Anderson localization of weakly interacting bosons Giovanni Modugno LENS and Dipartimento di Fisica, Università di Firenze 2° INSTANS Conference, September 11, 2008

The team G. Roati, L. Fallani, G. M., C. D’Errico C. Fort, M. Inguscio, M. Fattori, M. Modugno, M. Zaccanti

Disorder Disorder is ubiquitous in nature Superfluids in porous media Superconducting thin films Wave propagation in random media Still under investigation, despite of several decades of research; also applicative interests Wave propagation in engineered materials (photonic lattices)

Anderson localization No transport can occurr for D>J, due to the destructive interference of many possible paths

Anderson localization An essential feature is the absence of interactions between particles: Light in powders and disordered photonic crystals Van Albada & Lagendijk, Phys. Rev. Lett. 55, 2692 (1985) Wiersma, et al. , Nature 390, 671 (1997) Lahini, et al., Phys. Rev. Lett. 100, 013906 (2008). Microwaves Dalichaouch, et al, Nature 354, 53 (1991). Ultrasounds Weaver, Wave Motion 12, 129-142 (1990). Disordered electronic systems Akkermans & Montambaux Mesoscopic Physics of electrons and photons (Cambridge University Press,2006). Lee & Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985) Dynamical systems (kicked rotor) under study Chabè et al, arXiv:0709:4320.

What about ultracold atoms? Both Bose and Fermi gases available in traps Optical standing-waves realize perfect lattices with adjustable dimensionality

What about ultracold atoms? The atoms in a Bose-Einstein condensate are naturally interacting Bose glass in a disordered lattice? L. Fallani et al., Phys. Rev. Lett. 98, 130404 (2007) Mott insulator in an ordered lattice M. Greiner et al., Nature 415, 39 (2002)

What about ultracold atoms? Tuning of the atom-atom interaction (s-wave scattering length) is in some case possible through magnetically-tunable Feshbach resonances Moerdijk et al, Phys. Rev. A 51, 4852 (1995); Inouye et al, Nature 392, 151 (1998).

Our approach to Anderson localization A binary incommensurate lattice in 1D: quasi-disorder is easier to realize than random disorder, but shows the same phenomenology An ultracold Bose gas of 39K atoms: precise tuning of the interaction to zero Investigation of transport in space and of momentum distribution: direct observation of Anderson localization for matter-waves

localization transition at finite D = 2J Disorder models 1D Anderson model pure random localization for any D 1D Aubry-André model quasiperiodic localization transition at finite D = 2J incommensurate lattice speckle

Realization of the Aubry-Andrè model The first lattice sets the tunneling energy J The second lattice controls the site energy distribution D S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980); G. Harper, Proc. Phys. Soc. A 68, 674 (1965). Grempel, D. R., Fishman, S. & Prange, R. E., Phys. Rev. Lett. 49, 833-836 (1982).

Localization threshold in the A-A model localized states: extended states: Solution of the A-A model for the experimental parameters: b = 1030/860 = 1.1972..

Energy spectrum } 4J } 2D Localization takes place at energies well above the disorder

The weakly interacting Bose gas G. Roati et al. Phys. Rev. Lett. 99, 010403 (2007).

Experimental scheme Roati et al., Nature 453, 895 (2008)

Probing the transport properties The noninteracting BEC is initially confined in a harmonic trap and then left free to expand in the quasiperiodic lattice D/J Ballistic expansion: Ballistic expansion with reduced velocity Absence of diffusion:

Probing the transport properties D/J = 0 D/J = 1.8 D/J = 4.2 D/J = 7 0 ms ballistic expansion ballistic expansion at reduced speed time localization 750 ms

Scaling behavior: onset of localization Probing the transport properties Size of the condensate after 750 ms of expansion in the quasi-periodic lattice: Scaling behavior: onset of localization only depends on D/J

Probing the spatial distribution No disorder: wavefunction is delocalized on the whole system size ... … a harmonic trap is present: gaussian distribution With disorder: eigenstates are exponentially localized

Exponential localization gaussian exponential Fit of the density distribution with a generalized exponential function: D/J = 0 D/J = 1.8 D/J = 4.2 D/J = 7

Probing the momentum distribution narrow peaks in p(k) broad peaks in p(k) Long free expansion: xk

Scaling behavior with D/J Probing the momentum distribution experiment theory Density distribution after time-of-flight of the initial stationary state Scaling behavior with D/J Width of the central peak Visibility

Counting the localized states one localized state two localized states three localized states ~10 localized states

AL of matter-waves in random disorder (no lattice) Before expansion Semilog plot BEC (t=0) After expansion J. Billy et al., Nature 453, 891 (2008) (Bouyer-Aspect group, Orsay)

AL of photons in a quasi-disordered lattice Lahini. et al., arXiv (2008). (Group of Silberberg, Weizmann)

What’s next? Disorder with controllable interaction. The disordered Bose-Hubbard model: Higher dimensionality for disorder; ideal and superfluid Fermi gases; random disorder; …

Delocalization due to interaction: preliminary No interaction: few independent localized states With interaction: localized states get more extendend and lock in phase

Delocalization due to interaction: preliminary a=1.7 a0 U=0.15 D a=9.6 a0 U=0.8 D a=23 a0 U=2.0 D

Quantum gases experiments at LENS Three-body Efimov physics Quantum gases with dipolar interaction R q E Quantum interferometry, and fundamental forces close to surfaces