anderson localization in becs

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

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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

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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

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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