Optical lattices forultracold atomic gases

Sestri Levante, 9 June 2009

Andrea Trombettoni(SISSA, Trieste)

Outlook A brief introduction on ultracold atoms

Why using optical lattices?

Effective tuning of the interactions Experimental realization of interacting lattice Hamiltonians

Ultracold bosons on a disordered lattice: the shift of the critical temperature

Trapped ultracold atoms: Bosons

Bose-Einstein condensation of a dilute bosonic gas

Probe of superfluidity: vortices

System:- typically alkali gases (e.g., Rb or Li)- temperature order of 10-100 nK- number of particles: 103-106

- size order of 1-100 m

Trapped ultracold atoms: Fermions

Tuning the interactions…

… and inducing a fermionic “condensate”

A non-interacting Fermi gas

Ultracold atoms in an optical lattice

2 2 2( ) sin ( ) sin ( ) sin ( ) extV r V k x k y k z

a 3D lattice

It is possible to control:- barrier height- interaction term- the shape of the network- the dimensionality (1D, 2D, …)- the tunneling among planes or among tubes (in order to have a layered structure)…

Tuning the interactions with optical lattices

2

20

ˆ ˆ ˆ ˆ ˆ2 extH d r V r gm

bosonic field̂2

0

4 ag

m

s-wave scattering length

For large enough barrier height ˆ ˆi ii

r r a tight-binding Ansatz [Jaksch et al. PRL (1998)]

†

,

ˆ ˆ ˆ ˆ ˆ ˆ12 i i i i i j

i i i j

Un n n t a a

H

increasing the scattering lengthor

increasing the barrier height the ratio U/t increases

†ˆ ˆ ˆ( )i i in a a

Bose-HubbardHamiltonian

Ultracold fermions in an optical lattice (Fermi-)Hubbard Hamiltonian[Hofstetter et al., PRL (2002) – Chin et al., Nature (2006)]

Quantum phase transitions in bosonic arrays

Increasing V, one passes from a superfluid to a Mott insulator [Greiner et al., Nature (2001)]

1,

, ,

1 ˆ ˆˆ ˆ ˆcos( ) 2 i i j j J i j

i j i j

Q Q E H C ,ˆˆ ˆ ˆ, ; 2 i j i j j jn i Q e n

Similar phase transitions studied in superconducting arrays [see Fazio and van der Zant, Phys. Rep. 2001]:

ˆ iii ia n e

Why using optical lattices?Effective tuning of the interactions

Nonlinear discrete dynamics: negative mass, solitons, dynamical instabilities

Experimental realization of interacting lattice Hamiltonians: Study of quantum & finite temperature phase transitions

[A. Trombettoni, A. Smerzi and P. Sodano, New J. Phys. (2005)]

central peak of the momentum distribution:Good description at finite T by an XY model

thermally driven vortex proliferation

[Schweikhard et al., PRL (2007)]

In the continuous 2D Bose gas BKT transition

observed in the Dalibard group in Paris,

see Hadzibabibc et al., Nature (2006)

Finite temperature Berezinskii-Kosterlitz-Thouless transition in a 2D lattice

2D optical lattices “simulating” graphene

With three lasers suitably placed:

Zhu, Wang and Duan, PRL (2007)

Ultracold bosons and/or fermions in trapping potentials provide new experimentally realizable interacting systems on which to test well-known paradigms of the statistical mechanics:

-) in a periodic potential -> strongly interacting lattice systems

-) interaction can be enhanced/tuned through Feshbach resonances

(BEC-BCS crossover – unitary limit)

-) inhomogeneity can be tailored – defects/impurities can be added

-) effects of the nonlinear interactions on the dynamics

-) strong analogies with superconducting and superfluid systems

-) used to study 2D physics

-) predicted a Laughlin ground-state for 2D bosons in rotation: anyionic excitations

…

Trapped ultracold atoms

Outlook A brief introduction on ultracold atomsWhy using optical lattices?

Effective tuning of the interactions Experimental realization of interacting lattice Hamiltonians

Ultracold bosons on a disordered lattice: the shift of the critical temperature

Infinite-range model: Tc<0, and vanishing Tc for large filling f 3D lattice: ordered limit & connection with the spherical model 3D lattice with disorder: Tc>0 for large f - Tc<0 for small f

with: L. Dell’Anna, S. Fantoni (SISSA), P. Sodano (Perugia)[J. Stat. Mech. P11012 (2008)]

Bosons on a lattice with disorder

† †

,ij i j i i i

i j i

H t a a a a

T

S

Nf

Nfilling

total number of particles

number of sites

,ij it random variables: produced by a speckle or by an incommensurate bichromatic lattice

From the replicated action disorder is similar to an attractive interaction

Replicated action

Introducing N replicas =1,…,N

effective attraction

Shift of the critical temperature

in a continuous Bose gas due to the repulsion

For an ideal Bose gas, the Bose-Einstein critical temperature is

2/3

0 2

3/ 2c

nT

m

What happens if a repulsive interaction is present?

