Normal Modes of a Vortex in a Trapped Bose-Einstein Condensate

A hydrodynamic description is used to study the normal modes of a vortex in a zero-temperature Bose-Einstein condensate. In the Thomas-Fermi (TF) limit, the circulating superfluid velocity far from the vortex core provides a small perturbation that splits the originally degenerate normal modes of a vortex-free condensate. The relative frequency shifts are small in all cases considered (they vanish for the lowest dipole mode with |m|=1), suggesting that the vortex is stable. The Bogoliubov equations serve to verify the existence of helical waves, similar to those of a vortex line in an unbounded weakly interacting Bose gas. In the large-condensate (small-core) limit, the condensate wave function reduces to that of a straight vortex in an unbounded condensate; the corresponding Bogoliubov equations have no bound-state solutions that are uniform along the symmetry axis and decay exponentially far from the vortex core.Comment: 15 pages, REVTEX, 2 Postscript figures, to appear in Phys. Rev. A. We have altered the material in Secs. 3B and 4 in connection with the normal modes that have |m|=1. Our present treatment satisfies the condition that the fundamental dipole mode of a condensate with (or without) a vortex should have the bare frequency $\omega_\perp

Beyond the Thomas-Fermi approximation for a trapped condensed Bose-Einstein gas

Corrections to the zero-temperature Thomas-Fermi description of a dilute interacting condensed Bose-Einstein gas confined in an isotropic harmonic trap arise due to the presence of a boundary layer near the condensate surface. Within the Bogoliubov approximation, the various contributions to the ground-state condensate energy all have terms of order R^{-4}ln R and R^{-4}, where R is the number-dependent dimensionless condensate radius in units of the oscillator length. The zero-order hydrodynamic density-fluctuation amplitudes are extended beyond the Thomas-Fermi radius through the boundary layer to provide a uniform description throughout all space. The first-order correction to the excitation frequencies is shown to be of order R^{-4}.Comment: 12 pages, 2 figures, revtex. Completely revised discussion of the boundary-layer corrections to collective excitations, and two new figures added. To appear in Phys. Rev. A (October, 1998

Vortex Waves in a Cloud of Bose Einstein - Condensed, Trapped Alkali - Metal Atoms

We consider the vortex state solution for a rotating cloud of trapped, Bose Einstein - condensed alkali atoms and study finite temperature effects. We find that thermally excited vortex waves can distort the vortex state significantly, even at the very low temperatures relevant to the experiments.Comment: to appear in Phys. Rev.

Symmetry-projected variational approach for ground and excited states of the two-dimensional Hubbard model

We present a symmetry-projected configuration mixing scheme to describe ground and excited states, with well defined quantum numbers, of the two-dimensional Hubbard model with nearestneighbor hopping and periodic boundary conditions. Results for the half-filled 2{\times}4, 4{\times}4, and 6{\times}6 lattices, as well as doped 4 {\times} 4 systems, compare well with available results, both exact and from other state-of-the-art approximations. We report spectral functions and density of states obtained from a well-controlled ansatz for the (Ne {\pm} 1)-electron system. Symmetry projected methods have been widely used for the many-body nuclear physics problem but have received little attention in the solid state community. Given their relatively low (mean-field) computational cost and the high quality of results here reported, we believe that they deserve further scrutiny

Vortex stabilization in a small rotating asymmetric Bose-Einstein condensate

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 $\gamma$ and the applied external rotation speed $\Omega$. For a given $\gamma$, the transition lines between no-vortex and vortex states are shifted toward higher $\Omega$ relative to those for the symmetric case. We also find a re-entrant behavior, where the number of vortex cores can decrease for large $\Omega$. 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 $\Omega$ 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 $\Omega$

Off-axis vortices in trapped Bose condensed gases: angular momentum and frequency splitting

We consider non centered vortices and their arrays in a cylindrically trapped Bose-Einstein condensate at zero temperature. We study the kinetic energy and the angular momentum per particle in the Thomas Fermi regime and their dependence on the distance of the vortices from the center of the trap. Using a perturbative approach with respect to the velocity-field of the vortices, we calculate to first order the frequency shift of the collective low-lying excitations due to the presence of an off-center vortex or a vortex array, and compare these results with predictions which would be obtained by the application of a simple sum-rule approach, previously found to be very successful for centered vortices. It turns out that the simple sum-rule approach fails for off-centered vortices.Comment: 11 pages, LaTeX, 3 figures. Perturbative approach adde

Topological Landau-Ginzburg Theory for Vortices in Superfluid 4^4He

We propose a new Landau-Ginzburg theory for arbitrarily shaped vortex strings in superfluid $^4$He. The theory contains a topological term and directly describes vortex dynamics. We introduce gauge fields in order to remove singularities from the Landau-Ginzburg order parameter of the superfluid, so that two kinds of gauge symmetries appear, making the continuity equation and conservation of the total vorticity manifest. The topological term gives rise to the Berry phase term in the vortex mechanical actions.Comment: LATEX, 9 page

Levy distribution in many-particle quantum systems

Levy distribution, previously used to describe complex behavior of classical systems, is shown to characterize that of quantum many-body systems. Using two complimentary approaches, the canonical and grand-canonical formalisms, we discovered that the momentum profile of a Tonks-Girardeau gas, -- a one-dimensional gas of $N$ impenetrable (hard-core) bosons, harmonically confined on a lattice at finite temperatures, obeys Levy distribution. Finally, we extend our analysis to different confinement setups and demonstrate that the tunable Levy distribution properly reproduces momentum profiles in experimentally accessible regions. Our finding allows for calibration of complex many-body quantum states by using a unique scaling exponent.Comment: 7 pages, 6 figures, results are generalized, new examples are adde

Theory of coherent Bragg spectroscopy of a trapped Bose-Einstein condensate

We present a detailed theoretical analysis of Bragg spectroscopy from a Bose-Einstein condensate at T=0K. We demonstrate that within the linear response regime, both a quantum field theory treatment and a meanfield Gross-Pitaevskii treatment lead to the same value for the mean evolution of the quasiparticle operators. The observable for Bragg spectroscopy experiments, which is the spectral response function of the momentum transferred to the condensate, can therefore be calculated in a meanfield formalism. We analyse the behaviour of this observable by carrying out numerical simulations in axially symmetric three-dimensional cases and in two dimensions. An approximate analytic expression for the observable is obtained and provides a means for identifying the relative importance of three broadening and shift mechanisms (meanfield, Doppler, and finite pulse duration) in different regimes. We show that the suppression of scattering at small values of q observed by Stamper-Kurn et al. [Phys. Rev. Lett. 83, 2876 (1999)] is accounted for by the meanfield treatment, and can be interpreted in terms of the interference of the u and v quasiparticle amplitudes. We also show that, contrary to the assumptions of previous analyses, there is no regime for trapped condensates for which the spectral response function and the dynamic structure factor are equivalent. Our numerical calculations can also be performed outside the linear response regime, and show that at large laser intensities a significant decrease in the shift of the spectral response function can occur due to depletion of the initial condensate.Comment: RevTeX4 format, 16 pages plus 7 eps figures; Update to published version: minors changes and an additional figure. (To appear in Phys. Rev. A

On the Response Function Technique for Calculating the Random-Phase Approximation Correlation Energy

We develop a scheme to exactly evaluate the correlation energy in the random-phase approximation, based on linear response theory. It is demonstrated that our formula is completely equivalent to a contour integral representation recently proposed by Donau et al. being numerically more efficient for realistic calculations. Numerical examples are presented for pairing correlations in rapidly rotating nuclei.Comment: 4 pages, 4 figure