Many-Body Anderson Localization in Disordered Bose Gases
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1 Many-Body Anderson Localization in Disordered Bose Gases Laurent Sanchez-Palencia Laboratoire Charles Fabry - UMR8501 Institut d'optique, CNRS, Univ. Paris-sud 11 2 av. Augustin Fresnel, Palaiseau, France
2 Acknowledgements Quantum Matter Theory team Anderson localization in ultracold atoms M. Piraud, L. Pezzé (now in Florence), B. Hambrecht Many-body localization in disordered Bose Gases S. Lellouch, L.-K. Lim, T.-L. Dao, P. Lugan (now in Lausanne) Quantum Monte Carlo force G. Boeris, G. Carleo Collaboration and stimulating discussions A. Aspect's experimental group, T. Bourdel, P. Bouyer, V. Josse et al. B. van Tiggelen, T. Giamarchi, M. Lewenstein, and many others 2
3 Anderson Localization : A Brief Overview E. Abrahams et al., Phys. Rev. Lett. 42, 673 (1979) One-parameter scaling theory Renormalization Group flow : dln(g) / dlnl = (g) Anderson transition insulating fixed point insulator : g(l) ~ exp(-l/lloc) (g) ~ ln(g) - Ad metallic fixed point (g) d=3 d=2 d=1 ln(g) metal : g(l) ~ Ld-2 (g) ~ d Bd/g 1D : all states localized Lloc ~ l* 2D : all states localized Lloc ~ l*exp(kl*/b2) 3D : mobility edge at kl*= c~1 kl*> c: diffusion kl*< c: localization 3
4 Anderson Localization : A Brief Overview Interplay of disorder and interactions Relevant to many systems Electronic systems (Coulomb interaction) Dirty superconductors 4 He superfluid films in porous media Ultracold atoms An outstanding problem of notorious difficulty Strongly depends on the nature of quantum particles (bosons, fermions,...) the nature of interactions (attractive/repulsive, short/long range,...) the system dimension (1D, 2D, 3D,...) Competition or cooperation of disorder and interactions 4
5 Disordered, Interacting Bose Gas Hertz et al., Phys. Rev. Lett. 43, 942 (1979) Giamarchi and Schulz, Europhys. Lett. 3, 1287 (1987) Giamarchi and Schulz, Phys. Rev. B 37, 325 (1988) Lee and Gunn, J. Phys.: Cond. Matt. 38, 7753 (1990) Altman et al., Phys. Rev. Lett. 93, (2004) Lugan et al., Phys. Rev. Lett. 98, (2007) Falco et al., Europhys. Lett. 85, (2009) 1D case K=3/2 for counterpart in Bose lattice systems, see Fisher et al., Phys. Rev. B 40, 546 (1989) Scalettar et al., Phys. Rev. Lett. 66, 3140 (1991) Rapsch et al., Europhys. Lett. 46, 559 (1999) A highly nontrivial behavior weak interactions compete with the disorder strong interactions cooperate with the disorder 5
6 Disordered, Interacting Bose Gas Hertz et al., Phys. Rev. Lett. 43, 942 (1979) Giamarchi and Schulz, Europhys. Lett. 3, 1287 (1987) Giamarchi and Schulz, Phys. Rev. B 37, 325 (1988) Lee and Gunn, J. Phys.: Cond. Matt. 38, 7753 (1990) Altman et al., Phys. Rev. Lett. 93, (2004) Lugan et al., Phys. Rev. Lett. 98, (2007) Falco et al., Europhys. Lett. 85, (2009) 1D case K=3/2 for counterpart in Bose lattice systems, see Fisher et al., Phys. Rev. B 40, 546 (1989) Scalettar et al., Phys. Rev. Lett. 66, 3140 (1991) Rapsch et al., Europhys. Lett. 46, 559 (1999) A highly nontrivial behavior weak interactions compete with the disorder strong interactions cooperate with the disorder 6
7 Condensed Matter with Ultracold Atoms Flexibility control of parameters (dimensionality, shape, interactions, bosons/fermions/mixtures,...) High-precision measuments complementary to condensed-matter tools variety of diagnostic tools (direct imaging, velocity distribution, oscillations, Bragg spectroscopy,...) from J.R. Ensher et al., Science 269, 198 (1995) Model systems parameters known ab-initio direct comparison with theory towards quantum simulators controlled systems to study basic questions specific ingredients in ultracold atoms 7
