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arXiv:cond-mat/9807258 · Physical Review Letters

Quantum Spins Mixing in Spinor Bose-Einstein Condensates

C. K. Law, Han Pu, N. P. Bigelow

Published Dec 14, 1998• 627 citations
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Abstract

A set of collective spin states is derived for a trapped Bose-Einstein condensate in which atoms have three internal hyperfine spins. These collective states minimize the interaction energy among condensate atoms, and they are characterized by strong spin correlations. We also examine the internal dynamics of an initially spin-polarized condensate. The time scale of spin mixing is predicted.

DOI
10.1103/physrevlett.81.5257

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A set of collective spin states is derived for a trapped Bose-Einstein condensate in which atoms have three internal hyperfine spins. These collective states minimize the interaction energy among condensate atoms, and they are characterized by strong spin correlations. We also examine the internal dynamics of an initially spin-polarized condensate. The time scale of spin mixing is predicted.
Summary form only given. The optical trapping of sodium Bose condensates has opened the possibility to study spinor condensates which have their hyperfine spin as a new degree of freedom. They represent a superfluid system with a vector order parameter. A host of novel phenomena have been predicted. A spinor condensate such as the one realized by Stamper-Kurn et al. involves all three hyperfine states in the lowest energy f=1 manifold and thus poses challenges towards the understanding of the low energy properties and dynamics of such systems. We consider a Bose condensate which contains atoms with three internal spins in a cigar-shaped trap and approximate the wavefunction of the spin state.
Bose-Einstein condensation (BEC) is a phenomenon in which identical bosons occupy the same quantum state below a certain critical temperature. A hallmark of BEC is the coherence between particles every particle shares the same quantum wavefunction and phase. This coherence has been demonstrated for the external (motional) degrees of freedom of the atomic condensates by interfering two condensates. In this thesis, the coherence is shown to extend to the internal spin degrees of freedom of a spin-1 Bose gas evidenced by the observed coherent and reversible spin-changing collisions. The observed coherent dynamics are analogous to Josephson oscillations in weakly connected superconductors and represent a type of matter-wave four-wave mixing. Control of the coherent evolution of the system using magnetic fields is also demonstrated. The studies on spinor condensates begin by creating spinor condensates directly using all-optical approaches that were first developed in our laboratory. All-optical formation of Bose-Einstein condensates (BEC) in 1D optical lattice and single focus trap geometries are developed and presented. These techniques offer considerable flexibility and speed compared to magnetic trap approaches, and the trapping potential can be essentially spin-independent and are ideally suited for studying spinor condensates. Using condensates with well-defined initial non-equilibrium spin configuration, spin mixing of F = 1 and F = 2 spinor condensates of rubidium-87 atoms confined in an optical trap is observed. The equilibrium spin configuration in the F = 1 manifold confirms that 87Rb is ferromagnetic. The coherent spinor dynamics are demonstrated by initiating spin mixing deterministically with a non-stationary spin population configuration. Finally, the interplay between the coherent spin mixing and spatial dynamics in spin-1 condensates with ferromagnetic interactions is investigated.
The spinor dynamics of Bose-Einstein condensates of 87Rb^{87}\mathrm{Rb}87Rb atoms with hyperfine spins of 1 and 2 are investigated. A mixture of spinor condensates is generated in the transversely polarized spin-1 and the longitudinally polarized spin-2 states using simultaneous Ramsey interferometry. In the subsequent dynamics, temporal modulation of spin-1 magnetization is observed in the spinor mixture due to the spin-dependent interaction with the spin-2 component. By comparison with numerical simulations based on the Gross-Pitaevskii equation, we find that the observed nonferromagnetic dynamics reflect the ground-state properties of the spinor mixture.
The dynamics of F =1 spinor condensates initially prepared in a double-well potential is studied in the mean-field approach. It is shown that a small seed of m =0 atoms on a system with initially well-separated m =1 and m =-1 condensates has a dramatic effect on their mixing dynamics, acting as an effective barrier for a remarkably long time. We show that this effect is due to the spinor character of the system, and provides an observable example of the interplay between the internal spin dynamics and the macroscopic evolution of the magnetization in a spinor Bose-Einstein condensate.
An effective Hamiltionian with level splitting of the collective spin states is derived for a spinor Bose condensate of trapped atoms with hyperfine spin f = 1 beyond the single-mode approximation. The splitting of these spin states arises from quantum polarization between the mf = +1 and mf = -1 spin channels, and is characterized by small magnetization exhibited by the trapped gas. For big ferromagnetic 87 Rb and antiferromagnetic 23 Na Bose gases, the splitting fraction in total energy increases as -N-1 and decreases as N-2 when the atom number N varies, with typical order of 10-6 and 10-4, respectively. In the case of small condensates, the splitting fraction increases as -N-2, with typical order of 10-5~10-6, both for 87 Rb and 23 Na atoms. In both cases of the big and small condensates, the splitting energy is analogous to the anisotropic energy of a magnet, being proportional to the square of the pseudo-magnetization.
We report on the observation of a collective spin mode in a spinor Bose-Einstein condensate. Initially, all spins point perpendicular to the external magnetic field. The lowest energy mode consists of a sinusoidal oscillation of the local spin around its original axis, with an oscillation amplitude that linearly depends on the spatial coordinates. The frequency of the oscillation is set by the zero-point kinetic energy of the BEC. The observations are in excellent agreement with hydrodynamic equations. The observed spin mode has a universal character, independent of the atomic spin and spin-dependent contact interactions.