[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.
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