Anderson localization of ultracold atoms in a (laser speckle) disordered potential
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1 Pittsburgh, march 19, years of Anderson localization Anderson localization of ultracold atoms in a (laser speckle) disordered potential Alain Aspect Groupe d Optique Atomique Institut d Optique - Palaiseau 1
2 Anderson localization of ultra cold atoms in a laser speckle disordered potential 1. Anderson localization: the naïve view of an AMO experimentalist: 1 particle quantum interference effect 2. Anderson localization with cold atoms in laser speckle A well controlled system 3. 1D Anderson localization?! 4. 1D expansion of a cloud of ultracold atoms released in a speckle disordered potential: a first attempt 5. Suppression of the expansion of a cloud of ultracold atoms released in a weak speckle: Anderson localization 2
3 Anderson localization of ultra cold atoms in a laser speckle disordered potential 1. Anderson localization: the naïve view of an AMO experimentalist: 1 particle quantum interference effect 2. Anderson localization with cold atoms in laser speckle A well controlled system 3. 1D Anderson localization?! 4. 1D expansion of a cloud of ultracold atoms released in a speckle disordered potential: a first attempt 5. Suppression of the expansion of a cloud of ultracold atoms released in a weak speckle: Anderson localization 3
4 Anderson localization: a model for metal/insulator transition induced by disorder Classical model of metal: impurities hinder, do not cancel, ohmic conduction Classical particles bouncing on impurities diffusive transport (Drude) Matter waves scattered on impurities incoherent addition delocalized (extended) states (cf. radiative transfer) mean free path Anderson L. (1958): impurities totally cancel ohmic conduction Tight binding model of electrons on a lattice with impurities: if mean free path smaller than de Broglie wavelength: localized states: insulator Quantum effect Mobility edge (Ioffe Regel) λ db 4
5 Anderson localization: the point of view of an AMO physicist Coherent addition of waves scattered on impurities. If mean free path smaller than de Broglie wavelength: destructive interference in forward scattering, then in any direction: localized states: insulator Quantum (wavelike) effect Mobility edge (Ioffe Regel) λ db Main features: Interference of many scattered wavelets localization Single particle quantum effect (no interaction) 5
6 The experimental quest of AL in AMO physics Electromagnetic waves scattering on non absorptive impurities: not easy to discriminate from ordinary absorption Microwaves (cm) on dielectric spheres: discriminating localization from absorption by study of statistical fluctuations of transmission Chabonov et al., Nature 404, 850 (2000) Light on dielectric microparticles (TiO 2 ): Exponential transmission observed; questions about role of absorption Wiersma et al., Nature 390, 671 (1997) Discriminating localization from absorption by time resolved transmission Störzer et al., Phys Rev Lett 96, (2006) Difficult to obtain < λ / 2 π (Ioffe-Regel mobility edge) No direct observation of the exponential profile in 3D Most of these limitations do not apply to the 2D or 1D localization of light observed in disordered 2D or 1D photonic lattices: T. Schwartz et al. (M. Segev), Nature 446, 52 (2007). Lahini et al. (Silberberg), PRL 100, (2008). 6
7 Anderson localization of ultra cold atoms in a laser speckle disordered potential 1. Anderson localization: the naïve view of an AMO experimentalist: 1 particle quantum interference effect 2. Anderson localization with cold atoms in laser speckle A well controlled system 3. 1D Anderson localization?! 4. 1D expansion of a cloud of ultracold atoms released in a speckle disordered potential: a first attempt 5. Suppression of the expansion of a cloud of ultracold atoms released in a weak speckle: Anderson localization 7
8 Good features Ultra cold atoms (matter waves) Good candidate to observe AL Controllable dimensionality (1D, 2D, 3D) Wavelength λ db easily controllable over many orders of magnitude (1 nm to 10 μm) Pure potentials (no absorption), with easily controllable amplitude and statistical properties Many observation tools: light scattering or absorption, Bragg spectroscopy, A new feature: interactions between atoms A hindrance to observe AL (pure wave effect for single particle) New interesting problems (theoretical challenge) 8
