Quantitative study of two- and three-dimensional strong localization of matter waves by atomic scatterers

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 82, Issue 4, 2010, Pages

  Antezza, Mauro ^a   Castin, Yvan ^a   Hutchinson, David A W ^b  

^a LABORATOIRE KASTLER BROSSEL   (France)

^b UNIVERSITY OF OTAGO   (New Zealand)

Author keywords

[No Author keywords available]

Indexed keywords

3D SYSTEMS; BOUND STATE; DENSITY OF LOCALIZED STATE; DENSITY OF STATE; HIGH ENERGY; LOCALIZATION LENGTH; MATTER WAVES; POSITIVE ENERGIES; QUANTITATIVE STUDY; SCATTERING LENGTH; THREE-DIMENSIONAL (3D); TWO-DIMENSIONAL (2D) SYSTEMS;

HIGH ENERGY PHYSICS; SCATTERING; THREE DIMENSIONAL;

ATOMS;

EID: 77957840494     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.82.043602     Document Type: Article

References (39)

1. P. W. Anderson, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.109.1492 109, 1492 (1958).
2. P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.57.287 57, 287 (1985).
3. D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.66.261 66, 261 (1994).
4. F. Evers and A. D. Mirlin, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.80.1355 80, 1355 (2008).
5. J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature NATUAS 0028-0836 10.1038/nature07000 453, 891 (2008).
6. G. Roati, C. D'Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature NATUAS 0028-0836 10.1038/nature07071 453, 895 (2008).
7. J. Chabé, G. Lemarié, B. Grémaud, D. Delande, P. Szriftgiser, and J. C. Garreau, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.255702 101, 255702 (2008).
8. U. Gavish and Y. Castin, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.020401 95, 020401 (2005).
9. P. Massignan and Y. Castin, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.74.013616 74, 013616 (2006).
10. P. Vignolo, Z. Akdeniz, and M. P. Tosi, J. Phys. B JPAPEH 0953-4075 10.1088/0953-4075/36/22/013 36, 4535 (2003);
11. B. Horstmann, J. I. Cirac, and T. Roscilde, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.76.043625 76, 043625 (2007);
12. P. Buonsante, F. Massel, V. Penna, and A. Vezzani, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.79.013623 79, 013623 (2009);
13. K. V. Krutitsky, M. Thorwart, R. Egger, and R. Graham, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.77.053609 77, 053609 (2008).
14. G. Lamporesi, J. Catani, G. Barontini, Y. Nishida, M. Inguscio, and F. Minardi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.153202 104, 153202 (2010).
15. Multiply occupied sites can lead to three-body losses, and to the formation of weakly bound BB or AB dimers, leading to nonelastic scattering of the matter wave. There are several ways to avoid multiple occupancies: to use polarized fermions as scatterers, to use a very small filling factor to make the multiple occupancy statistically irrelevant, or to use experimental techniques such as the radiofrequency one of S. Fölling, A. Widera, T. Müller, F. Gerbier, and I. Bloch, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett. 97.060403 97, 060403 (2006) to filter out the doubly occupied sites.
16. The effective scattering length generally differs from the free space scattering length a, due to the effect of the confining potential experienced by the B atoms: a eff / a ho is a function of a / a ho and of the A - B mass ratio, as calculated in [9].
17. B. A. van Tiggelen, A. Lagendijk, A. Tip, and G. F. Reiter, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/15/5/011 15, 535 (1991).
18. E. Mandonnet, Ph.D. thesis of Université Paris 6, 2000, online at [http://tel.ccsd.cnrs.fr/tel-00011872];
19. F. A. Pinheiro, M. Rusek, A. Orlowski, and B. A. van Tiggelen, Phys. Rev. E EULEEJ 1539-3755 10.1103/PhysRevE.69.026605 69, 026605 (2004).
20. P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, Phys. Rev. B EULEEJ 1098-0121 10.1103/PhysRevB.22.3519 22, 3519 (1980);
21. P. Erdos and R. C. Herndon, Adv. Phys. ADPHAH 0001-8732 10.1080/00018738200101358 31, 65 (1982).
22. / 3 / m as units of length and energy, respectively.
23. The energy of the ABn bound states between an A atom and nB atomic scatterers can be calculated by using Eq. (23) in 3D, or its 2D equivalent ln(aeff/d)=-mi∞(E) from (41), where the eigenvalues m∞ of M∞can be calculated analytically in the case of n≤4.
24. +. Then to first order in δq, E-Edim=-2δq/(maeff). Replacing q with its zeroth order value 1/aeff in the off-diagonal terms of M(E), one obtains the eigenvalue problem (E-Edim)Di=Σj≠ ittrans(rij)Dj.
25. As there is no averaging over disorder, the spatial oscillations in the Green's function are not washed out. For values of ξ larger than the ones presented in the figure, that is for ξ getting larger than the lattice constant d, our fitting procedure based on Eq. (10) is no longer appropriate.
26. This picture is expected for a quadratic matter wave dispersion relation, as considered here. For more complicated dispersion relations, as in the Hubbard model, there may be several mobility edges [3].
27. The generalized Hellmann-Feynman theorem is dmi∞(z)/dz=vi *•[dM∞(z)/dz]u→i, where ui and vi are the right and left eigenvectors, respectively, corresponding to the eigenvalue mi∞(z), and normalized as vi*•ui=1. Since M∞(z) is complex symmetric, one has vi=ui* so that it suffices to calculate the right eigenvector numerically.
28. We note the presence of a narrow vertical band of resonances for small values of the energy E in the first energy pixel of Fig. 7(b). Indeed, for very small values of E an extra density of long-lived resonances appear, with values of a eff spreading over a large interval [including negative values out of the range of Fig. 7(b)]. Nonetheless, these resonances are not spatially localized, and are eliminated by the filtering used in Fig. 7(d). These long-lived extended states correspond to kR ≤ 1, which suggest that they are related to finite size effects.
29. For a uniformly distributed state in the sphere of radius R, one has Vp1/3/R=(4π/3)1/3 1.6 and σ/R=(3/5)1/2 0.77.
30. -1. The participation length lp=(∫Rdxψ(x)2) 2/(∫Rdxψ(x)4) is thus lP=2Nd/(N+1) and has a finite limit 2d when N→+∞
31. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.42.673 42, 673 (1979).
32. L. Pricoupenko, and M. Olshanii, J. Phys. B JPAPEH 0953-4075 10.1088/0953-4075/40/11/009 40, 2065 (2007).
33. I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, San Diego, 1994).
34. A. Lagendijk and B. A. van Tiggelen, Phys. Rep. PRPLCM 0370-1573 10.1016/0370-1573(95)00065-8 270, 143 (1996);
35. A. Derode, A. Tourin, and M. Fink, Phys. Rev. E PRPLCM 1539-3755 10.1103/PhysRevE.64.036605 64, 036605 (2001);
36. D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.99.253902 99, 253902 (2007).
37. Y. Castin, J. Phys. IV France JPICEI 1155-4339 10.1051/jp4:2004116004 116, 89 (2004).
38. In this regime, one could however imagine a highly improbable realization of disorder where a cavity of radius l with no scatterers exists inside the system. If 2 / (m l 2) < ρ g eff there may be a localized state inside this cavity with 0 < E < ρ g eff.
39. S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. Corcovilos, and R. G. Hulet, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.102.090402 102, 090402 (2009).