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


2
2
2m
g 
*
   V ext   g      
4 a 
2m
Gross Pitaevskii description works if
L   
2
, V ext  V trap

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 ( ky )  sin ( kz )]
2
2
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 and
act as an ideal system for studying solid state physics phenomenon, with
more tunability of parameters than in actual solids.
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
ˆ ( x) 


i
ai w ( x  xi )
The many boson hamiltonian is
Bose Hubbard Model
H  t

i, j

(aˆ i aˆ j  h .c ) 
U
2
 nˆ ( nˆ
i
i
 1)  (   V T )  nˆ i
i
i
Trapping potential confining frequency
10-200 Hz
Optical Lattice potential confining
frequency 10-40 KHz
U 
4 a s 
m
t
I Bloch, Nature (review article)
2
 | w( x) | d x
4
 w ( x  xi ) [
*

3
2
2m
  V 0 ]w ( x  xi ) d x
2
3
Bose Hubbard Model
Bose Hubbard Model : It describes an interacting boson gas in a lattice
potential, with only onsite interactions.
H  t
 (aˆ
i, j

i
aˆ j  h .c ) 
U
2
 nˆ
i
i
( nˆ i  1)    nˆ i
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
aˆ i  aˆ i   aˆ i
aˆ i   ,  aˆ i  aˆ i


aˆ i aˆ j   ( aˆ i  aˆ i )  
H
MF
i



U
2

i


n
 O( )
2
nˆ i ( nˆ i  1)   ( aˆ i  aˆ i )  

i
i

2
f
i
n
ni
2
  nˆ i
Gutzwiller variational
Wave function
Cold Atoms with long range Interaction
•Example 1 : dipolar cold gases
Example 2: Cold Polar Molecules
•(52Cr Condensate, T. Pfau’s
group Stutgart ( PRL, 2005)
U dd
2
C dd  1  3 cos  

 , C dd   0  m2

3


4 
r

Example 3: BEC coupled with
excited Rydberg states: ( Nath
et al., PRL 2010)
Add Optical lattice
Tight binding approximation
Extended Bose Hubbard Model
Extended Bose Hubbard Model
H  t
 (aˆ

i
aˆ j  h .c ) 
i, j
 V2
 nˆ nˆ
i
k
 V3
 i ,k 
 nˆ nˆ
i
l
U
2
 nˆ ( nˆ
i
i
i
i

 nˆ nˆ
i
j
i, j
NN
 i ,l 
NNN
 1)    nˆ i  V1
NNNN
K Goral et al. PRL,2002
Santos et al. PRL, 2003
Minimal EBH model-just add the nearest neighbor
interaction
H  t

i, j 

(aˆ i aˆ j  h .c ) 
U
2
 nˆ
i
i
( nˆ i  1)    nˆ i  V
i
 nˆ nˆ
i
j
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
DW
Kovrizhin , G. V. Pai, Sinha, EPL 72(2005)
G. G. Bartouni et al. PRL (2006)
Pai and Pandit (PRB, 2005)
(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)  2Vdn
and then to DW(n/2+1) at
  Un  2Vdn
d - being the dimension of
system.
Phase diagram of e-BHM with DW , SS, MI and SF phases
Density Wave Phase :
• Alternating number of particles at each site of the
form | n , n , n , n .... 
1
2
1
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
Supersolid Phase :
( Superfluid +Density wave )
Superfluid
Kim and Chan, Science (2004)
• Why Superfluid?
, there is macroscopic wave|  | 0
function showing superfluid behaviour, flows
effortlessly.
• Why Crystalline ?
Order parameter
shows an
oscillatory behaviour as a function of site coordinate
Soldiers marching along coherently
Magnetic field for neutral atoms
How to create artificial magnetic field for neutral atoms?
Hˆ 0 
2
pˆ
2m

1
2
mr
2
2
  2 1
1
2
2
2
ˆ
ˆ
ˆ
H rot  H 0   L z 
( pˆ  m (   r ))  m (    ) r
2m
2
 m  

B  2  zˆ , A 
(  r )

JILA, Oxford
G. Juzelineus et al.
PRA (2006)
Rotating Optical Lattice
Y J Lin et al. Nature(2009)
Bose Hubbard model in a magnetic field
Hˆ 1 


d r ˆ

4 a 
2

m
ˆ ( x) 



2





2
( r )( 
(   i A )  V o   )ˆ ( r )
2m

d r (ˆ

  2
( r )) ( ( r ))
2
ai w ( x  xi )e

i



x

   
dr  A ( r )


xi


i
H  t

i, j

(aˆ i aˆ j exp( i  ij )  h .c ) 
U
2
 nˆ
i
i
( nˆ i  1)    nˆ i  V
i
 nˆ nˆ
i
j
i, 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
( R.Sachdeva, S.Johri, S.Ghosh arXiv 1005.4391v1 )
magnetic field
H  t


(aˆ i aˆ j exp( i  ij )  h .c ) 
U
i, j
2
 nˆ
i
( nˆ i  1)    nˆ i  V
i
i
 nˆ nˆ
i
i, j 
• Ground state of the Hamiltonian is found by variational minimization
with a Gutzwiller wave function
|  

i
  | H | 
fn | n i
i
n
• For the Density wave phase we have two sublattices A & B
|   (| 
A
A

