Correlated States in Optical Lattices

Fei Zhou (PITP,UBC) Feb. 1, 2004

At Asian Center, UBC

Outline

S=0 bosons (Recent experiments)

S=1 bosons1) Polar condensates2) Mott states (Spin singlet and nematic Mott states)3) Low dimensional Mott states (Dimerized valence bond crystals)4) Spin singlet condensates or superfluid

Approach: Constrained quantum rotor model ( O(2), O(3) rotors and a constraint on an inversion or an Ising constraint)

Optical lattices

S=1/2 Fermions in optical lattices

(small hopping limit)

Neel OrderedGapless Spin liquid HTcS made of cold atoms?

S=0 bosons in lattices

In (a) and (b), one boson per site. t is the hopping and can be varied by tuning laser intensities of optical lattices; U is an intra-site interaction energy. In a Mott state, all bosons are localized.

M. P. A. Fisher et al., PRB 40, 546 (1989);On Mott states in a finite trap, seeJaksch et al., PRL. 81, 3108-3111(1998).

U

Mott states ( t << U)

Condensates (t >>U)

3D real space imagine of a condensate released from a lattice (TOF)

Greiner et al., Nature 415, 41(2002)

Absorption images of interference patterns

as the laser intensity is increased (from a to h). (a-d) BECs and (g-h) Mott insulating states.

Probing excitationsIn (c-f), width of interference peaks versus perturbations.

(c-d) BECs and (e-f) Mott insulators

.

/

/

),π

R:(,,φB:(

.n|α

|Sn,n|)(n:|

|iφθe

θ||iφθe

φ)(θn|

21

0

21

02

0

1

0

0

01

12

012

)

sincos

sin,

S=1 bosons with Anti-ferromagnetic interactions

.2,0,,4

)()(

02

,2121

FggM

ag

grrrrU

FF

FF

Ho, 98; Ohmi & Machida, 98; Law,98.

|1,0> state as a triplet cooper pair

)|d|d|u(|ux

en

);|u|d|d(|uZ

en

)n,/|Sn,/|Sn,/|Sn,/(|Sn,|S

;|diφeθ

|uiφeθ

n,/|S

|diφeθ

|uiφeθ

n,/|S

21212

1

21212

1

221

121

221

121

2

11

22

22

21

22

22

21

/cos/sin

/sin/cos

(This is the Anderson-Morel state!)

Pairs given here are not polarized, i.e. .0|| nSn

Condensates of spin one bosons (d>1)

kzyx

kC

TV

C

TVN

TVN

nQ

C

pBEC.,,

,1

,0;

)!(

),0

(~

N(Q)

Q

xy

z

Snap shots

Half vortices

In a half vortex, each atom makes a spin rotation; a half vortex carries one half circulation of an integer vortex. A half vortex ring is also a hedgehog.

circulation

y

spin rotation

Z

x

y

x

The vortex is orientated along the z-direction; the spin rotation and circulating current occur in an x-y plane.

Z

ring

S=1 bosons with anti-ferromagnetic interactions

in optical lattices (3D and 2D, N=2k)

Polar BEC (a)

Nematic MI (b)

Spin Singlet MI (c)

t: Hopping

).()

;)()

;)()

1

1

optVzt

SE

CE

Cc

SE

CE

CoptVzt

CEb

CE

optVzta

he critical value of is determined numerically.

.

03

)02

(4

03

)02

2(4

MN

aa

SE

MN

aa

CE

Each site is characterized by two unit vectors, blue and red ones. a) Polar BECs (pBEC); b) Nematic mott insulators (NMI); c) Spin singlet mott insulators (SSMI).

Ordering in states present in 3d ,2d lattices

.02:);31

(2:

.312

OSSMInnNONMI

CCCCO

Constrained quantum rotor model valid as the hopping is much less than “U”.

).()(,1)1)(2

.,2

)1

.'

]'

,[

,2)(2

kn

MIH

kn

MIHk

Nk

S

sE

cE

zt

cE

ztex

J

nkk

ik

nk

S

ln

kn

klexJ

kS

ksE

MIH

.,1;:Re SzSl

Sk

Skl

JHf exHAF

Calculated Spectra for given N

In a),b), (n)* (n ) (ground states) as a function of n.

