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arXiv:cond-mat/9905182 · Physical Review A

Spin-domain formation in spinor Bose-Einstein condensation

Tomoya Isoshima, Kazushige Machida, Tetsuo Ohmi

Published Dec 1, 1999• 84 citations
arXivHTMLPDF

Abstract

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 $0$ of the hyperfine state $F=1$ of ${}^{23}\mathrm{Na}$ 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.

DOI
10.1103/physreva.60.4857

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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 mF=\ifmmode±\else\textpm\fi1,{m}_{F}=\ifmmode\pm\else\textpm\fi{}1,mF​=\ifmmode±\else\textpm\fi1, 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.
Bose-Einstein condensates - a state of matter at low temperature in which a macroscopic amount of bosons occupy the quantum-mechanical ground state- were originally trapped magnetically. The drawback of this trapping method is that only the weak-field seeking spin state is trapped and we therefore only have a one component i.e. scalar condensate. In contrast, optical trapping confines atoms of all spin state and therefore allows for spin degrees of freedom, resulting in spinor Bose Einstein condensates. In this thesis, density profiles of these spinor condensates have been calculated in Thomas-Fermi approximation. The profiles show areas with strict phase separation, but also areas with spin mixtures. For sodium atoms we get a strict separation between the mF = ±1 and the mF = 0 spin state in Thomas-Fermi approximation. This however causes discontinuities in the transition between spin domains and thus these interfaces were studied further. We found that the density profile of a spin state behaves roughly like a hyperbolic tangent at such an interface. Finally, a differential equation is proposed that could be solved as a starting point of follow-up research.
We explore the time evolution of two-component Bose-Einstein condensates (BECs), quasi one dimensional with respect to their spinor dynamics, following a quench from one component BECs with a U(1)\mathrm{U}(1)U(1) order parameter into two-component condensates with a $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ order parameter. In our case, these two spin components have a propensity to phase separate, i.e., they are immiscible. Remarkably, these spin degrees of freedom can equivalently be described as a single-component attractive BEC. A spatially uniform mixture of these spins is dynamically unstable, rapidly amplifying any quantum or pre-existing classical spin fluctuations. This coherent growth process drives the formation of numerous spin-polarized domains, which are far from the system's ground state. At much longer times these domains grow in size, coarsening, as the system approaches equilibrium. The experimentally observed time evolution is consistent with our fully three-dimensional stochastic-projected Gross--Pitaevskii calculation.
Bose–Einstein Condensation (BEC) of atomic gases has attracted a renewed theoretical and experimental interest in quantum many-body systems at extremely low temperatures (Pethick & Smith; 2002). This excitement stems from two favorable features: (1) by applying magnetic fields and lasers, most of the system parameters, such as the shape, dimensionality, internal states of the condensates, and even the strength of the interatomic interactions, are controllable; (2) due to the diluteness, the mean-field theory explains experiments quite well. In particular, the Gross–Pitaevskii (GP) equation demonstrates its validity as a basic equation for the condensate dynamics. The GP equation is the counterpart of the nonlinear Schrodinger (NLS) equation in nonlinear optics. Thus, a study based on nonlinear analysis is possible and important. In nonlinear physics, a soliton is remarkable object not only for the fact that exact solutions can be obtained but also for its usefulness as a communications tool due to its robustness. In general, solitons are formed under the balance between nonlinearity and dispersion. For atomic condensates, the former is attributed to the interatomic interactions, while the latter comes from the kinetic energy. Either dark or bright solitons are allowable depending on the positive or negative sign of the interatomic coupling constants g, respectively, and indeed have been observed in a quasione-dimensional (q1D) optically constructed waveguide (Strecker et al.; 2002) (Khaykovich et al.; 2002). Such matter-wave solitons are expected in atom optics for applications in atom laser, atom interferometry, and coherent atom transport (Meystre; 2002). In this chapter, we extend the analysis of the matter-wave solitons to a multicomponent case by considering the so-called spinor condensate (Stenger et al.; 1998) whose spin degrees of freedom are liberated under optical traps. Based on theoretical and experimental results, we introduce a new integrable model which describes the dynamical properties of the matter-wave soliton of spinor condensates (Ieda et al.; 2004a). We employ the inverse scattering method to solve this model exactly. As a result, we predict the occurrence of undiscovered physical phenomena such as macroscopic spin precession and spin switching. The chapter is organized as follows. In Sec. 2, the mean field theory of condensate is briefly reviewed. Section 3 introduces an effective interatomic coupling in a q1D condensate. Using these results, we consider a spinor condensate in q1D regime in Sec. 4. Then, in Sec. 5, we
We study spin-polarized states and their stability in the antiferromagnetic phase of spinor (F=1)(F=1)(F=1) quasi-one-dimensional Bose-Einstein condensates. Using analytical approximations and numerical methods, we find various types of polarized states, including patterns of the Thomas-Fermi type, structures featuring a pulse in one component inducing a hole in the other components, states with holes in all three components, and domain walls (DWs). The stability analysis based on the Bogoliubov--de Gennes equations reveals intervals of weak oscillatory instability in families of these states, except for the DWs, which are always stable. The development of the instabilities is examined by means of direct simulations.
We theoretically and numerically study spin turbulence (ST) with small spin magnitude in spin-1 spinor Bose-Einstein condensates by using the spin-1 spinor Gross-Pitaevskii (GP) equations. This kind of ST is realized in two cases: (i) with antiferromagnetic interaction and (ii) with ferromagnetic interaction under a static magnetic field. The ST with small spin magnitude can exhibit two characteristic power laws in the spectrum of the spin-dependent interaction energy: $\ensuremath{-}1$ and $\ensuremath{-}7/3$ power laws in the low- and high-wave-number regions, respectively. These power laws are derived from a Kolmogorov-type dimensional scaling analysis for the equations of motion of the spin vector and nematic tensor. To confirm these power laws, we perform a numerical calculation of the spin-1 spinor GP equations in a two-dimensional uniform system. In case (i), the $\ensuremath{-}7/3$ power law tends to appear in the high-wave-number region, but the spectrum in the low-wave-number region deviates from the $\ensuremath{-}1$ power law. In contrast, both the $\ensuremath{-}1$ and $\ensuremath{-}7/3$ power laws tend to appear in case (ii).
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.