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

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

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

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 39K 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

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

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

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

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

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

Momentum distribution – observables width of central peak 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)

Probing the delocalization exponent momentum width 0.05D correlations

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

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

Expansion in a lattice Prepare interacting system in optical trap + lattice, then release from trap and change interactions radial confinement ≈ 50 Hz initial size 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)

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

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

Conclusion and Outlook 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) What’s 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

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

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 Jeff  0 G. Roati et al., Nature 453, 896 (2008)

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 0.05D

Theory density profiles Eint AG cutoff for evaluating different regimes fBEC BEC