Three-dimensional Bose-Einstein Condensate Research Articles

Vortex Quantum Droplets under Competing Nonlinearities

This concise review summarizes recent advancements in theoretical studies of vortex quantum droplets (VQDs) in matter-wave fields. These are robust self-trapped vortical states in two- and three-dimensional (2D and 3D) Bose–Einstein condensates (BECs) with intrinsic nonlinearity. Stability of VQDs is provided by additional nonlinearities resulting from quantum fluctuations around mean-field states, often referred to as the Lee–Huang–Yang (LHY) corrections. The basic models are presented, with emphasis on the interplay between the mean-field nonlinearity, LHY correction, and spatial dimension, which determines the structure and stability of VQDs. We embark by delineating fundamental properties of VQDs in the 3D free space, followed by consideration of their counterparts in the 2D setting. Additionally, we address stabilization of matter-wave VQDs by optical potentials. Finally, we summarize results for the study of VQDs in the single-component BEC of atoms carrying magnetic moments. In that case, the anisotropy of the long-range dipole-dipole interactions endows the VQDs with unique characteristics. The results produced by the theoretical studies in this area directly propose experiments for the observation of novel physical effects in the realm of quantum matter, and suggest potential applications to the design of new schemes for processing classical and quantum information.

Direct creation of interacting quasi-one-dimensional Bose–Einstein condensate through fast evaporative cooling

Preparation of atomic Bose–Einstein condensate (BEC) with tunable interactions in a quasi-one-dimensional (quasi-1D) optical trap is essential for both the observation of bright matter-wave solitons and the quantum simulation based on the discrete atomic momentum states. However, the quasi-1D BEC has been obtained by a complex process, which includes the creation of a three-dimensional BEC and its adiabatic transform into a quasi-1D trap. Here, we report the direct creation of a quasi-1D BEC of 133Cs atoms by the fast evaporative cooling of ultracold atoms prepared by the degenerated Raman sideband cooling. We produce the pure BEC of up to 5.5×104 atoms in a quasi-1D optical trap with an evaporative time of 6 s. We demonstrate the anisotropic expansion of the atomic cloud after the release from the quasi-1D trap and study the dependence of both the phase space density and the temperature on the number of atoms in the trap during the evaporative cooling. Our study facilitates the promising applications of quasi-1D interacting atomic gases.

Collapse inhibition in three-dimensional Bose–Einstein condensates: Internal repulsion and particle loss

We consider the dynamics of three-dimensional Bose–Einstein condensates of cold atoms in the presence of internal dynamical counter-pressure that may arise due to three-body atomic collisions. We demonstrate that, in the corresponding phenomenological approach, in addition to the condensate density losses, these three-body interactions may lead to an internal pressure resulting in a repulsive potential in the condensate. This repulsion, being nonlocal and expressed with a specific form of the integral dependence on the condensate density, similar to the manifestation of the Coulomb forces in charged systems, can determine the entire condensate evolution. We find that these effects can inhibit the collapse of a self-attracting condensate and even cause its explosion, and identify the corresponding conditions numerically.

Universal dynamics of a turbulent superfluid Bose gas

We study the emergence of universal scaling in the time-evolving momentum distribution of a harmonically trapped three-dimensional Bose-Einstein condensate, parametrically driven to a turbulent state. We demonstrate that the out-of-equilibrium dynamics post excitation is described by a single function due to nearby nonthermal fixed points. The observed behavior connects the dynamics of a quantum turbulent state to several far-from-equilibrium phenomena. We present a controllable protocol to explore universality in such systems, obtaining the associated scaling exponents. Our experimental results thus offer a promising route to investigate the complex dynamics of the quantum turbulent regime.

Quantum droplets in three-dimensional Bose–Einstein condensates

The properties of 3D Bose–Einstein condensate have been studied with variational and numerical methods. In the variational approach, we use the super-Gaussian trial function, and it is demonstrated that this trial function is a good approach for providing the descriptions of the quantum droplets. The analytical equations for the variational parameters are obtained. The frequency of small oscillations of quantum droplets near the equilibrium position is estimated. It is also found that periodic modulation of the coupling constants leads to resonance oscillations of the quantum droplets parameters or emission of waves depending on the amplitude of the modulations. The predictions are supported by direct numerical simulation of the governing equation.