8 Condensed Matter with Ultracold Atoms Superfluid-Mott insulator ttransition (Hänsch, Bloch, Esslinger) Tonks-Girardeau gas (Weiss, Bloch) BEC-BCS crossover (Jin, Ketterle, Grimm, Salomon) Quantized vortices in Fermion gases (Ketterle) Berezinskii-Kosterlitz-Thouless crossover (Dalibard, Phillips, Cornell) Spin super-exchange (Phillips, Bloch) Anderson localization (Aspect, Inguscio, Hulet, DeMarco) 8
9 Investigating Quantum Disordered Systems with Ultracold Atoms LSP and M. Lewenstein, Nat. Phys. 6, 87 (2010) Tackling many oustanding challenges on disordered systems with ultracold atoms - Anderson localization - interplay of interactions (non-linearities) and disorder - disordered, strongly-correlated systems (bosons, fermions) - simulators for disordered, spin systems 9
10 Anderson Localization in Ultracold Atom Gases 10
11 Anderson Localization in Ultracold Atom Gases 1D single-particle Anderson localization LSP et al., Phys. Rev. Lett. 98, (2007) Billy et al., Nature 453, 891 (2008) Roati et al., Nature 453, 895 (2008) Piraud et al., Phys. Rev. A 83, (R) (2011) 11
12 Anderson Localization in Ultracold Atom Gases 1D single-particle Anderson localization LSP et al., Phys. Rev. Lett. 98, (2007) Billy et al., Nature 453, 891 (2008) Roati et al., Nature 453, 895 (2008) Piraud et al., Phys. Rev. A 83, (R) (2011) 3D single-particle Anderson localization Kuhn et al., New J. Phys. 9, 161 (2007) Skipetrov et al., Phys. Rev. Lett. 100, (2008) Kondov et al., Science 334, 66 (2011) Jendrzejewski et al., Nature Phys. 8, 398 (2012) Piraud et al., Europhys. Lett. 99, (2012) 12
13 Anderson Localization in Ultracold Atom Gases 1D single-particle Anderson localization LSP et al., Phys. Rev. Lett. 98, (2007) Billy et al., Nature 453, 891 (2008) this workshop RoatiSee et al.,posters Nature 453,in 895 (2008) Piraud et al., Phys. Rev. A 83, (R) (2011) J. Richard et al., «Ultra-cold atoms in disorder: 3D localization and coherent backscattering» Jendrzejewski et al., Nature Phys. 8, 398 (2012) M. Piraud et al., «Matter wave transport and Anderson localization in anisotropic disorder» Piraud et al., Europhys. Lett. 99, (2012) Piraud et al., Phys. Rev. A 85, (2012) 3D single-particle Anderson localization Kuhn et al., New J. Phys. 9, 161 (2007) Skipetrov et al., Phys. Rev. Lett. 100, (2008) Kondov et al., Science 334, 66 (2011) Jendrzejewski et al., Nature Phys. 8, 398 (2012) Piraud et al., Europhys. Lett. 99, (2012) 13
14 Anderson Localization in Ultracold Atom Gases 1D single-particle Anderson localization LSP et al., Phys. Rev. Lett. 98, (2007) Billy et al., Nature 453, 891 (2008) Roati et al., Nature 453, 895 (2008) Piraud et al., Phys. Rev. A 83, (R) (2011) Effects of interactions in expanding gases Paul et al., Phys. Rev. Lett. 98, (2007) Kopidakis et al., ibid. 100, (2008) Pikovsky & Shepelyansky, ibid. 100, (2008) Flach et al., ibid. 102, (2009) Lucioni et al., ibid. 106, (2011) 3D single-particle Anderson localization Kuhn et al., New J. Phys. 9, 161 (2007) Skipetrov et al., Phys. Rev. Lett. 100, (2008) Kondov et al., Science 334, 66 (2011) Jendrzejewski et al., Nature Phys. 8, 398 (2012) Piraud et al., Europhys. Lett. 99, (2012) 14
