Transport of an Interacting Bose Gas in 1

Transport of an Interacting Bose Gas in 1 D Disordered Lattices Chiara D’Errico CNR-INO, LENS and Dipartimento di Fisica, Università di Firenze 15° International Conference on Transport in Interacting Disordered Systems, Sant Feliu , September 2013

Disorder in quantum systems There is a growing interest in determining exactly how disorder affects the properties of quantum systems. Superconducting thin films Superfluids in porous media Graphene Biological systems Light propagation in random media

Anderson localization • Non-interacting particles hopping in a the lattice • With random on-site energy A critical value of disorder is able to localize the particle wavefunction • The eigenstates are spatially localized with exponentially decreasing tails. •

Disorder and quantum gases also Shlyapnikov, Burnett, Roth, Sanchez-Palencia, Giamarchi, Natterman, Garcia-Garcia …. Urbana Hannover Rice U. Paris Florence L. Sanchez-Palencia and M. Lewenstein, Nat. Phys. 6, 87 (2010); G. Modugno, Rep. Prog. Phys. 73, 102401 (2010).

Interplay between disorder and interaction Many-body problem to investigate the interplay between disorder & interaction Theoretical interest on the investigation of 1 D bosons at T=0, which is a simple prototype of disordered interacting systems Giamarchi & Schultz, PRB 37 325 (1988) Fisher et al PRB 40, 546 (1989), … Rapsch, Schollwoeck, Zwerger EPL 46 559 (1999), …

A 1 D quasiperiodic lattice 1 D system in a quasiperiodic potential 4 J 2 D In the tight binding limit: Aubry-Andrè or Harper model Metal-insulator transition at D=2 J S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980). L. Fallani et al. , PRL 98, 130404 (2007). M. Modugno, New J. Phys. 11, 033023 (2009).

A 1 D quasiperiodic lattice Energy correlation function

A 1 D quasiperiodic lattice Miniband structure Short, uniform localization length:

Interplay between disorder and interaction 1 D system in a quasiperiodic potential 4 J 2 D In the tight binding limit: Aubry-Andrè or Harper model Metal-insulator transition at D=2 J S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980). L. Fallani et al. , PRL 98, 130404 (2007). M. Modugno, New J. Phys. 11, 033023 (2009). Tuned on the Feshbach resonance

Interplay between disorder and interaction Potassium-39 BEC G. Roati, et al. Phys. Rev. Lett. 99, 010403 (2007).

Interplay between disorder and interaction Disorder Anderson localization Glass? ? ? ? Superfluid Mott insulator Interaction

Anomalous diffusion with disorder, noise and interactions

Disorder Anomalous diffusion with disorder, noise and interactions D/J=4 D/J=2. 5 D/J=0 Interaction time

Disorder Anomalous diffusion with disorder, noise and interactions time Interaction

Anomalous diffusion with disorder, noise and interactions E. Lucioni et al. , Phys. Rev. Lett. 106, 230403 (2011). E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013). Eint=Un(x, t)

Anomalous diffusion with disorder, noise and interactions Levy flights Brownian motion Many classes of linear disordered systems Localized interacting systems? J-P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990) D. L. Shepelyansky, Phys. Rev. Lett. 70, 1787 (1993) S. Flach, et al, Phys. Rev. Lett. 102, 024101 (2009)

Coherent hopping between localized states s Instantaneous diffusion coefficient: Width-dependent diffusion coefficient: Standard Diffusion Equation with Gaussian solution: Subdiffusive behaviour, i. e. decreasing diffusion coefficient: E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).

Nonlinear diffusion equation What about the evolution of the distribution n(x, t)? Nonlinear Diffusion Equation: B. Tuck, Journal of Physics D: Applied Physics 9, 1559 (1976)

Nonlinear diffusion equation What about the evolution of the distribution n(x, t)? Solution of NDE: E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).

Noise- and interaction-assisted transport Can we learn something abouth the complex properties of the energy transport in biological systems with our ultracold atom system? Ø Disorder Ø Noise Ø Interactions ? Collaboration with F. Caruso and M. Plenio, Ulm University Chin et al. , New J. Phys. 12 065002 (2010)

Noise-assisted diffusion Our noise: sine modulation of the secondary lattice with a random frequency Frequencies are changed randomly with time step Td normal diffusion

Noise-assisted diffusion a 0. 5 increasing noise amplitude Also observed in atomic ionization (Walther), kicked rotor (Raizen) and photonic lattices (Segev&Fishman): M. Arndt et al, Phys. Rev. Lett. 67, 2435 (1991); D. A. Steck, et al, Phys. Rev. E 62, 3461 (2000).

