HRI Workshop on strong Correlation, Nov. 2010

Cold Atoms in rotating optical lattice

Sankalpa Ghosh, IIT Delhi Ref: Rashi Sachdev, Sonika Johri, SG arXiv: 1005.4391

Acknowledgement: G.V Pi, K. Sheshadri, Y. Avron, E. Altman

Bosons and Fermions

Nobel Prizes 1997, 2001

Bose Einstein Condensate of Cold Atoms

Bose Einstein condensate of cold atoms T=nK Characterized by a macroscopic wave function  

N

0

Described by Gross-Pitaevski equation



g

 2  2  2

m

 4 

a

 2 2

m



V ext

 ,

V ext



g



V trap

 *      Gross Pitaevskii description works if

L

    2 2

mg

• •

Optical Lattices

Optical lattices are formed by standing waves of counter propagating laser beams and act as a lattice for ultra cold atoms.

V

(

x

,

y

,

z

) 

V

0 [sin 2 (

kx

)  sin 2 (

ky

)  sin 2 (

kz

)] These systems are highly tunable : lattice spacing and depth can be varied by tuning the frequency and intensity of lasers. • Nature, Vol 388, 1997 These optical lattices thus are artificial perfect crystals for atoms more tunability of parameters than in actual solids.

and act as an ideal system for studying solid state physics phenomenon, with

Bose Hubbard Model

If the wavelength of the lattice potential is of the order of the coherence length then the Gross-Pitaevskii description breaks down .

Tight binding approximation

(

x

)  

a i w

(

x



x i i

The many boson hamiltonian is )

Bose Hubbard Model

H

 

t i

  ,

j

 (

a

ˆ

i



a

ˆ

j



h

.

c

) 

U

2 

i

ˆ

i

(

n

ˆ

i

 1 )  (  

V T

) 

i

ˆ

i

Trapping potential confining frequency 10-200 Hz Optical Lattice potential confining frequency 10-40 KHz

U

 4 

a s

 2

m

 |

w

(

x

) | 4

d

3

x t

 

w

(

x



x i

) * [   2 2

m

 2 

V

0 ]

w

(

x



x i

)

d

3

x

I Bloch, Nature (review article)

Bose Hubbard Model

Bose Hubbard Model

: It describes an interacting boson gas in a lattice potential, with only onsite interactions.

H

 

t



i

  ,

j

(

a

ˆ

i



a

ˆ

j



h

.

c

) 

U

2 

i

ˆ

i

( ˆ

i

 1 )   

i n

ˆ

i

Fisher et al. PRB (1989) Sheshadri et al. EPL(1993) Jaksch et.al, PRL (1998) t>> U Superfluid phase : sharp interference pattern Mott Insulator phase : phase coherence lost U >> t

Mean field treatment

Sheshadri et al. EPL (1993) Decouple the hopping term and retain the terms only linear in fluctuation ˆ

i

ˆ

i

 ˆ

i

  ˆ

i

  ,  ˆ

i

 ˆ

i

ˆ

i



j

  ( ˆ

i

 ˆ

i

 )   2 

O

( 2 )

H i MF

 

i

  

U

2  ˆ

i

 (

i i



n f n i

ˆ

i n i

 1 )   ( ˆ

i

 ˆ

i

)   2   ˆ

i

Gutzwiller variational Wave function

Cold Atoms with long range Interaction

• Example 1 : dipolar cold gases • ( 52 Cr Condensate, T. Pfau’s group Stutgart ( PRL, 2005) Example 2: Cold Polar Molecules Example 3: BEC coupled with excited Rydberg states: ( Nath et al., PRL 2010)

U dd



C dd

4  3 cos 2 

r

3   ,

C dd

  0 

m

2 Add Optical lattice Tight binding approximation Extended Bose Hubbard Model

Extended Bose Hubbard Model

H



V

2  

t



i

 ,

j

 (

a

ˆ

i



a

ˆ 

i

, 

k



n

ˆ

i n

ˆ

k



V

3

j

 

i

  ,

l h

.

c

) ˆ

i

ˆ

l



U

2   

i n

ˆ

i

(

n

ˆ

i

 1 )   

i n

ˆ

i



V

1

i

  ,

j

 ˆ

i n

ˆ

j

NN interaction NNN NNNN K Goral et al. PRL,2002 Santos et al. PRL, 2003 Minimal EBH model-just add the nearest neighbor

H

 

t



i

,  

j

(

a

ˆ

i



a

ˆ

j



h

.

c

) 

U

2 

i n

ˆ

i

(

n

ˆ

i

 1 )   

i n

ˆ

i



V



i

,  

j n

ˆ

i j

T D Kuhner et al. (2000) New Quantum Phases – Density wave and supersolid

Due to the competition between NN term and the onsite interaction, new phases such as

Density wave and supersolids are formed

Kovrizhin , G. V. Pai, Sinha, EPL 72(2005) G. G. Bartouni et al. PRL (2006) DW (½)=|1,0,1,0,1,0,......> MI( 1) =|1,1,1,1,……> At t=0, we have transitions between DW (n/2) to MI(n) at  

U

(

n

 1 )  2

Vdn

 

Un

 2

Vdn

at d - being the dimension of system.

