Transport of an Interacting Bose Gas in 1D Disordered Lattices

Chiara DErrico

CNR-INO, LENS and Dipartimento di Fisica, Universit di Firenze

15 International Conference on Transport in Interacting Disordered Systems, Sant Feliu , September 2013

Biological systemsThere is a growing interest in determining exactly how disorder affects the properties of quantum systems. Superfluids in porous mediaSuperconducting thin filmsLight propagation in random mediaGrapheneDisorder in quantum systems

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 gasesHannoverFlorenceParisUrbanaRice 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 .

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 systemsRapsch, Schollwoeck, Zwerger EPL 46 559 (1999), Interplay between disorder and interaction

In the tight binding limit: Aubry-Andr or Harper modelMetal-insulator transition at D=2J S. Aubry and G. Andr, 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 potentialA 1D quasiperiodic lattice

A 1D quasiperiodic latticeEnergy correlation function

Short, uniform localization length:Miniband structureA 1D quasiperiodic lattice

In the tight binding limit: Aubry-Andr or Harper modelMetal-insulator transition at D=2J Tuned on the Feshbach resonanceS. Aubry and G. Andr, 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 interaction1D system in a quasiperiodic potential

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

InteractionDisorderSuperfluidAnderson localizationGlass????Mott insulatorInterplay between disorder and interaction

Anomalous diffusion with disorder, noise and interactions

InteractionDisorderD/J=0D/J=2.5D/J=4timeAnomalous diffusion with disorder, noise and interactions

InteractionDisordertimeAnomalous 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).Anomalous diffusion with disorder, noise and interactionsEint=Un(x,t)

Levy flightsMany classes of linear disordered systemsBrownian motionJ-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

Coherent hopping between localized statessInstantaneous 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:

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

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

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

Noise-assisted diffusionOur noise: sine modulation of the secondary lattice with a random frequencyFrequencies are changed randomly with time step Tdnormal diffusion

Noise-assisted diffusiona 0.5

increasing noise amplitudeAlso 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 diffusionsC. DErrico et al., New J. Phys.15, 045007 (2013).Normal diffusion:General expectation:Our perturbative result for qp lattices:(works for both experiment and DNLSE)

Noise-assisted diffusionC. DErrico et al., New J. Phys.15, 045007 (2013).

Noise-assisted diffusionC. DErrico et al., New J. Phys.15, 045007 (2013).

Noise + interactions?Anderson localizationinteractions alonenoise alonenoise + interactions

Noise and interaction: generalized diffusion equationExperimentDNLSEnoise alone interactions alonenoise + interactions

D=0, U=J

nr=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):nmean ~ 2

D=0, U=JOptical lattices create an array of quasi one-dimensional systems:

t=0trap minimum is shiftedt=t*all fields are switched offTOF image (16.6 ms)System at equilibriumt*=0t*0Transport in 1D systemPolkovnikov et al. Phys. Rev. A 71, 063613 (2005); applied on Bose gases by DeMarco, Naegerl, Schneble.

Transport in the weakly interacting regime: clean systemDynamical 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:1303.1616Without disorder: D/J=0

Transport in the weakly interacting regime: clean systemWithout disorder: D/J=0pCAt p=pc we observe a sudden increase of the damping and of the width

Transport in the weakly interacting regime:clean systemWithout disorder: D/J=0J. 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

Transport in the weakly interacting regime:clean systemWithout disorder: D/J=0E. 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 pc and g on U suggest a quantum activation of phase slip

Fixed interaction energy: U/J=1.26pCpCTransport in the weakly interacting regime: with disorderThe damping rate is enhanced and the critical momentum is reduced by disorder

pCpCTransport in the weakly interacting regime: with disorderFixed interaction energy: U/J=1.26DCL. Tanzi et al., ArXiv:1307.4060, accepted by PRL

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

Conclusions & OutlookWe 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)

Massimo Inguscio Team The TeamEleonora LucioniLuca TanziLorenzo GoriAvinash KumarSaptarishi ChaudhuriC.D.

Giovanni ModugnoFor Noise-assisted transport: collaboration withF. Caruso B. Deissler (Ulm University) M. Moratti M. B. Plenio (Ulm University)

Thank you for the attention

*More specific discussion ***Simplify discsussion, better picture*Simplify discsussion, better picture*Add a sketch of the Feshbach*Put also a photo of the apparatus*Put also a photo of the apparatus*Put also a photo of the apparatus*Check for subdiffusionGarcia garcia?**Add DNLSE results*Add DNLSE results*Put also a photo of the apparatus*Put also a photo of the apparatus*Put also a photo of the apparatus**Put also a photo of the apparatus*Put also a photo of the apparatus**Put also a photo of the apparatus*Add initial superfluid preparation******