We consider a Bose-Einstein condensate (BEC), which is characterized by long-range and anisotropic dipole-dipole interactions and vanishing s-wave scattering length, in a double-well potential. The properties of this system are investigated as functions of the height of the barrier that splits the harmonic trap into two halves, the number of particles (or dipole-dipole strength) and the aspect ratio $λ$, which is defined as the ratio between the axial and longitudinal trapping frequencies $ω_z$ and $ω_ρ$. The phase diagram is determined by analyzing the stationary mean-field solutions. Three distinct regions are found: a region where the energetically lowest lying stationary solution is symmetric, a region where the energetically lowest lying stationary solution is located asymmetrically in one of the wells, and a region where the system is mechanically unstable. For sufficiently large aspect ratio $λ$ and sufficiently high barrier height, the system consists of two connected quasi-two-dimensional sheets with density profiles whose maxima are located either at $ρ=0$ or away from $ρ=0$. The stability of the stationary solutions is investigated by analyzing the Bogoliubov de Gennes excitation spectrum and the dynamical response to small perturbations. These studies reveal unique oscillation frequencies and distinct collapse mechanisms. The results derived within the mean-field framework are complemented by an analysis based on a two-mode model.
Within the formalism of the Gross-Pitaevskii equation, we derive effective one- and two-dimensional equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. These are based on an ansatz for the condensate wavefunction whose width in the tightly-confined direction/s is treated variationally. The equations constitute a coupled partial differential for the low-dimensional wavefunction and algebraic equations for the width parameters. This approach accurately predicts the ground state densities of cigar-shaped and pancake-shaped dipolar Bose-Eins…
We study the behavior of two coupled purely dipolar Bose-Einstein condensates, each located in a cylindrically symmetric pancake-shaped external confining potential, as the separation b between the traps along the tight confining direction is varied. The solutions of the coupled Gross-Pitaevskii and Bogoliubov-de Gennes equations, which account for the full dynamics, show that the system behavior is modified by the presence of the second dipolar BEC. For sufficiently small b, the presence of the second dipolar BEC destabilizes the system dramatically. In this regime, the coupled system collaps…
Faraday and resonant density waves emerge in Bose-Einstein condensates as a result of harmonic driving of the system. They represent nonlinear excitations and are generated due to the interaction-induced coupling of collective oscillation modes and the existence of parametric resonances. Using a mean-field variational and a full numerical approach, we studied density waves in dipolar condensates at zero temperature, where breaking of the symmetry due to anisotropy of the dipole-dipole interaction (DDI) plays an important role. We derived variational equations of motion for the dynamics of a dr…
The effect of dipolar orientation with respect to the condensate plane on the mean-field dynamics of dipolar Bose-Einstein condensates in a pancake-shaped confinement is discussed. The stability of a quasi-two-dimensional condensate, with respect to the tilting angle, is found to be different from a two-dimensional layer of dipoles, indicating the relevance of the transverse extension while characterizing two-dimensional dipolar systems. An anisotropic excitation spectrum exhibiting a highly tunable, rotonlike minimum can arise entirely from the dipole-dipole interactions, by tilting the dipol…
Pancake-shaped vortex droplets in dipolar molecular BECs
Foshan University（佛山大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Technion（以色列理工学院）；Dongguan University of Technology（东莞理工学院）；Tel Aviv University（特拉维夫大学）；Instituto de Alta Investigación, Universidad de Tarapacá（塔拉帕卡大学高级研究所）
Motivated by ongoing measurements at JILA, we calculate the recoil-free spectra of dipolar interacting fermions, for example ultracold heteronuclear molecules, in a one-dimensional lattice of two-dimensional pancakes, spectroscopically probing transitions between different internal (e.g., rotational) states. We additionally incorporate p-wave interactions and losses, which are important for reactive molecules such as KRb. Moreover, we consider other sources of spectral broadening: interaction-induced quasiparticle lifetimes and the different polarizabilities of the different rotational states …
We consider a $^{168}$Er-$^{164}$Dy dipolar mixture, trapped by a cigar shaped harmonic potential. We derive the quasi-1D inter-species effective potential exhibiting the tilting angles and show that it is a quite natural generalization of the situation of a single dipolar gas. By solving the coupled Gross-Pitaevskii equations, we observe a transition from miscible to immiscible mixture as the orientations of the magnetic moments are varied. The atom numbers are also shown to lead to noticeable effects on the mixture.
