1. From localization to coherence: A tunable Bose-Einstein condensate in disordered potentials
Benjamin Deissler
LENS and Dipartimento di Fisica, Università di Firenze
June 03, 2010

2. Introduction
Disorder is ubiquitous in nature. Disorder, even if weak, tends to inhibit transport and can destroy superfluidity.
Superfluids in porous media
Granular and thin-film superconductors
Light propagation in random media
Graphene
Still under investigation, despite several decades of research; also important for applications (e.g. wave propagation in engineered materials)
Ultracold atoms: ideal model system
Reviews:
Aspect & Inguscio: Phys. Today, August 2009
Sanchez-Palencia & Lewenstein: Nature Phys. 6, (2010)

3. Adding interactions – schematic phase diagram
Bosons with repulsive interactions
localization through disorder
localization through interactions
cf. Roux et al., PRA 78, (2008) Deng et al., PRA 78, (2008)

4. Our approach to disorder & localization
A binary incommensurate lattice in 1D: quasi-disorder is easier to realize than random disorder, but shows the same phenomenology (“quasi-crystal”)
An ultracold Bose gas of 39K atoms: precise tuning of the interaction to zero
Fine tuning of the interactions permits the study of the competition between disorder and interactions
Investigation of momentum distribution: observation of localization and phase coherence properties
Investigation of transport properties

5. Realization of the Aubry-André modelJ 4J 2D J 4J 2D J 4J
4.4 lattice sites
The first lattice sets the tunneling energy J
The second lattice controls the site energy
distribution D
quasiperiodic potential:
localization transition at finite D = 2J
S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980); G. Harper, Proc. Phys. Soc. A 68, 674 (1965)

6. Experimental scheme
G. Roati et al., Phys. Rev. Lett. 99, (2007)

7. Probing the momentum distribution – non-interacting
experiment
theory
Density distribution after
ballistic expansion of the initial
stationary state
Scaling behavior with D/J
Measure
Width of the central peak
exponent of generalized exponential
G. Roati et al., Nature 453, 896 (2008)

8. Adding interactions… Anderson ground-state Anderson glass
Fragmented BEC
Extended BEC

9. Quasiperiodic lattice: energy spectrum
4J+2Δ
Energy spectrum: Appearance of “mini-bands”
lowest “mini-band” corresponds to lowest lying energy eigenstates
width of lowest energies 0.17D
mean separation of energies 0.05D
cf. M. Modugno: NJP 11, (2009)

10. Momentum distribution – observables
width of central peak
2. Fourier transform :
average local shape of the wavefunction
Fit to sum of two generalized exponential functions
exponent
3. Correlations:
Wiener-Khinchin theorem
gives us spatially averaged correlation function
fit to same function, get spatially averaged correlation g(4.4 lattice sites)

11. Probing the delocalization
exponent
momentum width
0.05D
correlations

12. Probing the phase coherence
Increase in correlations and decrease in the spread of phase  number of phases in the system decreases
0.05D
0.17D

13. Comparison experiment - theory
independent exponentially localized states
Experiment
Theory
formation of fragments
0.05D
single extended state
B. Deissler et al., Nature Physics 6, 354 (2010)

14. Expansion in a lattice
Prepare interacting system in optical trap + lattice, then release from trap and change interactions
radial confinement ≈ 50 Hz
initial size
many theoretical predictions:
Shepelyansky: PRL 70, 1787 (1993)
Shapiro: PRL 99, (2007)
Pikovsky & Shepelyansky: PRL 100, (2008)
Flach et al.: PRL 102, (2009)
Larcher et al.: PRA 80, (2009)

15. Expansion in a lattice Characterize expansion by exponent a:
a = 1: ballistic expansion
a = 0.5: diffusion
a < 0.5: sub-diffusion
fit curves to

16. Expansion in a lattice Expansion mechanisms:
resonances between states (interaction energy enables coupling of states within localization volume)
but: not only mechanism for our system  radial modes become excited over 10s
reduce interaction energy, but enable coupling between states
(cf. Aleiner, Altshuler & Shlyapnikov: arXiv: )
 combination of radial modes and interactions enable delocalization

17. Conclusion and Outlook
control of both disorder strength and interactions
observe crossover from Anderson glass to coherent, extended state by probing momentum distribution
interaction needed for delocalization proportional to the disorder strength
observe sub-diffusive expansion in quasi-periodic lattice with non-linearity
B. Deissler et al., Nature Physics 6, 354 (2010)
What’s next?
Measure of phase coherence for different length scales
What happens for attractive interactions?
Strongly correlated regime  1D, 2D, 3D systems
Random disorder
Fermions in disordered potentials
…and much more

18. The Team Massimo Inguscio Giovanni Modugno Experiment: Ben Deissler
Matteo Zaccanti
Giacomo Roati
Eleonora Lucioni
Luca Tanzi
Chiara D’Errico
Marco Fattori
Theory:
Michele Modugno

20. Counting localized states
one localized state
two localized states
three localized states
many localized states
controlled by playing with harmonic confinement and loading time
reaching the Anderson-localized ground state is very difficult, since Jeff  0
G. Roati et al., Nature 453, 896 (2008)

21. Adiabaticity? Preparation of system not always adiabatic
 in localized regime, populate several states where theory expects just one
see non-adiabaticity as transfer of energy into radial direction
0.05D

22. Theory density profiles
Eint AG
cutoff for evaluating different regimes
fBEC
BEC