UCSB, May 11, 2004online.itp.ucsb.edu/online/gases_c04/zhou/pdf/Zhou.pdf · UCSB, May 11, 2004...

Introduction to spin correlated states in optical latticesIntroduction to spin correlated states in optical lattices

Fei Zhou, (Univ British Columbia (KITP Quantum Gases Conf 5/11/Fei Zhou, (Univ British Columbia (KITP Quantum Gases Conf 5/11/04)04) 11

Introduction to Spin Correlated States Introduction to Spin Correlated States Introduction to Spin Correlated States Introduction to Spin Correlated States in O ptical L atticesin O ptical L atticesin O ptical L atticesin O ptical L attices

Fei Zhou ( PITP and UBC, Vancouver)

KITP Conference on Quantum Gases, UCSB, May 11, 2004

Thanks: FOM (Nether lands), NSERC (Canada) and a grant from UBC

Outline• Introduction

i) Some general applications ii) S=1/2 Fermions and S=0 bosons in lattices

• Spin-one bosons with anti-ferromagnetic interaction i) Mott states (Nematic states)ii) Valence bond crystals and Spin singlet condensates

• Magnetically Stabilized Nematic Order

• Correlated states as Hamming codes and stabilizer codes



Inform ation processor and Inform ation storageInform ation processor and Inform ation storageInform ation processor and Inform ation storageInform ation processor and Inform ation storage

100…Ghz, 200…GbDVD/CD dr.$$$$(?)

Sim ulations of a cosm ic string netw orkSim ulations of a cosm ic string netw orkSim ulations of a cosm ic string netw orkSim ulations of a cosm ic string netw ork

(Allen & Shellard, 00)



L arge Scale Structure and C osm ic StringsL arge Scale Structure and C osm ic StringsL arge Scale Structure and C osm ic StringsL arge Scale Structure and C osm ic Strings

A lab for A lab for A lab for A lab for baby universe ?baby universe ?baby universe ?baby universe ?

(χ(2), (χ(2), (χ(2), (χ(2), n(2))

Creation of vor tices, monopoles and half vor tices.

(χ(χ(χ(χ(1), n(1))

2

21

ZSS

R××××====



S=1/2 F erm ions in optica l lattices S=1/2 F erm ions in optica l lattices S=1/2 F erm ions in optica l lattices S=1/2 F erm ions in optica l lattices (small hopping limit)

Neel OrderedGapless Spin liquid HTcS made of cold atoms?

S=0 bosons in latticesS=0 bosons in latticesS=0 bosons in latticesS=0 bosons in lattices

In (a) and (b), one boson per site. t is the hopping and can be var ied 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)



Quantum Spin Nematic Statesand spin singlet states

.

/

/

),

R:(,,B:(

.n||Sn,n|)(n:|

|i

e||

i e

)(

n|

−−−−⇔⇔⇔⇔

⇔⇔⇔⇔

>=>=>=>=<<<<>>>>−−−−>=>=>=>=−−−−

>>>>−−−−−−−−>>>>++++>>>>−−−−

>=>=>=>=

21

0

21

02

0

1

0

0

01

12

012

)

sincos

sin,

φ

θ

S=1 bosons w ith A ntiS=1 bosons w ith A ntiS=1 bosons w ith A ntiS=1 bosons w ith A nti----ferrom agnetic interactionsferrom agnetic interactionsferrom agnetic interactionsferrom agnetic interactions

.2,0,,4

)()(

02

,2121

====>>>>====

−−−−====−−−−

FggM

ag

grrrrU

FF

FF

ππππδδδδ

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



S=1 states as AM triplet Cooper pairs

.

).()(

)|d|d|u(|ux

en

);|u|d|d(|uz

en

ikkni)n|n|n|n(|n,|S yxy

21212

1

21212

1

21212

11

>>>>>>>>++++>>>>>>>>====

>>>>>>>>++++>>>>>>>>====

++++⋅⋅⋅⋅====∆∆∆∆⇔⇔⇔⇔>>>>>>>>−−−−++++>>>>−−−−>>>>>=>=>=>=====

αβαβαβαβαβαβαβαβ σσσσσσσσ

Pairs given here are unpolar ized, i.e. .0|| >=< nSn

α

φ

θ

C ondensates of sp in one bosons C ondensates of sp in one bosons C ondensates of sp in one bosons C ondensates of sp in one bosons (d>1)

====++++====++++××××

××××++++====

kzyx

kC

TV

CT

VN

TVN

nQ

C

pBEC.,,

,1

,0;

)!(

),0

(~ αααααααααααα

ααααααααψψψψ

N(Q)

Q

xy

z

φφφφ

θθθθ

Snap shots



H alf vortices H alf vortices H alf vortices H alf vortices

In a half vortex, each atom makes a ππππ spin rotation; a half vor tex car r ies one half circulation of an integer vor tex. A half vor tex r ing is also a hedgehog.

