Kicked rotor and Anderson localization

Transcription

1 Kicked rotor and Anderson localization Dominique Delande Laboratoire Kastler-Brossel Ecole Normale Supérieure et Université Pierre et Marie Curie (Paris, European Union) Boulder July 2013

2 Classical anisotropic diffusion Numerics along the 2 or 3 axis Gaussian fits Numerics along the 1 axis Diffusion tensor diagonal in the (1,2,3) axes: Approximate expressions:

3 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

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

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

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

7 How to identify unambiguously the transition? jã(p)j2 hp2 (t)i 3 increasing K values Time (number of kicks) At criticality, one expects an anomalous diffusion with 2 hp (t)i ' t with 2 = 3

8 Phase diagram of the Anderson transition (from numerics) 1000 kicks

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

10 Experimental momentum distributions localized critical distribution diffusive momentum (in units of 2 recoil momenta)

11 Experimental momentum distributions population localized (exponential) diffusive (Gaussian) momentum (in units of 2 recoil momenta)

12 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)

13 Critical regime of the quasi-periodic kicked rotor Excellent agreement with the one-parameter scaling law over 7.5 decades

14 Rescaled dynamics at various times (numerics) increasing time ln (K; t) Critical point K (kick strength)

15 Finite time scaling ) Diffusive 2 ln( ) = ln(hp (t)i=t 2=3 ln»(k) The displacement is proportional to»(k) K Localized 1=3 ln(1=t ) ln(»(k)=t1=3 )

16 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

17 Rescaled experimental results increasing K values diffusive (slope -1) hp2 (t)i 1 (K; t) = 2=3 ¼ 2 t 0 (t) t2=3 0 (t) : Population in the zero-velocity class localized (slope 2) 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.

18 Experimental measurement of the critical exponent Scaling function: 2 hp (t)i = 2=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, PRL, 108, (2012), arxiv:

19 Universality of the critical exponent: experimental test The critical exponent is universal Weighted average: º = 1:63 0:05 Table of data sets º = 1:58 M. Lopez et al, PRL, 108, (2012), arxiv:

20 Prediction of the self-consistent theory of localization Experimental points M. Lopez et al, NJP 15, (June 2013) arxiv: improved predictions simple prediction Very good agreement for the position of the critical point Only fair agreement for the critical conductance

21 Momentum distribution at the critical point 2 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)

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

23 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, Phys. Rev. Lett. 105, (2010)

24 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, (2010) Rescaled momentum p=t1=3