We develop a variational theory for a dipolar condensate in an elongated (cigar shaped) confinement potential. Our formulation provides an effective one-dimensional extended meanfield theory for the ground state and its collective excitations. We apply our theory to investigate the properties of rotons in the system comparing the variational treatment to a full numerical solution. We consider the effect of quantum fluctuations on the scattering length at which the roton excitation softens to zero energy.
We derive rigorous one- and two-dimensional mean-field equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. We show how the dipolar interaction modifies the contact interaction of the strongly confined atoms. In addition, our equations introduce a nonlocal potential, which is anisotropic for pancake-shaped condensates. We propose to observe this anisotropy via measurement of the condensate aspect ratio. We also derive analytically approximate density profiles from our equations. Both the numerical solutions of our reduced mean-field equat…
We investigate the ground-state and magnetic properties of a dipolar spin-1 Bose-Einstein condensate trapped in a symmetric double-well potential. In particular, we focus on the spin-vortex states by assuming that each potential well is highly pancake shaped. We show that the presence of the double-well potential gives rise to two different spin configurations for the spin-vortex pair states. We also study the response of the coupled spin-vortex pair to static transverse magnetic fields.
We study the Josephson oscillation and self trapping dynamics of a cigar-shaped dipolar Bose-Einstein condensate of $^{52}$Cr atoms polarized along the symmetry axis of an axially-symmetric double-well potential using the numerical solution of a mean-field model, for dominating repulsive contact interaction (large positive scattering length a) over an anisotropic dipolar interaction. Josephson-type oscillation emerges for small and very large number of atoms, whereas self trapping is noted for an intermediate number of atoms. The dipolar interaction pushes the system away from self trapping to…
We investigated the ground state wave function and free expansion of a trapped dipolar condensate. We find that dipolar interaction may induce both biconcave and dumbbell density profiles in, respectively, the pancake- and cigar-shaped traps. On the parameter plane of the interaction strengths, the density oscillation occurs only when the interaction parameters fall into certain isolated areas. The relation between the positions of these areas and the trap geometry is explored. By studying the free expansion of the condensate with density oscillation, we show that the density oscillation is de…
We consider a spin-1 Bose-Einstein condensate with Rashba spin-orbit coupling and dipole-dipole interaction confined in a cigar-shaped trap. Due to the combined effects of spin-orbit coupling, dipole-dipole interaction, and trap geometry, the system exhibits a rich variety of ground-state spin structures, including twisted spin vortices. The ground-state phase diagram is determined with respect to the strengths of the spin-orbit coupling and dipole-dipole interaction.
We derive rigorously one- and two-dimensional mean-field equations for cigar- and pancake-shaped Bose-Einstein condensates (BEC) with higher order interactions (HOI). We show how the higher order interaction modifies the contact interaction of the strongly confined particles. Surprisingly, we find that the usual Gaussian profile assumption for the strongly confining direction is inappropriate for the cigar-shaped BEC case, and a Thomas-Fermi type profile should be adopted instead. Based on the derived mean field equations, the Thomas-Fermi densities are analyzed in presence of the contact inte…
Symmetry-breaking magnetization dynamics of a spin-1 Bose-Einstein condensate (BEC) due to the dipole-dipole interaction are investigated using the mean-field and Bogoliubov theories. When a magnetic field is applied along the symmetry axis of a pancake-shaped BEC in the m = 0 hyperfine sublevel, transverse magnetization develops breaking the chiral or axial symmetry. A variety of magnetization patterns are formed depending on the strength of the applied magnetic field. The proposed phenomena can be observed in 87Rb and 23Na condensates.
