Localisation d'anderson et transition métal-isolant d'anderson dans les gaz atomiques froids

Transcription

1 Localisation d'anderson et transition métal-isolant d'anderson dans les gaz atomiques froids Dominique Delande Laboratoire Kastler-Brossel Ecole Normale Supérieure et Université Pierre et Marie Curie (Paris)

2 Outline What is Anderson localization? Anderson localization with cold atoms in a disordered optical potential The periodically kicked rotor: dynamical localization and Anderson localization Experimental realization of the periodically kicked rotor Anderson localization and Anderson transition for the three-color kicked rotor Critical regime

3 Anderson (a.k.a. Strong) localization Particle in a disordered (random) potential: One-dimensional system Two-dimensional system Particle with energy E Disordered potential V(z) (typical value V0) When E V0, the particle is classically trapped in the potential wells. When E À V0, the classical motion is ballistic in 1d, typically diffusive in dimension 2 and higher. Quantum interference may inhibit diffusion at long times => Anderson localization

4 Anderson localization in 1d Anderson localization is the generic behaviour, even for very small disorder! Simple Anderson model: discretized Schroedinger equation on a 1d lattice. H= X n ²n jnihnj + tjnihn + 1j + tjn + 1ihnj t=1: hopping Sites ²n : diagonal disorder Schroedinger equation at energy E: Ãn+1 + (²n E)Ãn + Ãn 1 = 0 µcan be rewritten µ Ãn+1 Ãn = Tn Ãn Ãn 1 with the µtransfer matrix: Tn = E ²n 1 1 0

5 Anderson localization in 1d Propagation of the transfer matrix: µ µ Ãn+1 Ã1 =T Ãn Ã0 T = Tn Tn 1 T1 = µ E ²n 1 µ 1 E ²n µ 1 E ² T is the product of independent random matrices with unit determinant => the eigenvalues of T typically behave like exp( n) at large n. Boundary conditions forbid exponential growth at large distance. µ jn n0 j Ãn» exp 2»loc»loc : localization length Exponential localization of the wavefunction»loc 4 E2 ¼ h²2n i (Advertisement: for numerical experiments on the 1d Anderson model, see )

6 Anderson localization in 1d Continuous Schroedinger equation: ~2 Ã(z) + V (z)ã(z) = EÃ(z) 2m Spatial discretization: Ã((n + 1)±) + Ã((n 1)±) 2Ã(n±) ±2 => back to Anderson model with correlated disorder Typical exponential localization of an eigenstate: zoom Superposition of forward and backward propagating waves Strong fluctuations on top of average exponential decay.

7 Anderson localization in 1d Propagation of an initially localized wave-packet: Initial Gaussian wavepacket jã(z; t = 10000)j2 Log scale jã(z; t = 20000)j2 Exponential localization at long time Both hz 2 i(t) and jã(z; t)j2 display signatures of Anderson localization

8 Anderson localization in 1d When averaged over time and/or different realizations of the disorder, the fluctuations are smoothed out: jã(z)j2 Averaged over 800 realizations of the disorder jã(z)j2 Log scale

9 Anderson localization in 1d and 2d Anderson localization is due to interference between various multiple scattering paths: requires perfect phase coherence. not restricted to quantum matter waves. It has been observed with several types of waves: microwaves, light, acoustic waves, seismic waves, etc. The importance of interference terms depends crucially on the geometry of multiple scattering paths, i.e. on the dimension. A detailed understanding can be obtained from the scaling theory of localization (Abrahams, Anderson, Licciardello, Ramakrishnan, 1979). Dimension 1: (almost) always localized. Localization length Dimension 2: marginally localized. Localization length ` : mean free path.»loc = 2`»loc ' ` exp µ ¼k` 2

10 Anderson localization in 3d (and beyond) For weak disorder, the quantum motion is diffusive (weak localization) => metallic behaviour. For strong disorder, Anderson localization takes place => insulator. The metal-insulator transition (Anderson transition) takes place at the mobility edge. It is controlled by the k` parameter. The critical point is approximately given by: (k`)c = 1 Ioffe-Regel criterion The Anderson transition is a second-order transition. On the insulating side, the localization length diverges like:»loc 1» [(k`)c k`]º º : critical exponent On the metallic side: D ' [k` (k`)c ]s with s = (d 2)º Numerical results suggest º ¼ 1:60. Experimental results on solid state samples are closer to º = 1 A simple mean field approach also finds º = 1

