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arXiv:cond-mat/9803231 · Physical Review Letters

Spinor Bose Condensates in Optical Traps

Tin-Lun Ho

Published Jul 27, 1998• 1,552 citations
arXivHTMLPDF

Abstract

We show that in an optical trap the ground states of spin-1 bosons such as ${}^{23}\mathrm{Na}$, ${}^{39}\mathrm{K}$, and ${}^{87}\mathrm{Rb}$ can be either ferromagnetic or ``polar" states, depending on the scattering lengths in different angular momentum channels. The collective modes of these states have very different spin character and spatial distributions. While ordinary vortices are stable in the polar state, only those with unit circulation are stable in the ferromagnetic state. The ferromagnetic state also has coreless (or Skyrmion) vortices like those of ${}^{3}\mathrm{He}\ensuremath{-}A$.

DOI
10.1103/physrevlett.81.742

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We show that in an optical trap the ground states of spin-1 bosons such as 23Na{}^{23}\mathrm{Na}23Na, 39K{}^{39}\mathrm{K}39K, and 87Rb{}^{87}\mathrm{Rb}87Rb can be either ferromagnetic or ``polar" states, depending on the scattering lengths in different angular momentum channels. The collective modes of these states have very different spin character and spatial distributions. While ordinary vortices are stable in the polar state, only those with unit circulation are stable in the ferromagnetic state. The ferromagnetic state also has coreless (or Skyrmion) vortices like those of 3He\ensuremath−A{}^{3}\mathrm{He}\ensuremath{-}A3He\ensuremath−A.
This thesis is concerned with the structure and stability of vortices in spin-1 atomic Bose-Einstein Condensates (BECs) in rotating, optical traps. We numerically study these vortex structures using a classical mean-field theory which allows atomic interactions to change the local expectation value of the atomic spin. Initially applying a model in which the atoms interact only via scattering which does not conserve an initial longitudinal magnetisation, we identify the energetically stable configurations of singular and nonsingular vortices via propagation in imaginary time in a rotating frame of reference. We find that the cores of singular vortices fill with atoms in the spinor BEC and show that this can be understood in terms of an energetic hierarchy of length scales. By refining the numerical model to explicitly conserve longitudinal magnetisation, we show that the conservation of a strong magnetisation can lead to a mixing of the two phases of the ground-state manifold (polar and ferromagnetic), which are characterised by the expectation value of the spin. This occurs as a result of the introduction of a new characteristic length scale determined by the longitudinal magnetisation. A surprising consequence is the stability of a ferromagnetic coreless vortex in the polar interaction regime, which otherwise is energetically unstable. We construct analytic spinor wavefunctions which parametrise the interpolation between the polar and ferromagnetic phases, exhibiting different vortex topologies in the respective phases. Finally by identifying stationary states of the system, we show how nonlocal dipoledipole interactions between atoms introduces an additional length scale determined by the strength of the dipolar interaction. The energetic hierarchy of length scales then determines whether the dipolar interactions have a significant effect on the stationary vortex structures. In particular we show how a BEC with polar interactions adopts the properties of a ferromagnetic condensate when the dipolar interactions dominate.
We report the observation of strongly ferromagnetic F = 1 spinor Bose-Einstein condensates of 7 Li atoms.The condensates are generated in an optical dipole trap without using magnetic Feshbach resonances, so that the condensates have internal spin degrees of freedom.Studying the nonequilibrium spin dynamics, we have measured the ferromagnetic spin interaction energy and determined the s-wave scattering length difference among total spin f channels to be a f =2a f =0 = -18(3) Bohr radius.This strong collision-channel dependence leads to a large variation in the condensate size with different spin composition.We were able to excite a radial monopole mode after a spin-flip transition between the |m F = 0 and |m F = 1 spin states.From the experiments, we estimated the scattering length ratio a f =2 /a f =0 = 0.27(6), and determined a f =2 = 7(2) and a f =0 = 25(5) Bohr radii, respectively.The results indicate the spin-dependent interaction energy of our system is as large as 46% of the condensate chemical potential.
We study the energetic and dynamic stability of coreless vortices in nonrotated spin-1 Bose-Einstein condensates, trapped with a three-dimensional optical potential and a Ioffe-Pritchard field. The stability of stationary vortex states is investigated by solving the corresponding Bogoliubov equations. We show that the quasiparticle excitations corresponding to axisymmetric stationary states can be taken to be eigenstates of angular momentum in the axial direction. Our results show that coreless vortex states can occur as local or global minima of the condensate energy or become energetically or dynamically unstable depending on the parameters of the Ioffe-Pritchard field. The experimentally most relevant coreless vortex state containing a doubly quantized vortex in one of the hyperfine spin components turned out to have very nontrivial stability regions, and especially a quasiperiodic dynamic instability region which corresponds to splitting of the doubly quantized vortex.
Abstract The ground-state phases of rotating spin-1 Bose-Einstein condensates in three different box traps are studied numerically. In circular or square box trap, influenced by the combined effects of spin-exchange interaction and rotation, traditional skyrmion vortex for a rotating spin-1 BEC in harmonic trap are replaced by the homogeneous coreless vortices layered with fourfold rotational symmetry, which are usually generated from the trap boundary every four. In a triangular box trap, one skyrmion vortex structure is firstly found in the trap center, while other vortices are all generated from the boundary and arrange parallel to the trap edge in the form of vortex chains .
We perform detailed close-coupling calculations for the rubidium isotopes 85Rb{}^{85}\mathrm{Rb}85Rb and 87Rb{}^{87}\mathrm{Rb}87Rb to ascertain the nature of their spinor Bose-Einstein condensates. These calculations predict that the spinor condensate for the spin-1 boson 87Rb{}^{87}\mathrm{Rb}87Rb has a ferromagnetic nature. The spinor condensates for the spin-2 bosons 85Rb{}^{85}\mathrm{Rb}85Rb and 87Rb,{}^{87}\mathrm{Rb},87Rb, however, are both predicted to be polar. The nature of a spin-1 condensate hinges critically on the sign of the difference between the s-wave scattering lengths for total spins 0 and 2 while the nature of a spin-2 condensate depends on the values of the differences between s-wave scattering lengths for the total spins 0, 2, and 4. These scattering lengths were extracted previously and found to have overlapping uncertainties for all three cases, thus leaving the nature of the spinor condensates ambiguous. The present study exploits a refined uncertainty analysis of the scattering lengths based on recently improved result from experimental work by Roberts et al., which permits us to extract an unambiguous result for the nature of the ground state spinor condensates.
We theoretically investigate vortex-lattice phases of rotating spinor Bose-Einstein condensates (BEC's) with a ferromagnetic spin interaction by numerically solving the Gross-Pitaevskii equation. The spinor BEC under slow rotation can sustain a rich variety of exotic vortices due to the multicomponent order parameters, such as Mermin-Ho and Anderson-Toulouse coreless vortices (the two-dimensional Skyrmion and meron) and nonaxisymmetric vortices with shifting vortex cores. Here, we present the spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field. In addition, the vortex phase diagram is constructed in the plane of the total magnetization MMM and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.