We study a Bose-Einstein condensate of 52Cr atoms confined in a toroidal trap with a variable strength of s-wave contact interactions. We analyze the effects of the anisotropic nature of the dipolar interaction by considering the magnetization axis to be perpendicular to the trap symmetry axis. In the absence of a central repulsive barrier, when the trap is purely harmonic, the effect of reducing the scattering length is a tuning of the geometry of the system: from a pancake-shaped condensate when it is large, to a cigar-shaped condensate for small scattering lengths. For a condensate in a tor…
We study the propagation of anisotropic sound and shock waves in dipolar Bose-Einstein condensate in three dimensions (3D) as well as in quasi-two (2D, disk shape) and quasi-one (1D, cigar shape) dimensions using the mean-field approach. In 3D, the propagation of sound and shock waves are distinct in directions parallel and perpendicular to dipole axis with the appearance of instability above a critical value corresponding to attraction. Similar instability appears in 1D and not in 2D. The numerical anisotropic Mach angle agrees with theoretical prediction. The numerical sound velocity in all …
We consider two separated pancake-shaped trapped gases interacting with a dipolar (either magnetic or electric) force. We study how the center of mass motion propagates from one cloud to the other as a consequence of the long-range nature of the interaction. The corresponding dynamics is fixed by the frequency difference between the in-phase and the out-of-phase center of mass modes of the two clouds, whose dependence on the dipolar interaction strength and the cloud separation is explicitly investigated. We discuss Fermi gases in the degenerate as well as in the classical limit and comment on…
We consider singly-quantized vortex states in a condensate of 52Cr atoms in a pancake trap. We obtain the vortex solutions by numerically solving the Gross-Pitaevskii equation in the rotating frame with no further approximations. The behavior of the condensate is studied under three different situations concerning the interactions: only s-wave, s-wave plus dipolar and only dipolar. The energy barrier for the nucleation of a vortex is calculated as a function of the vortex displacement from the rotation axis in the three cases. These results are compared to those obtained for contact interactio…
Gases of doubly-dipolar particles, with both magnetic and electric dipole moments, offer intriguing novel possibilities. We show that the interplay between doubly-dipolar interactions, quantum stabilization, and external confinement results in a rich ground-state physics of supersolids and incoherent droplet arrays in doubly-dipolar condensates. Our study reveals novel possibilities for engineering quantum droplets and droplet supersolids, including supersolid-supersolid transitions and the realization of supersolid arrays of pancake droplets.
We present a theory for the emergence of a supersolid state in a cigar-shaped dipolar quantum Bose gas. Our approach is based on a reduced three-dimensional (3D) theory, where the condensate wavefunction is decomposed into an axial field and a transverse part described variationally. This provides an accurate fully 3D description that is specific to the regime of current experiments and efficient to compute. We apply this theory to understand the phase diagram for a gas in an infinite tube potential. We find that the supersolid transition has continuous and discontinuous regions as the average…
Quantized vortices are the prototypical feature of superfluidity. Pervasive in all natural systems, vortices are yet to be observed in dipolar quantum gases. Here, we exploit the anisotropic nature of the dipole-dipole interaction of a dysprosium Bose-Einstein condensate to induce angular symmetry breaking in an otherwise cylindrically symmetric pancake-shaped trap. Tilting the magnetic field towards the radial plane deforms the cloud into an ellipsoid through magnetostriction, which is then set into rotation. At stirring frequencies approaching the radial trap frequency, we observe the genera…
By means of mean-field theory, we have studied the structure and excitation spectrum of a purely dipolar Bose gas in pancake-shaped trap where the confinement in the x-y plane is provided by a highly anharmonic potential resulting in an almost uniform confinement in the plane. We show that the stable condensates is characterized by marked radially structured density profiles. The stability diagram is calculated by independently varying the strength of the interaction and the trap geometry. By computing the Bogoliubov excitation spectrum near the instability line we show that soft "angular" rot…
We propose methods to create fractional vortices in the cyclic state of an F = 2 spinor Bose-Einstein condensate by manipulating its internal spin structure using pulsed microwave and laser fields. The stability of such vortices is studied as a function of the rotation frequency of the confining harmonic trap both in pancake and cigar shaped condensates. We find a range of parameters for which the so-called 1/3-vortex state is energetically favorable. Such fractional vortices could be created in condensates of 87Rb atoms using current experimental techniques facilitating probing of topological…
We perform a full three-dimensional study on miscible-immiscible conditions for coupled dipolar and non-dipolar Bose-Einstein condensates (BEC), confined within anisotropic traps. Without loosing general miscibility aspects that can occur for two-component mixtures, our main focus was on the atomic erbium-dysprosium ($^{168}$Er-$^{164}$Dy) and dysprosium-dysprosium ($^{164}$Dy-$^{162}$Dy) mixtures. Our analysis for pure-dipolar BEC was limited to coupled systems confined in pancake-type traps, after considering a study on the stability regime of such systems. In case of non-dipolar systems wit…
Based on the two-dimensional mean-field equations for pancake-shaped dipolar Bose-Einstein condensates in a rotating frame with both attractive and repulsive dipole-dipole interaction (DDI) as well as arbitrary polarization angle, we study the profiles of the single vortex state and show how the critical rotational frequency change with the s-wave contact interaction strengths, DDI strengths and the polarization angles. In addition, we find numerically that at the `magic angle' $\vartheta=\arccos(\sqrt{3}/3)$, the critical rotational frequency is almost independent of the DDI strength. By nume…
We study the properties of a magnetic dipolar Bose-Einstein condensate (BEC) in an elongated (cigar shaped) confining potential in the beyond quasi-one-dimensional (quasi-1D) regime. In this system the dipole-dipole interactions (DDIs) develop a momentum-dependence related to the transverse confinement and the polarization direction of the dipoles. This leads to density fluctuations being enhanced or suppressed at a length scale related to the transverse confinement length, with local atom number measurements being a practical method to observe these effects in experiments. We use meanfield th…
By numerical and variational analysis of the three-dimensional Gross-Pitaevskii equation we study the formation and dynamics of bright and vortex-bright solitons in a cigar-shaped dipolar Bose-Einstein condensate for large repulsive atomic interactions. Phase diagram showing the region of stability of the solitons is obtained. We also study the dynamics of breathing oscillation of the solitons as well as the collision dynamics of two solitons at large velocities. Two solitons placed side-by-side at rest coalesce to form a stable bound soliton molecule due to dipolar attraction.