Instabilities of a vortex-ring-bright soliton in trapped binary three-dimensional Bose-Einstein condensates

Instabilities of vortex-ring-bright coherent structures in harmonically trapped two-component three-dimensional Bose-Einstein condensates are studied numerically within the coupled Gross-Pitaevskii equations and interpreted analytically. Interestingly, the filled vortex core with a sufficiently large amount of the bright component is observed to reduce the parametric interval of stability of the vortex ring. We have identified the mechanisms of several linear instabilities and one nonlinear parametric instability in this connection. Two of the linear instabilities are qualitatively different from ones reported earlier, to our knowledge, and are associated with azimuthal modes of $m=0$ and $m=1$, i.e., deviations of the vortex from the stationary ring shape. Our nonlinear parametric resonance instability occurs between the $m=0$ and $m=2$ modes and signals the exchange of energy between them.

Dynamics of Bose-Einstein condensates under anharmonic trap

The dynamics of weakly interacting three-dimensional Bose-Einstein condensates (BECs), trapped in external axially symmetric plus anharmonic distortion potential are studied. Within a variational approach and time-dependent Gross-Pitaevskii equation, the coupled condensate width equations are derived. By modulating anharmonic distortion of the trapping potential, nonlinear features are studied numerically and illustrated analytically, such as mode coupling of oscillation modes, and resonances. Furthermore, the stability of attractive interaction BEC in both repulsive and attractive anharmonic distortion is examined. We demonstrate that a small repulsive and attractive anharmonic distortion is effective in reducing (extending) the condensate stability region since it decreases (increases) the critical number of atoms in the trapping potential.

Nonlinear dynamics of a Bose-Einstein condensate excited by a vortex ring phase imprinting

We study the nonlinear dynamics of a three-dimensional Bose-Einstein condensate (BEC) excited by a vortex ring phase imprinting. We identify independent, integrated, and stationary modes of the center-of-mass oscillation of the condensate with respect to the vortex ring movement. We show that the oscillation amplitude of the center-of-mass of the condensate depends strongly on the initial radius of the vortex ring, the nonlinear inter-atomic interaction, and the aspect ratio of the trap, while the oscillation frequency is fixed and equals to the frequency of the harmonic trap in the direction of the ring movement. However, when applying Kelvin wave perturbations on the vortex ring, the center-of-mass oscillation of the BEC is changed non-trivially with respect to the perturbation modes, the long-scale perturbation strength as well as the wave number of the perturbations. The parity of the wave number of the Kelvin perturbations plays an important role on the mode of the center-of-mass oscillation of the condensate.

Characterising arbitrary dark solitons in trapped one-dimensional Bose-Einstein condensates (a) (a)Contribution to the Focus Issue Turbulent Regimes in Bose-Einstein Condensates edited by Alessandra Lanotte, Iacopo Carusotto and Alberto Bramati.

We present a method to detect the presence and depth of dark solitons within repulsive one-dimensional harmonically trapped Bose-Einstein condensates. For a system with one soliton, we provide numerical evidence that the shift of the density in Fourier space directly maps onto the depth of the soliton. For multi-soliton systems, combining our spectral method with established imaging techniques, the character of the solitons present in the condensate can be determined. We verify that the detection of solitons by the spectral shift works in the presence of waves induced by density engineering methods. Finally we discuss implications for vortex detection in three-dimensional Bose-Einstein condensates.

Effective self-similar expansion of a Bose-Einstein condensate: Free space versus confined geometries

We compare the exact evolution of an expanding three-dimensional Bose-Einstein condensate with that obtained from the effective scaling approach introduced in D. Gu\'ery-Odelin [Phys. Rev. A 66, 033613 (2002)]. This approach, which consists in looking for self-similar solutions to be satisfied on average, is tested here in different geometries and configurations. We find that, in case of almost isotropic traps, the effective scaling reproduces with high accuracy the exact evolution dictated by the Gross-Pitaevskii equation for arbitrary values of the interactions, in agreement with the proof-of-concept of M. Modugno, G. Pagnini, and M. A. Valle-Basagoiti [Phys. Rev. A 97, 043604 (2018)]. Conversely, it is shown that the hypothesis of universal self-similarity breaks down in case of strong anisotropies and trapped geometries.

