1. Anderson localization in BECs
Graham Lochead
Journal Club 24/02/10

2. Anderson localization
Outline
Anderson localization
What is it?
Why is it important?
Recent experiments in BECs
Observation of localization in 1D
Future possibilities
Graham Lochead
Journal Club 24/02/10

3. Anderson localization
Ubiquitous in wave phenomenon
Phase coherence and interference
Exhibited in multiple systems
Conductivity
Magnetism
Superfluidity
EM and acoustic wave propagation
[P.W. Anderson, Phys. Rev. 109, 1492 (1958)]
Graham Lochead
Journal Club 24/02/10

4. Perfect crystal latticeV a
Electron-electron interactions are ignored
Bloch wavefunctions – electrons move ballistically
Delocalized (extended) electron states
Graham Lochead
Journal Club 24/02/10

5. Weakly disordered crystal lattice
Impurities cause electron to have a phase coherent mean free path
Wavefunctions still extended
Conductance decreased due to scattering
Graham Lochead
Journal Club 24/02/10

6. Weak localization Caused by multiple scattering events
Each scattering event changes phase of wave by a random amount
Only the original site has constructive interference
Most sites still have similar energies thus hopping occurs
Graham Lochead
Journal Club 24/02/10

7. Strongly disordered crystal lattice2Δ
Disorder energy is random from site to site
Mean free path at a minimum
[Ioffe and Regel, Prog. Semicond. 4, 237 (1960)]
Graham Lochead
Journal Club 24/02/10

8. Strong localization Hopping stops for critical value of disorder, Δ
Neighbouring electron energies too dissimilar – little wavefunction overlap
Electrons become localized – zero conductance
is the localization length
Transition from extended to localized states seen in all dimensions
[P.W. Anderson, Phys. Rev. 109, 1492 (1958)]
Graham Lochead
Journal Club 24/02/10

9. Non-periodic lattice Truly random potential
Hopping is suppressed due to poor energy and wavefunction overlap
Localization occurs due to coherent back scatter (same as weak localization)
Graham Lochead
Journal Club 24/02/10

10. Dimension effects of non-periodic lattice
All states are localized in one and two dimensions for small disorder
k is the wavevector of a particle in free space
Above two dimensions a phase transition (Anderson transition) occurs from extended states to localized ones for certain k
So-called mobiliity edge, kmob distinguishes between extended and localized states, k < kmob are exponentiallylocalized, k ~ lmfp
[Abrahams, E et. al. Phys. Rev. Lett. 42, 673–676 (1979) ]
Graham Lochead
Journal Club 24/02/10

11. Recent papers on cold atoms
[Nature 453, 895 (2008)]
[Nature 453, 891 (2008)]
Graham Lochead
Journal Club 24/02/10

12. Disorder can be controlled Interactions can be controlled
Why cold atoms?
Disorder can be controlled
Interactions can be controlled
Experimental observations easier
Quantum simulators of condensed matter
Graham Lochead
Journal Club 24/02/10

13. Roati et. al experimental setup
Condensed 39K in an optical trap
Applied a deep lattice perturbed by a second incommensurate lattice
Quantum degenerate gas
Thermal atoms
Magnetic coils
Lattice/
waveguide
Trapping potential
Graham Lochead
Journal Club 24/02/10

14. Lattice potential Interference of two counter-propagating lasers
of k1 leads to a periodic potential
Overlapping a second pair of counter-propagating lasers of k2 leads to a quasi--periodic potential
Graham Lochead
Journal Club 24/02/10

15. “Static scheme”
An interacting gas in a lattice can be modelled by the Hubbard Hamiltonian
Where J is the energy associated with hopping between sites, V is the depth of the potential, and U is the interaction potential
U is reduced via magnetic Feshbach resonance to ~10-5 J
V is recoil depth of lattice
Graham Lochead
Journal Club 24/02/10

16. Aubry-André model
Hubbard Hamiltonian is modified to the Aubry-André model
k2 = 1032 nm, k1 = 862 nm β = …
J and Δ can be controlled via the intensities of the two lattice lasers
Δ/J gives a measure of the disorder
[S. Aubry, G. André, Ann. Israel Phys. Soc. 3, 133 (1980)]
Graham Lochead
Journal Club 24/02/10

17. Localization! In situ absorption images of the condensate
Graham Lochead
Journal Club 24/02/10

18. Spatial widths Root mean squared size of the condensate at 750 μs
Dashed line is initial size of condensate
Graham Lochead
Journal Club 24/02/10

19. Spatial profile Spatial profile of the optical depth of the condensate
Δ/J = 1
Δ/J = 15
Tails of distribution fit with:
α = 2 corresponds to Gaussian
α = 1 corresponds to exponential
Graham Lochead
Journal Club 24/02/10

20. Momentum distribution
Δ/J = 0
Measured by inverting spatial distribution
Δ/J = 1.1
Δ/J = 7.2
Δ/J = 25
Graham Lochead
Journal Club 24/02/10

21. Interference of localized states
Δ/J = 10
One localized state
Two localized state
Three localized state
Several localized states formed from reducing size of condensate
States localized over spacing of approximately five sites
Graham Lochead
Journal Club 24/02/10

22. Billy et. al experimental setup
Condensed 87Rb in a waveguide
Applied a speckle potential to create random disorder
Graham Lochead
Journal Club 24/02/10

23. Speckle potentials Random phase imprinting – interference effect
Modulus and sign of V(r) can be controlled by laser intensity and detuning
Correlation length σR = 0.26 ± 0.03 μm
Graham Lochead
Journal Club 24/02/10

24. “Transport scheme” Gross-Pitaevskii equation
Expansion driven by interactions
Atoms given potential energy
Density decreases – interactions become negligable
Localization occurs
Graham Lochead
Journal Club 24/02/10

25. Localization again!
Tails of distribution fitted with exponentials again - localization
Graham Lochead
Journal Club 24/02/10

26. Temporal dynamics
Localization length becomes a maximum then flattens off – expansion stopped
Graham Lochead
Journal Club 24/02/10

27. Localization length
kmax is the maximum atom wavevector – controlled via condensate number/density
[Sanchez-Palencia, L. et. al Phys. Rev. Lett. 98, (2007)]
Graham Lochead
Journal Club 24/02/10

28. Beyond the mobility edge
Some atoms have more energy than can be localized
Power law dependence in wings
Measured valueof β = 1.95 ± 0.1
agrees with theory of β = 2
Graham Lochead
Journal Club 24/02/10

29. Expand both systems to 2D and 3D
Future directions
Expand both systems to 2D and 3D
Interplay of disorder and interactions
Simulate spin systems
Two-component condensates
Different “glass” phases (Bose, Fermi, Lifshitz)
[Damski, B. et. al Phys. Rev. Lett. 91, (2003)]
Graham Lochead
Journal Club 24/02/10

30. Anderson localization is where atoms become exponentially localized
Summary
Anderson localization is where atoms become exponentially localized
Cold atoms would be useful to act as quantum simulators of condensed matter systems
Localization seen in 1D in cold atoms in two different experiments
Graham Lochead
Journal Club 24/02/10