Cold Atoms in rotating optical lattice Sankalpa Ghosh, IIT Delhi Ref: Rashi Sachdev, Sonika Johri,...
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Transcript of Cold Atoms in rotating optical lattice Sankalpa Ghosh, IIT Delhi Ref: Rashi Sachdev, Sonika Johri,...
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
HRI Workshop on strong Correlation, Nov. 2010
Bosons and Fermions
Nobel Prizes 1997, 2001
Bose Einstein Condensate of Cold Atoms
Bose Einstein condensate of cold atoms
Characterized by a macroscopic wave function 0N
trapext
ext
VVm
ag
gVm
,2
4
22
*22
Described by Gross-Pitaevski equation
T=nK
Gross Pitaevskii description works if
mgL
2
2
Optical Lattices• Optical lattices are formed by standing waves of counter
propagating laser beams and act as a lattice for ultra cold atoms.
• These systems are highly tunable: lattice spacing and depth can be varied by tuning the frequency and intensity of lasers.
• 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.
)](sin)(sin)([sin),,( 2220 kzkykxVzyxV
Nature, Vol 388, 1997
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 ii
i xxwax The many boson hamiltonian is
i iiTiiji
ji
nVnnU
chaatH ˆ)()1ˆ(ˆ2
).ˆˆ(,
Bose Hubbard Model
Trapping potential confining frequency 10-200 Hz
Optical Lattice potential confining frequency 10-40 KHz
I Bloch, Nature (review article)
xdxxwVm
xxwt
xdxwm
aU
ii
s
30
22
*
342
)(]2
[)(
|)(|4
Bose Hubbard Model Bose Hubbard Model : It describes an interacting boson gas in a lattice potential, with only onsite interactions.
i iiiiji
ji
nnnU
chaatH ˆ)1ˆ(ˆ2
).ˆˆ(,
Fisher et al. PRB (1989)Sheshadri et al. EPL(1993)Jaksch et.al, PRL (1998)
Superfluid phase : sharp interference pattern
Mott Insulator phase : phase coherence lost
t>> U
U >> t
Mean field treatment
Sheshadri et al. EPL (1993)
Decouple the hopping term and retain the terms only linear in fluctuation
iiiiiMFi
iiji
iii
iii
naannU
H
Oaaaa
aaa
aaa
ˆ)ˆˆ()1ˆ(ˆ2
)()ˆˆ(ˆˆ
ˆˆ,ˆ
ˆˆˆ
2
22
in
ini
ii
nf
Gutzwiller variational Wave function
Cold Atoms with long range Interaction •Example 1 : dipolar cold gases •(52Cr Condensate, T. Pfau’s group Stutgart ( PRL, 2005)
203
2
,cos31
4 mdddd
dd Cr
CU
Example 2: Cold Polar Molecules
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
ji
jii i
iiijiji
nnVnnnU
chaatH,,
ˆˆˆ)1ˆ(ˆ2
).ˆˆ(
lli
ikki
i
jiji
i iiiiji
ji
nnVnnV
nnVnnnU
chaatH
ˆˆˆˆ
ˆˆˆ)1ˆ(ˆ2
).ˆˆ(
,3
,2
,1
,
NNN NNNN
NN
K Goral et al. PRL,2002Santos et al. PRL, 2003
Minimal EBH model-just add the nearest neighbor interaction
New Quantum Phases – Density wave and supersolid
T D Kuhner et al. (2000)
Phase diagram of e-BHM with DW , SS, MI and SF phases
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)Pai and Pandit (PRB, 2005)
At t=0, we have transitions betweenDW (n/2) to MI(n) at
and then to DW(n/2+1) at
d - being the dimension of system.
VdnnU 2)1(
VdnUn 2
DW (½)=|1,0,1,0,1,0,......>
MI( 1) =|1,1,1,1,……>
Density Wave Phase : • Alternating number of particles at each site of the form
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
....,,,| 2121 nnnn
Supersolid Phase :
• Why Superfluid? , there is macroscopic wave-function showing superfluid behaviour, flows effortlessly.
• Why Crystalline ? Order parameter shows an oscillatory behaviour as a function of site co-ordinate
0||
( Superfluid +Density wave )
Soldiers marching along coherently
Crystalline
Superfluid
Kim and Chan, Science (2004)
Magnetic field for neutral atoms
How to create artificial magnetic field for neutral atoms?
