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Studies in Applied Mathematics

Static States for Rotating Two‐Component Bose–Einstein Condensates

Hichem Hajaiej, Xiao Luo, Tao Yang

Published Jan 1, 2025• 0 citations

Abstract

ABSTRACT In this paper, we study static states for rotating two‐component Bose–Einstein condensates (BECs) in two and three dimensions. This leads to analyze normalized solutions for a coupled Schrödinger system with rotation. In dimension two, it corresponds to a mass‐critical problem, for which we obtain some existence and nonexistence results. In the three‐dimensional case, the problem becomes mass‐supercritical, where we prove a multiplicity result along with an accurately asymptotical analysis. Furthermore, a stability result is also established in both cases. We not only extend the main results in Ardila and Hajaiej (Journal of Dynamics and Differential Equations 35 (2023), 1643–1665), Arbunich et al. (Letters in Mathematical Physics 109 (2019), 1415–1432), and Luo and Yang (Journal of Differential Equations 304 (2021), 326–347) from the rotating one‐component BEC to rotating two‐component BECs, but we also partially extend the results of Guo et al. (Discrete and Continuous Dynamical Systems 37 (2017), 3749–3786; Journal of Differential Equations 264 (2018), 1411–1441) from nonrotating two‐component BECs to rotating two‐component BECs.

DOI: 10.1111/sapm.70013

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ABSTRACT In this paper, we study static states for rotating two‐component Bose–Einstein condensates (BECs) in two and three dimensions. This leads to analyze normalized solutions for a coupled Schrödinger system with rotation. In dimension two, it corresponds to a mass‐critical problem, for which we obtain some existence and nonexistence results. In the three‐dimensional case, the problem becomes mass‐supercritical, where we prove a multiplicity result along with an accurately asymptotical analysis. Furthermore, a stability result is also established in both cases. We not only extend the main results in Ardila and Hajaiej (Journal of Dynamics and Differential Equations 35 (2023), 1643–1665), Arbunich et al. (Letters in Mathematical Physics 109 (2019), 1415–1432), and Luo and Yang (Journal of Differential Equations 304 (2021), 326–347) from the rotating one‐component BEC to rotating two‐component BECs, but we also partially extend the results of Guo et al. (Discrete and Continuous Dynamical Systems 37 (2017), 3749–3786; Journal of Differential Equations 264 (2018), 1411–1441) from nonrotating two‐component BECs to rotating two‐component BECs.

The dynamics of two-component Bose-Einstein condensates in rotating traps is investigated. In the Thomas-Fermi limit, equations of motion are derived showing multiple static solutions for a vortex-free condensate. Dynamic stability analysis of these solutions and comparison to truncated Wigner simulations enable us to identify the regimes for which vortex states will occur. In addition, our analysis predicts center-of-mass oscillations that are induced by interspecies interactions and affect each component separately. For attractive interspecies interactions, these oscillations lead to a stable symmetry-broken state.

We study analytically and asymptotically, as well as numerically, ground states and dynamics of two-component spin-orbit-coupled Bose--Einstein condensates (BECs) modeled by the coupled Gross--Pitaevskii equations (CGPEs). In fact, due to the appearance of the spin-orbit (SO) coupling in the two-component BEC with a Raman coupling, the ground state structures and dynamical properties become very rich and complicated. For the ground states, we establish existence and nonexistence results under different parameter regimes and obtain their limiting behaviors and/or structures with different combinations of the SO and Raman coupling strengths. For the dynamics, we show that the motion of the center-of-mass is either nonperiodic or with different frequency than the trapping frequency when the external trapping potential is taken as harmonic and/or the initial data is chosen as a stationary state (e.g., ground state) with a shift, which is completely different from the case of a two-component BEC without the SO coupling, and obtain the semiclassical limit of the CGPEs in the linear case via the Wigner transform method. Efficient and accurate numerical methods are proposed for computing the ground states and dynamics, especially for the case of box potentials. Numerical results are reported to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomenon in the SO-coupled BECs.

We study the dynamics of rotating two-component Bose-Einstein condensates(BECs) toward vortex lattice formation by numerically solving the two-dimensional Gross-Pitaevskii equations.The dynamics are similar to that of single-component BEC, and the whole changes of physical quantities in two cases are qualitatively similar.However, there are some differences between two cases.First, the characteristic time toward vortex entering depends on the inter-component interaction.The time toward vortex entering becomes shorter with increasing the inter-component interaction.Second, the angular momenta exchange between two components with keeping the total angular momentum.

We consider a two-dimensional nonlinear Schrödinger equation proposed in Physics to model rotational binary Bose–Einstein condensates. The nonlinearity is a logarithmic modification of the usual cubic nonlinearity. The presence of both the external confining potential and rotating frame makes it difficult to apply standard techniques to directly construct ground states, as we explain in an appendix. The goal of the present paper is to analyze the orbital stability of the set of energy minimizers under mass constraint, according to the relative strength of the confining potential compared to the angular frequency. The main novelty concerns the critical case where these two effects compensate exactly (lowest Landau Level): orbital stability is established by using techniques related to magnetic Schrödinger operators.

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.