Anderson localization in BECs

Report
Anderson localization
in BECs
Graham Lochead
Journal Club 24/02/10
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
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
Perfect crystal lattice
V
a
Electron-electron interactions are ignored
Bloch wavefunctions – electrons move ballistically
Delocalized (extended) electron states
Graham Lochead
Journal Club 24/02/10
Weakly disordered crystal lattice
l
Impurities cause electron to have a phase coherent mean free path mfp
Wavefunctions still extended
Conductance decreased due to scattering
Graham Lochead
Journal Club 24/02/10
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
Strongly disordered crystal lattice
2Δ
Disorder energy is random from site to site
Mean free path at a minimum
lmfp  a
[Ioffe and Regel, Prog. Semicond. 4, 237 (1960)]
Graham Lochead
Journal Club 24/02/10
Strong localization
Hopping stops for critical value of disorder, Δ
Neighbouring electron energies too dissimilar – little wavefunction overlap
Electrons become localized – zero conductance
  exp r Lloc 
Lloc 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
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
Dimension effects of non-periodic lattice
All states are localized in one and two dimensions for small disorder
L  lmfp
1D
loc
2D
loc
L


 lmfp exp klmfp 
2

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
Recent papers on cold atoms
[Nature 453, 895 (2008)]
[Nature 453, 891 (2008)]
Graham Lochead
Journal Club 24/02/10
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
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
Trapping
potential
Magnetic
coils
Lattice/
waveguide
Graham Lochead
Journal Club 24/02/10
Lattice potential
Interference of two counter-propagating lasers
of k1 leads to a periodic potential
Vlattice  V1 sin 2 2k1x
Overlapping a second pair of counter-propagating
lasers of k2 leads to a quasi--periodic potential
Vlattice  V1 sin 2 2k1x
 V2 sin 2 2k2 x
Graham Lochead
Journal Club 24/02/10
“Static scheme”
An interacting gas in a lattice can be modelled by the
Hubbard Hamiltonian
1
†
†
ˆ
H   J  (aˆ j aˆl  h.c.)  V j aˆ j aˆ j  U , ' aˆ† , j aˆ† ', j aˆ ', j aˆ ', j
2  , ', j
j ,l
j
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
Aubry-André model
Hubbard Hamiltonian is modified to the Aubry-André model
Hˆ   J  (aˆ †j aˆl  h.c.)    cos2j   aˆ †j aˆ j
j ,l
j
k2

k1
k2 = 1032 nm, k1 = 862 nm
β = 1.1972…
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
Localization!
In situ absorption images of the condensate
Graham Lochead
Journal Club 24/02/10
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
Spatial profile
Spatial profile of the optical
depth of the condensate
a) Δ/J = 1
b) Δ/J = 15
Tails of distribution fit with:

f ( x)  A exp ( x  x0 ) / L


α = 2 corresponds to Gaussian
α = 1 corresponds to exponential
Graham Lochead
Journal Club 24/02/10
Momentum distribution
Δ/J = 0
Δ/J = 1.1
Δ/J = 7.2
Measured by
inverting spatial
distribution
Δ/J = 25

P(2k1 )  P(k1 ) 
Visibility 
P(2k1 )  P(k1 )
Graham Lochead
Journal Club 24/02/10
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
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
Speckle potentials
Random phase imprinting – interference effect

V (r)   E (r)  E
2
2

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
“Transport scheme”
Gross-Pitaevskii equation

 2 2
2
i

  V (r)  g  
t
2m
• Expansion driven by interactions
V (r )  
• Atoms given potential energy
• Density decreases – interactions become negligable
• Localization occurs
V (r )  
Graham Lochead
Journal Club 24/02/10
Localization again!
Tails of distribution fitted with exponentials again - localization
Graham Lochead
Journal Club 24/02/10
Temporal dynamics
Localization length becomes a maximum
then flattens off – expansion stopped
Graham Lochead
Journal Club 24/02/10
Localization length
2
2 4 kmax
Lloc 
m2V 2 (r) R 1  kmax R 
kmax is the maximum atom
wavevector – controlled via
condensate number/density
kmax R  1
[Sanchez-Palencia, L. et. al Phys. Rev. Lett. 98, 210401 (2007)]
Graham Lochead
Journal Club 24/02/10
Beyond the mobility edge
kmax R  1
kmax R  1
Some atoms have more energy
than can be localized
Power law dependence in wings
1 z

Measured valueof β = 1.95 ± 0.1
agrees with theory of β = 2
Graham Lochead
Journal Club 24/02/10
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, 080403 (2003)]
Graham Lochead
Journal Club 24/02/10
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

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