It has long been predicted that the scattering of ultracold atoms can be altered significantly through a so-called 'Feshbach resonance'. Two such resonances have now been observed in optically trapped Bose-Einstein condensates of sodium atoms by varying an external magnetic field. They gave rise to enhanced inelastic processes and a dispersive variation of the scattering length by a factor of over ten. These resonances open new possibilities for the study and manipulation of Bose-Einstein condensates.Bose-Einstein condensates of atomic gases offer new opportunities for studying quantum-degenerate fluids 1-5 . All the essential properties of Bose condensed systems-the formation and shape of the condensate, the nature of its collective excitations and statistical fluctuations, and the formation and dynamics of solitons and vortices-are determined by the strength of the atomic interactions. In contrast to the situation for superfluid helium, these interactions are weak, allowing the phenomena to be theoretically described from 'first principles'. Furthermore, in atomic gases the interactions can be altered, for instance by employing different species, changing the atomic density, or, as in the present work, merely by varying a magnetic field.At low temperatures, the interaction energy in a cloud of atoms is proportional to the density and a single atomic parameter, the scattering length a which depends on the quantum-mechanical phase shift in an elastic collision. It has been predicted that the scattering length can be modified by applying external magnetic 6-10 , optical 11,12 or radio-frequency 13 (r.f.) fields. Those modifications are only pronounced in a so-called ''Feshbach resonance'' 14 , when a quasibound molecular state has nearly zero energy and couples resonantly to the free state of the colliding atoms. In a timedependent picture, the two atoms are transferred to the quasibound state, 'stick' together and then return to an unbound state. Such a resonance strongly affects the scattering length (elastic channel), but also affects inelastic processes such as dipolar relaxation 6,7 and threebody recombination. Feshbach resonances have so far been studied at much higher energies 15 by varying the collision energy, but here we show that they can be 'tuned' to zero energy to be resonant for ultracold atoms. The different magnetic moments of the free and quasibound states allowed us to tune these resonances with magnetic fields, and as a result, minute changes in the magnetic field strongly affected the properties of a macroscopic system.Above and below a Feshbach resonance, the scattering length a covers the full continuum of positive and negative values. This should allow the realization of condensates over a wide range of interaction strengths. By setting a Ϸ 0, one can create a condensate with essentially non-interacting atoms, and by setting a Ͻ 0 one can make the system unstable and observe its collapse. Rapid tuning of an external magnetic field around a Feshbach resonance will lead to sudden changes of t...

Bose-Einstein condensates of dilute atomic gases, characterized by a macroscopic population of the quantum mechanical ground state, are a new, weakly interacting quantum fluid [1,2,3]. In most experiments condensates in a single weak field seeking state are magnetically trapped. These condensates can be described by a scalar order parameter similar to the spinless superfluid 4 He. Even though alkali atoms have angular momentum, the spin orientation is not a degree of freedom because spin flips lead to untrapped states and are therefore a loss process. In contrast, the recently realized optical trap for sodium condensates [4] confines atoms independently of their spin orientation. This opens the possibility to study spinor condensates which represent a system with a vector order parameter instead of a scalar. Here we report a study of the equilibrium state of spinor condensates in an optical trap. The freedom of spin orientation leads to the formation of spin domains in an external magnetic field. The structure of these domains are illustrated in spin domain diagrams. Combinations of both miscible and immiscible spin components were realized.A variety of new phenomena is predicted [5,6,7] for spinor condensates, such as spin textures, propagation of spin waves and coupling between superfluid flow and atomic spin. To date such effects could only be studied in superfluid 3 He, which can be described by Bose-Einstein condensation of Cooper pairs of quasi particles having both spin and orbital angular momentum [8]. Compared to the strongly interacting 3 He, the properties of weakly interacting BoseEinstein condensates of alkali gases can be calculated by mean field theories in a much more straightforward and simple way.Other systems which go beyond the description with a single scalar order parameter are condensates of two different hyperfine states of 87 Rb confined in magnetic traps. Recent experimental studies have explored the spatial separation of the two components [9,10] and their relative phase [11]. Several theoretical papers describe their structure [12,13,14,15,16,17,18] and their collective excitations [19,20,21,22].Compared to these two-component condensates, spinor condensates have several new features including the vector character of the order parameter and the changed role of spin relaxation collisions which allow for population exchange among hyperfine states without trap loss. In contrast, for 87 Rb experiments trap loss due to spin relaxation severely limits the lifetime.We consider an F = 1 spinor condensate subject to spin relaxation, in which two m F = 0 atoms can collide and produce an m F = +1 and an m F = −1 atom and vice versa. We investigate the distribution of hyperfine states and the spatial distribution in equilibrium assuming conservation of the total spin. The ground state spinor wave function is found by minimizing the free energy [5]where kinetic energy terms are neglected in the ThomasFermi approximation which is valid as long as the dimension of spin domains (typically 50 µm) is ...

Bose-Einstein condensates of sodium atoms have been confined in an optical dipole trap using a single focused infrared laser beam. This eliminates the restrictions of magnetic traps for further studies of atom lasers and Bose-Einstein condensates. More than five million condensed atoms were transferred into the optical trap. Densities of up to $3 \times 10^{15} cm^{-3}$ of Bose condensed atoms were obtained, allowing for a measurement of the three-body decay rate constant for sodium condensates as $K_3 = (1.1 \pm 0.3) \times 10^{-30} cm^6 s^{-1}$. At lower densities, the observed 1/e lifetime was more than 10 sec. Simultaneous confinement of Bose-Einstein condensates in several hyperfine states was demonstrated.Comment: 5 pages, 4 figure

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 > ...

Properties of a Bose-Einstein condensate were studied by stimulated, two-photon Bragg scattering. The high momentum and energy resolution of this method allowed a spectroscopic measurement of the mean-field energy and of the intrinsic momentum uncertainty of the condensate. The coherence length of the condensate was shown to be equal to its size. Bragg spectroscopy can be used to determine the dynamic structure factor over a wide range of energy and momentum transfers.[S0031-9007(99)

Rayleigh scattering off a Bose-Einstein condensate was studied. Exposing an elongated condensate to a single off-resonant laser beam resulted in the observation of highly directional scattering of light and atoms. This collective light scattering is caused by the coherent center-of-mass motion of the atoms in the condensate. A directional beam of recoiling atoms was built up by matter wave amplification.

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...

Sound propagation has been studied in a magnetically trapped dilute Bose-Einstein condensate. Localized excitations were induced by suddenly modifying the trapping potential using the optical dipole force of a focused laser beam. The resulting propagation of sound was observed using a novel technique, rapid sequencing of nondestructive phase-contrast images. The speed of sound was determined as a function of density and found to be consistent with Bogoliubov theory. This method may generally be used to observe high-lying modes and perhaps second sound.[S0031-9007(97)03665-X]