Dynamical excitation of maxon and roton modes in a Rydberg-dressed Bose-Einstein condensate

We investigate the dynamics of a three-dimensional Bose-Einstein condensate of ultracold atomic gases with a soft-core shape long-range interaction, which is induced by laser dressing the atoms to a highly excited Rydberg state. For a homogeneous condensate, the long-range interaction drastically alters the dispersion relation of the excitation, supporting both roton and maxon modes. Rotons are typically responsible for the creation of supersolids, while maxons are normally dynamically unstable in BECs with dipolar interactions. We show that maxon modes in the Rydberg-dressed condensate, on the contrary, are dynamically stable. We find that the maxon modes can be excited through an interaction quench, i.e. turning on the soft-core interaction instantaneously. The emergence of the maxon modes is accompanied by oscillations at high frequencies in the quantum depletion, while rotons lead to much slower oscillations. The dynamically stable excitation of the roton and maxon modes leads to persistent oscillations in the quantum depletion. Through a self-consistent Bogoliubov approach, we identify the dependence of the maxon mode on the soft-core interaction. Our study shows that maxon and roton modes can be excited dynamically and simultaneously by quenching Rydberg-dressed long-range interactions. This is relevant to current studies in creating and probing exotic states of matter with ultracold atomic gases.

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.

Direct Laser Cooling to Bose-Einstein Condensation in a Dipole Trap.

We present a method for producing three-dimensional Bose-Einstein condensates using only laser cooling. The phase transition to condensation is crossed with 2.5×10^{4} ^{87}Rb atoms at a temperature of T_{c}=0.6 μK after 1.4s of cooling. Atoms are trapped in a crossed optical dipole trap and cooled using Raman cooling with far-off-resonant optical pumping light to reduce atom loss and heating. The achieved temperatures are well below the effective recoil temperature. We find that during the final cooling stage at atomic densities above 10^{14} cm^{-3}, careful tuning of trap depth and optical-pumping rate is necessary to evade heating and loss mechanisms. The method may enable the fast production of quantum degenerate gases in a variety of systems including fermions.

Spectral and dynamical analysis of a single vortex ring in anisotropic harmonically trapped three-dimensional Bose-Einstein condensates

In the present work, motivated by numerous recent experimental developments we revisit the dynamics of a single vortex ring in anisotropic harmonic traps. At the theoretical level, we start from a general Lagrangian dynamically capturing the evolution of a vortex ring and not only consider its spectrum of linearized excitations, but also explore the full nonlinear dynamical evolution of the ring as a vortical filament. The theory predicts that the ring is stable for $1 \leq \lambda \leq 2$, where $\lambda=\omega_z/\omega_r$ is the ratio of the trapping frequencies along the $z$ and $r$ axes, i.e., for spherical to slightly oblate condensates. We compare this prediction with direct numerical simulations of the full 3D Gross-Pitaevskii equation (GPE) capturing the linearization spectrum of the ring for different values of the chemical potential and as a function of the anisotropy parameter $\lambda$. We identify this result as being only asymptotically valid as the chemical potential $\mu \rightarrow \infty$, revealing how the stability interval narrows and, in particular, its upper bound decreases for finite $\mu$. Finally, we compare at the dynamical level the results of the GPE with the ones effectively capturing the ring dynamics, revealing the unstable evolution for different values of $\lambda$, as well as the good agreement between the two.

Strong-coupling Bose polarons out of equilibrium: Dynamical renormalization-group approach

When a mobile impurity interacts with a surrounding bath of bosons, it forms a polaron. Numerous methods have been developed to calculate how the energy and the effective mass of the polaron are renormalized by the medium for equilibrium situations. Here we address the much less studied non-equilibrium regime and investigate how polarons form dynamically in time. To this end, we develop a time-dependent renormalization group approach which allows calculations of all dynamical properties of the system and takes into account the effects of quantum fluctuations in the polaron cloud. We apply this method to calculate trajectories of polarons following a sudden quench of the impurity-boson interaction strength, revealing how the polaronic cloud around the impurity forms in time. Such trajectories provide additional information about the polaron's properties which are challenging to extract directly from the spectral function measured experimentally using ultracold atoms. At strong couplings, our calculations predict the appearance of trajectories where the impurity wavers back at intermediate times as a result of quantum fluctuations. Our method is applicable to a broader class of non-equilibrium problems. We also apply it to calculate the spectral function and find good agreement with experimental results. At very strong couplings, we predict that quantum fluctuations lead to the appearance of a dark continuum with strongly suppressed spectral weight at low energies. While our calculations start from an effective Fr\"ohlich Hamiltonian describing impurities in a three-dimensional Bose-Einstein condensate, we also calculate the effects of additional terms in the Hamiltonian beyond the Fr\"ohlich paradigm. We demonstrate that the main effect of these additional terms on the attractive side of a Feshbach resonance is to renormalize the coupling strength of the effective Fr\"ohlich model.

