Experiments with ultracold, disordered atomic bosons Giovanni - PowerPoint PPT Presentation

Experiments with ultracold, disordered atomic bosons Giovanni Modugno LENS and Dipartimento di Fisica e Astronomia, Università di Firenze EXS2014, ICTP, Trieste

Disordered bosons: an open problem disorder many-body localization? insulator (Bose glass) insulator (Anderson) BEC normal interaction BEC normal temperature only partially understood in theory; very few experiments

Interacting bosons in 1D, at T  0 disorder insulator (Bose glass) insulator (Anderson) BEC normal interaction BEC normal temperature

Interacting bosons in 1D, at T  0 disorder insulator insulator in 1D (Bose glass) (Bose glass) insulator (Anderson) quasi-BEC BEC normal interaction BEC One dimension. Main results from theory: • Anderson localization depends only weakly on energy normal temperature • Bose-Einstein condensation is marginal • a small E int competes with disorder and tends to restore superfluidity • for E int /E kin >1 the bosons progressively behave like non-interacting fermions and get again localized

Disordered bosons at T=0 continuum lattice Giamarchi & Schulz , PRB 37 325 (1988), … Fisher et al PRB 40, 546 (1989), Rapsch, et al., EPL 46 559 (1999 ), … In a lattice: non-trivial competition between Bose glass, Mott insulator and superfluid, depending on the site occupation n

Disordered, interacting bosons: experiments Quantum magnets : thermodynamical systems tuning of disorder and interactions is hard Cold atoms : Yu et al., Nature 489 (2012) tuning of disorder and interactions is possible inhomogeneous, temperature control is hard Deissler et al., Nat. Phys. 6 (2010) Pasienski et al., Nat. Phys. 6 (2010); Fallani et al., PRL 98 (2007) Gadway et al., PRL 107 (2011)

The quasi-periodic lattice k   2 mod 1 k 1 J ~2 D     U      ˆ ˆ ˆ      D      ˆ ˆ ˆ . . cos 2 1 H J b b h c i n n n  1 i i i i i 2 i i i Aubry-Andrè model with metal-insulator transition at D =2 J Exponentially localized states with uniform x LOC   2 2    4 Interaction tuned via a Feshbach resonance U a dx m ( 39 K atoms) S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980). Theory by M. Modugno, A. Minguzzi, ...

One-dimensional lattices harmonic trap Quasi-1D: the radial trapping energy much larger than the other energy scales Longitudinal trapping: inhomogeneity

Coherence from momentum distribution FT          2 ( ) ( ) ( ) | ( ) | g x d x x x x k Spatially averaged TOF Momentum distribution correlation function   x ( ) exp( / ) g x x G » 1/ x

Coherence from momentum distribution G (units of  /d ) D / J 1 10 U / J D’Errico, Lucioni et al., Phys. Rev. Lett. 113, 095301 (2014) The small-U line is from P. Lugan, et al., Phys. Rev. Lett. 98, 170403 (2007), ...

Transport: mobility measurements free expansion shift, wait 0.8ms prepare in equilibrium  ideal fluid D / J U / J Incoherent regimes are also insulating

Excitation spectra main lattice modulation “energy” measurement prepare in equilibrium (15%, 200ms) D / J U / J D U Ströferle et al., Phys. Rev. Lett. 92, 130403 (2004), Iucci et al. Phys.Rev. A 73, 041608 (2006); Fallani et al., PRL 98, 130404 (2007).

Excitation spectra vs non-interacting fermions ___ excitation spectrum of non-interacting fermions Density by DMRG (correlation function of the hopping operator) D =0 D =6.3J D =9.5J D =6.3J G. Orso et al., Phys. Rev. A 80 033625 (2009) Theory by T. Giamarchi G. Pupillo et al, New. J. Phys. 8, 161 (2006). (Geneva), G. Roux (Orsay)

Γ Δ π Γ Δ Finite-T effects and comparison with theory π beyond Luttinger: fitted thermal length 1 / 2 x T = d /arcsinh ( / ) k B T Jn 25 7 Phenomenological broadening of TT 20 6 T exp = 3-6 J P ( k ) with exponential decay of the ξ T (units of d) 5 15 correlations Δ /J 4  x  ( / ) x 10 ( ) g x e T T 3 5 2 0 1 10 U/J DMRG, T=0 Exp.

Finite-T effects: large U Exact diagonalization for large U MI BG Thermal broadening appears only above a sizable crossover temperature. The strongly-correlated Bose glass survives at the experimental temperatures (an effect of the “Fermi energy” of fermionized bosons).

Finite-T effects: small U We have evidence of a large thermal broadening, but… k B T=3.1(4)J k B T=4.5(7)J ... the mobility does not show a relevant change with temperature. Relation with many-body localization? (Aleiner, Altshuler, Shlyapnikov, Nature Physics 6, 900 (2010); Michal, Altshuler, Shlyapnikov, arXiv.1402.4796.)

