Pohang, 2010

One-dimensional disordered Bose-Fermi mixtures in optical lattices

Gergely ZarándBudapest Univ. Technology and Economics

Collaborators:Francois Crépin & Pascal Simon (Orsay)

F. Crepin, G.Z., and P. Simon, to appear in PRL, arXiv:1005.2483;

Disordered 1D Bose-Fermi mixtures

Outline

The Bose Glass

Basic Experimental Methods

• Experiments• Creating disorder

• RG analysis and phase diagram• Replica Symmetry breaking• Physical observables

Conclusions

Interacting Bosons

Basic Experimental Methods

Trapping and (over)cooling atoms• Produce ion beam Rb, Na, Cs, K, Li,…• Laser cooling / Doppler Cooling / Zener slower / MOT traps

• Trapping the atoms (conservative traps)

• Evaporative cooling

• Detection:

Magnetic traps Dipol traps

below 1 μK (~20 kHz)

1-100 μK

Absorbtion imaging

(Hyperfine energy ~ 1 mK !)atoms10

K10~T9

Observation: Absorbtion imaging

TOF Detects momentum distribution of atoms :

Release atoms and then take a picture !

Spatial imaging

Absorption images after ballistic expansion of the cloud

(time-of-flight) from W. Ketterle

)/(~~)( 0 tmxpxt

Bose-Einstein condesation

Cold atoms: Highly tunable correlated systems

Tuning interactions:

Feschbach resonances:

Optical lattices 4/3~ sU

set 2~

REs

)(2 xV ma

)(Baa

1-3 dimensions

Bloch, Dalibard, and Zwerger, RMP 2008

Local interaction Scattering length

Recoil energy

Localizing atoms by disorder

Billy et al, Nature 453 2008

Map the wave functionMeasure localization length

Confining laser

speckle laser(disorder)

Rb87

The Bose Glass

Mott transition for Bosons

1ni

JU

UJ

Superfluid:

Insulator:

U

bosons on a lattice

1ni

sx nSFaaSF || 0

1in Superfluid for any J/U

Phase diagram

UJ /

1n

2n

n

SuperFluid

Mott Instulator

Incompressible state

nnnH U )1(2:0J

U/

0Jn

0 1

1

20T

Mott Insulator is incompressible !

Incompressible state

Mott Insulator is incompressible !

U/

0Jn

0 1

1

20T

01

nn:0J

Phase diagram

Bloch, et al. RMP 2008

Interacting bosons + disorder

Mott insulator

Bose glass

[Fisher, Weichman, Grinstein, Fisher, PRB 1989]

Bose-Glass

01

nn

01

nn

Measuring the Bose glass ?

[Fallani et al, PRL 98, 130404 (2007)]

Create quasirandom potential

Getting the Bose glass ?

[Fallani et al, PRL 98, 130404 (2007)]

Excitation spectrum

Bose glass ?

Mott insulatorSuperfluid condensate

disorder

Disordered Bosons in 1D

Bosonization of Bosons… [Haldane, Giamarchi]

Continuum field theory:

bH

Kinetic term compressibility

1n

.).()(1)( )(2 cheexxn xixibbx

bb

Density:

Bose field: )(2/1)( xibi

bexb

bK

Gas can be described in terms of density fluctuations

Luttinger liquid parameter bU 1bK

0bU bK

bKbb xx 2/1||

1~)0()(

Disorder in 1D Bose system

Disorder:

Forward scatteringBackward scattering

)(xb Gaussian static disorder field

)()( xnxVdxnH bbj

jjdis

Bosonize:

RG Analysis of Bosons

Giamarchi and Schulz, PRB 37, 325 (1988)

bK/13/2

bD

0

Luttinger liquid (superfluid) (quasi long-range order, weak repulsion)

Bose glassLocalized bosonsStrong repulsion

(non-perturbative)

*2/1||1~)0()(

bKbbx

x

?0/ tUb

tUb /

1D Disordered Bose-Fermi Mixtures

Interacting 1D Bose-Fermi systems

Hamiltonian:

)( fHfermions bosons interaction)( bH )( bfH

ftbt

bUbfU

Bosonization:

momentum densitybf

incommensurate

Remarks on bosonization of bosons and fermions… ( )

Fermions Bosons

1fK

1fK

bK non-interacting1bK Infinite repulsion

1fK no interactionrepulsive

attractive

Density: ....).()(1)( )(2,,,

,, cheexxnxixi

fbfbxfbfbfb

)()(2/1 .).()( xixixi

fffff echeex

)(2/1)( xi

bbbex

Fermion field Bose field

2/)( 1

||)sin(~)0()(

ff KK

Fff

x

xkx

bKbbx

x 2/1||1~)0()(

0bfU

Diagonalization:

(Wentzel-Bardeen instability)

Linear transformation

• sound velocity

• dimensionless coupling

• instability:• polaronic excitations/superconductivity?

