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

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Giovanni Modugno

LENS and Dipartimento di Fisica e Astronomia, Università di Firenze

EXS2014, ICTP, Trieste

Experiments with ultracold, disordered atomic bosons

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Disordered bosons: an open problem

disorder interaction temperature normal insulator (Anderson) BEC insulator (Bose glass) BEC normal many-body localization?

nly partially understood in theory; very few experiments

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insulator (Anderson)

Interacting bosons in 1D, at T 0

disorder interaction temperature normal BEC insulator (Bose glass) BEC normal

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insulator (Anderson)

Interacting bosons in 1D, at T 0

disorder interaction temperature normal BEC insulator (Bose glass) BEC normal insulator in 1D (Bose glass) quasi-BEC One dimension. Main results from theory:

Anderson localization depends only weakly on energy
Bose-Einstein condensation is marginal
a small Eint competes with disorder and tends to restore superfluidity
for Eint/Ekin>1 the bosons progressively behave like non-interacting

fermions and get again localized

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Disordered bosons at T=0

Giamarchi & Schulz, PRB 37 325 (1988), … Fisher et al PRB 40, 546 (1989), Rapsch, et al., EPL 46 559 (1999), …

continuum lattice In a lattice: non-trivial competition between Bose glass, Mott insulator and superfluid, depending on the site occupation n

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Yu et al., Nature 489 (2012) Fallani et al., PRL 98 (2007) Pasienski et al., Nat. Phys. 6 (2010); Gadway et al., PRL 107 (2011)

Disordered, interacting bosons: experiments

Quantum magnets: thermodynamical systems tuning of disorder and interactions is hard Cold atoms: tuning of disorder and interactions is possible inhomogeneous, temperature control is hard

Deissler et al., Nat. Phys. 6 (2010)

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1 mod

1 2

k k  

S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980). Theory by M. Modugno, A.

Minguzzi, ...

~2D

Aubry-Andrè model with metal-insulator transition at D=2J Exponentially localized states with uniform xLOC

The quasi-periodic lattice

 

   

  

   D    

  i i i i i i i i

n n U n i c h b b J H 1 ˆ ˆ 2 ˆ 2 cos . . ˆ ˆ ˆ

1

 



 dx a m U

4 2

2   

Interaction tuned via a Feshbach resonance (39K atoms) J

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Quasi-1D: the radial trapping energy much larger than the other energy scales

One-dimensional lattices

harmonic trap

Longitudinal trapping: inhomogeneity

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

     



) ( ) ( ) ( x x x x d x g

2

| ) ( | k 

Momentum distribution Spatially averaged correlation function

G » 1/x FT

) / exp( ) ( x x x g  

TOF

Coherence from momentum distribution

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Coherence from momentum distribution

G (units of /d) D/J U/J

The small-U line is from P. Lugan, et al., Phys. Rev. Lett. 98, 170403 (2007), ...

1 10

D’Errico, Lucioni et al., Phys. Rev. Lett. 113, 095301 (2014)

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D/J U/J

shift, wait 0.8ms free expansion prepare in equilibrium

 ideal fluid Incoherent regimes are also insulating

Transport: mobility measurements

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Excitation spectra

“energy” measurement prepare in equilibrium main lattice modulation (15%, 200ms)

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).

D/J U/J

U

D

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Excitation spectra vs non-interacting fermions

Density by DMRG excitation spectrum of non-interacting fermions (correlation function of the hopping operator) D=6.3J D=9.5J

G. Orso et al., Phys. Rev. A 80 033625 (2009)
G. Pupillo et al, New. J. Phys. 8, 161 (2006).

___

Theory by T. Giamarchi (Geneva), G. Roux (Orsay)

D=0 D=6.3J

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) / (

) (

T

x T

e x g

x 



Phenomenological broadening of P(k) with exponential decay of the correlations

1 10 25 20 15 10 5 7 6 5 4 3 2

Γ π Γ π ξT (units of d) U/J Δ/J Δ Δ

Finite-T effects and comparison with theory

fitted thermal length TT

xT = d/arcsinh

) / (

2 / 1

Jn T kB

Texp = 3-6 J

Exp. DMRG, T=0 beyond Luttinger:

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Thermal broadening appears only above a sizable crossover temperature. Exact diagonalization for large U

Finite-T effects: large U

MI BG

The strongly-correlated Bose glass survives at the experimental temperatures (an effect of the “Fermi energy” of fermionized bosons).

