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Discrete and Continuous Dynamical Systems - B

Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates

Dong Deng, Ruikuan Liu

Published Nov 9, 2018• 1 citation
Publisher

Abstract

The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.

DOI
10.3934/dcdsb.2018306

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The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.
In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.
We consider the statics and dynamics of F = 1 spinor Bose–Einstein condensates (BECs) confined in double-well potentials. We use a two-mode Galerkin-type quasi-analytical approximation to describe the stationary states of the system. This way, we are able to obtain not only earlier results based on the single-mode approximation (SMA) frequently used in studies of spinor BECs, but also additional modes that involve either two or all three spinor components of the F = 1 spinor BEC. The results based on this Galerkin-type decomposition are in good agreement with the analysis of the full system. We subsequently analyze the stability of these multi-component states, as well as their dynamics when we find them to be unstable. The instabilities of the symmetric or anti-symmetric states exhibit symmetry-breaking and recurrent asymmetric patterns. Our results yield qualitatively similar bifurcation diagrams both for polar (such as 23 Na) and ferromagnetic (such as 87 Rb) spinor BECs.
We are concerned with the three-component generalized Gross-Pitaevskii Equations (GGPE) in the spinor Bose-Einstein condensates with trapping potentials. Under certain conditions on trapping potentials and parameters, we establish the existence and uniqueness of corresponding standing wave solutions with behaviors tending to zero at infinity. The same issues for the corresponding Dirichlet boundary value problems on the ball centered at the origin are also concluded. Additionally, we provide a classification of various types of solutions to radially symmetric situations.
We propose a method to analytically solve the one-dimensional coupled nonlinear Gross-Pitaevskii equations which govern the motion of the spinor Bose-Einstein condensates. In a uniform external potential, the Hamiltonian comprises the kinetic energy, the linear and the quadratic Zeeman energies. Several classes of exact periodic and solitonic solutions, either in real or in complex forms, are obtained for both the F=1 and F=2 condensates. These solutions are general that contain neither approximations nor constraints on the system parameters.
The spatial structure of the spinor Bose-Einstein condensates with the spin degrees of freedom is analyzed based on the generalized Gross-Pitaevskii (GP) equation in light of the present spin-domain experiment on ${m}_{F}=\ifmmode\pm\else\textpm\fi{}1,$ and 000 of the hyperfine state F=1F=1F=1 of 23Na{}^{23}\mathrm{Na}23Na atom gases. The GP solutions in three- and one-spatial dimensional cases reproduce the observed spin-domain structures, revealing the length scale associated with the existence of the weak interaction of the spin-spin channel, other than the ordinary coherence length related to the density-density channel. The obtained domain structure in GP is compared with the result in the Thomas-Fermi approximation. The former solution is found to better describe the observed features than the latter.