1|| g

Cazalilla and Ho, PRL 2003; Mathey et al., PRL 2004

BF mixture + Disorder

Forward scattering can be guauged out

Add potential: )()()()( xxVdxxxVdx ffbb

backward scattering

Disorder correlations

Interaction-renormalized scaling dimensions

2/)0()(

,

,,

||1~

fb

fbfb ii ee

fbfb Kg ,,:0

• gsKb

f

vv

bbb ,/

Interaction generates effective attraction(~ phonons in superconductivity)

0g fbfb K ,,

Stabilizes SF + generates SC state

• for•

Renormalization group equation

fbfbX ,, 2

Relevance of disorder:

Feed-back of disorder:

Localization length:

Phase diagram

BFG

AG + SFB

LLBFG*

AG + SFB:

interacting Anderson glass +Bosonic Luttinger liquid (SF)

BFG:

Bosons and Fermions localized, but interacting

finitef

finite, bf

LL: Luttinger Liquid

BFG*:

Tiny bosonic localization length

Phase diagram (Rb-K mixture)

Capturing localized phase: Replica trick

Replica Trick:

replSn

a

n eZ afDa

bD

1

Replicated action:

nZZ n

n/)1(limlog

0

replica-interaction generated

Impurity average can be performed

Gaussian variational method

n

b

ba

n

c xxxG10

)( )0()(lim)0()(),(

Connected Green’s function:

Approximate Gaussian action:

),(),()(),(21 1* qqGqS n

bn

abn

aG

Find G that minimizes the Free energy

[Giamarchi and Le Doussal, PRB 53, 15206 (1996)]

GSGSSTeTGF G-Strln][

fnffnvKKK

vv

KK

vv

bnbbnvK

n

c

Iqvqg

qgIqvqG

ffbf

bf

bf

bf

bb

)()(

)()(),(][

221121

2122111)(

• Replica symmetry breaking (~ gap)

• ,

0b if bosons localizedif fermions localized0 f

0)(lim ,0

nfbI

n

2,, ~ fbfb

• and must be determined self-consistently

)(, nfbI fb,

Solution of resulting complicated integral equations:

AG + SFB:

BFG:

One-step replica symmetry breaking

two-step replica symmetry breaking

Bosonic localization: 2~ bb

RG

2-step RSB

How to measure ?

• TOF experiment ~),( Trnb)'()(

',

)'(RRe bb

RR

RRTriMb

*2/11

1~bKr

• Noise correlations

r

detects SF

Can detect SC pairing */2|'|1~)'()(

fKffRR

RR

• Bragg scattering

),( qSmeasures

Fully localized phase

Conclusions

• Disorder + BF interaction new phasesBose-Fermi glass, coexisting localized/LL phases

• For and mixtures parameters are within reach• Phase transitions can be detected by

• time of flight • noise correlations• Bragg scattering localized phases

LiLi 76

)0()( bb x

)0()( ff x

• Open question: bf

Composite fermion glass ?

bf

KRb 4087

F. Crepin, G.Z., and P. Simon, to appear in PRL, arXiv:1005.2483; and other papers under preparation

Localization of bosons by fermions

Ospelkaus et al. PRL 96, 180403 (2006)

Time of flight experimentRbK 8740 mixture

Without fermions

with fermions

Fourier transform of Wannier function '

',

)'()( RRRR

RRki aaekG

txmk /

Phase diagram

Bloch, et al. RMP 2008

Bosons in a trap:

potential )(zV

)(zV

eff

Shell structure

Interaction-renormalized scaling dimensions

fb

fbfb iiee

,

,,

2)0()(

||1~

4/:0 ,, fbfb Kg

2141

g

•• for

gsKb

f

v

v

bbb ,/

Interaction generates effective attraction…