Spin Correlated States in Optical Lattices

Fei Zhou (ITP, Utrecht) April 15, 2003

PIMS, Banff National Park, Canada

Acknowledgement:

E.Demler (Harvard), F. D. Haldane (Princeton), P. W. Wiegmann (Chicago) G. Barkema, M. Snoek, J. Wiemer (Utrecht)

Funded by the FOM, the Netherlands

Optical lattices

Information processor and Information storage

100…Ghz, 200…GbDVD/CD dr.$$$$(?)

A lab for baby universe ?

n(2))

Creation of vortices, monopoles and half vortices.

(1), n(1))

2

21

ZSS

R

Atoms in optical lattices Vs. electrons in Cuprates

a) Free from imperfections

b) Known interactions

c) Tunable coupling constants

d) ld, 2d and 3d lattices

e) S=0, 1/2, 1, 3/2 atoms

a) Defects or disorder

b) Material dependent

c) Barely changeable

d) layered structures

e) S=1/2 electrons

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)

Absorption images of interference patterns as the laser intensity is increased (from a to h).

(a-d) BECs and (g-h) Mott insulating statesGreiner et al., Nature 415, 39( 02)

.

2/1

0

2/1

)0,2

(:,

0

1

0

),0(:

.0||,|)1(:|

1|2

sin0|cos1|

2sin

),(|

RB

nSnnn

ie

ie

n

S=1 bosons with Anti-ferromagnetic interactions

.2,0

,,4

)()(

02

,2121

F

ggM

ag

grrrrU

FF

FF

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

Schematic of microscopic wave functions

a) NMI; b) SSMI (N=2k); c) SSMI (N=2k+1 in 1d).Each pair of blue and red dots with a ring is a spin singlet.

.|!

2/)(~,0;|

!

)(~,1;

vac

k N

Nk

Ck

C

SSMIvac

k N

Nnk

C

NMIEEtz

CS

.02:

);31

(2:

.312

OSSMI

nnNONMI

CCCCO

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

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

Spin one bosons in optical lattices

Work in progress

• Towards topological fault tolerant quantum information storage

We have found

1) Polar Condensates

2) Nematic Mott insulators

3) Spin singlet Mott insulators

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

5) Spin singlet condensates (1D)