Rotating Bose-Einstein Condensate Research Articles

On discrete ground states of rotating Bose–Einstein condensates

The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the L 2 L^2 - and H 1 H^1 -norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler–Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.

Energy spectra and fluxes of turbulent rotating Bose–Einstein condensates in two dimensions

We investigate the scaling of the energy cascade in a harmonically trapped, turbulent, rotating Bose-Einstein condensate in two dimensions. We achieve turbulence by injecting a localized perturbation into the condensate and gradually increasing its rotation frequency from an initial value to a maximum. The main characteristics of the resulting turbulent state depend on the initial conditions, rotation frequency, and ramp-up time. We analyze the energy and the fluxes of kinetic energy by considering initial profiles without vortices and with vortex lattices. In the case without initial vortices, we find the presence of Kolmogorov-like scaling (k−5/3) of the incompressible kinetic energy in the inertial range. However, with initial vortex lattices, the energy spectrum follows Vinen scaling (k−1) at transient iterations. For cases with high rotating frequencies, Kolmogorov-like scaling emerges at longer durations. We observe positive kinetic energy fluxes with both initial states across all final frequencies, indicating a forward cascade of incompressible and compressible kinetic energy.

Three-dimensional analysis of vortex-lattice formation in rotating Bose–Einstein condensates using smoothed-particle hydrodynamics

Recently, we presented a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit time-integrating scheme; our SPH scheme could reproduce the vortex lattices and their formation processes in rotating quasi-two-dimensional (2D) BECs trapped in a 2D harmonic potential. In this study, we have successfully demonstrated a simulation of rotating 3D BECs trapped in a 3D harmonic potential forming ‘cigar-shaped’ condensates. We have found that our scheme can reproduce the following typical behaviors of rotating 3D BECs observed in the literature: (i) the characteristic shape of the lattice formed in the cross-section at the origin and its formation process, (ii) the stable existence of vortex lines along the vertical axis after reaching the steady state, (iii) a ‘cookie-cutter’ shape, with a similar lattice shape observed wherever we cut the condensate in a certain range in the vertical direction, (iv) the bending of vortex lines when approaching the inner edges of the condensate, and (v) the formation of vortex lattices by vortices entering from outside the condensate. Therefore, we further validated our scheme by simulating rotating 3D BECs.

Numerical model of the Gross–Pitaevskii equation for rotating Bose–Einstein condensates using smoothed-particle hydrodynamics

This study proposed a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit real-time integration scheme. Specifically, the Gross–Pitaevskii equation was described as a complex representation to obtain a pair of time-dependent equations, which were then solved simultaneously following discretization based on SPH particle approximation. We adopt the fourth-order Runge–Kutta method for time evolution. We performed simulations of a rotating Bose gas trapped in a harmonic potential, showing results that qualitatively agreed with previously reported experiments and simulations. The geometric patterns of formed lattices were successfully reproduced for several cases, for example, the hexagonal lattice observed in the experiments of rotating BECs. Consequently, it was confirmed that the simulation began with the periodic oscillation of the condensate, which attenuated and maintained a stable rotation with slanted elliptical shapes; however, the surface was excited to be unstable and generated ripples, which grew into vortices and then penetrated inside the condensate, forming a lattice. We confirmed that each branch point of the phase of wavefunctions corresponds to each vortex. These results demonstrate our approach at a certain degree of accuracy. In conclusion, we successfully developed a new SPH scheme for the simulations of vortex lattice formation in rotating BECs.

Existence and Asymptotic Behavior of Ground States for Rotating Bose–Einstein Condensates

We study ground states of two-dimensional Bose–Einstein condensates with repulsive or attractive interactions in a trap rotating at velocity . It is known that there exist critical parameters and such that if , then there is no ground state for any ; if , then ground states exist if and only if . As a completion of the existing results, in this paper, we focus on the critical case where to classify the existence and nonexistence of ground states for any . Moreover, for a suitable class of radially symmetric traps , employing the inductive symmetry method, we prove that up to a constant phase, ground states must be real valued, unique, and free of vortices as , no matter whether the interactions of the condensates are repulsive or not.

