1. Anderson localization of weakly interacting bosons
Giovanni Modugno
LENS and Dipartimento di Fisica, Università di Firenze
2° INSTANS Conference, September 11, 2008

2. The team
G. Roati, L. Fallani, G. M., C. D’Errico C. Fort, M. Inguscio, M. Fattori, M. Modugno, M. Zaccanti

3. Disorder Disorder is ubiquitous in nature
Superfluids in porous media
Superconducting thin films
Wave propagation in random media
Still under investigation, despite of several decades of research; also applicative interests
Wave propagation in engineered materials (photonic lattices)

4. Anderson localization
No transport can occurr for D>J, due to the destructive interference of many possible paths

5. Anderson localization
An essential feature is the absence of interactions between particles:
Light in powders and disordered photonic crystals
Van Albada & Lagendijk, Phys. Rev. Lett. 55, 2692 (1985)
Wiersma, et al. , Nature 390, 671 (1997)
Lahini, et al., Phys. Rev. Lett. 100, (2008).
Microwaves
Dalichaouch, et al, Nature 354, 53 (1991).
Ultrasounds
Weaver, Wave Motion 12, (1990).
Disordered electronic systems
Akkermans & Montambaux Mesoscopic Physics of electrons and photons (Cambridge University
Press,2006).
Lee & Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985)
Dynamical systems (kicked rotor) under study
Chabè et al, arXiv:0709:4320.

6. What about ultracold atoms?
Both Bose and Fermi gases available in traps
Optical standing-waves realize perfect lattices with adjustable dimensionality

7. What about ultracold atoms?
The atoms in a Bose-Einstein condensate are naturally interacting
Bose glass in a disordered lattice?
L. Fallani et al., Phys. Rev. Lett. 98, (2007)
Mott insulator in an ordered lattice
M. Greiner et al., Nature 415, 39 (2002)

8. What about ultracold atoms?
Tuning of the atom-atom interaction (s-wave scattering length) is in some case possible through magnetically-tunable Feshbach resonances
Moerdijk et al, Phys. Rev. A 51, 4852 (1995); Inouye et al, Nature 392, 151 (1998).

9. Our approach to Anderson localization
A binary incommensurate lattice in 1D: quasi-disorder is easier to realize than random disorder, but shows the same phenomenology
An ultracold Bose gas of 39K atoms: precise tuning of the interaction to zero
Investigation of transport in space and of momentum distribution: direct observation of Anderson localization for matter-waves

10. localization transition at finite D = 2J
Disorder models
1D Anderson model
pure random
localization for any D
1D Aubry-André model
quasiperiodic
localization transition at finite D = 2J
incommensurate lattice
speckle

11. Realization of the Aubry-Andrè model
The first lattice sets the tunneling energy J
The second lattice controls the site energy distribution D
S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980); G. Harper, Proc. Phys. Soc. A 68, 674 (1965). Grempel, D. R., Fishman, S. & Prange, R. E., Phys. Rev. Lett. 49, (1982).

12. Localization threshold in the A-A model
localized states:
extended states:
Solution of the A-A model for the experimental parameters:
b = 1030/860 =

13. Energy spectrum
} 4J
} 2D
Localization takes place at energies well above the disorder

14. The weakly interacting Bose gas
G. Roati et al. Phys. Rev. Lett. 99, (2007).

15. Experimental scheme
Roati et al., Nature 453, 895 (2008)

16. Probing the transport properties
The noninteracting BEC is initially confined in a harmonic trap and then left free to expand in the quasiperiodic lattice
D/J
Ballistic expansion:
Ballistic expansion with reduced
velocity
Absence of diffusion:

17. Probing the transport properties
D/J = 0
D/J = 1.8
D/J = 4.2
D/J = 7
0 ms
ballistic expansion
ballistic expansion
at reduced speed
time
localization
750 ms

18. Scaling behavior: onset of localization
Probing the transport properties
Size of the condensate after 750 ms of expansion in the quasi-periodic lattice:
Scaling behavior: onset of localization
only depends on D/J

19. Probing the spatial distribution
No disorder: wavefunction is delocalized on the whole system size ...
… a harmonic trap is present: gaussian distribution
With disorder: eigenstates are exponentially localized

20. Exponential localization
gaussian
exponential
Fit of the density distribution with a generalized exponential function:
D/J = 0
D/J = 1.8
D/J = 4.2
D/J = 7

21. Probing the momentum distribution
narrow peaks in p(k)
broad peaks in p(k)
Long free expansion: xk

22. Scaling behavior with D/J
Probing the momentum distribution
experiment
theory
Density distribution after time-of-flight of the initial stationary state
Scaling behavior with D/J
Width of the central peak
Visibility

23. Counting the localized states
one localized state
two localized states
three localized states
~10 localized states

24. AL of matter-waves in random disorder (no lattice)
Before expansion
Semilog plot
BEC (t=0)
After expansion
J. Billy et al., Nature 453, 891 (2008) (Bouyer-Aspect group, Orsay)

25. AL of photons in a quasi-disordered lattice
Lahini. et al., arXiv (2008). (Group of Silberberg, Weizmann)

26. What’s next?
Disorder with controllable interaction. The disordered Bose-Hubbard model:
Higher dimensionality for disorder; ideal and superfluid Fermi gases; random disorder; …

27. Delocalization due to interaction: preliminary
No interaction: few independent localized states
With interaction: localized states get more extendend and lock in phase

28. Delocalization due to interaction: preliminary
a=1.7 a0
U=0.15 D
a=9.6 a0
U=0.8 D
a=23 a0
U=2.0 D

29. Quantum gases experiments at LENS
Three-body Efimov physics
Quantum gases with dipolar interaction R q E
Quantum interferometry, and fundamental forces close to surfaces