)(,7 nY m

0S

2

4

6

)(0,0 nY

)(,2 nY m

)(,4 nY m

)(,6 nY m

)(,3 nY m

)(,1 nY m

)(,5 nY m

1S

7

5

3

vacN

NnCn

ndn |

!

)()(

4)(

N=2kN=2k N=2k+1N=2k+1

Microscopic wave functionsa) N=2k; b) N=2k+1

Each pair of blue and red dots is a spin singlet. S=2,4 ,6….(in a) or S=3,5,… (in b) spin excitations can be created by breaking singlets.

Numerics I: Large N=2k limit

]0exp[~)( Qnnn

SSMI

NMI

vs. (proportional to hopping) is plotted here. Blue and Green lines represent metal stable states close to the critical point.

The energy Vs.

SSMI

NMI

10.85

Phase diagrams in high dimensions

t

NMIn=1

SSI NMI

n=3 NMI

SSI NMI

BEC E(k,x)

n=3

n=2

n=1

x

n

x

2

1

Mott states in a trap

Spin singlet quantum “condensates” in 1D optical lattices (SSQC)

tDVBC SSQC(“e”)

t

SSMI SSQC(“2e”)

(a)

(b)

]).[exp(ˆ,ˆ

2

;2)ˆ(.).(0.

2.

xklkb

Nik

Ck

C

xklklaZ

H

Nkb

NkC

Echl

bk

bzklkl

tm

H

ZH

mH

fqcH

S=1, “Q=e” bosons with AF interactions ===>S=0 , “Q=e” bosons interacting via Ising gauge fieldsN=2k+1

N=2k

A projected spin singlet Hilbert space

.,

.1,01,

;2

1,1

1,

,...1

,1,

,...|

evenlk

dk

N

kkd

xkk

kkd

kN

kkd

kN

In a projected spin singlet Hilbert space (states in a), b), d) and e)):i) An atom forms a singlet pair with another atom either at the same site or at a nearst neighboring site;ii) Each link is occupied by either one singlet pair of atoms or zero.

An effective Hamiltonian

.,

ˆ

2

;2)ˆ(.).(,0.

;2

.

lkd

klaZH

Nkb

NkC

Echl

bk

bzlkkl

tm

H

ZH

mH

fqcH

.1,

~1,

;...11

,1,

~,1...|

...1

,1,

,...|

kkd

kkd

kN

kkd

kN

kN

kkd

kN

Spin one bosons in optical lattices

Work in progress

• Towards fault tolerant quantum information storage

We have found

1) Half vortices in condensates

2) Nematic Mott insulators and spin singlet Mott insulators

3) Valence bond crystals (N=2k+1,1D)

4) Spin singlet condensates (1D)

References

Zhou, Phys. Rev. Lett. 87, 080401(2001).Zhou, Int. Jour. Mod. Phys. B 17, 2643(2003).

Demler and Zhou, Phys. Rev. Lett. 88,163001(2002).Snoek and Zhou, cond-mat/0306198 (to appear in PRB).Imambekov et al., PRA. 68, 063602 (2003).

Zhou, Euro. Phys. Lett. 63 (4), 505(2003).Zhou and Snoek, Ann. Phys. 309(2), 692(2003). Constrained quantum rotor models 1) Quantum dimers, or compact gauge theories, 2) The bilinear-biqudartic spin model (S=1),3) and fractionalization of atoms etc.

Hopping of S=1 Bosons

a) and d) SSMI states for even and

odd numbers of atoms.b) A kink-like S=0, Q=1 excitation

in an “odd” lattice.e) A string-like Q=1 excitation in

an “even” lattice.c) Hopping in an “odd” lattice

leads to kink-anti kink excitations.

f) In an “even” lattice hopping is suppressed because of a string of valence bonds between particle and hole excitations. Red dots are “charged”.

Excitations in 1D Mott insulators

a)S=1, Q=0; b) S=0, Q=-1 for N=2k+1; c) S=0, Q=-2; d) Q=-1 for N=2k.