9 Disordered potentials for cold atoms Magnetic potential on an atom chip Sub micron structures (near field), designed at will (e-beam lithography); uncontrolled randomness a problem, in spite of progress in understanding and controlling it (J. Estève et al. PRA 2004, JB Trebbia et al. PRL 2007) RF dressed potentials (Villetaneuse, Vienna, ) Fine enough structures to be demonstrated Impurity atoms in a lattice (Castin, Cirac, Sengstock experiment) Close to initial Anderson model, not easy to pin the impurities Optical dipole potential (R. Roth and K. Burnett, B. Damski et al., PRL 2003) laser beam transmitted through a random mask (Hanover): hard to make small structures (optical resolution) Optical disordered potential created by a laser speckle = our favorite 9
10 Optical dipole potential Inhomogeneous light field: Er (,) t = E()cos r [ ωt ϕ() r] 0 Induced atomic dipole: laser Interaction energy: D at () t = α E( r,) t r at at Far from atomic resonance, α real α < 0 above resonance α > 0 below resonance W = Er (, t) D ( t) = α at at r at [ E ( r )] 0 at 2 2 Atoms experience a (mechanical) potential proportional to light intensity U dip () r = α I () r Attracted towards large intensity regions below resonance Repelled out of large intensity regions above resonance 10
11 Speckle optical random potential Blue detuned light creates a repulsive potential for atoms proportional to light intensity V I δ Laser speckle: very well controlled random intensity pattern (Light amplitude = Gaussian random process, central limit theorem) Autocorrelation function rms width (speckle grain size) controlled by aperture λl σ R π D Calibrated for σ R > 1μm Extrapolated at σ R < 1μm Exponential intensity distribution 1 I PI ( ) = exp{ } I I Calibrated by RF spectroscopy (light shifts distribution) 11
12 Anderson localization of ultra cold atoms in a laser speckle disordered potential 1. Anderson localization: the naïve view of an AMO experimentalist: 1 particle quantum interference effect 2. Anderson localization with cold atoms in laser speckle A well controlled system 3. 1D Anderson localization?! 4. 1D expansion of a cloud of ultracold atoms released in a speckle disordered potential: a first attempt 5. Suppression of the expansion of a cloud of ultracold atoms released in a weak speckle: Anderson localization 12
13 1D Anderson localization? Theorist answer: all states localized in 1D Experimentalist question: Anderson like localization? AL: Interference effect between many scattered wavelets exponential wave function Localization in a strong disorder: particle trapped between two large peaks Classical localization, not Anderson E V E V = V dis z Localization in weak disorder (numerics): interference of many scattered wavelets E >> V dis numerics Looks like Anderson localization V z 13
14 1D localization in a weak disorder as Bragg reflection E Periodic potential V = V cos kz 0 p = k/2 k V p = k/2 even in the case of Bragg reflection of p ~ k/2 on cos kz No propagation in the gap at E No propagation of matter wave ψ = Aexp ± i p z if p ~ k/2 2 p E = >> V (weak perturbation) 0 2M 2 2 k = >> V 8M 0 14
15 1D localization in a weak disorder as Bragg reflection E V p = k/2 p = k/2 Periodic potential 0 k No propagation of matter wave ψ = Aexp ± i p z if p ~ k/2 2 p even in the case of E = >> V (weak perturbation) 0 2M Bragg reflection of p ~ k/2 on cos kz Disordered potential: many k components, act separately on various p components (Born approximation). Anderson localization: all p components Bragg reflected. Demands broad spectrum of disordered potential 15
16 1D Anderson localization in a weak uncorrelated disorder Disordered potential with a white spectrum of k vectors V dis V ( k) z k Anderson L.: all p components Bragg reflected What happens for a correlated potential (finite spectrum)? k 16
17 Case of a speckle disorder: cut off in the spatial frequency spectrum θ Speckle potential, created by diffraction from a scattering plate: no 2 sin k component beyond a cut off σ = R FT of auto correlation Only matter waves with 2p < 2 /σ R localize light Effective mobility edge k θ Spatial frequencies spectrum of speckle potential ck ˆ( ) 2/σ R Lyapunov coefficient First order calculation (phase formalism) 1 p γ ( p) = cˆ (2 ) γ ( p) = 0 for p > L ( p) σ lc o R 17