Effective two-mode model in Bose-Einstein condensates versus Gross-Pitaevskii simulations

We study the dynamics of three-dimensional Bose-Einstein condensates confined by double-well potentials using a two-mode model with an effective on-site interaction energy parameter. The effective on-site interaction energy parameter is evaluated for different numbers of particles ranging from a low experimental value to larger ones approaching the Thomas-Fermi limit, yielding important corrections to the dynamics. We analyze the time periods as functions of the initial imbalance and find a closed integral form that includes all interaction-driven parameters. A simple analytical formula for the self-trapping period is introduced and shown to accurately reproduce the exact values provided by the two-mode model. Systematic numerical simulations of the problem in 3D demonstrate the excellent agreement of the two-mode model for experimental parameters.

Dirty bosons in a three-dimensional harmonic trap

We study a three-dimensional Bose–Einstein condensate in an isotropic harmonic trapping potential with an additional delta-correlated disorder potential and investigate the emergence of a Bose-glass phase for increasing disorder strength. At zero temperature a first-order quantum phase transition from the superfluid phase to the Bose-glass phase is detected at a critical disorder strength, which agrees with the findings in the literature. Afterwards, we study the interplay between temperature and disorder fluctuations on the respective components of the particle density. In particular, we find for smaller disorder strengths that a superfluid region, a Bose-glass region, and a thermal region coexist. Furthermore, depending on the respective system parameters, three phase transitions are detected, namely, one from the superfluid to the Bose-glass phase, another one from the Bose-glass to the thermal phase, and finally one from the superfluid to the thermal phase. All these results are obtained by extending a quite recent Hartree–Fock mean-field theory for the dirty boson problem, which is based on the replica method, from the homogeneous case to a harmonic confinement.

Simultaneous dipole and quadrupole moment contribution in the Bogoliubov spectrum: Application of the non-integral Gross–Pitaevskii equation

We present the Gross–Pitaevskii equation for Bose–Einstein condensates (BECs) possessing the electric dipole and the electric quadrupole moments in a non-integral form. These equations are coupled with the Maxwell equations. The model under consideration includes the dipole–dipole, the dipole–quadrupole, and the quadrupole–quadrupole interactions in terms of the electric field created by the dipoles and quadrupoles. We apply this model to obtain the Bogoliubov spectrum for three-dimensional BECs with a repulsive short-range interaction. We obtain an extra term in the Bogoliubov spectrum in comparison with the dipolar BECs. We show that the quadrupole–quadrupole interaction gives a positive contribution in the Bogoliubov spectrum. Hence, this spectrum is stable.

Single and multiple vortex rings in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

In the present work, we explore the existence, stability and dynamics of single and multiple vortex ring states that can arise in Bose-Einstein condensates. Earlier works have illustrated the bifurcation of such states, in the vicinity of the linear limit, for isotropic or anisotropic three-dimensional harmonic traps. Here, we extend these states to the regime of large chemical potentials, the so-called Thomas-Fermi limit, and explore their properties such as equilibrium radii and inter-ring distance, for multi-ring states, as well as their vibrational spectra and possible instabilities. In this limit, both the existence and stability characteristics can be partially traced to a particle picture that considers the rings as individual particles oscillating within the trap and interacting pairwise with one another. Finally, we examine some representative instability scenarios of the multi-ring dynamics including breakup and reconnections, as well as the transient formation of vortex lines.

Reservoir interactions of a vortex in a trapped three-dimensional Bose-Einstein condensate

We simulate the dissipative evolution of a vortex in a trapped finite-temperature dilute-gas Bose-Einstein condensate using first-principles open-systems theory. Simulations of the complete stochastic projected Gross-Pitaevskii equation for a partially condensed Bose gas containing a single quantum vortex show that the transfer of condensate energy to the incoherent thermal component without population transfer provides an important channel for vortex decay. For the lower temperatures considered, this effect is significantly larger that the population transfer process underpinning the standard theory of vortex decay, and is the dominant determinant of the vortex lifetime. A comparison with the Zaremba-Nikuni-Griffin kinetic (two-fluid) theory further elucidates the role of the particle transfer interaction, and suggests the need for experimental testing of reservoir interaction theory. The dominance of this particular energetic decay mechanism for this open quantum system should be testable with current experimental setups, and its observation would have broad implications for the dynamics of atomic matter waves and experimental studies of dissipative phenomena.