11 Outline What is Anderson localization? Anderson localization with cold atoms in a disordered optical potential The periodically kicked rotor: dynamical localization and Anderson localization Experimental realization of the periodically kicked rotor Anderson localization and Anderson transition for the three-color kicked rotor Critical regime

12 Anderson localization with cold atoms Essential features of Anderson localization: Inhibition of transport direct measurement of the atomic wavefunction Due to quantum interference preserve phase coherence of the atomic wavefunction Zero-temperature effect cold atomic gas One-body physics dilute gas (no Bose-Einstein condensate) Driven by the amount of disorder create a tunable disorder using light-atom interaction

13 Quantum dynamics of the external motion of cold atoms Control of the dynamics with laser fields, magnetic fields, gravitational field. Orders of magnitude: Velocity: cm/s Very favorable! De Broglie wavelength: mm Time: ms-ms One-body (if sufficiently dilute, avoid BEC) zero-temperature quantum dynamics with small decoherence. The light-shift due to a quasi-resonant laser at frequency!l =!at ± is proportional to I/d, (with I the spacedependent laser intensity) and acts as an effective potential. A random disordered optical potential is created using a speckle pattern. Inelastic processes scale as I/d2, thus can be made negligible if a large detuning (and laser power!) is used.

14 Speckle optical potential Speckle created by shining a laser on a diffusive plate: P. Bouyer et al, Institut d'optique (Palaiseau) Speckle spot size ³¼ NA l : laser wavelength NA : Numerical Aperture The speckle electric field is a (complex) random variable with Gaussian statistics. All correlation functions can be computed. Depending on the sign of the detuning, the optical potential is bounded either from above or from below

15 1d Anderson localization of atomic matter waves Initial atomic density Laser beam for transverse confinement Final atomic density (after 1 second) Disordered potential (optical speckle potential) J. Billy et al, Institut d'optique (Palaiseau, France), Nature, 453, 891 (2008)

16 1d Anderson localization of atomic matter waves Log scale The final atomic density shows exponential localization with localization length of few 100µm! J. Billy et al, Institut d'optique (Palaiseau, France), Nature, 453, 891 (2008)

17 Temporal dynamics for 1d Anderson localization of atomic matter waves J. Billy et al, Institut d'optique (Palaiseau, France), Nature, 453, 891 (2008)

18 1d Anderson localization of atomic matter waves Initial atomic cloud produced from dilution of a Bose-Einstein condensate => non mono-energetic initial wave-packet. All k atomic wavectors in a range [{kmax,kmax] are populated => the long distance behaviour is dominated by kmax. Fluctuations are smoothed out by incoherent superposition of various k values. Lowest perturbation order (Born approximation) for the localization length: correlation length»loc 2 ~4 kmax = ¼m2 V02 ³(1 kmax ³) of the potential potential strength Observation of Anderson localization requires small kmax ³ Very cold atoms ³ = 0:26¹m kmax ³ = 0:65 Very small speckle spots Apparent mobility edge at kmax ³ = 1, but no real metalinsulator transition.

19 Comparison between the measured and calculated localization lengths Disordered potential strength J. Billy et al, Institut d'optique (Palaiseau, France), Nature, 453, 891 (2008)

20 Anderson localization of atomic matter waves in higher dimension? µ ¼k` In 2 dimensions:»loc ' ` exp ` : mean free path 2 In 3 dimensions, Ioffe-Regel criterion: (k`)c = 1 Practical observation of Anderson localization requires very strong scattering: k` ¼ few units k³ ¼ 1 Very cold atoms Powerful laser State of the art 2d (or, even worse, 3d) speckle pattern No experimental observation yet

21 Outline What is Anderson localization? Anderson localization with cold atoms in a disordered optical potential The periodically kicked rotor: dynamical localization and Anderson localization Experimental realization of the periodically kicked rotor Anderson localization and Anderson transition for the three-color kicked rotor Critical regime

