Ja n 20 07 Mixtures of Bose gases under rotation

One of the many interesting aspects of the field of cold atoms is that one may create mixtures of different species. The equilibrium density distribution of the atoms is an interesting problem by itself, since the different components may coexist, or separate, depending on the value of the coupling constants between the atoms of the same and of the different species. If this system rotates, the problem becomes even more interesting. In this case, the state of lowest energy may involve rotation of either one of the components, or rotation of all the components. Actually, the first vortex state in cold gases of atoms was observed experimentally in a two-component system [1], following the theoretical suggestion of Ref. [2]. More recently, vortices have also been created and observed in spinor Bose-Einstein condensates [3, 4]. Theoretically, there have been several studies of this problem [5, 6, 7], mostly in the case where the number of vortices is relatively large. Kasamatsu, Tsubota, and Ueda have also given a review of the work that has been done on this problem [8]. In this Letter, we consider a rotating two-component Bose gas in the limit of weak interactions and slow rotation, where the number of vortices is of order unity. Surprisingly, a number of exact analytical results exist for the energy of this system. The corresponding manybody wavefunction also has a relatively simple structure. We assume equal masses M for the two components, and a harmonic trapping potential Vt = M(ω ρ + ω zz )/2, with ρ = x + y. The trapping frequency ωz along the axis of rotation is assumed to be much higher than ω. In addition, we consider weak atomatom interactions, much smaller than the oscillator energy h̄ω, and work within the subspace of states of the lowest Landau level. The motion of the atoms is thus frozen along the axis of rotation and our problem becomes quasi-two-dimensional [9]. The relevant eigenstates are Φm(ρ, θ)φ0(z), where Φm(ρ, θ) are the lowestLandau-level eigenfunctions of the two-dimensional oscillator with angular momentum mh̄, and φ0(z) is the lowest harmonic oscillator eigenstate along the z axis. The assumption of weak interactions also excludes the possibility of phase separation in the absence of rotation [10], since the atoms of both species reside in the lowest state Φ0,0(r) = Φ0(ρ, θ)φ0(z), while the depletion of the condensate due to the interaction may be treated perturbatively. We label the two (distinguishable) components of the gas as A and B. In what follows the atomatom interaction is assumed to be a contact potential of equal scattering lengths for collisions between the same species and the different ones, aAA = aBB = aAB = a. The interaction energy is measured in units of v0 = U0 ∫