[14]. The atoms are confined in a magnetic trap that is produced by six permanent magnet cylinders. The magnets are arranged to produce a minimum at the trap center near which the field strength varies quadratically.

Quantum mechanics allows for many-particle wavefunctions that cannot be factorized into a product of single-particle wavefunctions, even when the constituent particles are entirely distinct. Such 'entangled' states explicitly demonstrate the non-local character of quantum theory, having potential applications in high-precision spectroscopy, quantum communication, cryptography and computation. In general, the more particles that can be entangled, the more clearly nonclassical effects are exhibited--and the more useful the states are for quantum applications. Here we implement a recently proposed entanglement technique to generate entangled states of two and four trapped ions. Coupling between the ions is provided through their collective motional degrees of freedom, but actual motional excitation is minimized. Entanglement is achieved using a single laser pulse, and the method can in principle be applied to any number of ions.

Local realism is the idea that objects have definite properties whether or not they are measured, and that measurements of these properties are not affected by events taking place sufficiently far away. Einstein, Podolsky and Rosen used these reasonable assumptions to conclude that quantum mechanics is incomplete. Starting in 1965, Bell and others constructed mathematical inequalities whereby experimental tests could distinguish between quantum mechanics and local realistic theories. Many experiments have since been done that are consistent with quantum mechanics and inconsistent with local realism. But these conclusions remain the subject of considerable interest and debate, and experiments are still being refined to overcome 'loopholes' that might allow a local realistic interpretation. Here we have measured correlations in the classical properties of massive entangled particles (9Be+ ions): these correlations violate a form of Bell's inequality. Our measured value of the appropriate Bell's 'signal' is 2.25 +/- 0.03, whereas a value of 2 is the maximum allowed by local realistic theories of nature. In contrast to previous measurements with massive particles, this violation of Bell's inequality was obtained by use of a complete set of measurements. Moreover, the high detection efficiency of our apparatus eliminates the so-called 'detection' loophole.

Bose-Einstein condensation of7 Li has been studied in a magnetically trapped gas. Because of the effectively attractive interactions between 7 Li atoms, many-body quantum theory predicts that the occupation number of the condensate is limited to about 1400 atoms. We observe the condensate number to be limited to a maximum value between 650 and 1300 atoms. The measurements were made using a versatile phase-contrast imaging technique. [S0031-9007(97) 7 Li atoms have a negative s-wave scattering length a, indicating that for a sufficiently cold and dilute gas the interatomic interactions are effectively attractive. Attractive interactions are thought to prevent BEC from occurring at all in a spatially homogeneous (i.e., untrapped) gas [3,4], and as recently as 1994, these interactions were believed to preclude BEC in a trap as well. Current theories predict that BEC can occur in a trap such as ours, but with no more than about 1400 condensate atoms [5][6][7][8][9][10][11]. Verification of this prediction would provide a sensitive test of many-body quantum theory. In our previous work [1], the condensate could not be directly observed, and the number of condensate atoms suggested by the measurements was overestimated. In this Letter we report quantitative measurements of the condensate number, which are consistent with the theoretical limit.The effects of interactions on a trapped condensate are studied using mean-field theory. For densities n such that na 3 ø 1, the mean-field interaction energy is given by U 4ph 2 an͞m, where m is the atomic mass. For 7 Li, a ͑214.5 6 0.4͒ Å [12]. Because a , 0, the interaction energy decreases with increasing n, so the condensate tends to collapse upon itself. When the confining potential is included in the theory, it is found that if U is sufficiently small compared to the trap energy-level spacing, the destabilizing influence of the interactions is balanced by the kinetic pressure of the gas, and a metastable condensate can form. This requirement for U leads to the prediction that the number of condensate atoms N 0 is limited. As the maximum N 0 is approached, the rate for inelastic collisions increases and the gas becomes progressively less stable with respect to thermal and quantum mechanical fluctuations [7][8][9][10].