)(,ˆ2
)(2
1))(ˆ(
2
1ˆˆˆ
2
1
2
ˆˆ
22220
222
0
rm
AzB
rmrmpm
LHH
rmm
pH
zrot
Rotate NIST SchemeJILA, Oxford
G. Juzelineus et al.PRA (2006)
Rotating Optical Lattice
Y J Lin et al. Nature(2009)
Bose Hubbard model in a magnetic field
x
ix
rArdi
ii
i
o
exxwax
rrrdm
a
rVAim
rrdH
)(
222
22
1
)()(ˆ
))(())(ˆ(4
)(ˆ))(2
)((ˆˆ
ji
jii i
iiiijjiji
nnVnnnU
chiaatH,,
ˆˆˆ)1ˆ(ˆ2
).)exp(ˆˆ(
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
ji
jii i
iiiijjiji
nnVnnnU
chiaatH,,
ˆˆˆ)1ˆ(ˆ2
).)exp(ˆˆ(
• Ground state of the Hamiltonian is found by variational minimization with a Gutzwiller wave function
• For the Density wave phase we have two sublattices A & B
|| Hii n
in nf ||
))(|(|| BA
2/
1
||N
i ni
inA
A
A
A nf
2/
1
||N
i mi
imB
B
B
B mf
0,nn
inAf 1,
0 nm
imBf
* Set m=n Mott Phase
( R.Sachdeva, S.Johri, S.Ghosh arXiv 1005.4391v1 )
Reduced Basis ansatz
Goldbaum et al ( PRA, 2008)Umucalilar et al. (PRA, 2007)
Close to the Mott or Density wave boundary only two neighboring Fock states are occupied
1||1|| 11 nfnfnf nnn
ji
jii i
iiiijjiji
nnVnnnU
chiaatH,,
ˆˆˆ)1ˆ(ˆ2
).)exp(ˆˆ(
1,1||1||
,1||1||
011
011
nmmfmfmf
nnnfnfnfBBBB
AAAA
im
im
im
i
in
in
in
i
MI-SF
DW-SS
Variational minimization of the energy gives
),1,(),,(
),1,(),,(
222
21111
222
21111
BBBBim
im
im
AAAAin
in
in
BBB
AAA
fff
fff
BAhp
BAhp
Bh
Bp
Ah
Ap
VnmVnm
VmnVmn
,,
,,
~]4)1[(,4
]4)1[(,4
DW BoundaryTime dependent variational mean field theory )cos(cos2)( yx kktk
Include Rotation
]),|||(|||1,[,,( 22
22
12
1)11AAAAAAAAAA iA
iiiiA
iA
iin
in
in fff
]),|||(|||1,[,,( 22
22
12
1)11BBBBBBBBBB iB
iiiiB
iB
iim
im
im fff
Substitute the variational parameters
Minimize with respect to the variational parameters
Two component superfluid order parameter
*1
*
*
1*
1ˆ
1ˆ
BB
AA
im
im
miB
iBB
i
nin
niA
iAA
ffma
ffna
BB
AA
iB
iB
iA
iA
,
Ai1 Bi
1G
j
iB
ji i
iAii
iB
iA Eccit BA
BA
BA
2
,
2,
*|
~||
~|).)exp(
~~(|~|
GiB
i
iA
i
T
ii
iB
iA
iB
iA Ent B
B
A
ABA
BABA
22
,
**|
~||
~|]
~~)[.ˆ](
~~[|~|
yxyxn yxiiii BABAˆˆ,ˆsinˆcosˆ ,,
),(
]41
41[
1
)4(
)(~,~
12
1
,,21
,,
21
mnnm
Vm
Vmnn
n
Vmn
tt BABA ii
BAiiBA
Harper Equation),(
~~1
),1(~
)1,(~
),1(~
),1(~
yxt
eyxeyxeyxeyx Ai
xiBi
xiBi
yiBi
yiBi ABBBB
),1(~
~1
)1,1(~
)1,1(~
),(~
),2(~
yxt
eyxeyxeyxeyx Bi
xiAi
xiAi
yiAi
yiAi BAAAA
BA
BABAii
TBi
Ai
TBi
Ai t
n,
]~~
[~1
]~~
)[.ˆ( Spinorial Harper Equation
Tiiii BABA iiyx )]2
exp()2
[exp(),(~ ,,
),(~
~1
),1(~
)1,(~
),1(~
),1(~
yxt
eyxeyxeyxeyx xixiyiyi
Where the spatial part of the wave function satisfies
Eigenvalues of Hofstadter butterfly can be mapped to t~1
Hofstadter Butterfly
),(),(),(),(),( yxayxeayxeyaxyax hc
ieBax
hc
ieBax
Hofstadter Equation in Landau gauge
),(),(),(),(),( 2222 yxayxeayxeeyaxeyax hc
ieBax
hc
ieBax
hc
ieBay
hc
ieBay
Color HF Avron et al.
Typically electron in a uniform magnetic field forms Landau Level each of is highly degenerate
e
hcABN
nE
d
c
0,.,
)2
1(
0
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 cellHofstadter butterfly
DW Phase Boundary
),(
]41
41[
1
)4(
)(~,~
12
1
,,21
,,
21
mnnm
Vm
Vmnn
n
Vmn
tt BABA ii
BAiiBA
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
Vortex in a supersolid
Vortex in a superfluid
Checker board vorticesSurrounding superfluid densityShows two sublattice modulation
Highest band of the Hofstadter butterfly
What about the other eigenvalues?
Density wave order parameter i
iii n
Nn )ˆ(
1ˆ[)1(
Good starting points for more general solutions within Gutzwiller approximation
Experimental detection
Time of flight imaging : interference pattern willbear signature of the sublattice modulated superfluiddensity around the core
Bragg Scattering : Structure factor, Phase sensitivity etc.
Real Space technique ?
Momentum space
Thanks for your attention