Transport revisited: clean system shift, wait a variable t free expansion prepare in equilibrium 0.5 Experiment no damping 0.4 low damping p 0 ( h /  1 ) high damping p C 0.3 Unstable regime 0.2 (interaction-enhanced dynamical instability) 0.1 Quantum phase slips 0.0 0 1 2 3 4 t (ms) L.Tanzi, et al. Phys. Rev. Lett. 111, 115301 (2013)

Transport revisited: disordered system D = 0 0.3 D = 3.6 J D = 10 J SF IN 0.2 p 0 ( h /  1 ) 0.1 0.0 0 1 2 3 t (ms) The critical momentum is reduced by disorder As p c  0 : the SF to IN crossover from a generalized Laundau criterion

Fluid-insulator crossover from transport G (units of  /d ) T=0 theory:  D   ( 2 ) / ( / ) J A nU J c A = 1.3  0.4  = 0.83  0.22 First steps towards a quantitative analysis of Bose glass and many-body localization physics in 1D Theory: P. Lugan, et al., Phys. Rev. Lett. 98, 170403 (2007), L. Fontanesi, et al., Phys. Rev. A 81, 053603 (2010), Altman et al. , ....

Non interacting particles in 3D disorder disorder insulator in 1D insulator (Bose glass) (Bose glass) insulator (Anderson) quasi-BEC BEC normal interaction BEC normal temperature energy There is a critical energy for localization (Anderson transition): P. W. Anderson, Phys. Rev. 109, 1492 (1958) , ... Not yet measured in experiments!

Simple picture of the mobility edge Localization length Diffusion coefficient x LOC   D m E c Energy    x       | | E E | | 1 . 6 D E E Critical behavior: LOC c c  D E Critical energy: c 50 years of theory of Anderson localization! e.g. E. Abrahams ed. World Scientific 2013

Experiments on Anderson localization in 3D Light waves : Sperling, et al. Nat. Photonics (2012), Wiersma et al, .... Sound waves : Hu et al, Nat. Physics 4, 945 (2008). Atomic kicked rotor : a momentum space version of the Anderson model Chabé et al. Phys. Rev. Lett. 101, 255702 (2008), ... Ultracold atoms : Interacting BEC Non-interacting fermions E E E c E c n(E) n(E) Kondov et al, Science 334,63 (2011) F. Jendrzejewski et al, Nat. Physics 8, 398 (2012)

3D speckles disorder Same coherent speckles as in Palaiseau z y x E R D s R but 39 K atoms with tunable interaction E R ≤  2 /m s R 2  70nK Semeghini, Landini et al., arXiv:1404.3528

Quasi-adiabatic preparation t= 0.2 s t= 0.1-5s trap interactions imaging speckles time Optimized by minimizing the kinetic energy

Time evolution of the spatial distribution 2 x  300 m m

From diffusion to localization D

Momentum distribution kinetic energy  10 nK much smaller than D =47 nK The momentum and energy distributions are related by the spectral function  ( , ) : probability of having a momentum k at an energy E E k

Energy distribution from momentum distribution      ( ) exp( / ) ( ) ( , ) ( ) f E E E n k E k f E dE m     ( ) ( , ) ( ) ( ) ( ) n E E k f E dk g E f E

Excitation spectroscopy trap + A=20% interaction D  D   ( , ) ( )( 1 cos( )) t A t r r speckles time    D 2      ( ) ( ) ( ) ( ) P f E f i E E In the linear regime: r i f i , i f   n(E) E c E

Excitation spectroscopy trap + A=20% interaction D  D   ( , ) ( )( 1 cos( )) t A t r r speckles time 1.00 N / N( ℏ  =0) 0.75 0.50 0.25 0.00 0 20 40 60 80 100 ℏ   k B (nK)

Excitation spectroscopy 16 Fitting model for the mobility edge: E c 14 ℏ   12 E     10 c ( ) ' ( , ) N n E dE 0 8 n (E) 6        ' ( , ) ( 1 ) ( ) ( ) n E p n E pn E 4 2 0 10 20 30 40 50 60 70 80 90 100 p and E c are fitting parameters E / k B (nK)

Excitation spectroscopy D = 47nK 1.00 N / N( ℏ  =0) 0.75 0.50 0.25 p  0.5 E c = 53(6) nK 0.00 0 20 40 60 80 100 ℏ   k B (nK)

The mobility edge 100 E = D diffusive 80 E = E R E / k B (nK) 60 localized 40 20 0 0 20 40 60 80 100 D / k B (nK)

Outlook disorder insulator in 1D insulator (Bose glass) (Bose glass) insulator (Anderson) quasi-BEC BEC normal interaction BEC normal temperature Open questions : • Anderson localization with interactions: many-body localization? • the Bose glass at finite temperature (without Mott physics) • BEC in disorder