π circulation

y

π spin rotation

Z

x

y

x

The vor tex is or ientated along the z-direction; the spin rotation and circulating current occur in an x-y plane.

Z

r ing

Schem atic of m icroscopic w ave functionsSchem atic of m icroscopic w ave functionsSchem atic of m icroscopic w ave functionsSchem atic of m icroscopic w ave functionsa) NMI; b) SSMI (N=2k); c) SSMI (N=2k+1 in 1d).

Each pair of blue and red dots with a r ing is a spin singlet.

.|!

2/)(~,0;|

!

)(~,1; >>>>∏∏∏∏

++++++++====>>>>∏∏∏∏

++++>>>>>>>>==== vac

k N

Nk

Ck

C

SSMIvac

k N

Nnk

C

NMIEEtz

CS

ααααααααψψψψηηηηααααααααψψψψηηηηηηηη

.:

);(:

.

023

12

3

12

====

−−−−====

>>>>++++<<<<−−−−>>>>++++=<=<=<=<

αβαβαβαβ

αβαβαβαβδδδδββββααααεεεεαβαβαβαβ

γγγγγγγγαβαβαβαβδδδδββββαααααβαβαβαβ

OSSMI

nnNONMI

CCCCO



Tw o particles per siteTw o particles per siteTw o particles per siteTw o particles per site

12||2||

];,2|0,1,2

0,0|[

====++++

>>>>±±±±±±±±====

++++>>>>∏∏∏∏∝∝∝∝ΨΨΨΨ

m

km

m mkk

ααααββββ

ααααββββ

NMI

SSMI

η =6.00η =6.00η =6.00η =6.00

ε ε ε ε vs. η η η η is plotted.

C onstrained quantum rotor m odelC onstrained quantum rotor m odelC onstrained quantum rotor m odelC onstrained quantum rotor m odelvalid when 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

−−−−========++++

−−−−

========

====

⋅⋅⋅⋅>>>><<<<

−−−−====

ηηηη

γγγγαβγαβγαβγαβγεεεεδδδδββββαααα

.,1;:Re SzSl

Sk

Skl

JHf exHAF========⋅⋅⋅⋅

>>>><<<<==== ηηηη



N um ericsN um ericsN um ericsN um erics I: L arge I: L arge I: L arge I: L arge N=2k lim itlim itlim itlim it

]0exp[~)( αβαβαβαβββββαααασσσσ Qnnn −−−−ΨΨΨΨ

η=10.91η=10.91η=10.91η=10.91

SSMI

NMI

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

Magnetically stabilized nematic order



O (2 ) O (2 ) O (2 ) O (2 ) nem aticnem aticnem aticnem atic order in external fieldsorder in external fieldsorder in external fieldsorder in external fields

φ

θ Hz

2S1S

zB

S

HE

tgµµµµ

θθθθ ====

θθθθ

L evel Crossing and L evel Crossing and L evel Crossing and L evel Crossing and N em aticN em aticN em aticN em atic O rder O rder O rder O rder (from a spin singlet state to a nematically ordered state)

0====S

210

2

±±±±±±±±========

,,

,

zS

S

On site Hilber t space

>>>>22,|

>>>>00,|

Pseudo spin Bloch sphere

all spin coherent states except nor th and south pole states havenonvanishing nematic order .



D evelopm ent of D evelopm ent of D evelopm ent of D evelopm ent of N em aticN em aticN em aticN em atic O rderO rderO rderO rder(mapping to a ferromagnetic XXZ model in a z-field )

ββββ

D.P.

U.P.Q

E

U.P.= |2,2> Mott state; D.P.= |0,0> Mott state; O(2) Ferro=Nematic state

O(2) Ferro.

cHh zz −−−−====

Repulsive Attractive

Tr icr i. o(3)

−−−−−−−−⋅⋅⋅⋅−−−−====>>>><<<<>>>><<<< k

kzzlzkl

kzkl

lk hH σσσσσσσσσσσσββββσσσσσσσσ

0

M agnetica lly stabilized M agnetica lly stabilized M agnetica lly stabilized M agnetica lly stabilized nem aticnem aticnem aticnem atic statesstatesstatesstates

2

2

122

2

1

12

c

z

z

h

h

Q

S

−−−−========ΩΩΩΩ

−−−−ΩΩΩΩΩΩΩΩ====

−−−−====

θθθθφφφφφφφφ

δδδδθθθθ

θθθθ

αβαβαβαβββββαααααβαβαβαβ

sin),sin,(cos

),(sin

),cos(

>>>>00,|

>>>>22,|

θθθθ



Fractioanlization of atoms in 1Dand Hamming/Stabilizer codes

Spin singlet quantum “condensates” in 1D optica l latticesSp in singlet quantum “condensates” in 1D optica l latticesSp in singlet quantum “condensates” in 1D optica l latticesSp in singlet quantum “condensates” in 1D optica l lattices(SSQ C)(SSQ C)(SSQ C)(SSQ C)

tDVBC SSQC(“ e” )

t

SSMI SSQC(“ 2e” )