Phase diagram of e-BHM with DW , SS, MI and SF phases

Density Wave Phase : Supersolid Phase :

( Superfluid +Density wave )

• Alternating number of particles at each site of the form |

n

1 ,

n

2 ,

n

1 ,

n

2 ....

 Superfluid order parameter or the macroscopic wave function vanishes. There is no coherence between the atomic wave functions at sites, on the other hand site states are perfect Fock states Crystalline Superfluid Kim and Chan, Science (2004) • Why |  |  Superfluid 0 effortlessly.

?

, there is macroscopic wave function showing superfluid behaviour, flows Soldiers marching along coherently • Why Crystalline ?

Order parameter shows an oscillatory behaviour as a function of site co ordinate

Magnetic field for neutral atoms

How to create artificial magnetic field for neutral atoms?

0  2 2

m

 1 2

m

 2 

r

2 

B rot

  2 

z

ˆ , 0 

A

  

m



z

 (   1 2

m r

 ) ( 

m

(  

r

 )) 2  1 2

m

(  2    2 )

r

2 G. Juzelineus et al.

PRA (2006) JILA, Oxford

Rotating Optical Lattice

Y J Lin et al. Nature(2009)

Bose Hubbard model in a magnetic field

 1   

d r

 ˆ  ( 

r

)(   2 2

m

(    

i A

) 2 4 

a

 2

m

(

x

)  

d r

(  ˆ  (

r

)) 2 ( ( 

r

)) 2  

i a i w

(

x



x i

)

e

   

i



x i

 

x



d r

 

A

( 

r

)     

V o

  )  ˆ ( 

r

)

H

 

t



i

,  

j

(

a

ˆ

i



a

ˆ

j

exp(

i



ij

) 

h

.

c

) 

U

2 

i n

ˆ

i

(

n

ˆ

i

 1 )   

i n

ˆ

i



V



i

, 

j

 ˆ

i n

ˆ

j

M. Niemeyer et al(1999), J Reijinders et al. (2004), C. Wu et al. (2004) M Oktel et al. (2007), D. GoldBaum et al. (2008) (2008), Sengupta and Sinha (2010), Das Sharma et al. (2010)

Topological constraint

Extended Bose Hubbard Model under magnetic field

( R.Sachdeva, S.Johri, S.Ghosh arXiv 1005.4391v1 )

H

 

t



i

  ,

j

( ˆ

i



a

ˆ

j

exp(

i



ij

) 

h

.

c

) 

U

2 

i n

ˆ

i

( ˆ

i

 1 )   

i n

ˆ

i



V



i

  ,

j n

ˆ

i j

• Ground state of the Hamiltonian is found by variational minimization with a Gutzwiller wave function |   

i n f n i

|

n



i

  |

H

|   • For the Density wave phase we have two sublattices A & B | 

A

|  

N i A

/ 2   1

n f n i A

 (| 

A f n i A

|

n i A

 

n

,

n

0   )(| 

B

 ) | 

B



f i m B N i B

/ 2   1

m

 

m

,

n f i B m

0  1 |

m i B



* Set m=n Mott Phase

Goldbaum et al ( PRA, 2008) Umucalilar et al. (PRA, 2007)

Reduced Basis ansatz

H

 

t



i

,  

j

(

a

ˆ

i



a

ˆ

j

exp(

i



ij

) 

h

.

c

) 

U

2 

i n

ˆ

i

(

n

ˆ

i

 1 )   

i n

ˆ

i



V



i

, 

j

 ˆ

i n

ˆ

j

Close to the Mott or Density wave boundary only two neighboring Fock states are occupied MI-SF |  

f n

 1 |

n

 1  

f n

|

n

 

f n

 1 |

n

 1  DW-SS | 

i A

| 

i B

 

f n i A

 1 |

n

 1  

f n i A

|

n

 

f n i A

 1 |

n

 1  ,

n



n

0

f i B m

 1 |

m

 1  

f m i B

|

m

 

f m i B

 1 |

m

 1  ,

m



n

0  1

(

f n i A

 1 ,

f n i A

,

f n i A

 1 )  (  1

A

, (

f i m B

 1 ,

f i B m

,

f i B m

 1 )  (  1

B

, 1   1 2

A

  2 2

A

,  2

A

) 1   1 2

B

  2 2

B

,  2

B

) Variational minimization of the energy gives 

A p



n

 4

Vm

, 

h A



B p

 

p A

, ,

h B m

 4

Vn

, 

h B

 

A

,

B p

,

h

    [(

n

 1 )  4

Vm

]   [(

m

 1 )  4

Vn

] Time dependent variational mean field theory  (

k

 DW Boundary )  2

t

(cos

k x

 cos

y

)

Include Rotation

Substitute the variational parameters (

f n i A

 1 ,

f n i A

,

f n i A

 1 )  [ 

i

1

A

 

A i A

 , 1  |  

i A A

| 2 (| 

i

1

A

| 2  | 

i

2

A

| 2 ), 

i

2

A

 

i A A

] (

f i B m

 1 ,

f i B m

,

f i B m

 1 )  [  1

i B

 

