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From localization to coherence: A tunable Bose- Einstein condensate in disordered potentials Benjamin Deissler LENS and Dipartimento di Fisica, Università di Firenze June 03, 2010

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Introduction Superfluids in porous media GrapheneLight propagation in random media Disorder is ubiquitous in nature. Disorder, even if weak, tends to inhibit transport and can destroy superfluidity. Still under investigation, despite several decades of research; also important for applications (e.g. wave propagation in engineered materials) Ultracold atoms: ideal model system Granular and thin- film superconductors Reviews: Aspect & Inguscio: Phys. Today, August 2009 Sanchez-Palencia & Lewenstein: Nature Phys. 6, 87-95 (2010)

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Adding interactions – schematic phase diagram localization through disorder localization through interactions cf. Roux et al., PRA 78, 023628 (2008) Deng et al., PRA 78, 013625 (2008) Bosons with repulsive interactions

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Our approach to disorder & localization A binary incommensurate lattice in 1D: quasi-disorder is easier to realize than random disorder, but shows the same phenomenology (quasi-crystal) An ultracold Bose gas of 39 K atoms: precise tuning of the interaction to zero Fine tuning of the interactions permits the study of the competition between disorder and interactions Investigation of momentum distribution: observation of localization and phase coherence properties Investigation of transport properties

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Realization of the Aubry-André model The first lattice sets the tunneling energy J The second lattice controls the site energy distribution S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980); G. Harper, Proc. Phys. Soc. A 68, 674 (1965) J 4J J 2 J 2 quasiperiodic potential: localization transition at finite = 2J 4.4 lattice sites

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Experimental scheme G. Roati et al., Phys. Rev. Lett. 99, 010403 (2007)

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Probing the momentum distribution – non-interacting experimenttheory Density distribution after ballistic expansion of the initial stationary state Measure Width of the central peak exponent of generalized exponential Scaling behavior with /J G. Roati et al., Nature 453, 896 (2008)

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Adding interactions… Anderson ground-state Anderson glass Extended BEC Fragmented BEC

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Quasiperiodic lattice: energy spectrum 4J+2Δ cf. M. Modugno: NJP 11, 033023 (2009) Energy spectrum: Appearance of mini-bands lowest mini-band corresponds to lowest lying energy eigenstates width of lowest energies 0.17 mean separation of energies 0.05

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Momentum distribution – observables 2. Fourier transform : average local shape of the wavefunction Fit to sum of two generalized exponential functions exponent 3. Correlations: Wiener-Khinchin theorem gives us spatially averaged correlation function fit to same function, get spatially averaged correlation g(4.4 lattice sites) 1.Momentum distribution width of central peak

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Probing the delocalization momentum width exponent correlations 0.05

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Probing the phase coherence Increase in correlations and decrease in the spread of phase number of phases in the system decreases 0.05 0.17

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Comparison experiment - theory ExperimentTheory 0.05 independent exponentially localized states formation of fragments single extended state B. Deissler et al., Nature Physics 6, 354 (2010)

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Expansion in a lattice Prepare interacting system in optical trap + lattice, then release from trap and change interactions radial confinement 50 Hz many theoretical predictions: Shepelyansky: PRL 70, 1787 (1993) Shapiro: PRL 99, 060602 (2007) Pikovsky & Shepelyansky: PRL 100, 094101 (2008) Flach et al.: PRL 102, 024101 (2009) Larcher et al.: PRA 80, 053606 (2009) initial size

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Expansion in a lattice Characterize expansion by exponent : = 1: ballistic expansion = 0.5: diffusion < 0.5: sub-diffusion fit curves to

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Expansion in a lattice Expansion mechanisms: resonances between states (interaction energy enables coupling of states within localization volume) but: not only mechanism for our system radial modes become excited over 10s reduce interaction energy, but enable coupling between states (cf. Aleiner, Altshuler & Shlyapnikov: arXiv:0910.4534) combination of radial modes and interactions enable delocalization

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Conclusion and Outlook Whats next? Measure of phase coherence for different length scales What happens for attractive interactions? Strongly correlated regime 1D, 2D, 3D systems Random disorder Fermions in disordered potentials …and much more control of both disorder strength and interactions observe crossover from Anderson glass to coherent, extended state by probing momentum distribution interaction needed for delocalization proportional to the disorder strength observe sub-diffusive expansion in quasi-periodic lattice with non-linearity B. Deissler et al., Nature Physics 6, 354 (2010)

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The Team Massimo Inguscio Giovanni Modugno Experiment: Ben Deissler Matteo Zaccanti Giacomo Roati Eleonora Lucioni Luca Tanzi Chiara DErrico Marco Fattori Theory: Michele Modugno

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Counting localized states one localized state two localized states three localized states many localized states controlled by playing with harmonic confinement and loading time reaching the Anderson- localized ground state is very difficult, since J eff 0 G. Roati et al., Nature 453, 896 (2008)

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Adiabaticity? Preparation of system not always adiabatic in localized regime, populate several states where theory expects just one see non-adiabaticity as transfer of energy into radial direction 0.05

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Theory density profiles E int cutoff for evaluating different regimes AG fBEC BEC

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