Analysis and efficient computation for the dynamics of two-component Bose-Einstein condensates

Abstract

Abstract. In this paper, we review some recent results on the analysis and efficient computation for the dynamics of rotating two-component Bose-Einstein condensates (BECs). We begin with the three-dimensional (3D) coupled GrossPitaevskii equations (GPEs) with an angular momentum rotation term and an external driving field, show how to scale it into dimensionless form, reduce it to a 2D and 1D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometric optics. Dynamic laws on the angular momentum expectation, density of each component, condensate widths and analytical solutions of stationary states with its center shifted from the trapping center are presented. In addition, efficient and accurate numerical methods for computing the dynamics of rotating two-component BEC are presented and some numerical results are reported to demonstrate the efficiency and accuracy of the numerical methods.

A Generalized-Laguerre–Fourier–Hermite Pseudospectral Method for Computing the Dynamics of Rotating Bose–Einstein Condensates

“…Fig. 3c), which confirms the analytical results in [4,46]; (ii) W 1 (t) and W 2 (t) are periodic functions when β 11 = β 12 = β 22 …”

“…Fig. 3a&c), which confirms the analytical results in [4,46]; (iii) the normalization N (Ψ) and energy E β,Ω (Ψ) are conserved well in the computation; (iv) the angular momentum expectation L z (t) is conserved when γ x = γ y , i.e. the trapping is radial symmetric, which again confirms the analytical results in [4,46].…”

Dynamical Laws of the Coupled Gross-Pitaevskii Equations for Spin-1 Bose-Einstein Condensates

“…The aim of this paper is to study mathematically the dynamical properties of spin-1 BEC based on the CGPEs (1.1)-(1.3) including time evolution of the density of each component, expected value of angular momentum (or angular momentum expectation) and condensate width as well as the construction of an analytical solution when the initial data is chosen as the stationary state with its center-of-mass shifted from the trap center. We remark that similar mathematical results have been carried out recently for GPE of single-component BEC [10,3] and CGPEs of two-component BEC [37,2].…”

“…In the literature, different spectrally accurate numerical methods were proposed to discretize GPE or CGPEs for studying the dynamics of BECs (cf. [7,2,3,4] and references therein). The key ideas of these methods are: (i) using a time-splitting technique to decouple the nonlinearity; (ii) adopting the spectral method to discretize the free Schrödinger equation without/with the angular rotation term; (iii) constructing the proper spectral basis functions such that the reduced ODE system is decoupled and thus can be integrated in time exactly.…”

Ground States of Two-component Bose-Einstein Condensates with an Internal Atomic Josephson Junction

“…The first experiment for two-component BEC was performed in JILA with |F = 2, m f = 2〉 and |1, −1〉 spin states of 87 Rb [28]. Since then, extensive experimental and theoretical studies of two-component BEC have been carried out in the last several years [3,10,19,27,35,38]. At temperature T much smaller than the critical temperature T c and after proper nondimensionalization and dimension reduction [29,38], a two-component BEC with an internal atomic Josephson junction (or an external driving field) can be well described by the following coupled Gross-Pitaevskii equations (CGPEs) in dimensionless form [29,37,38]:…”