Harmonic Trapping Potential Research Articles

Monopole dynamics of Bose–Fermi mixtures in a three-dimensional optical lattice on a harmonic trap

Motivated by recent developments in experimental studies of Bose–Fermi mixtures, we investigate monopole oscillation dynamics of Bose–Fermi mixtures in a three-dimensional (3D) optical lattice and an external isotropic harmonic trap potential. We use dynamical Gutzwiller approximation and calculate time dependence of the average spatial extent of particles in a coexisting phase where superfluid bosons and metal fermions coexist. With the trap potential, the bosons concentrate at and around the center of the potential and the fermions surround the bosons. They begin to demonstrate monopole oscillations when the trap potentials are suddenly changed to the ones with smaller curvature. In particular, the correlation between the oscillations of the bosons and fermions is different depending on the condition of the initial confinement. We also show the case where only one of the two trap potentials is changed.

Bose–Einstein condensates in an eightfold symmetric optical lattice**Project supported by the National Natural Science Foundation of China (Grant Nos. 11434011, 11522436, and 11774425), the National Key R&D Program of China (Grants No. 2018YFA0306501), the Beijing Natural Science Foundation, China (Grant No. Z180013), and the Research Funds of Renmin University of China (Grants Nos. 16XNLQ03 and 18XNLQ15).

We investigate the properties of Bose–Einstein condensates (BECs) in a two-dimensional quasi-periodic optical lattice (OL) with eightfold rotational symmetry by numerically solving the Gross–Pitaevskii equation. In a stationary external harmonic trapping potential, we first analyze the evolution of matter-wave interference pattern from periodic to quasi-periodic as the OL is changed continuously from four-fold periodic to eight-fold quasi-periodic. We also investigate the transport properties during this evolution for different interatomic interaction and lattice depth, and find that the BEC crosses over from ballistic diffusion to localization. Finally, we focus on the case of eightfold symmetric lattice and consider a global rotation imposed by the external trapping potential. The BEC shows vortex pattern with eightfold symmetry for slow rotation, becomes unstable for intermediate rotation, and exhibits annular solitons with approximate axial symmetry for fast rotation. These results can be readily demonstrated in experiments using the same configuration as in Phys. Rev. Lett. 122 110404 (2019).

Inducing and controlling magnetism in the honeycomb lattice through a harmonic trapping potential

We study strongly interacting ultracold spin-1/2 fermions in a honeycomb lattice in the presence of a harmonic trap. Tuning the strength of the harmonic trap we show that it is possible to confine the fermions in artificial structures reminiscent of graphene nanoflakes in solid state. The confinement on small structures induces magnetic effects which are absent in a large graphene sheet. Increasing the strength of the harmonic potential we are able to induce different magnetic states, such as a N\'eel-like antiferromagnetic or ferromagnetic states, as well as mixtures of these basic states. The realization of different magnetic patterns is associated with the terminations of the artificial structures, in turn controlled by the confining potential.

Dynamic phase fluctuations in potential-driven Bose–Einstein condensate

We report on the dynamics of a Bose–Einstein condensate in one and two dimensions driven by the time-dependent harmonic trapping potential. Without the inter-particle interaction, the condensate exhibits the coherent behavior with the time-oscillating density distribution. When the inter-particle interaction is taken into consideration, the phase fluctuations, the multi-peak structure of the density distribution, and the coherence revival phenomenon, apart from the well-studied density oscillation behavior, can be observed in the condensate. Furthermore, it is demonstrated that due to the dimensional restriction these effects are more stable in the two and three-dimensional system, if compared to the one-dimensional case.

Third- and fourth-order virial coefficients of harmonically trapped fermions in a semiclassical approximation

Using a leading-order semiclassical approximation, we calculate the third- and fourth-order virial coefficients of nonrelativistic spin-1/2 fermions in a harmonic trapping potential in arbitrary spatial dimensions, and as functions of temperature, trapping frequency and coupling strength. Our simple, analytic results for the interaction-induced changes $\Delta b_3$ and $\Delta b_4$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.