18 1D Anderson localization in a weak speckle potential? First order calculations*: exponentially localized wave functions provided that p < / σ R Localization results from interference of many scattered wavelets (not a classical localization, weak disorder) Effective mobility edge Same features as genuine (3D) Anderson localization Worth testing it * Beyond Born approximation? Ask question! 18
19 Anderson localization of ultra cold atoms in a laser speckle disordered potential 1. Anderson localization: the naïve view of an AMO experimentalist: 1 particle quantum interference effect 2. Anderson localization with cold atoms in laser speckle A well controlled system 3. 1D Anderson localization?! 4. 1D expansion of a cloud of ultracold atoms released in a speckle disordered potential: a first attempt 5. Suppression of the expansion of a cloud of ultracold atoms released in a weak speckle: Anderson localization 19
20 A 1D random potential for 1D guided atoms Atoms tightly confined in x-y plane, free along z: 1D matterwaves Cylindrical lens = anisotropic speckle, elongated along x-y fine along z V V(z) z σ, σ 50μm ; σ 1 μm x y z z BEC elongated along z and confined (focussed laser) transversely to z TF TF 2R 300μm ; 2 3μm Imaged with resonant light z R 1 D situation for the elongated BEC. Many speckle grains covered (self averaging system = ergodic) 1D situation: invariant transversely to z 20
21 Ballistic expansion of a 1D BEC Cloud of trapped ultracold atoms (dilute BEC) directly observable by absorption imaging: N atoms with the same confined wave function. Release of trapping potential along z: expansion in the 1D atom guide R(t) Initial interaction energy μ in converted into kinetic energy After a while, interaction free ballistic expansion Superposition of plane waves with p p = 2Mμ max in 21
22 Suppression of the expansion of a 1D BEC released in a 1D speckle potential (2005) D. Clément et al., PRL 95, [2005] 2L V dis = 0 z Vdis = 0.3μin V dis Ballistic expansion = 0.7μ in Anderson localization? Expansion stops in disordered potential 22
23 Suppression of the expansion of a 1D BEC released in a 1D speckle potential (2005) D. Clément et al., PRL 95, [2005] 2L V dis = 0 z Vdis = 0.3μin V dis = 0.7μ in Not Anderson localization: atom energy of the order of height of potential barriers: classical localization. No localization observed in the case of weak disorder Analogous results in Florence (C. Fort et al., PRL 95, [2005]) BEC in dis. potential also studied in Hanover, Rice University, Illinois 23
24 Why didn t we observe AL? Expansion of the BEC: initial interaction energy converts into kinetic energy Distribution of atomic matter waves with maximum p value p max = 2Mμ in D(p) 0 p σ R p max p Efficient localization happens only for matter waves such that p σ R effective mobility edge! In the 2005 experiment p max > (same situation σ in Florence) R 24
25 Anderson localization of ultra cold atoms in a laser speckle disordered potential 1. Anderson localization: the naïve view of an AMO experimentalist: 1 particle quantum interference effect 2. Anderson localization with cold atoms in laser speckle A well controlled system 3. 1D Anderson localization?! 4. 1D expansion of a cloud of ultracold atoms released in a speckle disordered potential: a first attempt 5. Suppression of the expansion of a cloud of ultracold atoms released in a weak speckle: Anderson localization 25
26 Search for Anderson localization in a weak speckle potential: new experiment As previously: expansion of a BEC in a wave guide for matter waves. But Better optical access allowing finer speckle (σ R = 0.26 μm) Residual longitudinal potential well compensated: balistic expansion over 4 mm Initial density small enough that max velocity well below the effective mobility edge p max = 0.65 / σ R (fluorescence imaging:1 at / μm) Deep in weak disorder regime V R = 0.1 μ ini 26
27 Anderson localization in a weak speckle: below the effective mobility edge p max σ R = 0.65 Expansion stops. Exponential localization? 27
28 Anderson localization in a weak speckle below the effective mobility edge Direct observation of the wave function (squared modulus) p max σ R = 0.65 Nature, june 12, 2008 Exponential localization in the wings Exponential fit Localization length Is this measured localization length meaningful? 28