15 Anderson Localization in Ultracold Atom Gases 1D single-particle Anderson localization LSP et al., Phys. Rev. Lett. 98, (2007) Billy et al., Nature 453, 891 (2008) Roati et al., Nature 453, 895 (2008) Piraud et al., Phys. Rev. A 83, (R) (2011) Effects of interactions in expanding gases Paul et al., Phys. Rev. Lett. 98, (2007) Kopidakis et al., ibid. 100, (2008) Pikovsky & Shepelyansky, ibid. 100, (2008) Flach et al., ibid. 102, (2009) Lucioni et al., ibid. 106, (2011) Effects of interactions in trapped gases 3D single-particle Anderson localization Kuhn et al., New J. Phys. 9, 161 (2007) Skipetrov et al., Phys. Rev. Lett. 100, (2008) Kondov et al., Science 334, 66 (2011) Jendrzejewski et al., Nature Phys. 8, 398 (2012) Piraud et al., Europhys. Lett. 99, (2012) Damski et al., Phys. Rev. Lett. 91, (2003) Roth & Burnett, Phys. Rev. A 68, (2003) Lugan et al., Phys. Rev. Lett. 98, (2007) Lugan et al., Phys. Rev. Lett. 99, (2007) Falco et al., Phys. Rev. A 75, (2007) Fallani et al., Phys. Rev. Lett. 98, (2007) Deissler et al., Nat. Phys. 6, 354 (2010) 15 Pasienski et al., Nat. Phys. 6, 677 (2010)
16 Disordered, Interacting Bose Gas Bose gas at thermal equilibrium in a disordered potential with repulsive interactions any dimension d Ĥint = (g/2) dr (r) (r) (r) (r), 2 with 0 < g (ℏ /m) n 2/d-1 g (ℏ /m) n VR 2 Disordered potential Ĥdis = dr V(r) (r) (r) R V(r) = VRv (r/ R), where VR is the amplitude and R the correlation length 16
17 Disordered, Weakly interacting Bose Gas fixed value of R=ℏ2/2m R2VR Lugan et al., Phys. Rev. Lett. 98, (2007) LSP, Phys. Rev. A 74, (2006) Generic quantum state diagram: The various regimes result from the interplay of kinetic energy, interactions, and disorder. For bosons, strong (mean field) interactions delocalize Boundaries between regimes are straight lines 17
18 Disordered, Weakly interacting Bose Gas fixed value of R=ℏ2/2m R2VR Lugan et al., Phys. Rev. Lett. 98, (2007) LSP, Phys. Rev. A 74, (2006) Lifshits-Anderson glass Interactions determine the populations of single-particle localized states 18
19 Disordered, Weakly interacting Bose Gas Lugan et al., Phys. Rev. Lett. 98, (2007) Lifshits-Anderson regime Single-particle eigenstates : energies P = 1/ dr (r) 4 : participation length N( ) : DoS (stretched exponential) 19
20 Disordered, Weakly interacting Bose Gas Lugan et al., Phys. Rev. Lett. 98, (2007) Lifshits-Anderson regime Single-particle eigenstates : energies P = 1/ dr (r) 4 : participation length N( ) : DoS (stretched exponential) Minimize energy with ansatz 20
21 Disordered, Weakly interacting Bose Gas fixed value of R=ℏ2/2m R2VR Lifshits-Anderson glass Interactions determine the populations of single-particle localized states Lugan et al., Phys. Rev. Lett. 98, (2007) LSP, Phys. Rev. A 74, (2006) fragmented BEC Interplay of interactions and disorder. The interactions tends to merge the fragments 21
22 Disordered, Weakly interacting Bose Gas fixed value of R=ℏ2/2m R2VR Lugan et al., Phys. Rev. Lett. 98, (2007) LSP, Phys. Rev. A 74, (2006) (quasi-)bec Disorder only weakly affects the density profile: single connected fragment Lifshits-Anderson glass Interactions determine the populations of single-particle localized states fragmented BEC Interplay of interactions and disorder. The interactions tends to merge the fragments 22
23 Disordered, Weakly interacting Bose Gas LSP, Phys. Rev. A 74, (2006) (quasi-)bec regime In meanfield approach, The healing length, sets the minimal, varition scale of nc(r) 23
24 Disordered, Weakly interacting Bose Gas LSP, Phys. Rev. A 74, (2006) (quasi-)bec regime In meanfield approach, The healing length, sets the minimal, varition scale of nc(r) For / R 1 (Thomas-Fermi) not fragmented if VR 24
25 Disordered, Weakly interacting Bose Gas P. Lugan et al., Phys. Rev. Lett. 98, (2007) 25
