Angular Momentum Rotation Term Research Articles

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.

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.

Exact Splitting Methods for Kinetic and Schrödinger Equations

In (Bernier in Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators. arXiv:1912.13219, (2019)), some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, Fokker–Planck equations, and Schrödinger type equations with an angular momentum rotation term. In this work, these exact splittings are used combined with pseudo-spectral methods in space. High accuracy and efficiency of exact splitting methods are illustrated and comparison are performed with the numerical methods in literature. We show that our methods can be used to improve significantly some classical splitting methods for some nonlinear or non-quadratic equations.

Quantized vortices in spinor Bose–Einstein condensates with time–space modulated interactions and stability analysis**Project supported by the Beijing Natural Science Foundation, China (Grand No. 1182009) and the National Natural Science Foundation of China (Grant No. 11471182).

The three-component Gross–Pitaevskii equation with an angular momentum rotational term can be served as a model to study spinor Bose–Einstein condensates (BECs) with time–space modulated interactions. Vortex solutions of the spinor BECs with spatiotemporally modulated interactions are worked out by similarity transformation. Theoretical analysis and numerical simulation of vortex states are demonstrated. Stable vortex states are obtained by adjusting the frequency of the external potential and the spatiotemporally modulated interaction.

Operator-compensation methods with mass and energy conservation for solving the Gross-Pitaevskii equation

In this work, operator-compensation based high order methods are presented to solve the Gross-Pitaevskii equation modeling Bose-Einstein condensation (BEC). We begin with the high-order approximation to the Laplacian by the central finite difference method combined with operator-compensation technique, and the Crank-Nicolson and time-splitting methods are then proposed for the time discretization. We show that the Crank-Nicolson operator-compensation method keeps the mass and energy conservation, while the time-splitting operator-compensation method only conserves the mass. Then by extending the high-order operator-compensation methods for the first order derivative, the time-splitting finite difference method is developed to solve the Gross-Pitaevskii equation with an angular momentum rotation term. Numerical results are given to test accuracy and verify the conservation properties.

Unconditional convergence of linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

Optimal H<sup>1</sup>-Error Estimates for Crank-Nicolson Finite Difference Scheme for Gross-Pitaevskii Equation with Angular Momentum Rotation Term

Abstract Optimal H1-error estimates for a Crank-Nicolson finite difference scheme for 2D-Gross-Pitaevskii equation with angular momentum rotation term are derived. The analysis is based on classical energy estimate method and on the lifting technique. With no constraint on the grid ratio, we show that the convergence rate of approximate solutions is equivalent to O (τ2 + h2), consistent with numerical results of the existing studies.

Unconditional and optimal H1 error estimate of a Crank–Nicolson finite difference scheme for the Gross–Pitaevskii equation with an angular momentum rotation term

This paper is concerned with the time-step condition (or grid ratio condition) of finite difference method for solving the Gross–Pitaevskii equation with an angular momentum rotation term. An optimal error estimate of the finite difference method in H1 norm is established without any constraints of the grid ratios. Besides the standard energy method, a ‘cut-off’ function technique and a ‘lifting’ technique is introduced in analyzing the proposed scheme. The analysis method used in this paper can be applied to many other finite difference schemes for solving the nonlinear Schrödinger-type equations for which previous works often require certain restriction on the grid ratios. Numerical results are reported to verify the error estimates and conservation laws.

An efficient spectral method for computing dynamics of rotating two-component Bose–Einstein condensates via coordinate transformation

In this paper, we propose an efficient and accurate numerical method for computing the dynamics of rotating two-component Bose–Einstein condensates (BECs) which is described by the coupled Gross–Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field. By introducing rotating Lagrangian coordinates, we eliminate the angular momentum rotation term from the CGPEs, which allows us to develop an efficient numerical method. Our method has spectral accuracy in all spatial dimensions and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in the literature. Numerical results show that to achieve the same accuracy, our method takes much shorter computing time. We also apply our method to study issues such as dynamics of vortex lattices and giant vortices in rotating two-component BECs. Furthermore, we generalize our method to solve the vector Gross–Pitaevskii equations (VGPEs) which is used to study rotating multi-component BECs.

Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

In this paper, we begin with the nonlinear Schrödinger/Gross–Pitaevskii equation (NLSE/GPE) for modeling Bose–Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, and dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported.

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose--Einstein condensates (BECs) in a rotational frame with or without long-range dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross--Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable, and very efficient in computation. It is spectral-order accurate in space and second-order accurate in time and conserves the mass on the discrete level. We compare our method with some representative methods in the literature to demonstrate its efficiency and accuracy. In addition, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation, and center of mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without or with the long-range DDI.

