Twin Peaks: momentum-space dynamics of ultracold matter waves in random potentials

Transcription

1 Twin Peaks: momentum-space dynamics of ultracold matter waves in random potentials T. Karpiuk N. Cherroret K.L. Lee C. Müller B. Grémaud C. Miniatura IHP, 7 Nov 2012

2 Experimental and Numerical scenario (1) An ultracold-atom wave packet is launched at t=0 inside a random potential with initial momentum k0. k0 (2) The wave packet evolves during a time t in the speckle potential. Then all fields are switched off. (3) The momentum distribution at time t is extracted and averaged over many disorder realizations. In experiments, it is measured by the time of flight technique.

3 Theoretical Framework (1) 1-particle Schrödinger equation: apple ~ 2 2m r2 + V (r) V (r )V (r + r) = V 2 C(r/ (k,t)= (k,t) 2 (k,t)= F kk 0(t) = Z Initial momentum distribution dk 0 (2 ) Z 2 F kk 0 (t) 0 (k 0 ) Z d! de 2 e i!t kk 2 0 E(!) Disorder strength k 0 Correlation length 2-point correlator Width k k 0 Intensity propagation kernel from k to k via energy shell E L = 1 k Coherence length

4 Theoretical Framework (2) kk 0 E(!) =G E+!/2 (k, k 0 )G E!/2 (k, k0 ) Green s function of Schrödinger equation in Fourier space obeys Bethe-Salpeter equation Diagrammatics... Time evolution of the disorder-averaged momentum distribution? Time scales Ballistic Isotropization Diffusion Localization 0 s loc scattering time transport time localization time coherence time

5 2D Numerical experiment: Early-time dynamics t 0 s k 0 =(2/, 0) Depletion of the initial mode (k,t) ' e t/ s(k 0 ) (k,t= 0) Progressive isotropization of the momentum distribution See Cord Müller s talk CBS peak emerges around k = k 0

6 Intermediate-time dynamics: emblematic CBS peak 0 s loc = 2 loc/d t t loc Isotropic background Fast (diffusive) randomization of the direction of propagation Disorder-broadened ring-shaped structure of mean radius k0 k 0 k CBS Peak at k = k 0 Coherent Enhancement of the backscattering probability

7 Intermediate-time dynamics: CBS theory (k,t)= D (k)+ CBS (k,t) CBS kk 0 E (!) = Max-Crossed D kk 0 E (!) = Ladder [Langer & Neal (1966)] [Gorkov, Larkin, Khmelnitskii (1979)] Diffusive isotropic background D (k)=a(e k E k0 )/2 0 CBS peak CBS (k,t)= D (k)exp Dt (k + k 0 ) 2 See also Acoustics [Tourin, Derode, Roux, van Tiggelen, Fink, PRL 1997] 7

8 CBS Theory: Comparison numerics/analytics Incorporating 0 (k) ) CBS (k,t ) = D (k) exp[ (k + k 0) 2 /(2 q(t) 2 )] (1 + t/ ) Properties of the peak: Angular width Contrast (t) = 0 q1+ /t Initial angular width 0 = k/k 0 C(t) =(1+t/ ) 1 =(2D k 2 ) 1 Coherence time OK with numerics

9 With real experiments too! Jendrzejewski et al. (2012) Labeyrie et al. (2012) See Vincent Josse s poster

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11 Long-time dynamics: Localization and CFS 0 s t = 10 t loc H t' =2 (E) 2 loc Heisenberg time k loc loc Localization triggers a second coherent peak at k = k0 t t = 90 H This coherent forward scattering (CFS) peak twins the CBS peak t = configurations

12 Diagrammatic description of CFS Twin-like structure suggests «Double» CBS effect k 0! k 0! k 0 CFS kk 0 E (!) = = At «lowest order» Diffusive regime Enhancement 1 k 0` 1 Localized regime CFS (k,t)= D (k) t H F ( k k 0 loc ) + Hikami boxes corrections... (Linear) growth of the peak for loc t H «OK at the precision of numerics»...

13 CBS and CFS long-time limit? 0 s loc H t Contrast = Peak maximum Background CONTRAST 0.8 CBS? CFS CBS and CFS peaks look like mirror images TIME Τ Ζ Twin peaks structure for t Simple explanation? H

14 A hand-waving argument (1) Our setup: Infinite system, but wave packet expansion bounded by localization. Effective system size L! loc Associated Heisenberg time H =2 ~ L 2! 2 ~ loc 2 t H Use modal approach (t)i = X n L c n ni e i nt/~ Peaked initial distribution: c n = n (t = 0) n(k 0 )

15 (k,t) 2 = X n,m n(k 0 ) m (k 0 ) n (k) m(k)e i( n m)t/~ = X n n 2 (k 0 ) n 2 (k)+ X n6=m( ) t H n(r) real ) n(k) = n ( k) ) (k,t H )= ( k,t H ) Time-reversal invariance (k = ±k 0,t H ) 2 X n n (k 0 ) 2 n (k) 2 A Constant background ( k 0,t H ) 2 = (k 0,t H ) 2 = X n n(k 0 ) 4 = A Enhancement Factor = n 4 /( n 2 ) 2 1 =2 ( for Gaussian statistics )

16 hvi = ~ m Z A hand-waving argument (2) dkk (k,t) If = D + CBS D (k,t) )hvi =0 D Then ( isotropic) However CBS (k,t) )hvi 6=0 CBS ( peaked at -k0) But localization imposes hvi =0 Hence, either CBS! 0 as time increases... or a forward peak compensates. That s CFS.

17 Remarks, conclusions, outlook (1) Short-time dynamics of the momentum distribution of the wave packet gives access to scattering and transport times High visibility CBS interference effect proves phase coherence. Localization triggers a twin peak. Positive characterisation of Anderson localization in 1D, 2D and 3D. Access to localization length and to localization and Heisenberg times. Related Past Works: Diffusive Recurrent Coherent Forward Peak B. van Tiggelen et al., J. Phys: Condens. Matter 2, 7653 (1990) Dynamical echo and CBS factor 3 Prigodin et al., PRL 72, 546 (1994) Weaver and Lobkis, PRL 84, 4942 (2000)

18 Remarks, conclusions, outlook (2) More hard work awaits! Theoretical description of the full dynamics beyond diagrammatics? Is CFS visible in open geometries? (classical waves setups) Behavior of CFS in 3D media when crossing the mobility edge? 18

19 CBS and CFS Theory: N. Cherroret, T. Karpiuk, C.A. Müller, B. Grémaud, C. Miniatura PRA 85, (2012) T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C.A. Müller, C. Miniatura arxiv , Accepted to PRL (2012) CBS Experiments: F. Jendrzejewski, K. Müller, J. Richard, A. Date, T. Plisson, P. Bouyer, A. Aspect, V. Josse, PRL 109, (2012) Viewpoint by S. Skipetrov, Physics 5, 123 (2012) G. Labeyrie, T. Karpiuk, J.-F. Schaff, B. Grémaud, Ch. Miniatura, D. Delande, arxiv: Thank You!