(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

σσσσππππ

σσσσ

σσσσ

++++++++====ΨΨΨΨ====ΨΨΨΨ

>>>><<<<ΓΓΓΓ====

−−−−++++++++++++>>>><<<<

−−−−====

++++====

S=1, “Q =e” bosons w ith A F interactions =>S=1, “Q =e” bosons w ith A F interactions =>S=1, “Q =e” bosons w ith A F interactions =>S=1, “Q =e” bosons w ith A F interactions =>S=0 , “Q =e” bosons interacting via S=0 , “Q =e” bosons interacting via S=0 , “Q =e” bosons interacting via S=0 , “Q =e” bosons interacting via IsingIsingIsingIsing gauge fieldsgauge fieldsgauge fieldsgauge fields

N=2k+1

N=2k



A projected sp in singlet H ilbert spaceA projected sp in singlet H ilbert spaceA projected sp in singlet H ilbert spaceA projected sp in singlet H ilbert space

.,

.1,01,

;2

1,1

1,

,...1

,1,

,...|

evenlk

dk

N

kkd

xkk

kkd

kN

kkd

kN

====++++++++

====++++++++

−−−−====++++

>>>>++++++++

σσσσ

In a projected spin singlet Hilber t 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 neighbor ing site;ii) Each link is occupied by either one singlet pair of atoms or zero.

A n effective H am ilton ianA n effective H am ilton ianA n effective H am ilton ianA n effective H am ilton ian

.,

ˆ

2

;2)ˆ(.).(,0.

;2

.

lkd

klaZH

Nkb

NkC

Echl

bk

bzlkkl

tm

H

ZH

mH

fqcH

>>>><<<<ΓΓΓΓ====

−−−−++++++++++++>>>><<<<

−−−−====

++++====

σσσσ

.1,

~1,

;...11

,1,

~,1...|

...1

,1,

,...|

++++≠≠≠≠++++

>>>>−−−−++++++++++++

>>>>++++++++

kkd

kkd

kN

kkd

kN

kN

kkd

kN



H opping of S=1 B osonsH opping of S=1 B osonsH opping of S=1 B osonsH opping of S=1 B osons

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 str ing-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 str ing of valence bonds between par ticle and hole excitations. Red dots are “ charged” .

Tow erTow erTow erTow er---- like H ilbert space for “odd” M ott stateslike H ilbert space for “odd” M ott stateslike H ilbert space for “odd” M ott stateslike H ilbert space for “odd” M ott states

2,1,| ====>>>> igE iαααα

>>>>2| g>>>>1| g



A 3-bit Hamming code [3,1,3]

0000 →→→→),,( 1111 →→→→),,(

0

1

Q uantum E rror Correction C odeQ uantum E rror Correction C odeQ uantum E rror Correction C odeQ uantum E rror Correction C ode

>>>>2| g >>>>3| g>>>>1| g

>>>>11 |ˆ gE >>>>13 |ˆ gE>>>>12 |ˆ gE >>>>32 |ˆ gE>>>>31 |ˆ gE>>>>23 |ˆ gE>>>>22 |ˆ gE>>>>21 |ˆ gE >>>>33 |ˆ gE

αβαβαβαβββββαααα δδδδ CgEEg ijji >=>=>=>=<<<< ++++ |ˆ|

Code Space

Error Space

Requirement for a QECC1) Errors in different code words are distinguishable; 2) Subspace of er rors are indistinguishable so that there will be no information leakage.



Spin one bosons in optical latticesSp in one bosons in optical latticesSp in one bosons in optical latticesSp in one bosons in optical lattices

Work in progress

• Magnetically stabilized nematic order

• Towards fault tolerant quantum information storage

We have discussed

1) Half vortices in condensates

2) Nematic Mott insulators and spin singlet Mott insulators

3) Valence bond crystals (N=2k+1,1D) and Spin singlet condensates

R eferencesR eferencesR eferencesR eferences

1) Zhou, Phys. Rev. Lett. 87, 080401(2001); Int.Jour. Mod. Phys.B17, 2643-2698 (June, 2003).

2) Demler and Zhou, Phys. Rev. Lett. 88, 163001(2002).Snoek and Zhou, cond-mat/0306198 (2003), PRB (2004) A. Imambekov et al., cond-mat/0306204(2003), PRA (2004)

3) Zhou, cond-mat/0207041, Euro. Phys. Lett. 63,505 (July, 2003). Zhou and Snoek, Annals of Physics 308, 692(2003).

4) A. Imambekov et al., cond-mat/0401526; Zhou et al., PITPpreprint (2004).

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