26 Disordered, Weakly interacting Bose Gas Bose gas at thermal equilibrium in a disordered potential with repulsive interactions any dimension d Ĥint = (g/2) dr (r) (r) (r) (r), 2 with 0 < g (ℏ /m) n 2/d-1 g (ℏ /m) n VR 2 Disordered potential Ĥ = dr V(r) (r) (r)interactions dis Strong-enough repulsive R destroy V(r) = localization VRv (r/ R), where VR is the amplitude and R the correlation length 26
27 Disordered, Weakly interacting Bose Gas Bose gas at thermal equilibrium in a disordered potential with repulsive interactions any dimension d Ĥint = (g/2) dr (r) (r) (r) (r), 2 with 0 < g (ℏ /m) n 2/d-1 g (ℏ /m) n VR 2 Disordered potential Ĥ = dr V(r) (r) (r)interactions dis Strong-enough repulsive R destroy V(r) = localization VRv (r/ R), of background! where VR is the amplitude and R the correlation length Let us consider many-body excitations, ie Bogoliubov quasiparticles 27
28 Many-Body Anderson Localization in a Disordered Bose Gas P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) see also N. Bilas and N. Pavloff, Eur. Phys. J. 40, 387 (2006) V. Gurarie et al., Phys. Rev. Lett. 101, (2008) C. Gaul and C.A. Müller, Europhys. Lett. 83, (2008); Phys. Rev. A 83, (2011) Consider the many-body Hamiltonian Apply the standard Bogolyubov-Popov approach Then, determine 1) The (quasi-)bec background nc 2) The Bogoliubov quasi-particles (excitations) 28
29 (Quasi-)BEC Background LSP, Phys. Rev. A 74, (2006) P. Lugan and LSP, Phys. Rev. A 84, (2011) 1) The (quasi-)bec background nc : smoothing solution In meanfield approach, For / R 1 (Thomas-Fermi) not fragmented if VR 29
30 (Quasi-)BEC Background LSP, Phys. Rev. A 74, (2006) P. Lugan and LSP, Phys. Rev. A 84, (2011) 1) The (quasi-)bec background nc : smoothing solution In meanfield approach, For arbitrary value of / R, apply Green-function perturbation not fragmented if VR V(r) : smoothed potential 30
31 (Quasi-)BEC Background LSP, Phys. Rev. A 74, (2006) P. Lugan and LSP, Phys. Rev. A 84, (2011) 1) The (quasi-)bec background nc : smoothing solution V(r) : smoothed potential 31
32 (Quasi-)BEC Background LSP, Phys. Rev. A 74, (2006) P. Lugan and LSP, Phys. Rev. A 84, (2011) 1) The (quasi-)bec background nc : smoothing solution V(r) : smoothed potential 32
33 (Quasi-)BEC Background LSP, Phys. Rev. A 74, (2006) P. Lugan and LSP, Phys. Rev. A 84, (2011) 1) The (quasi-)bec background nc : smoothing solution V(r) should never be approximated by V(r), even for / R 1 V(r) : smoothed potential 33
34 Anderson Localization of Bogoliubov Quasiparticles P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) 2) Collective excitations Insert Then, look for the modes that diagonalize Ĥ 34
35 Anderson Localization of Bogoliubov Quasiparticles P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) 2) Collective excitations 35
36 Anderson Localization of Bogoliubov Quasiparticles P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) 2) Collective excitations 36
37 Anderson Localization of Bogoliubov Quasiparticles P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) To solve the problem, use the auxiliary functions with 37
38 Anderson Localization of Bogoliubov Quasiparticles P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) To solve the problem, use the auxiliary functions with Retain the lowest order terms (in V) maps the BdGE into a single-wave equation with a screened potential Advantages of this formulation : - easier to solve than the coupled BdGEs! - clear physical picture 38
