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
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