Interferences and solitons in the Bose–Einstein condensates with two- and three-body interactions

The Bose–Einstein condensates in a harmonic trapping potential are studied numerically in one dimension by the symplectic method. The conservations of invariants are tested and the eigenvalues of different nonlinear coefficients are presented. Condensates with attractive two-body interaction and repulsive three-body interaction are investigated, and the interference and soliton behaviors of the condensates are discussed both in free space and in a harmonic trapping potential. It is demonstrated that the dynamic behaviors of condensates with attractive two-body interaction are quite different to those with repulsive two-body interactions, and the addition of a repulsive three-body interaction has significant effect on the stability of the state and the dynamical behavior of the system.

Cyclotron dynamics of neutral atoms in optical lattices with additional magnetic field and harmonic trap potential**Project supported by the National Natural Science Foundation of China (Grant Nos. 11764039, 11847304, 11865014, 11475027, 11305132, and 11274255), the Natural Science Foundation of Gansu Province, China (Grant No. 17JR5RA076), and the Scientific Research Project of Gansu Higher Education Department, China (Grant No. 2016A-005).

We analytically and numerically discuss the stability and dynamics of neutral atoms in a two-dimensional optical lattice subjected to an additional harmonic trap potential and artificial magnetic field. The harmonic trap potential plays a key role in modifying the equilibrium state properties of the system and stabilizing the cyclotron orbits of the condensate. Meanwhile, the presence of the harmonic trap potential and lattice potential results in rich cyclotron dynamics of the condensate. The coupling effects of lattice potential, artificial magnetic field, and harmonic trap potential lead to single periodic, multi-periodic or quasi-periodic cyclotron orbits of the condensate. So we can control the cyclotron dynamics of neutral atoms in optical lattice by manipulating the strength of harmonic confinement, artificial magnetic field, and initial conditions. Our results provide a direct theoretical evidence for the cyclotron dynamics of neutral atoms in optical lattices exposed to the artificial gauge magnetic field and harmonic trap potential.

GSGPEs-v1.1: A MATLAB code for computing the ground state of systems of Gross–Pitaevskii equations

GSGPEs is a MATLAB/GNU Octave suite of programs for the computation of the ground state of systems of Gross–Pitaevskii equations. It can compute the ground state in the defocusing case, for any number of equations with harmonic or quasi-harmonic trapping potentials, in spatial dimension one, two or three. The computation is based on a spectral decomposition of the solution into Hermite functions and direct minimization of the energy functional through a Newton-like method with an approximate line-search strategy.This new version is due to a change in the function eig of Matlab® which requires a new way to compute Gauss–Hermite quadrature nodes and weights. Program summaryProgram Title: GSGPEsProgram Files doi:http://dx.doi.org/10.17632/3rn4z5dzwj.1Licensing provisions: GPLv3Programming language: MATLAB/GNU OctaveJournal reference of previous version: M. Caliari and S. Rainer, Comput. Phys. Commun. 184 (2013) 812–823Does the new version supersede the previous version?: YesReasons for the new version: The computation of eigenpairs (by [V, D] = eig (A)) was used in order to get the Gauss–Hermite quadrature nodes and weights via the Golub–Welsch algorithm. A change in the function eig of Matlab® occurred after version R2011b, which still gives a norm of AV−VD of the order of machine precision, produces inaccurate quadrature weights.Summary of revisions: Gauss–Hermite quadrature nodes and weights are now computed by the fast algorithm by Glaser, Liu and Rokhlin [1], as implemented by Hale and Trefethen [2].Nature of problem: A system of Gross–Pitaevskii Equations (GPEs) is used to mathematically model a Bose–Einstein Condensate (BEC) for a mixture of different interacting atomic species. The equations can be used both to compute the ground state solution (i.e., the stationary order parameter that minimizes the energy functional) and to simulate the dynamics. For particular shapes of the traps, three-dimensional BECs can be also simulated by lower dimensional GPEs.Solution method: The ground state of a system of Gross–Pitaevskii equations is computed through a spectral decomposition into Hermite functions and the direct minimization of the energy functional.

