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.) cos2j 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