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 the hydrodynamic flow in a Bose-Einstein condensate stirred by a macroscopic object, a blue detuned laser beam, using nondestructive in situ phase contrast imaging. A critical velocity for the onset of a pressure gradient has been observed, and shown to be density dependent. The technique has been compared to a calorimetric method used previously to measure the heating induced by the motion of the laser beam.PACS 03.75.Fi, 67.40.Vs, 67.57.De Beginning with the London conjecture [1], BoseEinstein condensation has been considered crucial for the understanding of superfluidity. Since then, the weakly interacting Bose gas has served as an idealized model for a superfluid [2]. It has a phonon-like energy-momentum dispersion relation that does not allow for the generation of elementary excitations below a critical velocity, thus implying dissipationless flow at lower velocities. The onset of dissipation has been treated in the framework of the nonlinear Schrödinger equation [3][4][5][6][7] and shows an intriguing richness. Experiments on liquid helium could not test these theories, because superfluidity and dissipation are even more complex in this system due to the presence of strong interactions and surface effects [8].The creation of Bose-Einstein condensates in dilute gases has dramatically changed this situation, allowing for quantitative tests of microscopic theories using the tools and precision of atomic physics experiments [9]. A number of recent experiments have examined phenomenological features of superfluidity in gaseous BoseEinstein condensates. These include the observation of vortices [10,11], a non-classical moment of inertia [12], and suppression of collisions from microscopic impurities [13]. In previous work we found evidence for a critical velocity in a stirred condensate [14]. In this Letter we study the onset of dissipation with higher sensitivity using repeated in situ non-destructive imaging of the condensate. These images show the distortion of the density distribution around the moving object, thus directly probing the dynamics of the flow field that has been recently treated with different models [15][16][17][18][19][20]. The experimental setup was similar to the one used in our previous work [14]. Improvements in the evaporation strategy and decompression techniques allowed us to produce pure condensates with up to 5×10 7 sodium atoms, with densities ranging from 8.4×1013 to 3.5×10 14 cm −3 , corresponding to chemical potentials from 60 to 250 nK. We determined the Thomas-Fermi radius R z along the axial direction through in situ phase contrast imaging. The sound velocity at the center of the condensate was then evaluated through the relationship c s = 2πν z R z / √ 2, with ν z =20.1Hz being the axial trapping frequency. The macroscopic moving object was a 514 nm laser beam blue-detuned with respect to the sodium transitions, thereby creating an effective repulsive potential for the atoms. The beam is focused on the center of the condensate to a Gaussian 1/e 2 diameter o...

We studied the nucleation of vortices in a Bose-Einstein condensate stirred by a laser beam. The vortex cores were observed using time-of-flight absorption imaging. Depending on the stirrer size, either discrete resonances or a broad response was visible as the stir frequency was varied. Stirring beams small compared to the condensate size generated vortices below the critical rotation frequency for the nucleation of surface modes, suggesting a local mechanism of generation. In addition, we observed the centrifugal distortion of the condensate due to the rotating vortex lattice and found evidence for bent vortices.

By colliding two Bose-Einstein condensates, we have observed strong bosonic stimulation of the elastic scattering process. When a weak input beam was applied as a seed, it was amplified by a factor of 20. This large gain atomic four-wave mixing resulted in the generation of two macroscopically occupied pair-correlated atomic beams.

Phonons with wavevector q/h were optically imprinted into a Bose-Einstein condensate. Their momentum distribution was analyzed using Bragg spectroscopy with a high momentum transfer. The wavefunction of the phonons was shown to be a superposition of +q and −q free particle momentum states, in agreement with the Bogoliubov quasiparticle picture.PACS numbers: 05.30.Jp, 03.75.Fi, 67.40.Db The pioneering work of Bogoliubov in 1947 constitutes the first microscopic theory that attributes superfluidity to Bose-Einstein condensation [1]. As described by Einstein noninteracting bosons condense [2], but they are not superfluid. However, Bogoliubov showed that with weak interactions the condensate will exhibit superfluidity. Repulsive interactions change the elementary excitations at long wavelengths from free particles into phonons which, according to the Landau criterion, leads to superfluidity [3]. The main step in the non-perturbative treatment is the Bogoliubov transformationwhich expresses the creation and annihilation operators b † q and b q for Bogoliubov quasiparticles in terms of creation and annihilation operators a † q and a q for free particles in momentum states q. This transformation also plays a crucial role in general relativity, where it connects the particle operators in different reference frames [4]. In this paper we experimentally verify the Bogoliubov transformation by generating quasiparticles with momentum +q and observing that they are a superposition of +q and −q momentum states of free particles.Following a recent suggestion of Brunello et al.[5], we perform an experiment that combines the two regimes of Bragg spectroscopy, where the momentum imparted to the atoms is either smaller [6] or larger [7] than the sound velocity c times the mass m of the atoms. An optical lattice moving through the condensate at the sound velocity imprinted phonons with wavelengths equal to the spatial period of the lattice [6]. This was accomplished by intersecting two laser beams at a small angle and choosing their frequency difference to be equal to the phonon frequency. The momentum analysis of the phonon wavefunction was performed by a second Bragg pulse consisting of counterpropagating laser beams that * Current Address: School of Physics Georgia Institute of Technology, Atlanta, GA 30332 † Group website: http://cua.mit.edu/ketterle_group/ transferred a large momentum Q (two photon recoils) to atoms with initial momentum p. In the limit Q ≫ p, mc, the resonance frequency ν is equal to the kinetic energy transferred: hν = Q 2 /2m+Qp/m, where the second term is simply the Doppler shift [7,8,9]. The resulting frequency spectrum shows three peaks corresponding to the three momentum components (0, +q and −q) of a condensate with phonons (see Fig. 2) [5].The Bogoliubov spectrum for the energy of elementary excitations is [10]where c = √ 4πh 2 an/m, a is the scattering length, and n is the density. The amplitudes u q and v q in Eq. 1 are given byThe non-trivial aspect of the Bogoliubov transformation manifests itse...

We observe the formation of shock waves in a Bose-Einstein condensate containing a large number of sodium atoms. The shock wave is initiated with a repulsive blue-detuned light barrier, intersecting the BoseEinstein condensate, after which two shock fronts appear. We observe breaking of these waves when the size of these waves approaches the healing length of the condensate. At this time, the wave front splits into two parts and clear fringes appear. The experiment is modeled using an effective one-dimensional Gross-Pitaevskiilike equation and gives excellent quantitative agreement with the experiment, even though matter waves with wavelengths two orders of magnitude smaller than the healing length are present. In these experiments, no significant heating or particle loss is observed.

Phase contrast imaging is used to observe Bose-Einstein condensates at finite temperature in situ. The imaging technique is used to accurately derive the absolute phase shift of a probe laser beam due to both the condensate and the thermal cloud. The accuracy of the method is enhanced by using the periodicity of the intensity signal as a function of the accumulated phase. The measured density profiles can be described using a two-relevant-parameter fit, in which only the chemical potential and the temperature are to be determined. This allows us to directly compare the measured density profiles to different mean-field models in which the interaction between the condensed and the thermal atoms is taken into account to various degrees.