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Finite-T effects: small U

We have evidence of a large thermal broadening, but… ... 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.)

kBT=3.1(4)J kBT=4.5(7)J

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1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5

Experiment no damping

low damping

high damping

p0 (h/1)

t (ms)

Transport revisited: clean system

shift, wait a variable t free expansion prepare in equilibrium

pC

Quantum phase slips Unstable regime (interaction-enhanced dynamical instability)

L.Tanzi, et al. Phys. Rev. Lett. 111, 115301 (2013)

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1 2 3 0.0 0.1 0.2 0.3

D = 0 D = 3.6 J D = 10 J

p0 (h/1) t (ms)

Transport revisited: disordered system

The critical momentum is reduced by disorder As pc 0 : the SF to IN crossover from a generalized Laundau criterion

SF IN

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Fluid-insulator crossover from transport

G (units of /d)



) / ( / ) 2 ( J nU A J

c

  D

A = 1.3  0.4  = 0.83  0.22

Theory: P. Lugan, et al., Phys. Rev. Lett. 98, 170403 (2007), L. Fontanesi, et al., Phys. Rev. A 81, 053603 (2010), Altman et al. , ....

First steps towards a quantitative analysis of Bose glass and many-body localization physics in 1D T=0 theory:

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Non interacting particles in 3D disorder

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

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Simple picture of the mobility edge

Energy

xLOC Diffusion coefficient

Ec

Localization length Critical behavior:  x    | |

c LOC

E E 6 . 1 | |     

c

E E D Critical energy:

D 

c

E

m D  

50 years of theory of Anderson localization! e.g. E. Abrahams ed. World Scientific 2013

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Experiments on Anderson localization in 3D

Ultracold atoms: Non-interacting fermions E Ec n(E)

Kondov et al, Science 334,63 (2011)

F. Jendrzejewski et al, Nat. Physics 8, 398 (2012)

Interacting BEC E Ec n(E) 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), ...

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z y x

3D speckles disorder

Same coherent speckles as in Palaiseau ER ≤ 2/msR

270nK

sR D

ER but 39K atoms with tunable interaction

Semeghini, Landini et al., arXiv:1404.3528

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Quasi-adiabatic preparation

t=0.1-5s speckles trap time imaging interactions t=0.2 s

Optimized by minimizing the kinetic energy

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Time evolution of the spatial distribution

2

x

 300 mm

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From diffusion to localization

D

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Momentum distribution

kinetic energy 10 nK much smaller than D =47 nK

The momentum and energy distributions are related by the spectral function

) , ( k E 

: probability of having a momentum k at an energy E

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Energy distribution from momentum distribution

) / exp( ) (

m

E E E f  



 dE E f k E k n ) ( ) , ( ) ( 

) ( ) ( ) ( ) , ( ) ( E f E g dk E f k E E n    

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Excitation spectroscopy

A=20%

speckles trap + interaction time

)) cos( 1 )( ( ) , ( t A t   D  D r r

In the linear regime: E Ec n(E)



) ( ) ( ) ( ) (

2 ,

      D  

i f i f i

E E i f E f P r

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Excitation spectroscopy

A=20%

speckles trap + interaction time

20 40 60 80 100 0.00 0.25 0.50 0.75 1.00

ℏ  kB (nK) N / N(ℏ=0)

)) cos( 1 )( ( ) , ( t A t   D  D r r

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Excitation spectroscopy

Fitting model for the mobility edge:

) ( ) ( ) 1 ( ) , ( '        E pn E n p E n





c

E

dE E n N ) , ( ' ) (  

10 20 30 40 50 60 70 80 90 100 2 4 6 8 10 12 14 16

Ec

n (E) E / kB (nK)

ℏ

p and Ec are fitting parameters

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Excitation spectroscopy

20 40 60 80 100 0.00 0.25 0.50 0.75 1.00

ℏ  kB (nK) N / N(ℏ=0)

D = 47nK p0.5 Ec= 53(6) nK

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20 40 60 80 100 20 40 60 80 100

diffusive

E = ER E / kB (nK) D / kB (nK) E = D

localized

The mobility edge

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Outlook

disorder interaction temperature normal BEC insulator (Bose glass) BEC normal insulator in 1D (Bose glass) quasi-BEC Open questions:

Anderson localization with interactions: many-body localization?
the Bose glass at finite temperature (without Mott physics)
BEC in disorder

insulator (Anderson)

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One dimension: Chiara D’Errico Eleonora Lucioni Luca Tanzi Lorenzo Gori Avinash Kumar, Saptarishi Chaudhuri Theory: Guillaume Roux, Ian Mc Culloch, Thierry Giamarchi

The team

Three dimensions: Manuele Landini Giulia Semeghini Patricia Castilho, Sanjukta Roy, Andreas Trenkwalder, Giacomo Spagnolli, Marco Fattori, Massimo Inguscio