We have investigated motional heating of laser-cooled 9 Be + ions held in radio-frequency (Paul) traps. We have measured heating rates in a variety of traps with different geometries, electrode materials, and characteristic sizes.The results show that heating is due to electric-field noise from the trap electrodes which exerts a stochastic fluctuating force on the ion. The scaling of the heating rate with trap size is much stronger than that expected from a spatially uniform noise source on the electrodes (such as Johnson noise from external circuits), indicating that a microscopic uncorrelated noise source on the electrodes (such as fluctuating patch-potential fields) is a more likely candidate for the source of heating.

The theory of quantum mechanics applies to closed systems. In such ideal situations, a single atom can, for example, exist simultaneously in a superposition of two different spatial locations. In contrast, real systems always interact with their environment, with the consequence that macroscopic quantum superpositions (as illustrated by the 'Schrodinger's cat' thought-experiment) are not observed. Moreover, macroscopic superpositions decay so quickly that even the dynamics of decoherence cannot be observed. However, mesoscopic systems offer the possibility of observing the decoherence of such quantum superpositions. Here we present measurements of the decoherence of superposed motional states of a single trapped atom. Decoherence is induced by coupling the atom to engineered reservoirs, in which the coupling and state of the environment are controllable. We perform three experiments, finding that the decoherence rate scales with the square of a quantity describing the amplitude of the superposition state.

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We study the prospects for observing superfluidity in a spin-polarized atomic gas of 6 Li atoms, using state-of-the-art interatomic potentials. We determine the spinodal line and show that a BCS transition to the superfluid state can indeed occur in the (meta)stable region of the phase diagram if the densities are sufficiently low. We also discuss the stability of the gas due to exchange and dipolar relaxation and conclude that the prospects for observing superfluidity in a magnetically trapped atomic 6 Li gas are particularly promising for magnetic bias fields larger than 10 T. PACS numbers: 03.75.Fi, 32.80.Pj, 42.50.Vk Ultracold atomic gases have received much attention in recent years, because of their novel properties. For instance, these gases are well suited for high-precision measurements of single-atom properties and for the observation of collisional and optical phenomena that reflect the (Bose or Fermi) statistics of the constituent particles. Moreover, a large variety of experimental techniques are available to manipulate the atomic gas samples by means of electromagnetic fields, which offers the exciting possibility to achieve the required conditions for quantum degeneracy and to study macroscopic quantum effects in their purest form.At present, most experimental attempts towards quantum degeneracy have been performed with bosonic gases and have been aimed at the achievement of Bose-Einstein condensation. In particular, most of the earlier experiments used atomic hydrogen [1,2]. These experiments provided crucial ingredients for the recent attempts with alkali vapors, for which the experimental advances towards the degeneracy regime were so rapid that Bose-Einstein condensation has actually been reported now for the isotopes 87 Rb [3] and 7 Li [4].In view of these exciting developments it seems timely to investigate theoretically also the properties of spinpolarized atomic 6 Li, since 6 Li is a stable fermionic isotope of lithium that can be trapped and cooled in much the same way as its bosonic counterpart. Therefore, magnetically trapped 6 Li promises to be an ideal system to study degeneracy effects in a weakly interacting Fermi gas, thus providing valuable complementary information on the workings of quantum mechanics at the macroscopic level. Moreover, using a combination of theoretical analysis and experimental results [5][6][7], accurate knowledge of the interparticle (singlet and triplet) potential curves of lithium have recently been obtained, which lead to the prediction of a large and negative s-wave scattering length a of 24.6 3 10 3 a 0 (a 0 is the Bohr radius) for a spin-polarized 6 Li gas. This is important for two reasons: First, the fact that the scattering length is negative implies that at the low temperatures of interest [L ¿ r V , where L ͑2ph 2 ͞mk B T͒ 1͞2 is the thermal de Broglie wavelength of the atoms and r V is the range of the interaction] the effective interaction between the lithium atoms is attractive, and we expect a BCS-like phase transition to a superfluid state ...