22 Anderson localization with cold atoms Essential features of Anderson localization: Inhibition of transport direct measurement of the atomic wavefunction Due to quantum interference preserve phase coherence of the atomic wavefunction Zero-temperature effect cold atomic gas One-body physics dilute gas (no Bose-Einstein condensate) Driven by the amount of disorder use chaotic temporal dynamics instead of static disorder Depends on the dimension effective dimension can be easily adjusted by varying the temporal sequence

23 The (periodically) kicked rotor +1 Hamiltonian p2 H= + k cos µ 2 Map from kick n to kick n+1: ½ X n= 1 ±(t nt ) T pn+1 = T pn + kt sin µn Standard map stochasticity parameter µn+1 = µn + T pn+1 K = kt The kicked rotor is periodic in θ, the spatially unfolded version is not => requires the inclusion of a (constant) quasi-momentum in the quantum treatment (Bloch theorem). kicked rotor spatially unfolded version

24 Classical dynamics of the kicked rotor Depends on a single parameter K=kT.

25 Classical dynamics of the kicked rotor Depends on a single parameter K=kT. Mainly regular for small K => momentum growth is bounded. ½ In+1 = In + K sin µn µn+1 = µn + In+1

26 Classical dynamics of the kicked rotor Depends on a single parameter K=kT. Mainly regular for small K => momentum growth is bounded. Chaos appears around K=0.5, then progressively invades the whole phase space. Unbounded momentum growth above K=1. I=Tp µ

27 Classical dynamics of the kicked rotor Depends on a single parameter K=kT. Mainly regular for small K => momentum growth is bounded. Chaos appears around K=0.5, then progressively invades the whole phase space. Unbounded momentum growth above K=1. Almost globally chaotic above K=4. I=Tp µ

28 Classical dynamics of the kicked rotor Depends on a single parameter K=kT. Mainly regular for small K => momentum growth is bounded. Chaos appears around K=0.5, then progressively invades the whole phase space. Unbounded momentum growth above K=1. Almost globally chaotic above K=4. pn+1 = pn + k sin µn At large K, random walk in momentum space => chaotic diffusion hp2n i hp20 i ¼ nk 2 =2 Diffusion constant: 2 k D= 2T µ

29 The quantum kicked rotor +1 2 X p H= + k cos µ ±(t nt ) Hamiltonian 2 n= 1 Evolution operator over one period: U = exp µ 2 ip T 2~ exp free evolution over one period µ ik cos µ ~ kick Very easy to numerically generate the quantum evolution.

30 Quantum dynamics of the kicked rotor Numerical experiment: compare classical and quantum dynamics (averaged over an ensemble of initial conditions). Start from a state well localized in momentum space. jã(p)j2 (log scale) hp2 (t)i Time t (number of kicks)

31 Quantum dynamics of the kicked rotor Numerical experiment: compare classical and quantum dynamics (averaged over an ensemble of initial conditions). Start from a state well localized in momentum space. At short time, the quantum and classical dynamics are equivalent. jã(p)j2 (log scale) Classical dynamics Quantum dynamics hp2 (t)i Time t (number of kicks)

32 Quantum dynamics of the kicked rotor Numerical experiment: compare classical and quantum dynamics (averaged over an ensemble of initial conditions). Start from a state well localized in momentum space. At long time, the quantum dynamics freeze. Equivalent to Anderson localization in momentum space (Fishman et al, 1982) jã(p)j2 (log scale) Quantum dynamics: dynamical localization Casati et al. (1979) 2 hp (t)i Classical dynamics: chaotic diffusion Time t (number of kicks)

33 Dynamical vs. Anderson localization The Hamiltonian is time-periodic => use Floquet theorem. Floquet states: eigenstates of the one-period evolution operator: µ 2¼~ iei T > Ei 0 U jái i = exp jái i T ~ In the momentum eigenbasis (labelled by integer m, wavefunction components cm), the eigen-equation becomes: ²m  m + X r6=0 Wr Âm r = W0 Âm where W (µ) = tan µ k cos µ 2~ = 1 X Fishman, Grempel, Prange (1982) Wr exp (irµ) r= 1 represents the coupling between various momentum states. ²m 2³ E = tan 4 m ~ 2 2 2~ 2 T 3 5 is the on-site energy.

