The atomic quasi-periodic kicked rotor: Experimental test of the universality of the Anderson transition, Critical behavior and Large fluctuations

Transcription

1 The atomic quasi-periodic kicked rotor: Experimental test of the universality of the Anderson transition, Critical behavior and Large fluctuations Dominique Delande Laboratoire Kastler-Brossel Ecole Normale Supérieure et Université Pierre et Marie Curie (Paris, European Union) Pohang August 011

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

3 Outline Anderson localization with cold atoms The periodically kicked rotor: dynamical localization and Anderson localization Anderson localization and Anderson transition for the quasi-periodic three-color kicked rotor Experimental test of the universality of the Anderson transition Critical regime of the Anderson transition Multifractality and fluctuations of the critical wavefunctions

4 Anderson localization in 3d (and beyond) Weak disorder => diffusive motion, metallic behaviour. Strong disorder, Anderson localization takes place => insulator. The metal-insulator transition (Anderson transition) takes place at the mobility edge, controlled by the k` parameter: (k`)c = 1 Ioffe-Regel criterion 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 Numerical results suggest º ¼ 1:60. with s = (d )º

5 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

6 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: µm Time: µs-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. Inelastic processes scale as I/d, can be made negligible.

7 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

8 Outline Anderson localization with cold atoms The periodically kicked rotor: dynamical localization and Anderson localization Anderson localization and Anderson transition for the quasi-periodic three-color kicked rotor Experimental test of the universality of the Anderson transition Critical regime of the Anderson transition Multifractality and fluctuations of the critical wavefunctions

9 The (periodically) kicked rotor +1 Hamiltonian p H= + k cos µ 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 Classical dynamics fully chaotic for K>4. Analogous to a random walk in momentum space => chaotic diffusion hp (t)i ¼ Dt The quantum dynamics displays dynamical localization (Anderson localization in momentum space). kicked rotor The kicked rotor is periodic in θ, the spatially unfolded spatially unfolded version is not => inclusion of a (constant) quasiversion momentum in the quantum treatment.

10 Dynamical vs. Anderson localization Time-periodic Hamiltonian => µ use Floquet eigenstates: iet ¼~ U jâi = exp jâi >E 0 ~ T In the momentum eigenbasis (labelled by integer m), it writes: ² m Âm + X r6=0 where W (µ) = tan Fishman, Grempel, Prange (198) Wr Âm r = W0 Âm µ k cos µ ~ = 1 X Wr exp (irµ) r= 1 Wr describes hopping and decreases fast at large r => ³ Anderson model Pseudo-random on-site energy. ²m ²m = tan 4 E m ~ ~ m K is the control parameter of the Anderson model. Various Floquet states correspond to different realizations of disorder, with the same localization length. T 3 5

11 Outline Anderson localization with cold atoms The periodically kicked rotor: dynamical localization and Anderson localization Anderson localization and Anderson transition for the quasi-periodic three-color kicked rotor Experimental test of the universality of the Anderson transition Critical regime of the Anderson transition Multifractality and fluctuations of the critical wavefunctions

12 The kicked rotor using cold atoms λl/ Optical potential created by standing laser wave. Temporal modulation of the laser intensity => kicked rotor with adjustable effective Plank constant (!r : atomic recoil frequency, T: kicking period). Experimental setup: ~e = 8!r T 1. Prepare a cold atomic cloud in a Magneto-Optical Trap;. Switch off MOT and magnetic field; 3. Apply the sequence of kicks (10-00 kicks); 4. Analyze momentum distribution along the laser axis. Gravitation and decoherence (spontaneous emission) limit the phase coherence time to kicks.

13 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 Spatial dynamics one-dimensional, but introduce one or several additional temporal dimensions. X p H= + k cos µ (1 + ² cos! t cos!3 t) ±(t nt ) n Time Quasi-periodic kicked rotor Equivalent to the Anderson model in higher dimension (Casati, Guarneri and Shepelyansky, PRL, 1989). Localized-delocalized Anderson transition when K=kT is increased in a 3-color system.

14 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 p H= + K cos µ (1 + ² cos! t cos!3 t) ±(t n) n is identical to the one of the state: ª(x1 ; x ; x3 ) = Ã(x1 = µ)±(x )±(x3 ) of the 3D periodically kicked pseudo-rotor : X p1 H= +! p +!3 p3 + K cos x1 [1 + ² cos x cos x3 ] ±(t n) n 3D pseudo-rotor Kick The initial state is completely delocalized in the transverse p and p3 directions. Classical dynamics of the 3D periodic kicked rotor is an anisotropic chaotic diffusion in momentum space.

