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

VVmag

gVm

,24

22

*22

Described by Gross-Pitaevski equation

T=nK

Gross Pitaevskii description works if mg

L2

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

nVnnUchaatH 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

xdxwmaU

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

nnnUchaatH 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

naannUH

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

,cos314 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

nnVnnnUchaatH,,

ˆˆˆ)1ˆ(ˆ2

).ˆˆ(

lli

ikki

i

jiji

i iiiiji

ji

nnVnnV

nnVnnnUchaatH

ˆˆˆˆ

ˆˆˆ)1ˆ(ˆ2

).ˆˆ(

,3

,2

,1

,

NNN NNNNNNK 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

)(21))(ˆ(

21ˆˆˆ

21

2ˆˆ

22220

222

0

rmAzB

rmrmpm

LHH

rmmpH

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

rrrdma

rVAim

rrdH

)(

222

22

1

)()(ˆ

))(())(ˆ(4

)(ˆ))(2

)((ˆˆ

ji

jii i

iiiijjiji

nnVnnnUchiaatH,,

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

nnVnnnUchiaatH,,

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 ansatzGoldbaum 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

nnVnnnUchiaatH,,

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

in

in

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 ,,

),(

]4141[

1)4(

)(~,~

12

1

,,21

,,

21

mnnmVmVmnn

nVmn

tt BABA iiBA

iiBA

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 hcieBax

hcieBax

Hofstadter Equation in Landau gauge

),(),(),(),(),( 2222 yxayxeayxeeyaxeyax hcieBax

hcieBax

hcieBay

hcieBay

Color HF Avron et al.

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

ehcABN

nE

d

c

0,.,

)21(

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 cell

Hofstadter butterfly

DW Phase Boundary

),(

]4141[

1)4(

)(~,~

12

1

,,21

,,

21

mnnmVmVmnn

nVmn

tt BABA iiBA

iiBA

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