34 Dynamical vs. Anderson localization (II) ²m  m + X r6=0 on-site random energy Wr Âm r = W0 Âm hopping eigenenergy The Wr decrease rapidly at large r and the ²m are pseudorandom variables => similar to Anderson model. ²m m Various Floquet states correspond to various realizations of the disorder, but have the same localization length. The hopping integrals Wr increase with the kick strength K => K plays the role of a control parameter of the Anderson model. More an analogy than a real proof.

35 Outline What is Anderson localization? Anderson localization with cold atoms in a disordered optical potential The periodically kicked rotor: dynamical localization and Anderson localization Experimental realization of the periodically kicked rotor Anderson localization and Anderson transition for the three-color kicked rotor Critical regime

36 The kicked rotor collaboration (Paris-Lille) Dominique Delande Benoît Grémaud Jean-Claude Garreau Pascal Szriftgiser Gabriel Lemarié Laboratoire Kastler-Brossel Université Pierre et Marie Curie and Ecole Normale Supérieure (Paris) Hans Lignier Julien Chabé Laboratoire PHLAM Université de Lille

37 The kicked rotor using cold atoms λl/2 Create a spatially periodic optical potential using a standing laser wave. Temporal modulation of the laser intensity (with δ-kicks) => kicked rotor with adjustable effective Plank constant ~e = 8!r T (!r : atomic recoil frequency, T: kicking period). Experimental setup: 1. Prepare a cold Cs cloud in a Magneto-Optical Trap; 2. Switch off MOT and magnetic field; 3. Apply the sequence of kicks ( kicks); 4. Analyze momentum distribution along the laser axis (using velocity selective Raman transitions); Atoms fall down because of gravitational field => 200 kicks maximum. Sources of decoherence: spontaneous emission, residual gravitational field along the laser axis are small enough over 200 kicks.

38 Experimental observation of dynamical localization with cold atoms Initial momentum distribution (Gaussian) Final momentum distribution (exponentially localized) Time (number of kicks) M. Raizen et al (1995) Momentum (in units of 2 recoil momenta)

39 Experimental observation of dynamical localization with cold atoms Energy Classical chaotic diffusion Dynamical localization Localization time of the order of 10 kicks M. Raizen et al (1995) (Number of kicks)

40 Outline What is Anderson localization? Anderson localization with cold atoms in a disordered optical potential The periodically kicked rotor: dynamical localization and Anderson localization Experimental realization of the periodically kicked rotor Anderson localization and Anderson transition for the three-color kicked rotor Critical regime

41 How to observe the Anderson transition in 3D? Kick strength Dynamical vs. Anderson localization Anderson localization: 1d disordered static x-space Dynamical localization: 1d chaotic time-periodic p-space Simple idea: keep the spatial dynamics one-dimensional, but introduce one or several additional temporal dimensions. X p2 H= + k cos µ (1 + ² cos!2 t cos!3 t) ±(t nt ) 2 n Time Substantially equivalent with the Anderson model in higher dimension (Casati, Guarneri and Shepelyansky, PRL, 1989). Prediction of a localized-delocalized Anderson transition when K=kT is increased in a 3-color system.

42 Mapping of the quasiperiodic kicked rotor on the 3D Anderson model The temporal evolution of the state Ã(µ) of the 1D quasiperiodic kicked rotor with Hamiltonian: X p2 H= + K cos µ (1 + ² cos!2 t cos!3 t) ±(t n) 2 n is identical to the one of the state: ª(x1 ; x2 ; x3 ) = Ã(x1 = µ)±(x2 )±(x3 ) of the 3D periodic kicked rotor : X p21 H= +!2 p2 +!3 p3 + K cos x1 [1 + " cos x2 cos x3 ] ±(t n) 2 n Pseudo 3D rotor Kick The initial state is completely delocalized in the transverse p2 and p3 directions. The classical dynamics of the 3D periodic kicked rotor is an anisotropic chaotic diffusion in momentum space.