Vortices number and the critical atoms number of a rotating Bose–Einstein condensates in two-dimensional optical lattice

In this paper, the vortices number and the critical atoms number are calculated. Calculations are performed for ultracold trapped Bose gas in two-dimensional deep optical lattice. Hellmann–Feynman theorem is used here to calculate the expectation values of the angular momentum and thermal average of square radius (in situ size). The above-mentioned parameters are calculated from the thermodynamic grand potential [Formula: see text]. The latter is calculated using a conventional semi-classical approximation. This thermodynamic grand potential allows us to investigate the impacts of the finite size and inter-atomic interaction effects. The obtained result for the vortices number has a peak with increase in the rate of rotation. While the critical atoms number has increase in nature with increase in the critical temperature. These parameters show a major dependence on the inter-atomic interaction, depth of optical lattice and number of particles.

Fragmentation and correlations in a rotating Bose–Einstein condensate undergoing breakup

The theoretical investigation of rotating Bose–Einstein condensates has mainly focused on the emergence of quantum vortex states and the condensed properties of such systems. In the present work, we concentrate on other facets by examining the impact of rotation on the ground state of weakly interacting bosons confined in anharmonic potentials computed both at the mean-field level and particularly at the many-body level of theory. For the many-body computations, we employ the well-established many-body method known as the multiconfigurational time-dependent Hartree method for bosons. We present how various degrees of fragmentation can be generated following the breakup of the ground state densities in anharmonic traps without ramping up a potential barrier for strong rotations. The breakup of the densities is found to be associated with the acquisition of angular momentum in the condensate due to the rotation. In addition to fragmentation, the presence of many-body correlations is examined by computing the variances of the many-particle position and momentum operators. For strong rotations, the many-body variances become smaller than their mean-field counterparts, and one even finds a scenario with opposite anisotropies of the mean-field and many-body variances. Further, it is observed that for higher discrete symmetric systems of order k, namely three-fold and four-fold symmetry, breakup to k sub-clouds and emergence of k-fold fragmentation take place. All in all, we provide a thorough many-body investigation of how and which correlations build up when a trapped Bose–Einstein condensate breaks up under rotation.

Second-order flows for computing the ground states of rotating Bose-Einstein condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modeled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. In comparison with the gradient flow type approaches: When explicit temporal discretization strategies are adopted, the proposed methods allow for larger stable time step sizes; While for semi-implicit discretization, using the same step size, a much smaller number of iterations are needed for the proposed methods to reach the stopping criterion, and every time step encounters almost the same computational complexity. Rich and detailed numerical examples are documented for verification and comparison.

Thomas–Fermi profile of a fast rotating Bose–Einstein condensate

We study the minimizers of a magnetic 2D non-linear Schr\"odinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is linked to the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.

Parametric triggering of vortices in toroidally trapped rotating Bose–Einstein condensates

We study the creation of vortices by triggering the rotating Bose–Einstein condensates in a toroidal trap with trap parameters such as laser beam waist and Gaussian potential depth. By numerically solving the time-dependent Gross–Pitaevskii equation in two dimensions, we observe a change in vortex structure and a considerable increase in the number of vortices when the waist of the irradiated laser beam is in consonance with the area of the condensate as we vary the Gaussian potential depth. By computing the root mean square radius of the condensate, we confirm the variation in the number of vortices generated as a function of the ratio between the root-mean-square radius of the condensate and the laser beam waist. In particular, the number of hidden vortices reaches the maximum value when the above ratio is close to the value 0.7. We find the variation in the number of vortices is rapid for deeper Gaussian potentials, and we conclude that the larger beam waist and deeper Gaussian potentials generate more vortices. Further, we calculate the number of vortices using the Feynman rule with Thomas Fermi approximation and compare them with the numerical results. We also observe that the critical rotation frequency decreases with an increase in depth of Gaussian potential.

Second-Order Flows for Computing the Ground States of Rotating Bose-Einstein Condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modelled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. This is judged by the fact that larger size of time steps can be applied in explicit methods, and much smaller number of time steps are needed in both explicit and semi-implicit discretizations, while every time step encounters almost the same computational complexity in comparison with the respective gradient flow type approaches. Rich and detailed numerical examples are documented for verification and comparison.

SU(3) Spin–Orbit Coupled Rotating Bose–Einstein Condensate Subject to a Gradient Magnetic Field

We consider a harmonically trapped rotating spin-1 Bose–Einstein condensate with SU(3) spin–orbit coupling subject to a gradient magnetic field. The effects of SU(3) spin–orbit coupling, rotation, and gradient magnetic field on the ground-state structure of the system are investigated in detail. Our results show that the interplay among SU(3) spin–orbit coupling, rotation, and gradient magnetic field can result in a variety of ground states, such as a vortex ring and clover-type structure. The numerical results agree well with our variational analysis results.

