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 observed Bose-Einstein condensation of molecules. When a spin mixture of fermionic 6Li atoms was evaporatively cooled in an optical dipole trap near a Feshbach resonance, the atomic gas was converted into 6Li2 molecules. Below 600 nK, a Bose-Einstein condensate of up to 900 000 molecules was identified by the sudden onset of a bimodal density distribution. This condensate realizes the limit of tightly bound fermion pairs in the crossover between BCS superfluidity and Bose-Einstein condensation.

Stimulated small-angle light scattering was used to measure the structure factor of a Bose-Einstein condensate in the phonon regime. The excitation strength for phonons was found to be significantly reduced from that of free particles, revealing the presence of correlated pair excitations and quantum depletion in the condensate. The Bragg resonance line strength and line shift agreed with predictions of a local density approximation.Spectroscopic studies have been used to assemble a complete understanding of the structure of atoms and simple molecules. Similarly, neutron and light scattering have long been used to probe the microscopic excitations of liquid helium [1][2][3][4], and can be regarded as the spectroscopy of a many-body quantum system. With the experimental realization of gaseous Bose-Einstein condensates, the spectroscopy of this new quantum fluid has begun.The character of excitations in a weakly-interacting Bose-Einstein condensed gas depends on the relation between the wavevector of the excitation q and the inverse healing length ξ −1 = √ 2mc s /h which is the wavevector related to the speed of Bogoliubov sound c s = µ/m where µ = 4πh 2 an 0 /m is the chemical potential, a the scattering length, n 0 the condensate density, and m the atomic mass. For large wavevectors (q ≫ ξ −1 ), the excitations are particlelike with a quadratic dispersion relation. Excitations in the free-particle regime have been accessed by near-resonant light scattering [5]. For small wavevectors (q ≪ ξ −1 ), the gas responds collectively and density perturbations propagate as phonons at the speed of Bogoliubov sound. Such quasi-particle excitations have been observed at wavelengths comparable to the size of the trapped gas [6] and thus were strongly influenced by boundary conditions.In this Letter, we describe the use of Bragg spectroscopy to probe excitations in the phonon regime. Using two laser beams intersecting at a small angle, excitations in a Bose-Einstein condensate were created with wavevector q < ξ −1 , thus optically "imprinting" phonons into the gas. The momentum imparted to the condensate in the form of quasi-particles was directly measured by a time-of-flight analysis. This study is the first to explore phonons with wavelengths much smaller than the size of the trapped sample, allowing a direct connection to the theory of the homogeneous Bose gas. By direct comparison, we show the excitation of phonons to be significantly weaker than that of free particles. This provides dramatic evidence for the presence of correlated momentum excitations in the many-body condensate wavefunction.In optical Bragg spectroscopy, an atomic sample is illumined by two laser beams with wavevectors k 1 and k 2 and a frequency difference ω which is much smaller than their overall detuning ∆ from an atomic resonance. The intersecting beams create a periodic, traveling intensity modulation I mod (r, t) = I cos(q · r − ωt) where q = k 1 − k 2 . Atoms exposed to this intensity modulation experience a potential due to the ac Stark effect o...

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We have produced a macroscopic quantum system in which a 6Li Fermi sea coexists with a large and stable 23Na Bose-Einstein condensate. This was accomplished using interspecies sympathetic cooling of fermionic 6Li in a thermal bath of bosonic 23Na. The system features rapid thermalization and long lifetimes.

Radio-frequency techniques were used to study ultracold fermions. We observed the absence of mean-field "clock" shifts, the dominant source of systematic error in current atomic clocks based on bosonic atoms. This absence is a direct consequence of fermionic antisymmetry. Resonance shifts proportional to interaction strengths were observed in a three-level system. However, in the strongly interacting regime, these shifts became very small, reflecting the quantum unitarity limit and many-body effects. This insight into an interacting Fermi gas is relevant for the quest to observe superfluidity in this system.

We have produced Bose-Einstein condensates in a ring-shaped magnetic waveguide. The fewmillimeter diameter non-zero bias ring is formed from a time-averaged quadrupole ring. Condensates which propagate around the ring make several revolutions within the time it takes for them to expand to fill the ring. The ring shape is ideally suited for studies of vorticity in a multiply-connected geometry and is promising as a rotation sensor. Scalar superfluids are characterized by a complex order parameter Ψ(r) which is uniquely defined throughout the fluid. This implies the irrotational motion of the fluid in the space where Ψ(r) = 0, leading to the Meissner effect in charged superfluids and to the Hess-Fairbank effect in neutral ones. Given this constraint, rotational motion of superfluids (or magnetic flux density in type-II superconductors) is accommodated by lines of quantized vorticity which disrupt the simple connectivity of the fluid. Multiple connectivity can also be imposed by the proper design of containers for the fluids. Such geometries are enlisted to translate phase variations of Ψ(r) into sensors of external fields. For example, a SQUID magnetometer makes use of a superconducting ring interrupted by Josephson junctions to allow continuous sensitivity to magnetic fields. A similar geometry was used in a superfluid 3 He gyroscope [1].Dilute gas superfluids enable novel forms of matterwave interferometry. Precise sensors of rotation, acceleration, and other sources of quantal phases [2, 3] using trapped or guided atoms have been envisioned. In particular, the sensitivity of atom-interferometric gyroscopes is proportional to the area enclosed by the closed loop around which atoms are guided [4]. Such considerations motivate the development of closed-loop atom waveguides which enclose a sizeable area.A number of multiply-connected trapping geometries for cold atoms have been discussed. Optical traps using high-order Gauss-Laguerre beams were proposed [5,6], and hollow light beams were used to trap non-degenerate atoms in an array of small-radius rings [7]. Large-scale magnetic "storage rings" were developed for cold neutrons [8] and discussed for atomic hydrogen [9]. More recently, closed-loop magnetic waveguides were demonstrated for laser cooled atoms [10,11]. Unfortunately, these guides are characterized by large variations in the potential energy along the waveguide and by high transmission losses at points where the magnetic field vanishes.In this Letter, we report the creation of a smooth, stable circular waveguide for ultracold atoms. A simple arrangement of coaxial electromagnetic coils was used to produce a static ring-shaped magnetic trap, which we call the quadrupole ring (Qring), in which strong transverse confinement is provided by a two-dimensional quadrupole field. Atoms trapped in the Qring experience large Majorana losses, but we can eliminate such losses with a timeorbiting ring trap (TORT) [12]. In this manner, stable circular waveguides with diameters ranging from 1.2 to 3 mm were produced. F...