43 Mapping of the quasiperiodic kicked rotor on the 3D Anderson model Similarly to the 1D periodic kicked rotor, the Schroedinger equation for the Floquet states of the 3D periodic kicked rotor can be written as: X ² m m + r 6=0 Wr m r = W0 m m = (m1 ; m2 ; m3 ) with diagonal disorder : ²m ½ µ ¾ 2 1 m1 = tan! ~ +! 2 m2 +! 3 m and the hopping coefficients Wm are the Fourier components of: W (x1 ; x2 ; x3 ) = tan [K cos x1 (1 + " cos x2 cos x3 )/ 2~] Effective 3D Anderson model, with a metal-insulator transition. For details, see Lemarié et al, PRA, 80, (2009), arxiv:

44 Schematic view of the experiment Kicks (amplitude quasi-periodically modulated with time) Final atomic cloud Time Initial atomic cloud X p2 H= + k cos µ (1 + ² cos!2 t cos!3 t) ±(t nt ) 2 n

45 Numerical results for the three-color kicked rotor Momentum distribution K (kick strength)

46 Numerical results for the three-color kicked rotor Momentum distribution Diffusive regime Localized regime K (kick strength)

47 How to identify unambiguously the transition? jã(p)j2 hp2 (t)i 3 increasing K values Time (number of kicks)

48 How to identify unambiguously the transition? jã(p)j 2 2 hp (t)i 3 increasing K values Time (number of kicks) At criticality, one expects an anomalous diffusion with (see below) 2 hp (t)i ' t with 2 = 3

49 Localized or diffusive? Instead of measuring hp2 (t)i, it is simpler to measure the population in the zero-momentum class 0 (t) Both in the localized and the diffusive regime, one has (t) / 2 hp (t)i In the experiments, one uses 0 (t) for detecting transition from localized to diffusive.

50 From localization to diffusive regime: experimental results 1= 20 (t) / hp2 (t)i diffusive <p2(t)> ~ t critical regime <p2(t)> ~ t2/3 localized

51 From localized to diffusive regime Experimental results 1= 20 (t) / hp2 (t)i log scale Numerical results diffusive regime (slope 1) critical regime (slope 2/3) localized regime (slope 0) time (number of kicks, log scale)

52 Scaling laws in the vicinity of the Anderson transition Characteristic lengths º Localization length ploc» (Kc K) Diffusion constant D» (K Kc )s Unified description of the localized and diffusive regimes (oneparameter scaling law): hp2 (t)i» tk1 F [(K Kc )tk2 ] Asymptotic regimes for F: Localized: k1 2ºk2 = 0 Diffusive: k1 + sk2 = 1 Together with Wegner's law s = (d 2)º, it gives k1 = 2=3 k2 = 1=3º Numerical and experimental data agree perfectly well with the 2/3 exponent. Thus, we study 1 (K; t) = 2 0 (t) t2=3 versus t and K.

53 Rescaled numerical results diffusive (slope -1) ln (K; t) 1 (K; t) = 2 0 (t) t2=3 increasing K values 0 (t) : Population in the zero-velocity class localized (slope 2) increasing time ln(1=t1=3 ) The critical regime is the horizontal line. Problem: it requires very long times to accurately measure the position of the transition as well as the critical exponent.

54 Finite-time scaling Directly transpose the ideas of finite size scaling (in usual configuration space) to the temporal dynamics of the kicked rotor => finite time scaling. Assume the one-parameter scaling law 1 (t) = 2 =F 2=3 0 (t) t µ»(k) t1=3 where the localization length (actually localization time)» depends only on the kick strength K. By simultaneously fitting various values of K and t, one can extract both the localization length»(k) and the scaling function.

55 ln( ) = ln(1= 20 (t)t2=3 ) Finite time scaling µ 1»(K) (t) = 2 =F 2=3 0 (t) t t1=3 ln(1=t1=3 )

56 ln( ) = ln(1= 20 (t)t2=3 ) Diffusive Finite time scalingµ 1»(K) (t) = 2 =F 2=3 0 (t) t t1=3 ln»(k) Localized ln(1=t1=3 ) ln(»(k)=t1=3 ) K The displacement is proportional to»(k)

57 Finite time scaling analysis of numerical results 1 Scaling function (t) = 2 0 (t) t2=3 Localization length Critical point Kc=6.6»» jk Kc j º Critical exponent º = 1:60 0:05 Chabé et al, PRL, 101, (2008) Numerical data up 106 kicks, latest result: º = 1:58 0:02