Homogeneous quantum gases open up new possibilities for studying many-body phenomena and have now been realised for a variety of systems. For gases with short-range interactions the way to make the cloud homogeneous is, predictably, to trap it in an ideal (homogeneous) box potential. We show that creating a close to homogeneous dipolar gas in the roton regime, when long-range interactions are important, actually requires trapping particles in soft-walled (inhomogeneous) box-like potentials. In particular, we numerically explore a dipolar gas confined in a pancake trap which is harmonic along t…
We have solved numerically the ground states of a Bose-Einstein condensate in the presence of dipolar interparticle forces using a semiclassical approach. Our motivation is to model, in particular, the spontaneous spin textures emerging in quantum gases with large dipole moments, such as 52Cr or Dy condensates, or ultracold gases consisting of polar molecules. For a pancake-shaped harmonic (optical) potential, we present the ground state phase diagram spanned by the strength of the nonlinear coupling and dipolar interactions. In an elongated harmonic potential, we observe a novel helical spin …
Roton excitations constitute a key feature of dipolar gases, connecting these gases with superfluid helium. We show that the density dependence of the roton minimum results in a spatial roton confinement, particularly relevant in pancake dipolar condensates with large aspect ratios. We show that roton confinement plays a crucial role in the dynamics after roton instability, and that arresting the instability may create a trapped roton gas revealed by confined density modulations. We discuss the local susceptibility against density perturbations, which we illustrate for the case of vortices. Ro…
We suggest and study the stable disk- and cigar-shaped gap solitons of a dipolar Bose-Einstein condensate of $^{52}$Cr atoms localized in the lowest band gap by three optical-lattice (OL) potentials along orthogonal directions. The one-dimensional version of these solitons of experimental interest confined by an OL along the dipole moment direction and harmonic traps in transverse directions is also considered. Important dynamics of (i) breathing oscillation of a gap soliton upon perturbation and (ii) dragging of a gap soliton by a moving lattice along axial $z$ direction demonstrates the stab…
Vortex lattice patterns with transitions from regular to other variety vortex shapes are predicted in rotating binary mixtures of dipolar Bose-Einstein condensates loaded in squared optical lattice. We focus our investigation in the experimentally accessible dipolar isotopes of dysprosium ($^{162,164}$Dy), erbium ($^{168}$Er), chromium ($^{52}$Cr), and rubidium ($^{87}$Rb), by considering the binary mixtures ($^{164}$Dy-$^{162}$Dy, $^{168}$Er-$^{164}$Dy, $^{164}$Dy-$^{52}$Cr and $^{164}$Dy-$^{87}$Rb), which are confined in strong pancake-shaped trap and loaded in squared two-dimensional optica…
We investigate the stability properties and the dynamics of Bose-Einstein condensates with axial symmetry, especially with dipolar long-range interaction, using both simulations on grids and variational calculations. We present an extended variational ansatz which is applicable for axial symmetry and show that this ansatz can reproduce the lowest eigenfrequencies of the Bogoliubov spectrum, and also the corresponding eigenfunctions. Our variational ansatz is capable of describing the roton instability of pancake-shaped dipolar condensates for arbitrary angular momenta. After investigating the …
By considering symmetric and asymmetric dipolar coupled mixtures (with dysprosium and erbium isotopes), we report a study on relevant anisotropic effects, related to spatial separation and miscibility, due to dipole-dipole interactions (DDIs) in rotating binary dipolar Bose-Einstein condensates. The binary mixtures are kept in strong pancake-like traps, with repulsive two-body interactions modeled by an effective two-dimensional (2D) coupled Gross-Pitaevskii equation. The DDI are tuned from repulsive to attractive by varying the dipole polarization angle. A clear spatial separation is verified…
