Abstract. 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. 1. Introduction. Since its realization in dilute bosonic atomic gases [3,20,21], Bose-Einstein condensation of alkali atoms and hydrogen has been produced and studied extensively in the laboratory [45] and has permitted an intriguing glimpse into the macroscopic quantum world. In view of potential applications [26,43,44], the study of quantized vortices, which are well-known signatures of superfluidity, is one of the key issues. Different research groups have obtained quantized vortices in Bose-Einstein condensates (BEC) experimentally, e.g., the JILA group [39], the ENS group [36,37], and the MIT group [45]. Currently, there are at least two typical ways to generate quantized vortices from the ground state of BEC: (i) impose a laser beam rotating with an angular velocity on the magnetic trap holding the atoms to create an harmonic anisotropic potential [17,33,1,13]; (ii) add to the stationary magnetic trap a narrow, moving Gaussian potential, representing a far-blue detuned laser [29,7]. The recent experimental and theoretical advances in the exploration of quantized vortices in BEC have spurred great excitement in the atomic physics community and renewed interest in studying superfluidity.The properties of BEC in a rotational frame at temperature T much smaller than the critical condensation temperature T c are well described by the macroscopic

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In this paper, we study dynamics of the ground state and central vortex states in Bose–Einstein condensation (BEC) analytically and numerically. We show how to define the energy of the Thomas–Fermi (TF) approximation, prove that the ground state is a global minimizer of the energy functional over the unit sphere and all excited states are saddle points in linear case, derive a second-order ordinary differential equation (ODE) which shows that time-evolution of the condensate width is a periodic function with/without a perturbation by using the variance identity, prove that the angular momentum expectation is conserved in two dimensions (2D) with a radial symmetric trap and 3D with a cylindrical symmetric trap for any initial data, and study numerically stability of central vortex states as well as interaction between a few central vortices with winding numbers ±1 by a fourth-order time-splitting sine-pseudospectral (TSSP) method. The merit of the numerical method is that it is explicit, unconditionally stable, time reversible and time transverse invariant. Moreover, it conserves the position density, performs spectral accuracy for spatial derivatives and fourth-order accuracy for time derivative, and possesses "optimal" spatial/temporal resolution in the semiclassical regime. Finally we find numerically the critical angular frequency for single vortex cycling from the ground state under a far-blue detuned Gaussian laser stirrer in strong repulsive interaction regime and compare our numerical results with those in the literatures.

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Abstract. 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 longrange 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 twodimensional (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. 1. Introduction. Bose-Einstein condensation (BEC), first observed in 1995 [4,18,23], has provided a platform to study the macroscopic quantum world. Later, with the observation of quantized vortices [2,19,34,35,37,39,50], rotating BECs have been extensively studied in the laboratory. The occurrence of quantized vortices is a hallmark of the superfluid nature of BECs. In addition, condensation of bosonic atoms and molecules with significant dipole moments whose interaction is both nonlocal and anisotropic has recently been achieved experimentally in trapped 52 Cr and 164 Dy gases [1,22,27,32,33,36,48].At temperatures T much smaller than the critical temperature T c , the properties of BEC in a rotating frame with long-range dipole-dipole interaction (DDI) are well