We use a variational method to investigate the ground-state phase diagram of a small, asymmetric Bose-Einstein condensate with respect to the dimensionless interparticle interaction strength γ and the applied external rotation speed Ω. For a given γ, the transition lines between no-vortex and vortex states are shifted toward higher Ω relative to those for the symmetric case. We also find a re-entrant behavior, where the number of vortex cores can decrease for large Ω. In addition, stabilizing a vortex in a rotating asymmetric trap requires a minimum interaction strength. For a given asymmetry, the evolution of the variational parameters with increasing Ω shows two different types of transitions (sharp or continuous), depending on the strength of the interaction. We also investigate transitions to states with higher vorticity; the corresponding angular momentum increases continuously as a function of Ω. PACS number(s): 03.75. Fi, 67.40.Vs, 32.80.Pj

A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity Ωc for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order expansion of the Bogoliubov equations determines the (negative) frequency ωa of the anomalous mode. Although Ωc = −ωa through first order, the second-order contributions ensure that the absolute value |ωa| is always smaller than the critical angular velocity Ωc. With increasing external rotation Ω, the dynamical instability of the condensate with a vortex disappears at Ω * = |ωa|, whereas the vortex state becomes energetically stable at the larger value Ωc. Both second-order contributions depend explicitly on the axial anisotropy of the trap. The appearance of a local minimum of the free energy for a vortex at the center determines the metastable angular velocity Ωm. A variational calculation yields Ωm = |ωa| to first order (hence Ωm also coincides with the critical angular velocity Ωc to this order). Qualitatively, the scenario for the onset of stability in the weak-coupling limit is the same as that found in the strong-coupling (Thomas-Fermi) limit.

The low-energy fluctuations of a trapped, interacting quasi one-dimensional Bose gas are studied. Our considerations apply to experiments with highly anisotropic traps. We show that under suitable experimental conditions the system can be described as a Luttinger liquid. This implies that the correlation function of the bosons decays algebraically preventing Bose-Einstein condensation. At significantly lower temperatures a finite size gap destroys the Luttinger liquid picture and BoseEinstein condensation is again possible. PACS: 03.75.Fi, 05.30.jp, 32.80.Pj, 67.90.+z, 71.10.Pm The experimental realization of Bose-Einstein condensation (BEC) in atomic vapors of 87 Rb [1] and 23 Na [2,3] has attracted a lot of interest [4]. Recently, a highly anisotropic, quasi one-dimensional trap has been designed [5]. Up to now, the possibility of BEC in one dimension has mainly been discussed for the noninteracting Bose gas [6,7]. The role of dimensionality has been carefully examined for the ideal bose gas by van Druten and Ketterle [8]. In one dimension the interaction between bosons plays an essential role due to the strong constraint in phase space [9]. The question of BEC in a quasi one-dimensional system is therefore more complicated. The purpose of this paper is to demonstrate that under suitable experimental conditions the low energy excitations of this system are described by a Luttinger liquid (LL) [10] model. The superfluid correlations of a LL decay algebraically and the system is not Bose condensed. At much lower temperatures which are determined by the extension of the trap in the longitudinal direction the spectrum of the phase fluctuations is again cut off by finite size effects and the bosons could condense again.The realization of a Luttinger liquid in a onedimensional Bose gas would be a highly non-trivial example of an interacting quantum liquid. Fermionic systems which are believed to be described by a Luttinger liquid include quasi one-dimensional organic metals [11], magnetic chain compounds,quantum wires and edge states in the Quantum Hall Effect. While these systems are always embedded in a three-dimensional matrix and thus show a crossover to a three-dimensional behavior at low temperatures, the trapped one-dimensional Bose gas would provide a clean testing ground for the concept of a Luttinger liquid.The paper is organized as follows: First we discuss the circumstances under which a trapped Bose gas can be considered as a one-dimensional quantum system. Next we demonstrate in an explicit calculation that there is a gapless mode with a linear dispersion. We show that the Hamiltonian of the low-lying excitations can be identified as that of a Luttinger liquid and therefore the densitydensity correlation function decays algebraically. In the reminder of the paper we discuss the implication of the algebraic decay of the particle-particle correlation function for BEC and review the properties of a Luttinger liquid. We consider the Bose gas in a cylindrical symmetric trap confined to the z-axis by a ti...

We consider a cylindrically symmetric trap containing a small Bose-Einstein condensate with a singly quantized vortex on the axis of symmetry. A time-dependent variational Lagrangian analysis yields the small-amplitude dynamics of the vortex and the condensate, directly determining the equations of motion of the coupled normal modes. As found previously from the Bogoliubov equations, there are two rigid dipole modes and one anomalous mode with a negative frequency when seen in the laboratory frame.