic phase transition of a trapped spin-1 Bose–Einstein condensate is investigated based on mean-field theory. In a harmonic trapping potential, an inhomogeneous atomic density leads to spatial variations of the critical point, magnetization time, and spin correlation length. The Kibble–Zurek phenomena are shown to emerge even in such inhomogeneous spinor condensates, when the quench of the quadratic Zeeman energy is fast enough. For slow quench, the magnetized region gradually expands from the center of the trap, pushing out spin vortices, which hinders the Kibble–Zurek mechanism from occurring. The case of a toroidal trapping potential is also discussed.