The critical temperature increases for a small (repulsive) interaction…

…and finally decreases

[see Blaizot, arXiv:0801.0009]

Long-range limit (I)

†iji j

i j S

tH a a

N

Without random-bond disorder ijt t

1 0S

S

N eigenvalues

one eigenvalue N

The relation between the number of particles and the chemical potential is

10

0 1

11; 0

11S

T EE

NN E t E

ee

The critical temperature is then

ft

Tke

f cBTkt cB /11ln1

1 )0(

/ )0(

1111

1111

Long-range limit (II)

2 2 2;ij ijt t t v t With random-bond disorder

Using results from the theory of random matrices

0

0

2 0 0 11 2

Xc

Xc

T e X Xv

T e

(0)

0 / ln 1 1/B cX f t k T f

[in agreement with the results for the spherical spin glass

by Kosterlitz, Thouless, and Jones, PRL (1976)]

1

0)0(

ffor

TTT ccc

3D lattice without disorder

†

,i j

i j

H t a a

The relation between the number of particles and the chemical potential is

single particle energies

2 cos cos cosx y zkE t k k k

(0)0

( ) 3 ( ) /

1 1

1 2 1k B CkT E E E k TBZ

k

dkN f

e e

For large filling

)1(

6

)2()0(

0

)0(

3 W

tfTk

EE

Tkkdf cBBZ

k

cB

516.1)1( W

3D lattice, with random-bond and on-site disorder:

3D lattice with disorder

• Introducing N replicas of the system and computing the effective replicated action

• Disorder (both on links and on-sites) is equivalent to an effective

attraction among replicas

• Diagram expansion for the Green’s functions for N 0

• Computing the self-energy

• New chemical potential (effective t larger, larger density of states)

† †

,ij i j i i i

i j i

H t a a a a

3D lattice with disorder:

Results for random-bond disorder

For large filling

When both random-bond and random on-site disorder are present

20

2)0(

01.016.0 vvT

T

c

c

2)0(2

2)0( 16.011)1(9

1

)1(4

9

4

31 vTW

WvTT ccc

3D lattice with disorder: numerical results

results for the continuous (i.e., no optical lattice)Bose gas [Vinokur & Lopatin, PRL (2002)]

2)0(

16.0 vT

T

c

c

20

262

)0(3

)0(2/

6mat

TkmK

T

T cB

c

c

A (very) qualitative explanation

Continuous Bose gas:

Repulsion critical temp. Tc

increasesDisorder “attraction” Tc decreases

Lattice Bose gas: Disorder “attraction”

Small filling continuous limit Tc decreases

Large filling all the band is occupied effective “repulsion”

Tc increases

Thank you!

Some details on the diagrammatic expansion (I)

Green’s functions:

N -> 0

At first order in v02

Some details on the

diagrammatic expansion (II)

24

11

12

1)1(1;

2

1)1(2:

25.06

)1(:

2220

2

20

20

vtW

vtttvW

tvrandomtFor

tvWtvrandomFor

ccij

ci

For large filling, the critical temperature coincides with the critical temperature of the spherical model

,

( )i j ii j

H t S S S real

Connection with the spherical model

The ideal Bose gas is in the same universality class of the spherical model [Gunton-Buckingham, PRL (1968)]

with the (generalized) constraint

2i S T

i

S f N N

Long-range limit (I)

†iji j

i j S

tH a a

N

Without random-bond disorder ijt t

The matrix to diagonalize is

S

th

N

0 1 1 1 1 1 1 1

1 0 1 1 1 1 1 1

1 1 0 1 1 1 1 1

1 1 1 0 1 1 1 1h I

where

1 0S

S

N eigenvalues

one eigenvalue N

The relation between the number of particles and the chemical potential is

10

0 1

11; 0

11S

T EE

NN E t E

ee

The critical temperature is then

1/11ln1

1 )0()0(

/ )0(

ffortfTkf

tTk

ef cBcBTkt cB

ftfTk cB 2/11/)0(

3D lattice with disorder:

Results for an incommensurate potential

Two lattices:

2 2 20 0 0 0

21 1

1 0

sin sin sin

sin cos

/

main incomm

main

incomm i

V r V r V r

V r V k x k y k z

V r V k x v qi

q k k

830 /1076q

206 /1000q

Stabilization of solitons by an optical lattice (I)

Recent proposals to engineer 3-body interactions [Paredes et al., PRA 2007 -Buchler et al., Nature Pysics 2007]

In 1D with attractive 3-body contact interactions: no Bethe solution is available – in mean-field [Fersino et al.,

PRA 2008]:2 2

0 0 0 022c

m x

2 2

3 4

N

N

/ 2TcN const in order to have a finite energy per particle

1/ 22

*0

3 / 30

8cosh 2 2

cx for every for c c

x

Stabilization of solitons by an optical lattice (II)

Problem: a small (residual) 2-body interaction make unstable such soliton solutions

Adding an optical lattice :

2 22 4

0 0 0 0 0 0 02

2

2

( ) sin

opt

opt

g c Vm x

V x qx

Soliton solutions stable for

0.96crit Agq A

for small q

2-Body Contact Interactions

N=2 Lieb-Liniger modelit is integrable and the ground-state energy E can be determined by Bethe ansatz:

2 2

2

1

24T

TT

mc NEN total number of particles

N

Mean-field works for 0c [3]:

[3] F. Calogero and A. Degasperis, Phys. Rev. A 11, 265 (1975)

1 0,...,GS N iix x x

0 x is the ground-state of the nonlinear Schrodinger equation2 2

2

0 0 0 022c

m x

with energy2 2

224T

T

mc NE

N

in order to have a finite energy per particle

TcN const

N-Body Attractive Contact Interactions

We consider an effective attractive 3-body contact interaction and, more generally, an N-body contact interaction:

2 2†

2

† †1 1 1 1

ˆ ˆ ˆ( ) ( )2

1 ˆ ˆ ˆ ˆ... ( )... ( ) ( ,..., ) ( )... ( )! N N N N

H dx x xm x

dx dx x x V x x x xN

1

1 11

( ,..., ) ( )N

N i ii

V x x c x x

With

2 2

† †2

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )2 !

N NcH dx x x dx x x

m x N

contact interactionN-bodyattractive (c>0)