In an optical trap, the ground state of spin-1 Bosons such as $^{23}$Na, $^{39}$K, and $^{87}$Rb can be either a ferromagnetic or a "polar" state, depending on the scattering lengths in different angular momentum channel. 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 superfluid $^{3}$He-A. Current estimates of scattering lengths suggest that the ground states of $^{23}$Na and $^{87}$Rb condensate are a polar state and a ferromagnetic state respectively.Comment: 11 pages, no figures. email : Ho@mps.ohio-state.ed

We show that binary mixtures of Bose condensates of alkali atoms have a great variety of ground state and vortex structures which can be accessed experimentally by varying the particle numbers of different alkalis. We have constructed a simple algorithm to determine the density profiles of the mixtures within Thomas-Fermi approximation. Many structures of the alkali binary contain a coexisting region, which is the analog of the long sought 3 He-4 He interpenetrating superfluids in ultralow temperature physics.[S0031-9007 (96)01390-7] PACS numbers: 03.75.Fi, 05.30.JpThe search of Bose condensate in alkali atoms [1-3] has a deep root in ultralow temperature physics. Since the discovery of superfluid 3 He, the searches of the next elemental superfluid have been focusing on spin polarized hydrogen and 3 He-4 He mixture. The former promises another Bose superfluid besides the only known example of 4 He, the latter, the first example of interpenetrating superfluids. The recent discoveries of alkali Bose condensates [1-3] have in essence achieved the goal of the superfluid hydrogen search. Since there are no intrinsic difficulties in loading more than one alkali element and having them cooled in the same trap, it appears highly promising that interpenetrating superfluids may be realized for the first time within the same experimental setting.In this paper, we shall discuss binary mixtures of alkali condensates. Such mixtures may consist of different alkalis such as 87 Rb-23 Na, or different isotopes such as 87 Rb-85 Rb, or different hyperfine states of the same alkali such as the (F 2, M F 2) and (F 1, M F 1) states of 87 Rb. We shall denote the two different alkalis as 1 and 2, and their particle numbers as N 1 and N 2 . Unlike single component systems which are characterized by a single scattering length, alkali binaries are characterized by three scattering lengths a 1 , a 2 , and a 12 , representing interactions between like and unlike alkalis. At present, the scattering lengths between many like alkali atoms are known, whereas those between unlike alkalis have not been measured. As we shall see, this moderate increase in energy scales leads to a proliferation of ground state and vortex structures.In the following, we shall present (a) a simple algorithm for determining the density profiles of the mixtures, (b) the evolution of the ground states and vortex states as a function of N 1 , N 2 . For length reasons, we shall limit ourselves to the vortex states where alkali 1 contains a 2p vortex and alkali 2 is vortex free. Our algorithm, however, can be applied to an arbitrary number of vortices in 1 and 2. As we shall see, the structure of the mixture depends on the ratio of g factors of the two alkalis and the ratios of their interaction parameters. These ratios determine whether alkali 2 when added to an existing cloud of 1 will stay at its exterior or interior. Another general feature of the mixture is that when N 1 ϳ N 2 , it generally contains a large coexisting region of 1 and 2. This is the analog of the long s...

Motivated by recent experiments carried out by Spielman's group at NIST, we study a general scheme for generating families of gauge fields, spanning the scalar, spin-orbit, and non-Abelian regimes. The NIST experiments, which impart momentum to bosons while changing their spin state, can in principle realize all these. In the spin-orbit regime, we show that a Bose gas is a spinor condensate made up of two non-orthogonal dressed spin states carrying different momenta. As a result, its density shows a stripe structure with a contrast proportional to the overlap of the dressed states, which can be made very pronounced by adjusting the experimental parameters.

We show that there are three possible phases for a spin-2 spinor Bose condensate, one more compared to the spin-1 case. The order parameters of these phases are the spontaneous magnetization and the singlet pair amplitude. Current estimates of scattering lengths show that all three phases have realizations in optically trapped alkali atoms. There is also a one-to-one correspondence between the structure of a spin-2 spinor Bose condensate and that of a d-wave BCS superfluid.PACS number͑s͒: 03.75.Fi, 05.30.Jp One of the recent major developments in Bose-Einstein Condensation ͑BEC͒ in atomic gases is the study of dilute Bose gases with internal degrees of freedom. The first realization of such system is found in optically trapped 23 Na, which is a spin-1 Bose gas ͓1͔. Recently, JILA has also created a ''spin-1/2'' Bose gas by continually cycling between the Fϭ1 and Fϭ2 states of magnetically trapped 87 Rb ͓2͔.In the case of spin-1 Bose gas, the nature of the spinor condensate depends crucially on the magnetic interaction. In zero magnetic field, the spinor condensate can be either ferromagnetic or ''polar,'' which has very different properties ͓3,4͔.Generally, only atoms in the low lying hyperfine multiplet are confined in the optical trap. Those in the higher hyperfine multiplet will leave the trap by spin-flip scattering. In the case of 23 Na and 87 Rb, their hyperfine multiplets (Fϭ2 and Fϭ1) are regular, i.e., the higher spin state (Fϭ2) has higher energy. Since spin-flip scattering is strong in 23 Na, it may be difficult to produce a spin-2 Bose gas in this system. On the other hand, 87 Rb has much weaker spin-flip scattering and is a candidate for optically trapped spin-2 Bose gas. In the case of 85 Rb, the lowest multiplet has spin Fϭ2.It also has a negative s-wave scattering length in zero field. Should the current effort to Bose condense 85 Rb in magnetic traps be successful, it is conceivable that an Fϭ2 spinor condensate can be trapped optically in low fields, if the three particle losses when the field is reduced through the Feshbach resonance is not too large.In this paper, we study the ground state structure of a spin-2 Bose gas within the single condensate approximation. In the case of spin-1 Bose gas, it has been realized recently that the ground state can be ''fragmented'' ͑i.e., containing more than one condensate͒ ͓5,6͔. Despite this fact, the phase diagram for single spinor condensates remains highly valuable and in fact gives the best agreement with experiments so far ͓1͔. This is because the spin-1 fragmented state is delicate with respect to spin-nonconserving perturbations, which will drive the system toward a single condensate state. For these reasons, we shall first focus on the ground states of single spinor condensates. We shall consider only linear Zeeman effect, which is already much more subtle than the spin-1 case. The actual fragmented structures as well as quadratic Zeeman effects will be discussed elsewhere.Because of the increase in spin value, spin-2 Bose gas has one more inte...

We present the theory of bosonic systems with multiple condensates, unifying disparate models which are found in the literature, and discuss how degeneracies, interactions, and symmetries conspire to give rise to this unusual behavior. We show that as degeneracies multiply, so do the types of fragmentation, eventually leading to strongly correlated states with no trace of condensation.