15 Mapping of the quasiperiodic kicked rotor on the 3D Anderson model Similar to the 1D periodic kicked rotor: the Schrödinger equation for the Floquet states of the 3D periodic kicked pseudo-rotor can be written as: X ²m m + Wr m r = W0 m m = (m1 ; m ; m3 ) r6=0 Diagonal disorder : ²m ½ µ ¾ 1 m1 = tan! ~ +! m +!3 m3 Hopping coefficients Wm : Fourier components of: W (x1 ; x ; x3 ) = tan [K cos x1 (1 + ² cos x cos x3 )/ ~] Effective 3D Anderson model, with a metal-insulator transition. For details, see Lemarié et al, PRA 80, (009), arxiv:

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

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

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

19 How to identify unambiguously the transition? jã(p)j hp (t)i 3 increasing K values Time (number of kicks)

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

21 From localization to diffusive regime: experimental results 1= 0 (t) / hp (t)i diffusive <p(t)> ~ t critical regime <p(t)> ~ t/3 localized

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

23 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): hp (t)i» tk1 F [(K Kc )tk ] Asymptotic regimes for F: Localized: k1 ºk = 0 Diffusive: k1 + sk = 1 Together with Wegner's law s = (d )º, it gives k1 = =3 k = 1=3º Numerical and experimental data agree perfectly well with the /3 exponent. hp (t)i Thus, we study (K; t) = versus t and K. t=3

24 Rescaled experimental results increasing K values diffusive (slope -1) hp (t)i 1 (K; t) = =3 ¼ t 0 (t) t=3 0 (t) : Population in the zero-velocity class localized (slope ) increasing time 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.

25 Finite-time scaling Transpose finite size scaling (in configuration space) to the temporal dynamics of the kicked rotor => finite time scaling. Assume the one-parameter scaling law hp (t)i (t) = =3 = F t µ»(k) t1=3 where the localization length» 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.

26 Experimental measurement of the critical exponent Scaling function: hp (t)i = =3 = F t µ»(k) t1=3 Fit using: Experimental points 1 = (K Kc )º +»(K) : cut-off taking into account experimental imperfections º = 1:64 0:08 M. Lopez et al, submitted to PRL arxiv:

27 Outline Anderson localization with cold atoms The periodically kicked rotor: dynamical localization and Anderson localization Anderson localization and Anderson transition for the quasi-periodic three-color kicked rotor Experimental test of the universality of the Anderson transition Critical regime of the Anderson transition Multifractality and fluctuations of the critical wavefunctions

28 Universality of the Anderson transition (numerics) Universality: the critical exponent ν does not depend on the microscopic details. Test for the quasi-periodically kicked rotor, by varying the effective Planck's constant, irrational ratio between the three frequencies, and the depth ε of the modulation. From numerics, we always observe (G. Lemarié et al, Europhys. Lett. 87, (009), arxiv: ) º = 1:58 0:0 Identical to the exponent numerically measured on the Anderson model (Slevin and Ohtsuki, 1999).

29 Universality of the critical exponent: experimental test YES, the critical exponent is universal! Weighted average: º = 1:63 0:05 Table of data sets º = 1:58 M. Lopez et al, submitted to PRL arxiv:

30 Outline Anderson localization with cold atoms The periodically kicked rotor: dynamical localization and Anderson localization Anderson localization and Anderson transition for the quasi-periodic three-color kicked rotor Experimental test of the universality of the Anderson transition Critical regime of the Anderson transition Multifractality and fluctuations of the critical wavefunctions

31 Momentum distribution at the critical point Very localized initial state => hjã(p; t)j i is 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)

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

33 Experimental measurements in the critical regime Characterized by a specific scaling: p / t1=3 p / t0 p / t1=3 p / t1= Raw experimental data Rescaled data 0 p=t p=t 1=3 p=t1= Lemarié et al, Phys. Rev. Lett. 105, (010)

34 Critical wavefunction At the critical point, the diffusion constant scales like: 1=3 D(!) / ( i!) Self-consistent theory of localization (à la Vollhardt-Wölfle) predicts at the critical point: ³ ³ 1=3 1=3 3 jã(p; t)j = 3½3= t Ai 3½3= t jpj (ρ: parameter related to Λc) Asymptotically like exp( jpj3= ) Excellent agreement with numerics and experimental results at short times (less than 1000 kicks)

35 Experimentally measured critical Green function Analytical prediction (Airy function) Experimental points (with error bars) Residual /Airy function Residual /Gaussian Residual /Exponential Lemarié et al, PRL 105, (010) Rescaled momentum p=t1=3

36 Outline Anderson localization with cold atoms The periodically kicked rotor: dynamical localization and Anderson localization Anderson localization and Anderson transition for the quasi-periodic three-color kicked rotor Experimental test of the universality of the Anderson transition Critical regime of the Anderson transition Multifractality and fluctuations of the critical wavefunctions

37 Momentum distribution at long time (critical regime) Extra probability near p=0! Airy function Million of kicks The population at zero velocity sligthly deviates from the t-1/3 prediction of the scaling theory of localization

38 Extra peak near zero-momentum at long time Momentum distribution Numerics Best fit (Airy function) Airy function from the wings Residual p=t1=3 The extra peak is very narrow and grows with time

39 Growth of the extra peak near the starting point Numerical experiment jã(p; t)j Airy function p=t1=3

40 Extra peak near zero-momentum at long time Momentum distribution p=t1=3 p=t1=3 The extra central peak grows and becomes narrower at long time, but the shape is the same, except near the tip.