B
)
N /2
N /2
|
 )(| 

i A 1
n
f
iA
n
* Set m=n
f
iA
n
| n iA 
  n ,n
|
B


i B 1
f mB | m i B 
i
m
f mB   m , n
i
0
0
1
Mott Phase
j
Goldbaum et al ( PRA, 2008)
Umucalilar et al. (PRA, 2007)
Reduced Basis ansatz
H  t


(aˆ i aˆ j exp( i  ij )  h .c ) 
i, j
U
2
 nˆ
i
( nˆ i  1)    nˆ i  V
i
i
 nˆ nˆ
i
j
i, j 
Close to the Mott or Density wave boundary only
two neighboring Fock states are occupied
MI-SF
DW-SS
|   f n 1 | n  1   f n | n   f n 1 | n  1 
|
iA
 f n A1 | n  1   f n A | n   f n A1 | n  1  , n  n 0
|
iB
 f mB1 | m  1   f mB | m   f mB 1 | m  1  , m  n 0  1
i
i
i
i
i
i
( f n A1 , f n A , f n A1 )  (  1 A ,
i
i
i
( f mB1 , f mB , f mB1 )  (  1 B ,
i
i
i
1  1A   2 A ,  2 A )
2
2
1   1B   2 B ,  2 B )
2
2
Variational minimization of the energy gives

A
p
 n  4Vm ,  h   [( n  1)  4Vm ]

B
p
 m  4Vn ,  h   [( m  1)  4Vn ]
A
~ pA, h, B  
B
A,B
p ,h
 
Time dependent variational mean field theory
DW Boundary

 ( k )  2 t (cos k x  cos k y )
Include Rotation
Substitute the variational parameters
(f
iA
n 1
, f
iA
n
, f
iA
n 1)
 [  
iA
1
iA 
A
, 1 |   AA | (|  1A |  |  2A | ),  2A   AA ]
i 
i
i
2
i
2
i
2
i
( f mB1 , f mB , f mB 1 )  [ 1 B   BB , 1 |   BB | (|  1 B |  |  2 B | ),  2 B   BB ]
i
i
i
i
i
2
i
2
i
2
i
i
Two component superfluid order parameter
~
t 
1 
t
 1 2
~ i ,i
,  A ,AB B 
 A  aˆ iA 
iA
 1 (  2 ) A , B
i A ,i B

i *
n
( n    4Vm )
n 1
[1  n
n    4Vm
1    4Vm
 B  aˆ iB 
iB
]

m  1 f mB f mB 1
i *
 A   A ,
iA
Minimize with respect to
the variational parameters

~
 |t |
~
 |t |


iB
1

~i *~i
( A A  B B exp( i  i A , i B )  c .c ) 
i, j
i *
m
 2   1 (m  n, n  m )
iA
1
i *
n  1 f n A f n A1
 [ A  B
 i A ,i B 

~ i A ~ iB
]( nˆ . )[  A  B
T
] 
 
iB
B
~ iA 2
~ iB 2
|

|

|

 A
 B |  EG
i
~ i A * ~ iB *
iB
B
iA
j
~ iA
~ iB
 | A |   | B |  E G
iA
2
2
iB

nˆ  cos  i A , i B xˆ  sin  i A , i B yˆ ,    x xˆ   y yˆ
Harper Equation
~B
 i ( x  1, y ) e
i  y
B
1 ~A
~B
~B
~B
 i  y
 i  x
i  x
  i B ( x  1, y ) e
  i B ( x , y  1) e
  i B ( x  1, y ) e
 ~  iA ( x, y )
t
1 ~B
~A
~A
~A
~A
i  y
 i  y
 i  x
i  x
 i ( x  2, y )e
  i ( x, y )e
  i ( x  1, y  1) e
  i ( x  1, y  1) e
 ~  i ( x  1, y )
t
A
A
A
A
B
Spinorial Harper
Equation
 ~A~B T
1 ~A~B T
ˆ
(
n
.

)[


]


iA
iB
~ [ i A  i B ]
t
 i A ,i B 
~
 ( x , y )  [exp(  i
i
A
,i B
2
) exp(  i
i
A
,i B
)]
T
2
Where the spatial part of the wave function satisfies
~
 ( x  1, y ) e
i  y
1 ~
~
~
~
 i  y
 i  x
i  x
  ( x  1, y ) e
  ( x , y  1) e
  ( x  1, y ) e
 ~  ( x, y )
t
Eigenvalues of Hofstadter butterfly can be mapped to
1
~
t
Hofstadter Butterfly
Hofstadter Equation in Landau gauge
 ( x  a, y)  (x  a, y)  e
ieBay
 ( x  a , y )e
2 hc
  ( x  a , y )e


Color HF Avron et al.
ieBax
hc
ieBax
 ( x, y  a )  e
ieBay
2 hc
e

hc
 ( x , y  a )   ( x , y )
ieBax
2 hc
ieBax
 ( x, y  a )  e
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
Nd 

0
) c
,   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 nΦ sublevels where nΦ=Ba2/φ0
namely the number of fluxes through each unit cell
Hofstadter butterfly
DW Phase Boundary
~
t 
1 
t
 1 2
~ i ,i
,  A ,AB B 
( n    4Vm )
n 1
 1 (  2 ) A , B
i A ,i B
[1  n
n    4Vm
1    4Vm
]
 2   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
i
(  1) [ nˆ i 
1
N
(  nˆ i )
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.