Multipath interference from large trapped ion chains

The demonstration of optical multipath interference from a large number of quantum emitters is essential for the realization of many paradigmatic experiments in quantum optics. However, such interference remains still unexplored as it crucially depends on the sub-wavelength positioning accuracy and stability of all emitters. We present the observation of controlled interference of light scattered from strings of up to 53 trapped ions. The light scattered from ions localized in a harmonic trapping potential is collected along the ion crystal symmetry axis, which guarantees the spatial indistinguishability and allows for an efficient scaling of the contributing ion number. We achieve the preservation of the coherence of scattered light observable for all the measured string sizes and nearly-optimal enhancement of phase sensitivity. The presented results will enable realization and control of directional photon emission, direct detection of enhanced quadrature squeezing of atomic resonance fluorescence, or optical generation of genuine multi-partite entanglement of atoms.

Collective oscillation modes of a superfluid Bose–Fermi mixture

In this work, we present a theoretical study for the collective oscillation modes, i.e. quadrupole, radial and axial mode, of a mixture of Bose and Fermi superfluids in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a molecular Bose–Einstein condensate (BEC) in harmonic trapping potentials with cylindrical symmetry of experimental interest. To this end, we start from the coupled superfluid hydrodynamic equations for the dynamics of Bose–Fermi superfluid mixtures and use the scaling theory that has been developed for a coupled system. The collective oscillation modes of Bose–Fermi superfluid mixtures are found to crucially depend on the overlap integrals of the spatial derivations of density profiles of the Bose and Fermi superfluids at equilibrium. We not only present the explicit expressions for the overlap density integrals, as well as the frequencies of the collective modes provided that the effective Bose–Fermi coupling is weak, but also test the valid regimes of the analytical approximations by numerical calculations in realistic experimental conditions. In the presence of a repulsive Bose–Fermi interaction, we find that the frequencies of the three collective modes of the Bose and Fermi superfluids are all upshifted, and the change speeds of the frequency shifts in the BCS–BEC crossover can characterize the different groundstate phases of the Bose–Fermi superfluid mixtures for different trap geometries.

The dynamics of a matter-wave soliton under the effect of a two-dimensional constant external force field

Using the time-dependent variational approach and the numerical solution of the Gross–Pitaevskii (GP) equation, the dynamics of a matter-wave soliton are investigated in a two-dimensional constant external force field. Firstly, the time-dependent variational approach is used to study the movements of the center-of-mass position in a simple harmonic trap and a two-dimensional weakly transmissive reflective barrier. Because the mathematical model for describing soliton dynamics in simple harmonic traps is simple, the analytic dynamical results of a soliton wave packet center-of-mass position is given directly. Since the model describing the weakly transmissive reflective barrier is difficult to solve analytically, we give the Newton’s equation of the center-of-mass position of motion, and then use the numerical solution method to study the movement of the center-of-mass in the case of several parameters. To study the motion of the soliton in the harmonic potential trap in the two-dimensional external force field under rotation, we use the split-step finite difference method to solve the two-dimensional GP equation; the motion of the center-of-mass position is discussed under the conditions of several parameter combinations. The results show that the rotation has a very important influence on the movement of the center-of-mass. When there is no rotation effect, the center-of-mass position acts as a simple harmonic motion under the combined action of the external force field and the potential well. When there is a rotation effect, the center-of-mass position is almost static.

Validity of Gross–Pitaevskii solutions of harmonically confined BEC gases in reduced dimensions

By exact numerical solutions of the Gross–Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose–Einstein condensates confined in harmonic trap potentials. Typically, these approximations are performed when one or more of the harmonic frequencies are much greater than the remaining ones, using arguments based on the adiabatic evolution of the initial approximated state. Deviations from the 3D solution are evaluated as a function of both the effective interaction strength and the ratio between the trap frequencies that define the reduced dimension where the condensate is confined. The observables analyzed are both of stationary and dynamical character, namely, the chemical potential, the wave function profiles, and the time evolution of the approximated 1D and 2D stationary states, considered as initial states in the 3D GP equation. Our study, besides setting quantitative limits on approximations previously developed, should be useful in actual experimental studies where quasi-1D and quasi-2D conditions are assumed. From a qualitative perspective, 1D and 2D approximations certainly become valid when the anisotropy is large, but in addition the interaction strength needs to be above a certain threshold.

