1. Transport of an Interacting Bose Gas in 1D 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

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

3. 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.

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

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

6. 4J4J 22 In the tight binding limit: Aubry-Andrè or Harper model Metal-insulator transition at  =2J 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). 1D system in a quasiperiodic potential A 1D quasiperiodic lattice

7. Energy correlation function

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

9. 4J4J 22 In the tight binding limit: Aubry-Andrè or Harper model Metal-insulator transition at  =2J Tuned on the Feshbach resonance 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). Interplay between disorder and interaction 1D system in a quasiperiodic potential

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

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

12. Anomalous diffusion with disorder, noise and interactions

13. Interaction Disorder  /J=0  /J=2.5  /J=4 time Anomalous diffusion with disorder, noise and interactions

14. Interaction Disorder time Anomalous diffusion with disorder, noise and interactions

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

16. Levy flights Many classes of linear disordered systems Brownian motion 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) Localized interacting systems? Anomalous diffusion with disorder, noise and interactions

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

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

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

20. 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?  Disorder  Noise  Interactions ? Chin et al., New J. Phys. 12 065002 (2010) Collaboration with F. Caruso and M. Plenio, Ulm University

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

22. Noise-assisted diffusion   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).

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

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

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

26. Noise + interactions? Anderson localizationinteractions alone noise alonenoise + interactions

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

28. k 2k12k1 -2k 1 0  =0, U=J r =50 kHz; J/h=100 Hz Experimental scheme and parameters for 1D system Strong 2D lattice (s=30) with weak 3D harmonic trapping + weaker 1D q.p. lattice (s=10) Inhomogeneous filling factor (3D Thomas-Fermi): n mean ~ 2  =0, U=J Optical lattices create an array of quasi one-dimensional systems:

29. t=0 trap minimum is shifted t=t* all fields are switched off TOF image (16.6 ms) System at equilibrium t*=0 t*≠0 kk Transport in 1D system A.Polkovnikov et al. Phys. Rev. A 71, 063613 (2005); applied on Bose gases by DeMarco, Naegerl, Schneble.

30. Transport in the weakly interacting regime: clean system Dynamical instability driven by quantum and thermal fluctuations. 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, ArXiv:1303.1616 Without disorder:  /J=0

31. Transport in the weakly interacting regime: clean system Without disorder:  /J=0 pCpC At p=p c we observe a sudden increase of the damping and of the width

32. Transport in the weakly interacting regime:clean system Without disorder:  /J=0 J. Mun et al., Phys. Rev. Lett. 99, 150604 (2007). L. Tanzi et al., ArXiv:1307.4060, accepted by PRL Also in 1D the onset of the Mott regime can be detected from a vanishing of pc, as in 3D

33. Transport in the weakly interacting regime:clean system Without disorder:  /J=0 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., ArXiv:1307.4060, accepted by PRL The observed dependences of p c and  on U suggest a quantum activation of phase slip

34. Fixed interaction energy: U/J=1.26 pCpC pCpC Transport in the weakly interacting regime: with disorder The damping rate is enhanced and the critical momentum is reduced by disorder

35. pCpC pCpC Transport in the weakly interacting regime: with disorder Fixed interaction energy: U/J=1.26 CC CC L. Tanzi et al., ArXiv:1307.4060, accepted by PRL

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

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

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

39. Thank you for the attention