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Anderson localization in BECs. Outline. Anderson localization What is it? Why is it important? Recent experiments in BECs Observation of localization in 1D Future possibilities. Anderson localization. Ubiquitous in wave phenomenon Phase coherence and interference - PowerPoint PPT Presentation
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Graham LocheadJournal Club 24/02/10
Anderson localization in BECs
Graham LocheadJournal 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 LocheadJournal 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 LocheadJournal Club 24/02/10
Perfect crystal lattice
a
Bloch wavefunctions – electrons move ballistically
Electron-electron interactions are ignored
Delocalized (extended) electron states
V
Graham LocheadJournal Club 24/02/10
Weakly disordered crystal lattice
Impurities cause electron to have a phase coherent mean free path
Wavefunctions still extended
Conductance decreased due to scattering
mfpl
Graham LocheadJournal 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 LocheadJournal Club 24/02/10
Strongly disordered crystal lattice
Mean free path at a minimum
[Ioffe and Regel, Prog. Semicond. 4, 237 (1960)]
2Δ
Disorder energy is random from site to site
almfp
Graham LocheadJournal Club 24/02/10
Strong localization
locLr exp
Electrons become localized – zero conductance
Neighbouring electron energies too dissimilar – little wavefunction overlap
[P.W. Anderson, Phys. Rev. 109, 1492 (1958)]
Hopping stops for critical value of disorder, Δ
is the localization length
Transition from extended to localized states seen in all dimensions
locL
Graham LocheadJournal 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 LocheadJournal Club 24/02/10
Dimension effects of non-periodic lattice
mfpDloc lL 1
All states are localized in one and two dimensions for small disorder
[Abrahams, E et. al. Phys. Rev. Lett. 42, 673–676 (1979) ]
Above two dimensions a phase transition (Anderson transition) occurs from extended states to localized ones for certain k
k is the wavevector of a particle in free space
So-called mobiliity edge, kmob distinguishes between extended and localized states, k < kmob are exponentiallylocalized, k ~ lmfp
mfpmfp
Dloc kllL
2exp2
Graham LocheadJournal Club 24/02/10
Recent papers on cold atoms
[Nature 453, 895 (2008)]
[Nature 453, 891 (2008)]
Graham LocheadJournal Club 24/02/10
Why cold atoms?
• Disorder can be controlled
• Interactions can be controlled
• Experimental observations easier
• Quantum simulators of condensed matter
Graham LocheadJournal 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 LocheadJournal Club 24/02/10
Lattice potential
xkVVlattice 12
1 2sin
Interference of two counter-propagating lasersof k1 leads to a periodic potential
Overlapping a second pair of counter-propagating lasers of k2 leads to a quasi--periodic potential
xkVVlattice 12
1 2sin xkV 22
2 2sin
Graham LocheadJournal Club 24/02/10
“Static scheme”
jjjjj
jjj
jllj
j aaaaUaaVchaaJH ,','†,'
†,
,',',
†
,
† ˆˆˆˆ2
1ˆˆ.).ˆˆ(ˆ
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 LocheadJournal Club 24/02/10
Aubry-André model
jjjlj
lj aajchaaJH ˆˆ2cos.).ˆˆ(ˆ †
,
†
Hubbard Hamiltonian is modified to the Aubry-André model
1
2
k
k
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)]
k2 = 1032 nm, k1 = 862 nm β = 1.1972…
Graham LocheadJournal Club 24/02/10
Localization!
In situ absorption images of the condensate
Graham LocheadJournal Club 24/02/10
Spatial widths
Root mean squared size of the condensate at 750 μs
Dashed line is initial size of condensate
Graham LocheadJournal Club 24/02/10
Spatial profile
Spatial profile of the opticaldepth of the condensate
a) Δ/J = 1b) Δ/J = 15
LxxAxf /)(exp)( 0
Tails of distribution fit with:
α = 2 corresponds to Gaussianα = 1 corresponds to exponential
Graham LocheadJournal Club 24/02/10
Momentum distribution
Measured by inverting spatial distribution
Δ/J = 0
Δ/J = 1.1
Δ/J = 7.2
Δ/J = 25
)()2(
)()2(
11
11
kPkP
kPkPVisibility
Graham LocheadJournal Club 24/02/10
Interference of localized states
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
Δ/J = 10
Graham LocheadJournal Club 24/02/10
Billy et. al experimental setup
• Condensed 87Rb in a waveguide
• Applied a speckle potential to create random disorder
Graham LocheadJournal Club 24/02/10
Speckle potentials
Random phase imprinting – interference effect
22)()( EEV rr
Modulus and sign of V(r) can be controlled by laser intensity and detuning
Correlation length σR = 0.26 ± 0.03 μm
Graham LocheadJournal Club 24/02/10
“Transport scheme”
222
)(2
gVmt
i
r
)(rV
Gross-Pitaevskii equation
• Expansion driven by interactions
• Atoms given potential energy
• Density decreases – interactions become negligable
• Localization occurs )(rV
Graham LocheadJournal Club 24/02/10
Localization again!
Tails of distribution fitted with exponentials again - localization
Graham LocheadJournal Club 24/02/10
Temporal dynamics
Localization length becomes a maximum then flattens off – expansion stopped
Graham LocheadJournal Club 24/02/10
Localization length
[Sanchez-Palencia, L. et. al Phys. Rev. Lett. 98, 210401 (2007)]
RRloc kVm
kL
max22
2max
4
1)(
2
r
kmax is the maximum atom wavevector – controlled via condensate number/density
1max Rk
Graham LocheadJournal Club 24/02/10
Beyond the mobility edge
1max Rk Some atoms have more energy than can be localized
1max Rk
Power law dependence in wings
agrees with theory of β = 2
z1
Measured valueof β = 1.95 ± 0.1
Graham LocheadJournal 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 LocheadJournal 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