Quantized vortices play a key role in superfluidity and superconductivity. We have observed the formation of highly ordered vortex lattices in a rotating Bose-condensed gas. These triangular lattices contained over 100 vortices with lifetimes of several seconds. Individual vortices persisted up to 40 seconds. The lattices could be generated over a wide range of rotation frequencies and trap geometries, shedding light on the formation process. Our observation of dislocations, irregular structure, and dynamics indicates that gaseous Bose-Einstein condensates may be a model system for the study of vortex matter.

Bose-Einstein condensates of sodium atoms have been prepared in optical and magnetic traps in which the energy-level spacing in one or two dimensions exceeds the interaction energy between atoms, realizing condensates of lower dimensionality. The cross-over into two-dimensional and onedimensional condensates was observed by a change in aspect ratio and saturation of the release energy when the number of trapped atoms was reduced.New physics can be explored when the hierarchy of physical parameters changes. This is evident in dilute gases, where the onset of Bose-Einstein condensation occurs when the thermal deBroglie wavelength becomes longer than the average distance between atoms. Dilutegas condensates of density n in axially-symmetric traps are characterized by four length scales: Their radius R ⊥ , their axial half-length R z , the scattering length a which parameterizes the strength of the two-body interaction, and the healing length ξ = (4πna) −1/2 . In almost all experiments on Bose-Einstein condensates, both the radius and length are determined by the interaction between the atoms and thus, R ⊥ , R z ≫ ξ ≫ a. In this regime, a BEC is three-dimensional and is well-described by the socalled Thomas-Fermi approximation [1]. A qualitatively different behavior of a BEC is expected when the healing length is larger than either R ⊥ or R z since then the condensate becomes restricted to one or two dimensions, respectively. New phenomena that may be observed in this regime are for example quasi-condensates [2-4] and a Tonk's gas of impenetrable bosons [4][5][6].In this Letter, we report the experimental realization of cigar-shaped one-dimensional condensates with R z > ξ > R ⊥ and disk-shaped two-dimensional condensates with R ⊥ > ξ > R z . The cross-over from 3D to 1D or 2D was explored by reducing the number of atoms in condensates which were trapped in highly elongated magnetic traps (1D) and disk-shaped optical traps (2D) and measuring the release energy. In harmonic traps, lower dimensionality is reached when µ 3D = 4π 2 a n/m < ω t . Here, ω t is the trapping frequency in the tightly confining dimension(s) and µ 3D is the interaction energy of a weakly interacting BEC, which in 3D corresponds to the chemical potential. Other experiments in which the interaction energy was comparable to the level spacing of the confining potential include condensates in onedimensional optical lattices [8] and the cross-over to an ideal-gas (zero-D) condensate [7], both at relatively low numbers of condensate atoms.Naturally, the number of interacting atoms in a lowerdimensional condensate is limited. The peak interaction energy of a 3D condensate of N atoms with mass m is given by1/2 are the oscillator lengths of the harmonic potential. The cross-over to 1D and 2D, defined by µ 3D = ω t or equivalently ξ = l t occurs if the number of condensate atoms becomeswhere we have used the scattering length (a = 2.75 nm) and mass of 23 Na atoms to derive the numerical factor. Our traps feature extreme aspect ratios resulting in N 1D > ...

We have studied dissipation in a Bose-Einstein condensed gas by moving a blue detuned laser beam through the condensate at different velocities. Strong heating was observed only above a critical velocity.PACS 03.75.Fi, 67.40.Vs,67.57.De Macroscopic quantum coherence and collective excitations are key features in our understanding of the phenomenon of superfluidity. The superfluid velocity is proportional to the gradient of the phase of a macroscopic wavefunction. Collective excitations determine a critical velocity below which the flow is dissipationless. This velocity is given by Landau's criterion [1],where ε is the energy of an excitation with momentum p. [4]. Previous work has explored some aspects related to superfluidity such as the macroscopic phase [5] and the phonon nature of low-lying collective excitations [4,6]. In this Letter we report on the measurement of a critical velocity for the excitation of a trapped BoseEinstein condensate. In analogy with the well known argument by Landau and the vibrating wire experiments in superfluid helium [7], we study dissipation when an object is moved through the fluid. Instead of a massive macroscopic object we used a blue detuned laser beam which repels atoms from its focus to create a moving boundary condition.The experiment was conducted in a new apparatus for the production of Bose-Einstein condensates of sodium atoms. The cooling procedure is similar to previous work [8]-the new features have been described elsewhere [9]. Briefly, laser cooled atoms were transferred into a magnetic trap in the Ioffe-Pritchard configuration and further cooled by rf evaporative cooling for 20 seconds, resulting in condensates of between 3 and 12 ×10 6 atoms. After the condensate was formed, we reduced the radial trapping frequency to obtain condensates which were considerably wider than the laser beam used for stirring. This decompression was not perfectly adiabatic, and heated the cloud to a final condensate fraction of about 60%. The final trapping frequencies were ν r = 65 Hz in the radial and ν z = 18 Hz in the axial direction. The resulting condensate was cigar-shaped with Thomas-Fermi diameters of 45 and 150 µm in the radial and axial directions, respectively. The final chemical potential, transition temperature T c and peak density n 0 of the condensate were 110 nK, 510 nK and 1.5 × 10 14 cm −3 , respectively.The laser beam for stirring the condensate had a wavelength of 514 nm and was focused to a Gaussian 1/e 2 beam diameter of 2w = 13µm. The repulsive optical dipole force expelled the atoms from the region of highest laser intensity. A laser power of 400 µW created a 700 nK barrier resulting in a cylindrical hole ∼ 13µm in diameter within the condensate. The laser barrier created a soft boundary, since the Gaussian beam waist was more than 10 times wider than the healing length ξ = (8πan 0 ) −1/2 = 0.3µm, a being the two-body scattering length. The laser was focused on the center of the cloud. Using an acousto-optic deflector, it was scanned back and forth along t...