Deep-learning-based quantum vortex detection in atomic Bose–Einstein condensates

Quantum vortices naturally emerge in rotating Bose–Einstein condensates (BECs) and, similarly to their classical counterparts, allow the study of a range of interesting out-of-equilibrium phenomena, such as turbulence and chaos. However, the study of such phenomena requires the determination of the precise location of each vortex within a BEC, which becomes challenging when either only the density of the condensate is available or sources of noise are present, as is typically the case in experimental settings. Here, we introduce a machine-learning-based vortex detector motivated by state-of-the-art object detection methods that can accurately locate vortices in simulated BEC density images. Our model allows for robust and real-time detection in noisy and non-equilibrium configurations. Furthermore, the network can distinguish between vortices and anti-vortices if the phase profile of the condensate is also available. We anticipate that our vortex detector will be advantageous for both experimental and theoretical studies of the static and dynamic properties of vortex configurations in BECs.

BEC2HPC: A HPC spectral solver for nonlinear Schrödinger and rotating Gross-Pitaevskii equations. Stationary states computation

We present BEC2HPC which is a parallel HPC spectral solver for computing the ground states of the nonlinear Schrödinger equation and the Gross-Pitaevskii equation (GPE) modeling rotating Bose-Einstein condensates (BEC). Considering a standard pseudo-spectral discretization based on Fast Fourier Transforms (FFTs), the method consists in finding the numerical solution of the energy functional minimization problem under normalization constraint by using a preconditioned nonlinear conjugate gradient method. We present some numerical simulations and scalability results for the 2D and 3D problems to obtain the stationary states of BEC with fast rotation and large nonlinearities. The code takes advantage of existing HPC libraries and can itself be leveraged to implement other numerical methods like e.g. for the dynamics of BECs. Program summaryProgram title: BEC2HPCCPC Library link to program files:https://doi.org/10.17632/mdzpw4dr4t.1Licensing provisions: GPLv2Programming language: C++, PythonNature of problem: This software computes the stationary states of rotating Bose–Einstein condensates (BEC) modeled by the Gross–Pitaevskii equation (GPE). It implements a numerical method that is particularly effective for BEC with fast rotation and large nonlinearities. The parallel implementation allows to perform large-scale simulations of 2D or 3D problems on parallel computing platforms.Solution method: The stationary states are computed using an iterative pseudo-spectral method based on Fast Fourier Transforms. The computation takes the form of a constrained minimization problem solved using a preconditioned nonlinear conjugate gradient method. This solver is implemented in distributed memory using MPI and a decomposition of the computational domain.Additional comments including restrictions and unusual features: The algorithms are implemented in C++ and MPI but a Python interface is provided for defining the physics of the problem. Results can be exported to HDF5 files and visualized with external tools such as ParaView. The code can be used to implement other spectral methods in parallel or to solve problems related to the dynamics of BECs.

Adjustable half-skyrmion chains induced by SU(3) spin–orbit coupling in rotating Bose–Einstein condensates* *Project supported by the National Key Research and Development Program of China (Grant No. 2016YFA0301500), the National Natural Science Foundationof China (Grant Nos. 61835013 and 11971067), the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant Nos. XDB01020300and XDB21030300), the Beijing Natural Science Foundation, China (Grant No.

The ground state properties of the rotating Bose–Einstein condensates (BECs) with SU(3) spin–orbit coupling (SOC) in a two-dimensional harmonic trap are studied. The results show that the ferromagnetic and antiferromagnetic systems present three half-skyrmion chains at an angle of 120° to each other along the coupling directions. With the enhancement of isotropic SU(3) SOC strength, the position of the three chains remains unchanged, in which the number of half-skyrmions increases gradually. With the increase of rotation frequency and atomic density–density interaction, the number of half-skyrmions on the three chains and in the regions between two chains increases gradually. The relationships of the total number of half-skyrmions on the three chains with the increase of SU(3) SOC strength, rotation frequency and atomic density–density interaction are also given. In addition, changing the anisotropic SU(3) SOC strength can regulate the number and morphology of the half-skyrmion chains.