Normalized standing waves for the Hartree equations

In this paper, we study the existence and stability/instability of standing waves with a prescribed L2-norm for the following Hartree equation(0.1)iut+Δu−a|x|2u+(|x|−λ⁎|u|2)u=0,(t,x)∈R+×RN, where t is time, N≥3, λ∈(0,N) and a≥0 is a constant. To get such solutions, we look for normalized solutions of the associated stationary equation(0.2)−Δu+a|x|2u+ωu−(|x|−λ⁎|u|2)u=0,x∈RN. Two cases are considered: (i) the system is free (a=0) and (ii) a harmonic trapping potential is added in the system (a>0). In case (i), by constructing a suitable submanifold of a L2-sphere, we prove the existence of a normalized solution for problem (0.2) with least energy in the L2-sphere, which corresponds to a normalized ground state standing wave of problem (0.1). Then, we show that each normalized ground state of problem (0.2) coincides a ground state of problem (0.2) in the usual sense. Furthermore, we give a global existence result for problem (0.1) and also show that any normalized ground state standing wave is strongly unstable. In case (ii), for small a>0, we prove the existence of two different standing waves, one corresponds to a topological local minimizer of the constrained energy functional, the other corresponds to a mountain pass type solution of problem (0.2) with its energy strictly larger than that of the normalized ground state. Moreover, we show that the set of normalized ground state solutions for problem (0.2) is stable under the associated flow. Finally, to connect case (i) and case (ii), we analyze the asymptotic behavior of the two solutions to problem (0.2) obtained in case (ii) as the trapping frequency a vanishes.

Site-resolved imaging of beryllium ion crystals in a high-optical-access Penning trap with inbore optomechanics.

We present the design, construction, and characterization of an experimental system capable of supporting a broad class of quantum simulation experiments with hundreds of spin qubits using 9Be+ ions in a Penning trap. This article provides a detailed overview of the core optical and trapping subsystems and their integration. We begin with a description of a dual-trap design separating loading and experimental zones and associated vacuum infrastructure design. The experimental-zone trap electrodes are designed for wide-angle optical access (e.g., for lasers used to engineer spin-motional coupling across large ion crystals) while simultaneously providing a harmonic trapping potential. We describe a near-zero-loss liquid-cryogen-based superconducting magnet, employed in both trapping and establishing a quantization field for ion spin-states and equipped with a dual-stage remote-motor LN2/LHe recondenser. Experimental measurements using a nuclear magnetic resonance (NMR) probe demonstrate part-per-million homogeneity over 7 mm-diameter cylindrical volume, with no discernible effect on the measured NMR linewidth from pulse-tube operation. Next, we describe a custom-engineered inbore optomechanical system which delivers ultraviolet (UV) laser light to the trap and supports multiple aligned optical objectives for topview and sideview imaging in the experimental trap region. We describe design choices including the use of nonmagnetic goniometers and translation stages for precision alignment. Furthermore, the optomechanical system integrates UV-compatible fiber optics which decouple the system's alignment from remote light sources. Using this system, we present site-resolved images of ion crystals and demonstrate the ability to realize both planar and three-dimensional ion arrays via control of rotating wall electrodes and radial laser beams. Looking to future work, we include interferometric vibration measurements demonstrating root-mean-square trap motion of ∼33 nm (∼117 nm) in the axial (transverse) direction; both values can be reduced when operating the magnet in free-running mode. The paper concludes with an outlook toward extensions of the experimental setup, areas for improvement, and future experimental studies.

Conductivity Spectrum of Ultracold Atoms in an Optical Lattice.

We measure the conductivity of neutral fermions in a cubic optical lattice. Using insitu fluorescence microscopy, we observe the alternating current resultant from a single-frequency uniform force applied by displacement of a weak harmonic trapping potential. In the linear response regime, a neutral-particle analog of Ohm's law gives the conductivity as the ratio of total current to force. For various lattice depths, temperatures, interaction strengths, and fillings, we measure both real and imaginary conductivity, up to a frequency sufficient to capture the transport dynamics within the lowest band. The spectral width of the real conductivity reveals the current dissipation rate in the lattice, and the integrated spectral weight is related to thermodynamic properties of the system through a sum rule. The global conductivity decreases with increased band-averaged effective mass, which at high temperatures approaches a T-linear regime. Relaxation of current is observed to require a finite lattice depth, which breaks Galilean invariance and enables damping through collisions between fermions.