B i B

 , 1  |  

i B B

| 2 (|  1

i B

| 2  | 

i B

2 | 2 ),  2

i B

 

B i B

]

Two component superfluid order parameter  

t

~  1 2  

t

 1  2 (

n

  1 (

m

,  ~

A i A

,

B

,

i B

 

n

,

n

 

m

) 1 ( 

n

  1 4

Vm

) [ 1 

n

 2 ) 

i A A

,

B

,

i B n

 1      4

Vm

] 4

Vm



A iA

 

B iB



a

ˆ

iA a

ˆ

iB

  

n



m n

 1

f n i A

*

f n i A

 1 *

m

 1

f i B m

*

f i m B

 * 1 

A i A

  

A i A

,  Minimize with respect to the variational parameters  

i

1

A

 |

t

~ | 

i

 , 

j

1

i B

 (  ~

A i A

*  ~

B i B

exp(

i



i A

,

i B

) 

c

.

c

) 

i

 |  ~

A i A



B i B

| 2  

j

|  ~

B i B

  

B i B

| 2 

E G

   |

t

~ | 

i A

  ,

i B

[  ~

A i A

*  ~

B i B

* ]( ˆ .

  )[  ~

A i A

 ~

B i B

]

T



i



A

|  ~

A i A

| 2  cos 

i A

,

i B



i



B

|  ~

B i B

| 2 

E G

 sin 

i A

,

i B

ˆ ,    

x

 

y

Harper Equation

 ~

i B B

(

x

 1 ,

y

)

e i



y

  ~

i B B

(

x

 1 ,

y

)

e



i



y

  ~

i B B

(

x

,

y

 1 )

e



i



x

  ~

i B B

(

x

 1 ,

y

)

e i



x

 1

t

~  ~

i A A

(

x

,

y

)  ~

i A A

(

x

 2 ,

y

)

e i



y

  ~

i A A

(

x

,

y

)

e



i



y

  ~

i A A

(

x

 1 ,

y

 1 )

e



i



x

  ~

i A A

(

x

 1 ,

y

 1 )

e i



x

 1

t

~  ~

i B B

(

x

 1 ,

y

) 

i A

  ,

i B

( ˆ .

  )[ 

i A A



i B B

]

T

 1

t

~ [ 

i A A



i B B

]

T

Spinorial Harper Equation  ~ (

x

,

y

)  [exp( 

i



i A

,

i B

2 ) exp( 

i



i A

,

i B

2 )]

T

Where the spatial part of the wave function satisfies  ~ (

x

 1 ,

y

)

e i



y

  ~ (

x

 1 ,

y

)

e



i



y

  ~ (

x

,

y

 1 )

e



i



x

  ~ (

x

 1 ,

y

)

e i



x

 1

t

~  ~ (

x

,

y

) Eigenvalues of Hofstadter butterfly can be mapped to 1 ~

Hofstadter Butterfly

Hofstadter Equation in Landau gauge  (

x



a

,

y

)   (

x



a

,

y

) 

e



ieBax hc

 (

x

,

y



a

) Color HF Avron et al. 

e ieBax hc

 (

x

,

y



a

)   (

x

,

y

)  (

x



a

,

y

)

e ieBay

2

hc

  (

x



a

,

y

)

e



ieBay

2

hc



e



ieBax

2

hc

 (

x

,

y



a

) 

e ieBax

2

hc

 (

x

,

y



a

)   (

x

,

y

)

Typically electron in a uniform magnetic field forms Landau Level each of is highly degenerate

E

 (

n

 1 2 )  

c N d

   0 ,  

B

.

A

,  0 

hc e

A plot of such energy levels as a function of Increasing strength of magnetic field will be a set Of straight line all starting from origin If a periodic potential is added as an weak perturbation then it lifts this degeneracy and splits each Landau level into n Φ sublevels where n Φ =Ba 2 /φ 0 namely the number of fluxes through each unit ce ll Hofstadter butterfly

DW Phase Boundary



t

~  2 1   (

n t

 1  2  

n

,  ~

A

,

i A B

,

i B

  1   1 4

Vm

) [ 1 

n

( 

n

2 ) 

i A A

,

B

,

i B

 1      4

Vm

] 4

Vm

  1 (

m



n

,

n



m

) Boundary of the DW & MI phase related to edge eigen value of Hofstadter Butterfly

Modification of the phase boundary due to the rotation or artificial magnetic field

Plot of Eigenfunction

Highest band of the Hofstadter butterfly Vortex in a supersolid Vortex in a superfluid Checker board vortices Surrounding superfluid density Shows two sublattice modulation

What about the other eigenvalues ?

Good starting points for more general solutions within Gutzwiller approximation Density wave order parameter (  1 )

i

[ ˆ

i

 1

N

( 

i

ˆ

i

)

Experimental detection

Real Space technique ?

Time of flight imaging : interference pattern will bear signature of the sublattice modulated superfluid density around the core Momentum space Bragg Scattering : Structure factor, Phase sensitivity etc.