Noise-assisted diffusion s Normal diffusion: General expectation: Our perturbative result for qp lattices: (works for both experiment and DNLSE) C. D’Errico et al. , New J. Phys. 15, 045007 (2013).

Noise-assisted diffusion C. D’Errico et al. , New J. Phys. 15, 045007 (2013).

Noise + interactions? Anderson localization noise alone interactions alone noise + interactions

Noise and interaction: generalized diffusion equation Experiment noise alone interactions alone noise + interactions DNLSE

Experimental scheme and parameters for 1 D system Strong 2 D lattice (s=30) with weak 3 D harmonic trapping + weaker 1 D q. p. lattice (s=10) Optical lattices create an array of quasi one-dimensional systems: nr=50 k. Hz; J/h=100 Hz D=0, U=J Inhomogeneous filling factor (3 D Thomas-Fermi): nmean ~ 2 -2 k 1 k 0 2 k 1 D=0, U=J

Transport in 1 D system t*=0 System at equilibrium t=0 trap minimum is shifted t=t* all fields are switched off A. Polkovnikov et al. Phys. Rev. A 71, 063613 (2005); applied on Bose gases by De. Marco, Naegerl, Schneble. t*≠ 0 Dk TOF image (16. 6 ms)

Transport in the weakly interacting regime: clean system Without disorder: D/J=0 A. Smerzi et al. , Phys. Rev. Lett. 89, 170402 (2002) E. Altman et al. , Phys. Rev. Lett. 95, 020402 (2005) L. Fallani et al. , Phys. Rev. Lett. 93, 140406 (2004) J. Mun et al. , Phys. Rev. Lett. 99, 150604 (2007) I. Danshita, Ar. Xiv: 1303. 1616 Dynamical instability driven by quantum and thermal fluctuations.

Transport in the weakly interacting regime: clean system Without disorder: D/J=0 p. C At p=pc we observe a sudden increase of the damping and of the width

Transport in the weakly interacting regime: clean system Without disorder: D/J=0 J. Mun et al. , Phys. Rev. Lett. 99, 150604 (2007). L. Tanzi et al. , Ar. Xiv: 1307. 4060, accepted by PRL Also in 1 D the onset of the Mott regime can be detected from a vanishing of pc, as in 3 D

Transport in the weakly interacting regime: clean system Without disorder: D/J=0 The observed dependences of pc and g on U suggest a quantum activation of phase slip E. Altman et al. , PRL 95, 020402 (2005) A Polkovnikov et al. , PRA 71 063613 (2005) I. Danshita and A Polkovnikov, PRA 85, 023638 (2012) I. Danshita, PRL 111, 025303 (2013) L. Tanzi et al. , Ar. Xiv: 1307. 4060, accepted by PRL

Transport in the weakly interacting regime: with disorder p. C The damping rate is enhanced and the critical momentum is reduced by disorder Fixed interaction energy: U/J=1. 26

Transport in the weakly interacting regime: with disorder p. C DC Fixed interaction energy: U/J=1. 26 L. Tanzi et al. , Ar. Xiv: 1307. 4060, accepted by PRL

Transport in the weakly interacting regime: with disorder P. Lugan, et al. , Phys. Rev. Lett. 98, 170403 (2007); L. Fontanesi, et al. , Phys. Rev. A 81, 053603 (2010).

Conclusions & Outlook ü We have studied the diffusion of a localized disordered system, assisted by interaction and noise ü We have studied the momentum-dependent transport for a weakly interacting disordered Bose gas on the BG – SF transition Ø Study a strongly correlated, disordered Bose gas in 1 D: correlations, excitations, compressibility, and transport Ø Investigation of a quantum quench on a strongly correlated system and effect of the disorder on thermalization of a closed system Ø Exploration of the role of temperature on the many-body fluidinsulator transition at large T I. L. Aleiner, B. L. Altshuler, G. V. Shlyapnikov, Nat. Phys. 6, 900 (2010)

The Team Massimo Inguscio Giovanni Modugno Eleonora Lucioni Luca Tanzi Lorenzo Gori Avinash Kumar Saptarishi Chaudhuri C. D. For Noise-assisted transport: collaboration with F. Caruso B. Deissler (Ulm University) M. Moratti M. B. Plenio (Ulm University)

Thank you for the attention