39 Anderson Localization of Bogoliubov Quasiparticles P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) To solve the problem, use the auxiliary functions with Retain the lowest order terms (in V) maps the BdGE into a single-wave equation with a screened potential note : Advantages of this formulation : - easier to solve than the coupled BdGEs! - clear physical picture 39
40 Anderson Localization of Bogoliubov Quasiparticles P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) To solve the problem, use the auxiliary functions with Retain the lowest order terms (in V) maps the BdGE into a single-wave equation with a screened potential Can be solved much easier than the BdGE! note : 40
41 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 41
42 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 42
43 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 43
44 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 44
45 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 45
46 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential For k 1/ : strong screening ~ k2 for k 0 (elastic media) For k 1/ : no screening ~ 1/k2 for k (free particles) 46
47 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 47
48 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 48
49 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential 1D speckle potential Analytics : screened formalism (solid lines : order 2 ; dotted lines : order 3) Numerics : direct diagonalization of the coupled BdGEs Very good agreement R/ =3.70 R/ =1.22 R/ =
50 Anderson Localization of Bogoliubov Quasiparticles : 1D P. Lugan et al., Phys. Rev. Lett. 99, (2007) P. Lugan and LSP, Phys. Rev. A 84, (2011) Many-body AL in a 1D disordered potential For fixed parameters, maximum of localization at k ~ min(1/ ; 1/ R) 1D speckle potential Absolute maximum of localization for ~ R Lloc ~ 280μm experimentally accessible 50
51 Conclusions and Perspectives Disorder and interactions in Bose gases Meanfield level (Gross-Pitaevskii) Repulsive interactions delocalize the density background Equation of state easily obtained Relevance of the «smoothed disordered potential» Many-body excitations (Bogoliubov quasiparticles) Anderson localization Localization properties can be calculated Perspectives Extension to higher dimensions, beyond standard self-consistent theory 51
52 Recent Work Disordered spins with two-component ultracold gases A. Sanpera et al., Phys. Rev. Lett. 93, (2004) V. Ahufinger et al., Phys. Rev. A 72, (2005) J. Wehr et al., Phys. Rev. B 74, (2006) A. Niederberger et al., Phys. Rev. Lett. 100, (2008) Anderson localization in 1D BECs LSP et al., Phys. Rev. Lett. 98, (2007) LSP et al., New J. Phys. 10, (2008) J. Billy et al., Nature 453, 891 (2008) P. Lugan et al., Phys. Rev. A 80, (2009) M. Piraud et al., Phys. Rev. A 83, (R) (2011) M. Piraud and LSP, arxiv: Disordered trapped fermions L. Pezzé et al., Europhys. Lett. 88, (2009) L. Pezzé and LSP, Phys. Rev. Lett. 106, (2011) Anderson localization in dimension d>1 F. Jendrzejewski et al., Nat. Phys. 8, 398 (2012) M. Piraud, L. Pezzé and LSP, Euophys. Lett. 99, (2012) M. Piraud, A. Aspect and LSP, Phys. Rev. A 85, (2012) Disordered, interacting Bose gases and many-body localization LSP, Phys. Rev. A 74, (2006) P. Lugan et al., Phys. Rev. Lett. 98, (2007) P. Lugan et al., Phys. Rev. Lett. 99, (2007) D. Clément et al., Phys. Rev. A 77, (2008) P. Lugan and LSP, Phys. Rev. A 84, (2011) Transport in 2D gases R. Martin-de-Saint-Vincent et al., Phys. Rev. Lett. 104, (2010) 52 L. Pezzé et al., New J. Phys. 13, (2011)
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