Université Lille 1 Sciences et Technologies, Lille, France Lille Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique

Université Lille 1 Sciences et Technologies, Lille, France Lille Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique FRISNO 11 Aussois 1/4/011 Quantum simulators: The Anderson transition and the quantum kicked rotor Jean-Claude Garreau Aussois Matthias Lopez, Julien Chabé ( 8/008), PhDs Gabriel Lemarié, PhD ( 9/009) Hans Lignier ( 9/009) post-doc Jean-François Clément, Pascal Szriftgiser, J. C. G. Benoît Grémaud, Dominique Delande 1

Simulating condensed matter with cold atoms 198 R. P. Feynman Simulating Physics with computers, Int. J. Th. Phys. 1 467 488 (198) A. Hemmerich and T. W. Hänsch, Two-dimensional atomic cristal bound by light, Phys. Rev. Lett. 70, 410 413 (1993) G. Grynberg et al., Quantized motion of cold cesium atoms in two- and threedimensional optical potentials, Phys. Rev. Lett. 70, 49 5 (1993) M. Ben Dahan et al., Bloch Oscillations of Atoms in an Optical Potential, Phys. Rev. Lett. 76, 4508 4511 (1996) S. R. Wilkinson et al., Observation of atomic Wannier-Stark ladders in an accelerating optical potential, PRL 76, 451 4515 (1996) M. Greiner et al., Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39 44 (00) J. Billy et al., Direct observation of Anderson localization of matter-waves in a controlled disorder, Nature 453, 891 894 (008) 011 J. Chabé et al., Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves, Phys. Rev. Lett. 101, 5570 (008) /6

The Anderson localization Tmum + Vr um + r = Eum Anderson crystal r W W Tm 1D: All eigenstates are localized whatever the disorder 3D: Mobility edge : For low enough disorder diffusion30restored 1 30 5 1 5 0 0 0 0 0-1 -1 15 55 10 10 10 15 15-5 5 30 030 15 10 5 5 0 0 0 0 - P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 149 (1958) 3 /6

Anderson transition in 1D A pedestrians view of the Anderson transition L R L D. J. Thouless, Electrons in disordered systems and the theory of localization, Phys. Rep. 13, 93-14 (1974) 4/6

The Anderson transition in 3D L L L Insulator Conductor 5 /6

The Anderson transition The Anderson transition in various dimensions Insulator Conductor α 1 0-1 - -3 1D D 3D 4D 5D ln r 6/6

Experiments in condensed matter Not easy to control decoherence No access to wavefunctions Strong interactions 7/6

Experiments in other wave systems Localization has been observed with sound waves, microwaves, light and with a Bose-Einstein condensate, in 1D J. Billy et al., Direct observation of Anderson localization of matter-waves in a controlled disorder, Nature 453, 891 (008) Transition observed with light M. Störzer et al., Observation of the Critical Regime Near Anderson Localization of Light, PRL 96, 063904 (006) and sound waves H. Hu et al., Localization of ultrasound in a three-dimensional elastic network, Nature Physics 4, 945 (008) experiments plagued by absorption: same signature as localization 8/6

Experiments with cold atoms Control of decoherence Direct access to wavefunctions Negligible particle-particle interactions How to implement disorder? Optical speckle (IOTA) Quasi-periodic spatial modulation (Florence) Temporal disorder: Quantum chaos (Lille) 9/6

The kicked rotor P H = + K cos x δ ( t n) n 10/6

The optical potential p after = p before + k L V ( x) I( x) sin( x) 11/6

The atomic kicked rotor (schematic) Acoustooptical modulator Cold-atom cloud Mirror P H = + K cos x δ ( t n) n F. L. Moore et al., Atom optics realization of the quantum δ-kicked rotator, Phys. Rev. Lett. 75, 4598 (1995) 1/6

Dynamical localization Classical x quantum p p ~Dt p ~cte «Dynamical localization» t G. Casati et al., Stochastic behavior of a quantum pendulum under periodic perturbation, Lect. Notes Phys. 93, 334 (1979) 13/6

Anderson x dynamical localization T n n V n + r n = E n Anderson + r r n x = na Kicked rotor p0 p0 + n exp( ik sin x / ) n = ( i) J ( K / ) n + r r ( k) r r p = n k Random Eq. (1) tan( K cos x / ) = r V r e ikr T n ε n = tan n / (1) Floquet s quasi-energy (deterministic!) Pseudo disorder Each Floquet state is a realization of the fixed disorder ~ W = cte K controls the tunneling ~ V Increasing K decreases W/V S. Fishman et al., Chaos, quantum recurrences, and Anderson localization, PRL 49, 509 (198) 14/6

The 3D kicked rotor How to realize a 3D kicked rotor? H = P + K cos x ( 1+ ε cos( ω t)cos( ω t) ) δ ( t 3 n n) substantially equivalent to an anisotropic 3D Anderson model The underlying unit of nature : different systems described by the same equations (The Feynman Lectures in Physics, vol. ch. 1) 15/6 G. Casati et al., Anderson transition in a one-dimensional system with three incommensurate frequencies, PRL 6, 345 (1989)

The (real) experiment 16/6

Sweeping the Anderson transition H = P + K cos x ( 1+ ε cos( ω t)cos( ω t) ) δ ( t 3 n n) Diffusive 0.8 Metal ε Critical Localized Insulator 0.1 4 K 9 17/6

Experimental momentum distributions Linear scale 150 kicks Log scale ψ ( p) ψ ( p) K = 5.0 Π 0 K = 5.0 K = 9.0 K = 9.0 p p G. Lemarié et al. Observation of the Anderson metal-insulator transition with atomic matter waves: Theory and experiment 18/6 PRA 80, 04366 (009)

Determination of the critical point Theory: at criticality Recipe: make a log-log plot of and measure its slope 19/6

Observing the transition experimentally Localized log(t 1/3 Π 0 ) Critical Diffusive log(t) 0/6

Finite size/time effects Small samples : no singular behavior Example: Bose-Einstein condensation 0.5 0.4 N N 0 /N 0.3 0. N = 1000 0.1 N = 100 0.0 0.8 0.9 1.0 1.1 1. T/T c 1/6

Extracting a critical exponent ξ K Kc ν Numerical Experimental KR (num.) K c = 6.9 v =1.59 ± 0.01 K c v = 6.7 ± =1.5 ± 0. 0. J. Chabé et al., PRL 101, 5570 (008) G. Lemarié et al., PRA 80, 04366 (009) Anderson (num) v =1.6 ± 0.01 /6

Scaling of the critical wavefunction Localized Critical Diffusive G. Lemarié et al. Critical state of the Anderson transition: Between a metal and an insulator, PRL 105, 090601 (010) 3/6

Form of the critical wavefunction Airy fit ρ fit ρ th 1.60.04 P(p,t)t 1/3 Airy Exponential χ =1.1 χ = 4.5 Gaussian χ = 8.8 PRL Editor s reading suggestion and Synopsis 4/6

Conclusion Prospects for future work Is the critical exponent really universal? How decoherence affects the transition? Other dimensions: D, 4D, 5D? ε More generally What is the effect of interactions on the transition? Use a Bose-Einstein condensate and Feshbach resonances Still more generally Simulation of condensed matter systems by dynamical, cold atom systems For example: Harper model : J. Wang and J. Gong, Proposal of a cold-atom realization of quantum maps with Hofstadter's butterfly spectrum, PRA 77, 031405(R) (008) 5/6

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