58 Experimental results The temporal dynamics is more restricted => usage of finite time scaling is mandatory ln (K; t) Diffusive increasing K values Localized increasing time ln(1=t1=3 )

59 Finite time scaling analysis of experimental results 1 Scaling function (t) = 2 (t) t2=3 0 Critical point Kc=6.4 Localization length»» jk Kc j º Critical exponent Excellent agreement with numerics without any adjustable parameter º = 1:4 0:3 Chabé et al, PRL, 101, (2008)

60 Universality of the Anderson transition The Anderson transition is universal, i.e. its characteristic properties such as the critical exponent - do not depend on the microscopic details of the system. We have numerically checked this universality for the quasiperiodically kicked rotor, by varying the effective Planck's constant, irrational ratio between the three frequencies, and the depth ε of the modulation. We always observe (G. Lemarié et al, Europhys. Lett. 87, (2009), arxiv: ) º = 1:58 0:02 It is identical to the exponent numerically measured on the Anderson model (Slevin and Ohtsuki, 1999).

61 Outline What is Anderson localization? Anderson localization with cold atoms in a disordered optical potential The periodically kicked rotor: dynamical localization and Anderson localization Experimental realization of the periodically kicked rotor Anderson localization and Anderson transition for the three-color kicked rotor Critical regime

62 Momentum distribution at the critical point The initial state is almost perfectly localized in momentum space => the momentum distribution at time t is nothing but a direct measure of the average intensity Green function G(0; p; t) Numerical experiment at the critical point: Momentum Green Function (log scale) (millions of kicks) Time invariant shape (neither Gaussian, nor exponential)

63 Momentum distributions at criticality Distributions at various times Distributions at various times rescaled by the critical t1/3 law

64 Experimental measurements in the critical regime Characterized by a specific scaling: p / t1=3 p / t0 p / t1=3 p / t1=2 Raw experimental data Rescaled data 0 p=t p=t 1=3 p=t1=2 Lemarié et al, to be published (2010)

65 Critical wavefunction At the critical point, the diffusion constant scales as: D(!) / ( i!)1=3 Self-consistent theory of localization (à la Vollhardt-Wölfle) predicts at the critical point of the Anderson metal-insulator transition: ³ ³ 1=3 1=3 3 2 jã(p; t)j = 3½3=2 t Ai 3½3=2 t jpj 2 Behaves asymptotically as exp( jxj3=2 ) In excellent agreement with numerics and experimental results

66 Experimentally measured critical Green function Analytical prediction (Airy function) Experimental points (with error bars) Residual /Airy function Residual /Gaussian Residual /Exponential Rescaled momentum p=t1=3

67 Summary Cold atoms are good systems for studying quantum interference, quantum chaos and the quantum dynamics of complex systems. Experimental observation of 1d Anderson localization of atomic matter waves in a disordered potential created by an optical speckle. Experimental observation of a transition from the localized to the diffusive regime, in a one-body system at zero-temperature, due to quantum interference, driven by the amount of disorder in the system => Anderson transition First experimental measurement of the critical exponent with atomic matter waves. No characteristic scale at the critical point. Self-consistent theory of localization accurately predicts the average intensity Green function.

68 Perspectives Fluctuations of the wave function at the critical point (multifractality). Can be measured in principle, but... What about two dimensions? Experiment in progress. Adding additional temporal quasi-periods => Anderson model in dimension 4, 5... Numerics in d=4 (May 2010): º = 1:15 0:03 Good agreement with Anderson model Most probably, the upper critical dimension is infinite. Controlled addition of decoherence. Interaction between atoms (dynamical localization of a BoseEinstein condensate) => possible modification of the critical exponent. Effect of quantum statistics (bosons vs. fermions).

69 Rescaled dynamics at various times (numerics) ln (K; t) The critical exponent can be extracted from the variations of the slope at the critical point vs. time Critical point K (kick strength) increasing time

70 Choice of parameters p 5 p 13 ² diffusive localized X p2 H= + k cos µ (1 + ² cos!2 t cos!3 t) ±(t nt ) 2 n

71 Experimental setup Standard Magneto-Optical Trap