No abstract

for trapping as 6 decreases is reflective of the incorporation of periodic components into the sequence of numbers generated.To summarize the motivation and principal conclusion of this Letter, we restate' that for values of b where numerically generated sequences~P Pear to be chaotic, it has not been settled whether those sequences "are truly chaotic, or whether, in fact, they are really periodic, but with exceedingly large periods and very long transients required to settle down. " On the one hand, Grossman and Thomae" have suggested that (only) the parameter value b =1 generates pure chaos [see the discussion following Eq. (31) of Ref. 5 and the correlations plotted in their Fig. 9]. On the other hand, for certa, in other values of b, numberical results of Lorenz (reported in Ref. 1) "strongly suggest that the sequences are truly chaotic. " The purpose of this communication was to use an independent and exact result from the statistical-mechanical theory of d =1 random walks to test the randomness of the parabolic map for parameter values where the existence of "true chaos" is still an open question.Our results strongly support the conclusions of Grossmann and Thomae.Correlations of linear polarizations of pairs of photons have been measured with time-varying analyzers. The analyzer in each leg of the apparatus is an acousto-optical switch followed by two linear polarizers. The switches operate at incommensurate frequencies near 50 MHz. Each analyzer amounts to a polarizer which jumps between two orientations in a time short compared with the photon transit time. The results are in good agreement with quantum mechanical predictions but violate Bell's inequalities by 5 standard deviations.

In 1958, P.W. Anderson predicted the exponential localization 1 of electronic wave functions in disordered crystals and the resulting absence of diffusion. It has been realized later that Anderson localization (AL) is ubiquitous in wave physics 2 as it originates from the interference between multiple scattering paths, and this has prompted an intense activity. Experimentally, localization has been reported in light waves 3,4,5,6,7 , microwaves 8,9 , sound waves 10 , and electron 11 gases but to our knowledge there is no direct observation of exponential spatial localization of matter-waves (electrons or others). Here, we report the observation of exponential localization of a Bose-Einstein condensate (BEC) released into a one-dimensional waveguide in the presence of a controlled disorder created by laser speckle 12 . We operate in a regime allowing AL: i) weak disorder such that localization results from many quantum reflections of small amplitude; ii) atomic density small enough that interactions are negligible. We image directly the atomic density profiles vs time, and find that weak disorder can lead to the stopping of the expansion and to the formation of a stationary exponentially localized wave function, a direct signature of AL. Fitting the exponential wings, we extract the localization length, and compare it to theoretical calculations. Moreover we show that, in our one-dimensional speckle potentials whose noise spectrum has a high spatial frequency cut-off, exponential localization occurs only when the de Broglie wavelengths of the atoms in the expanding BEC are larger than an effective mobility edge corresponding to that cut-off. In the opposite case, we find that the density profiles decay algebraically, as predicted in ref 13. The method presented here can be extended to localization of atomic quantum gases in higher dimensions, and with controlled interactions.