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

Abstract

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

Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

“…We now focus on the spatial discretization of system (27). We consider the case of the dimension d = 2, the generalization to d = 1 and d = 3 being direct by adapting the notations.…”

“…We consider the case of the dimension d = 2, the generalization to d = 1 and d = 3 being direct by adapting the notations. Since problem (27) is set in the whole space, the computational domain has to be truncated. Because there is no physical boundary, it is natural to choose a rectangular computational domain O :=] − a x , a x [×] − a y , a y [.…”

“…Finally, the finite difference discretization of problem (27) leads to the finite dimensional approximation: compute the sequence of vector fields…”

“…In this framework, a function ϕ (that can be considered as an approximation of the solutionφ of problem (27)) is defined on the uniform grid O J,K by ϕ j,k , for any indices ( j, k) ∈ P J,K , i.e. excluding j = 0 and k = 0.…”

“…We discuss some other schemes that we do not recommend (Section 4.4). We also present a recent idea [27] based on a change of frame for a rotational BEC that should be further investigated in the future since it simplifies the implementation of the standard schemes. We extend the time-splitting and relaxation schemes to multi-components GPEs (Section 4.5).…”