Transport of an Interacting Bose Gas in 1D Disordered Lattices

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

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 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).

5. Interplay between disorder and interaction 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 Giamarchi & Schultz, PRB 37 325 (1988) Fisher et al PRB 40, 546 (1989), … Rapsch, Schollwoeck, Zwerger EPL 46 559 (1999), …

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

7. A 1D quasiperiodic lattice Energy correlation function

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

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

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

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

12. Anomalous diffusion with disorder, noise and interactions

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

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

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

16. 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)

17. 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).

18. 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)

19. 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).

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

21. 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

22. 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).

23. 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).

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 alone noise + interactions

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

28. Experimental scheme and parameters for 1D system 2k1 k 0 -2k1 Strong 2D lattice (s=30) with weak 3D harmonic trapping + weaker 1D q.p. lattice (s=10) Optical lattices create an array of quasi one-dimensional systems: nr=50 kHz; J/h=100 Hz D=0, U=J D=0, U=J Inhomogeneous filling factor (3D Thomas-Fermi): nmean ~ 2

29. Transport in 1D system t*=0 System at equilibrium t=0 trap minimum is shifted t*≠0 t=t* all fields are switched off Dk TOF image (16.6 ms) • 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 Without disorder: D/J=0 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

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

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

33. 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., ArXiv:1307.4060, accepted by PRL

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

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

36. 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).

37. Conclusions & Outlook • Wehavestudied the diffusion of a localizeddisorderedsystem, assisted by interaction and noise • Wehavestudied the momentum-dependenttransport for a weaklyinteractingdisordered Bose gas on the BG – SF transition • Study a stronglycorrelated, disordered Bose gas in 1D: correlations, excitations, compressibility, and transport • Investigation of a quantum quench on a stronglycorrelatedsystem and effect of the disorder on the thermalization of a closedsystem • Exploration of the role of temperature on the many-body fluid-insulatortransitionat large T • I. L. Aleiner, B. L. Altshuler, G. V. Shlyapnikov, Nat. Phys. 6, 900 (2010)

38. The Team 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)

39. Thank you for the attention