A local refinement purely meshless scheme for time fractional nonlinear Schrödinger equation in irregular geometry region* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11501495, 51779215, and 11672259), the Postdoctoral Science Foundation of China (Grant Nos. 2015M581869 and 2015T80589), and the Jiangsu Government Scholarship for Overseas Studies, China (Grant No. JS-2017-227).

A local refinement hybrid scheme (LRCSPH-FDM) is proposed to solve the two-dimensional (2D) time fractional nonlinear Schrödinger equation (TF-NLSE) in regularly or irregularly shaped domains, and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross–Pitaevskii equation (TF-GPE) with the rotating Bose–Einstein condensate. It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge. The proposed LRCSPH-FDM (which is based on a local refinement corrected SPH method combined with FDM) is derived by using the finite difference scheme (FDM) to discretize the Caputo TF term, followed by using a corrected smoothed particle hydrodynamics (CSPH) scheme continuously without using the kernel derivative to approximate the spatial derivatives. Meanwhile, the local refinement technique is adopted to reduce the numerical error. In numerical simulations, the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method. The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D (where 1D stands for one-dimensional) analytical TF-NLSEs in a rectangular region (with regular or irregular particle distribution) or in a region with irregular geometry. The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain, and the results from the posed method are compared with those from the FDM. All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.

Numerical method for the projected Gross-Pitaevskii equation in an infinite rotating two-dimensional Bose gas.

We present a method for evolving the projected Gross-Pitaevskii equation in an infinite rotating Bose-Einstein condensate, the ground state of which is a vortex lattice. We use quasiperiodic boundary conditions to investigate the behavior of the bulk superfluid in this system, in the absence of boundaries and edge effects. We also give the Landau gauge expression for the phase of a BEC subjected to these boundary conditions. Our spectral representation uses the eigenfunctions of the one-body Hamiltonian as basis functions. Since there is no known exact quadrature rule for these basis functions we approximately implement the projection associated with the energy cutoff, but we show that by choosing a suitably fine spatial grid the resulting error can be made negligible. We show how the convergence of this model is affected by simulation parameters such as the size of the spatial grid and the number of Landau levels. Adding dissipation, we use our method to find the lattice ground state for N vortices. We can then perturb the ground-state, to investigate the melting of the lattice.

The Nonexistence of Vortices for Rotating Bose–Einstein Condensates with Attractive Interactions

This article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap $V(x)$ rotating at the velocity $\Omega $. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity $0<\Omega^*:=\Omega^*(V)\leq \infty$, depending on the general trap $V(x)$, such that for any rotational velocity $0\leq \Omega <\Omega ^*$, minimizers (i.e., ground states) exist if and only if $a<a^*=\|w\|^2_2$, where $a>0$ denotes the absolute product for the number of particles times the scattering length, and $w>0$ is the unique positive solution of $\Delta w-w+w^3=0$ in $\mathbb{R}^2$. If $V(x)=|x|^2$ and $ 0<\Omega <\Omega^*(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as $a \nearrow a^*$, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of $w$.

Influence of Dissipation on the Vortex Motion in Rotating Bose–Einstein Condensates

Within the framework of the Gross–Pitaevskii model, we deduce a system of equations that describes the motion of quantized vortices in Bose–Einstein condensates. We consider a strongly anisotropic magnetic trap whose trapping potential is much higher in the direction z than in the transverse direction. Under the action of this potential, the condensate takes the form of a plane disk. The transition to the two-dimensional case enables us to apply the method of matched asymptotic expansions and obtain the equations of vortex motion in the explicit form. We take into account the rotation of the condensate as a whole and the effect of dissipative processes as a result of which the system of vortices comes to the equilibrium state. Some examples of the vortex motion are presented for different values of the external parameters.

Multisymplectic structure-preserving scheme for the coupled Gross–Pitaevskii equations

In this paper, we study numerically the dynamics of the rotating Bose–Einstein condensates (BECs) modelled by the coupled Gross–Pitaevskii (CGP) equations with angular momentum rotating terms. First, the multisymplectic structure of the CGP equations is investigated by introducing some canonical momenta which allows us to construct the corresponding multisymplectic schemes. Then applying the midpoint rule in both temporal and spatial directions, a multisymplectic scheme is proposed for the CGP equations. The conservative properties and the convergence analysis are discussed for the multisymplectic scheme. The cut-off function technique is utilized for the error estimation. Numerical examples are carried out to verify the conservative property and the convergence rate.