Transport of Strongly Correlated Bosons in an Optical Lattice

The transport of strongly correlated bosons in a three‐dimensional optical lattice is studied within the Bose–Hubbard approximation. The transport is induced by a small displacement of the overall harmonic trapping potential. The subsequent relaxation dynamics is monitored by high precision density measurements with the help of scanning electron microscopy. Good agreement with a real space time‐dependent Gutzwiller mean‐field description is found.

Bose polaron in spherically symmetric trap potentials: Ground states with zero and lower angular momenta

Single-atomic impurities immersed in a dilute Bose gas in the spherically symmetric harmonic trap potentials are studied at zero temperature. In order to find the ground state of the polarons, we present a conditional variational method with fixed expectation values of the total angular momentum operators, $\hat{J}^2$ and $\hat{J}_z$, of the system, using a cranking gauge-transformation for bosons to move them in the frame co-rotating with the impurity. In the formulation, the expectation value $\langle \hat{J^2}\rangle$ is shown to be shared in impurity and bosons, but the value $\langle \hat{J}_z\rangle$ is carried by the impurity due to the rotational symmetry. We also analyze the ground-state properties numerically obtained in this variational method for the system of the attractive impurity-boson interaction, and find that excited boson distributions around the impurity overlap largely with impurity's wave function in their quantum-number spaces and also in the real space because of the attractive interaction employed.

Are smooth pseudopotentials a good choice for representing short-range interactions?

When seeking a numerical representation of a quantum-mechanical multiparticle problem it is tempting to replace a singular short-range interaction by a smooth finite-range pseudopotential. Finite basis set expansions, e.g.~in Fock space, are then guaranteed to converge exponentially. The need to faithfully represent the artificial length scale of the pseudopotential, however, places a costly burden on the basis set. Here we discuss scaling relations for the required size of the basis set and demonstrate the basis set convergence on the example of a two-dimensional system of few fermions with short-range $s$-wave interactions in a harmonic trapping potential. In particular we show that the number of harmonic-oscillator basis functions needed to reach a regime of exponential convergence for a Gaussian pseudopotential scales with the fourth power of the pseudopotential length scale, which can be improved to quadratic scaling when the basis functions are re-scaled appropriately. Numerical examples for three fermions with up to a few hundred single-particle basis functions are presented and implications for the feasibility of accurate numerical multi-particle simulations of interacting ultra-cold atom systems are discussed.

Ground-state vortex structures of a rotating binary dipolar Bose–Einstein condensate confined in harmonic plus quartic potential**Project supported by the Sichuan Province Education Department Key Natural Science Fund, China (Grant No. 17ZA339), the Chongqing Research Program of Basic Research and Frontier Technology, China (Grant No. cstc2014jcyjA50016), and the National Natural Science Foundation of China (Grant No. 61504016).

We consider a binary dipolar Bose–Einstein condensate confined in a rotating harmonic plus quartic potential trap. The ground-state vortex structures are numerically obtained as a function of the contact interactions and the dipole–dipole interaction in both slow and rapid rotation cases. The results show that the vortex configurations depend strongly on the strength of the contact interactions, the relative strength between dipolar and contact interactions, as well as on the orientation of the dipoles. A variety of exotic ground-state vortex structures, such as pentagonal and hexagon vortex lattice, square vortex lattice with a central vortex, annular vortex lines, and straight vortex lines, are observed by turning such controllable parameters. Our results deepen the understanding of effects of dipole–dipole interaction on the topological defects.

Three-dimensional splitting dynamics of giant vortices in Bose-Einstein condensates

We study the splitting dynamics of giant vortices in dilute Bose-Einstein condensates by numerically integrating the three-dimensional Gross-Pitaevskii equation in time. By taking advantage of tetrahedral tiling in the spatial discretization, we decrease the error and increase the reliability of the numerical method. An extensive survey of vortex splitting symmetries is presented for different aspect ratios of the harmonic trapping potential. The symmetries of the splitting patterns observed in the simulated dynamics are found to be in good agreement with predictions obtained by solving the dominant dynamical instabilities from the corresponding Bogoliubov equations. Furthermore, we observe intertwining of the split vortices in prolate condensates and a split-and-revival phenomenon in a spherical condensate.