41 Extra peak near zero-momentum at long time (log scale) jã(p; t)j Universal shape with a short-distance cut-off ~ equal to the mean free path log(p=t1=3) Intensity seems to increase like logarithm of time t

42 Extra peak off criticality Momentum distribution The extra peak vanishes off criticality Momentum p Momentum p

43 Momentum distribution Extra peak off criticality kicks Momentum p At long time in the diffusive regime, no extra peak on top of the Gaussian distribution => extra peak exists only in the very vicinity of the critical point

44 Possible experimental observation of the extra peak Momentum distribution Numerics Airy function 1=3 Scaled momentum p=t Small extra peak at short time => experimental observation would require excellent signal/noise

45 Beyond the average Green function : multifractality of the wavefunctions At the critical point, the eigenstates display strong fluctuations: Regions where the wavefunction is either exceptionally large or exceptionally small => typical multifractal behavior. Can be characterized by the generalized inverse participation Z ratio: q d Pq = jã(r)j d r Scaling with system size L: hpq i / L q In a metal: q = d(q 1) In general: q = d(q 1) + q 1=3 characterization of multifractality ( 0 = 1 = 0) In our case, t plays the role of the system size L, and momentum p the role of position r.

46 Multifractality spectrum for the quasi-periodic kicked rotor No direct access to the eigenstates => study the dynamics of wavepackets as a function of time Time plays the role of the system size. The superposition of Floquet states reduces the fluctuations: jã(t)i = Wavepacket At long time: But X n µ En t cn exp i ján (t)i ~ Floquet eigenstates ján (t + T )i = ján (t)i hjã(r; t)j i ' cn / Án (r = 0) The return probability X n X n jcn j ján (r)j hjã(0; t)j i is proportional to ján (0)j4 => Inverse participation ratio with q=

47 Mapping of the quasiperiodic kicked rotor on the 3D Anderson model Hamiltonian of the 1D quasiperiodic kicked rotor: X p1 H= + K cos x1 (1 + ² cos! t cos!3 t) ±(t n) n Hamiltonian of the 3D periodic kicked pseudo-rotor : X p1 H= +! p +!3 p3 + K cos x1 [1 + ² cos x cos x3 ] ±(t n) n Temporal evolution: Quasiperiodic 1D rotor Periodic 3D pseudo-rotor Ã(x1 ; t = 0) Ã(x1 ; t = 0)±(x )±(x3 ) Ã(x1 ; t) Ã(x1 ; t)±(x! t)±(x3!3 t) The Floquet states of the periodic 3D pseudo-rotor are delocalized in all directions (x1 ; x ; x3 ) => how are multifractal fluctuations affected when delocalized Floquet states are recombined in a wavepacket localized in x and x3?

48 Generalized inverse participation ratio for the quasiperiodic kicked rotor Clear manifestation of multifractality Scales as expected at least up to 100,000 kicks. Compatible with the growth of the extra peak near p=0. Very long time behavior is not clear. P. Akridas-Morel et al., preliminary results See also J. Martin et al., PRE 8, (010)

49 Singularity spectrum Legendre transformation of q : = q0 f ( ) = q q0 q Useful for describing the distribution of jãj Lf ( ) is the measure of the set of points where jã(r)j / L For the quasiperiodic kicked rotor: P. Akridas-Morel et al., preliminary results See also J. Martin et al., PRE 8, (010)

50 Summary Cold atoms are good systems for studying quantum interference, quantum chaos and the quantum dynamics of complex systems. 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 Experimental test of universality => the critical exponent ν is universal. No characteristic scale at the critical point. Self-consistent theory of localization accurately predicts the average intensity Green function at short time. Excellent agreement with experimental observations. Small deviations from the scaling laws are numerically observed at the critical point. They are probably related to strong multifractal fluctuations at criticality: Anomalous scaling of the zero-velocity population Extra narrow peak appearing at very long time near zero velocity. Multifractality spectrum of wavepackets at criticality.

51 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 º = 1:15 0:03 Experiment might be feasible. 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 multifractality??? Effect of quantum statistics (bosons vs. fermions).

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

53 Experimental setup Standard Magneto-Optical Trap