Radial basis function collocation method for a rotating Bose-Einstein condensation with vortex lattices

We study a radial basis function collocation method (RBFCM) to discretize a coupled nonlinear Schr<TEX>$\ddot{o}$</TEX>dinger equation (CNLSE) that governs a two dimensional rotating Bose-Einstein condensate (BEC) with an angular momentum rotation term. We exploit a RBFCM-continuation method (RBFCM-CM) to trace the solution curve of the CNLSE. We compare the performance of the RBFCM-CM with the FEM-CM. We observe that the RBFCM-CM is very robust in a coarse grid for resolving the ground state solution with many vortices when the angular momentum rotation is close to the limit. Numerical results demonstrate the efficiency and accuracy of the RBFCM-CM for computing the superfluid density of the ground level of the BEC.

On the Cauchy problem for nonlinear Schrödinger equations with rotation

We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model forsuperfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities withoutany restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case.

Numerical Simulations on Stationary States for Rotating Two-Component Bose-Einstein Condensates

We numerically study stationary states such as ground, symmetric and central vortex states as well as their energy diagrams for rotating two-component Bose-Einstein condensates (BECs), which are stationary solutions of time-independent coupled Gross-Pitaevskii equations with an angular momentum rotational term. We compute these stationary states by using normalized gradient flows with a backward Euler finite difference discretization, which is proved to be efficient and accurate. By using this discretization, we find various ground state configurations with several vortices for the two-component BECs, which are novel but not found in rotating two-component BECs; we also find that the critical angular velocity at which the ground states lose symmetry.

Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions

In this paper, we establish the global well posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space R3.

Dynamics of rotating two-component Bose–Einstein condensates and its efficient computation

In this paper, we investigate the dynamics of rotating two-component Bose–Einstein condensates (BEC) based on the coupled Gross–Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field, and propose an efficient and accurate method for numerical simulations. We prove the conservation of the angular momentum expectation, derive the dynamic laws for the density of each component and condensate widths, and analyze the dynamics of a stationary state with its center shifted from the trap center. By formulating the CGPEs in either 2D (two-dimensional) polar coordinate or 3D cylindrical coordinate system, the angular momentum rotation term becomes a term with constant coefficients. This allows us to develop an efficient time-splitting method which is time reversible, time transverse invariant, unconditionally stable, efficient and accurate for the problem. Moreover, it conserves the total position density in the discretized level. The numerical method is applied to verify our analytical results and study the dynamics of quantized vortex lattices in rotating two-component BEC with/without an external driving field.

Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term

AbstractIn this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space ℝ2. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Dynamics of the center of mass in rotating Bose–Einstein condensates

In this paper, we derive the analytical solution for a second-order ordinary differential system which governs the motion of the center of mass in the dynamics of a stationary state with its center shifted. A leap-frog Fourier pseudospectral (LFFP) method is presented for efficient and accurate numerical simulations of the Gross–Pitaevskii equation (GPE) with an angular momentum rotation term. Different motion patterns for the center of mass are observed and classified from the analytical solution and confirmed by directly simulating the GPE. To show the effectiveness of the LFFP method, the dynamics of vortex lattices are studied, and the numerical results demonstrate the efficiency and extremely high resolution of our method.

An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose–Einstein condensates

In this paper, we propose an efficient and spectrally accurate numerical method for computing the dynamics of rotating Bose–Einstein condensates (BEC) in two dimensions (2D) and 3D based on the Gross–Pitaevskii equation (GPE) with an angular momentum rotation term. By applying a time-splitting technique for decoupling the nonlinearity and properly using the alternating direction implicit (ADI) technique for the coupling in the angular momentum rotation term in the GPE, at every time step, the GPE in rotational frame is decoupled into a nonlinear ordinary differential equation (ODE) and two partial differential equations with constant coefficients. This allows us to develop new time-splitting spectral methods for computing the dynamics of BEC in a rotational frame. The new numerical method is explicit, unconditionally stable, and of spectral accuracy in space and second-order accuracy in time. Moreover, it is time reversible and time transverse invariant, and conserves the position density in the discretized level if the GPE does. Extensive numerical results are presented to confirm the above properties of the new numerical method for rotating BEC in 2D and 3D.

Dynamics of Rotating Bose--Einstein Condensates and its Efficient and Accurate Numerical Computation

In this paper, we study the dynamics of rotating Bose-Einstein condensates (BEC) based on the Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and present an efficient and accurate algorithm for numerical simulations. We examine the conservation of the angular momentum expectation and the condensate width and analyze the dynamics of a stationary state with a shift in its center. By formulating the equation in either the two-dimensional polar coordinate system or the three-dimensional cylindrical coordinate system, the angular momentum rotation term becomes a term with constant coefficients. This allows us to develop an efficient time-splitting method which is time reversible, unconditionally stable, efficient, and accurate for the problem. Moreover, it conserves the position density. We also apply the numerical method to study issues such as the stability of central vortex states and the quantized vortex lattice dynamics in rotating BEC.