29 Anderson localization in a weak speckle potential below the effective mobility edge Profile stops evolving: wings well fitted by an exponential Fitted localization length stationary: meaningful 29
30 Comparison to perturbative calculation L loc vs V R k max σ R = 0.65 μ ini = 220 Hz Magnitude and general shape well reproduced by theory without any adjustable parameter 30
31 What happens beyond the mobility edge? Theoretical prediction in our situation: BEC with large initial interaction energy D(p) /σ R Waves with p values between /σ R and p = 2M μ do not localize max Waves with p values below /σ R localize with different L loc power law wings ~ z -2 in L. Sanchez-Palencia et al., PRL 98, (2007) 0 p max p 31
32 What happens beyond the mobility edge? Theoretical prediction in our situation: BEC with large initial interaction energy p max > / σ R power law wings ~ z 2 L. Sanchez-Palencia et al., PRL 98, (2007) D(p) /σ R 0 p max p Experiment at p max σ R = 1.15 log-log scale Not exponential wings Power law wings ~ z -2 Can we really tell the difference between exponential and z 2? 32
33 Crossover from exponential to algebraic tails k max. σ R = 1.15 : algebraic k max. σ R = 0.65 : exponential V dis = 0.16 μ in (weak disorder) Log-Log plots allow us to tell the difference! 33
34 Conclusion Evidence of Anderson localization of Bosons in 1D laser speckle disordered potential Crossover from exponential to algebraic profiles (effective mobility edge) Good agreement with perturbative ab initio calculations (no adjustable parameter) Related results in Florence (Inguscio s group): Localization of bosons with suppressed interaction in a bichromatic potential with incommensurate periods (Aubry-André model) 34
35 Assets of our system Outlook Well controlled and well understood disordered potential (laser speckle = gaussian process) Cold atoms with controllable kinetic and interaction energy Direct imaging of atomic density (~wave function) Unambiguous distinction between algebraic and exponential Future plans: more 1D studies (tailored disorder) role of interactions 2D & 3D studies fermions vs bosons Theory far from complete. A quantum simulator? 35
36 Anderson localisation in Atom Optics group A collective work Vincent Juliette Philippe Pierre Alain Alain Ben Laurent Experimental teams (Philippe Bouyer) 1. David Clément, A. Varon, Jocelyn Retter 2. Vincent Josse, Juliette Billy, Alain Bernard, Zangchung Zuo and our electronic wizards: André Villing and Frédéric Moron Theory team (Laurent Sanchez Palencia): Pierre Lugan, Ben Hambrecht Collaborations: Dima Gangardt, Gora Shlyapnikov, Maciej Lewenstein 36
37 References to our work on quantum gases in (speckle) random potential Not Anderson Localization of an expanding BEC D. Clément et al., PRL 95, (2005); NJP 8, 165 (2006) D. Clément et al., NJP 8, 165 (2006) States of repulsive Bose Gases (coll. M. Lewenstein) L. Sanchez-Palencia, PRA 74, (2006) P. Lugan et al., PRL 98, (2007) Localization of quasiparticles of an interacting BEC P. Lugan et al., cond mat , PRL 2007 Anderson localization of an expanding BEC L. Sanchez-Palencia et al., PRL 98, (2007): theory J. Billy et al., june 2008 experiment 37
38 1 D BEC ATOM LASER Vincent Josse David Clément Juliette Billy William Guérin Chris Vo Zhanchun Zuo THEORY L. Sanchez-Palencia Pierre Lugan Groupe d Optique Atomique du Laboratoire Charles Fabry de l Institut d Optique Philippe Bouyer Chris Westbrook Welcome to Palaiseau Fermions Bosons mixtures Thomas Bourdel Gaël Varoquaux Jean-François Clément J.-P. Brantut Rob Nyman BIOPHOTONICS Karen Perronet David Dulin Nathalie Wesbrook He* BEC Denis Boiron A. Perrin V Krachmalnicoff Hong Chang Vanessa Leung ATOM CHIP BEC Isabelle Bouchoule Jean-Baptiste Trebia Carlos Garrido Alzar IFRAF experiments BIARO: T. Botter, S. Bernon ELECTRONICS André Villing Frédéric Moron OPTO-ATOMIC CHIP Karim el Amili Sébastien Gleyzes 38
39 Localization beyond the effective mobility edge Calculations (P. Lugan) beyond the Born approximation (4th order) (agreement with numerics, D. Delande, and diagrams, C. Müller) Lyapunov coefficient γ not exactly zero but crossover to a much smaller value at effective mobility edge cond mat.other * V dis Sharper crossover for weaker disorder γ Effective transition in a finite size system Effective m e p * analogous results by E. Gurevich,
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