Vortices are expected to exist in a supersolid but experimentally their detection can be difficult because the vortex cores are localized at positions where the local density is very low. We address here this problem by performing numerical simulations of a dipolar Bose-Einstein Condensate (BEC) in a pancake confinement at $T=0$ K and study the effect of quantized vorticity on the phases that can be realized depending upon the ratio between dipolar and short-range interaction. By increasing this ratio the system undergoes a spontaneous density modulation in the form of an ordered arrangement o…
We are pointing out relevant anisotropic effects, related to spatial separation, miscibility and mass-symmetry, due to dipole-dipole interactions in rotating binary dipolar Bose-Einstein condensates, by considering symmetric ($^{164}$Dy-$^{162}$Dy) and asymmetric ($^{168}$Er-$^{164}$Dy, $^{164}$Dy-$^{87}$Rb) dipolar mixtures. The binary mixtures are kept in strong pancake-shaped trap, modeled by an effective two-dimensional coupled Gross-Pitaevskii equation. The anisotropy of the dipolar interactions, on miscibility and vortex-lattice structures, is studied by tuning the polarization angle of …
We consider the fluctuations in atom number that occur within finite-sized measurement cells in a trapped Bose-Einstein condensate (BEC). This approximates the fluctuation measurements made in current experiments with finite resolution in situ imaging. A numerical scheme is developed to calculate these fluctuations using the quasiparticle modes of a cylindrically symmetric three-dimensionally trapped condensate with either contact or dipole-dipole interactions (DDIs). We use this scheme to study the properties of a pancake shaped condensate using cylindrical cells. The extension of the theory …
A model of the Bose-Einstein condensate (BEC) of dipolar atoms, confined in a combination of a cigar-shaped trap and optical lattice acting in the axial direction, is studied in the framework of the one-dimensional (1D) nonpolynomial Schrödinger equation (NPSE) with additional terms describing long-range dipole-dipole (DD) interactions. The NPSE makes it possible to describe the collapse of localized modes, which was experimentally observed in the self-attractive BEC confined in tight traps, in the framework of the 1D description. We study the influence of the DD interactions on the dynamics o…
Many of the static and dynamic properties of an atomic Bose-Einstein condensate (BEC) are usually studied by solving the mean-field Gross-Pitaevskii (GP) equation, which is a nonlinear partial differential equation for short-range atomic interaction. More recently, BEC of atoms with long-range dipolar atomic interaction are used in theoretical and experimental studies. For dipolar atomic interaction, the GP equation is a partial integro-differential equation, requiring complex algorithm for its numerical solution. Here we present numerical algorithms for both stationary and non-stationary solu…
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part), but also those which are purely imaginary but have negative Krein signature. These latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues, i.e., they are a necessary building block in the mechanism leading to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a general theory for constructing a meromorphic matrix-valued function, t…
We study the structure and stability of vortex lattices in two-component rotating Bose-Einstein condensates with intrinsic dipole-dipole interactions (DDIs) and contact interactions. To address experimentally accessible coupled systems, we consider $^{164}$Dy-$^{162}$Dy and $^{168}$Er-$^{164}$Dy mixtures, which feature different miscibilities. The corresponding dipole moments are $μ_{\mathrm{Dy}}=10μ_{\mathrm{B}}$ and $μ_{\mathrm{Er}}= 7μ_{\mathrm{B}}$, where $μ_{\mathrm{B}}$ is the Bohr magneton. For comparison, we also discuss a case where one of the species is non dipolar. Under a large asp…
Chemical reaction rates often depend strongly on stereodynamics, namely the orientation and movement of molecules in three-dimensional space. An ultracold molecular gas, with a temperature below 1 uK, provides a highly unusual regime for chemistry, where polar molecules can easily be oriented using an external electric field and where, moreover, the motion of two colliding molecules is strictly quantized. Recently, atom-exchange reactions were observed in a trapped ultracold gas of KRb molecules. In an external electric field, these exothermic and barrierless bimolecular reactions, KRb+KRb -> …
