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Graham LocheadJournal Club 24/02/10

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

• Exhibited in multiple systems– Conductivity– Magnetism– Superfluidity– EM and acoustic wave propagation

[P.W. Anderson, Phys. Rev. 109, 1492 (1958)]


Perfect crystal lattice

a

Bloch wavefunctions – electrons move ballistically

Electron-electron interactions are ignored

Delocalized (extended) electron states

V


Weakly disordered crystal lattice

Impurities cause electron to have a phase coherent mean free path

Wavefunctions still extended

Conductance decreased due to scattering

mfpl


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


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


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


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)


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


Recent papers on cold atoms

[Nature 453, 895 (2008)]

[Nature 453, 891 (2008)]


Why cold atoms?

• Disorder can be controlled

• Interactions can be controlled

• Experimental observations easier

• Quantum simulators of condensed matter


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


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


“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


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…


Localization!

In situ absorption images of the condensate


Spatial widths

Root mean squared size of the condensate at 750 μs

Dashed line is initial size of condensate


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


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


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


Billy et. al experimental setup

• Condensed 87Rb in a waveguide

• Applied a speckle potential to create random disorder


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


“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


Localization again!

Tails of distribution fitted with exponentials again - localization


Temporal dynamics

Localization length becomes a maximum then flattens off – expansion stopped


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


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


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)]


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