B.Reznik

IncollaborationwithJ.IgnacioCirac andErez Zohar

MPQ

1

WorkshoponComputationalComplexityandHighEnergyPhysicsAugust2017

UniversityofMaryland

SIMULATINGLATTICEGAUGETHEORIESWITHCOLDATOMS

OUTLINE

QUANTUMSIMULATIONCOLDATOMSINOPTICALLATTICESHAMILTONIANLATTICEGAUGETHEORY(LGT)

ANALOGSIMULATION:LGT– REQUIREMENTS– EXACTANDEFFECTIVELOCALGAUGEINVARIANCE– LINKSANDPLAQUETTES– Examples:cQED,SU(2)DIGITALSIMULATION

CURRENTEXPERIMENTSOUTLOOK.

QUANTUMSIMULATIONANALOG

COLDATOMS

COLDATOMS

COLDATOMSOPTICALLATTICES

COLDATOMSOPTICALLATTICES

Inthepresence𝑬 𝑟, 𝑡 theatomshasatimedependentdipolemoment𝑑 𝑡 = 𝛼 𝜔 𝑬 𝑟, 𝑡 ofsomenonresonantexcitedstates.Starkeffect:

V r ≡ ΔE r = 𝛼 𝜔 ⟨ 𝑬 𝑟, 𝑡 𝑬 𝑟, 𝑡 ⟩/𝛿

𝛿Atom

COLDATOMSOPTICALLATTICES

(a)2darrayofeffective1dtraps(b)3dsquarelattice

M.Lewenstein et.al,AdvancesinPhysics,2010.

COLDATOMSOPTICALLATTICES

COLDATOMSQUANTUMSIMULATIONS

COLDATOMSQUANTUMSIMULATIONS

24<ORDERSOFMAGNITUDE!!

LONGRANGEFORCES?

LONGRANGEFORCES?

REQUIREFORCECARRIER

‘NEWFIELD’

REQUIREFORCECARRIER

‘GAUGEFIELD’

THESTANDARDMODEL

• Matter:=fermions(QuarksandLeptons w.

mass,spin1/2,flavor,charge)

• Interactionsmediators:=YMgaugefields(spin1bosons).

Electromagnetic:masslesschargeless photon,(1),U(1)Weakinteraction:massive,chargedZ,W’s,(3),SU(2)Stronginteraction:masslessGluons,(8),SU(3)

GAUGEFIELDS

Abelian FieldsMaxwell

Non-Abelian fieldsYang-Mills

Massless Massless

Long-rangeforces Confinement

Chargeless Carrycharge

Lineardynamics Selfinteracting& NL

𝛼345 ≪ 1,𝑉345 𝑟 ∝ :;

We(ordinarily)don’tneedQFTquantumfieldtheorytounderstandthestructureofatoms:

𝑚=𝑐? ≫ 𝐸BCDE=;F ≃ 𝛼345? 𝑚=𝑐?

Butalsohigherenergieseffectsarewelldescribedusingperturbationtheory- (Feynmandiagrams)workswell.

QED

• QuantumChromodynamicsasymptoticfreedom:athighenergies,couplingconstant‘goes’tozero.

• Thenucleus,areseenasbuiltof‘free’point-likeparticles=quarks.

QCD:ATHIGHENERGYASYMPTOTICFREEDOM

r

V(r)

“StrongCoulombpotential”

QCD:ATLOWENERGIESASYMPTOTICFREEDOM

𝛼3H5 > 1, 𝑉3H5 𝑟 ∝ 𝑟Non-perturbativeconfinementeffect

Nofreequarks!theyconstructHadrons:Mesons(twoquarks),Baryons(threequarks),…ColorElectricflux-tubes:“anon-abelianMeissner effect”. r

V(r)Staticpot.forapairofheavyquarks

Coulomb

Confinement

Q Q

Q Q

(some)OPENPROBLEMS

–MassgapofYang-Mills(puregauge)theories.–Phasesofnon-Abeliantheorieswithfermionicmatter–Colorsuperconductivity?– Quark-gluonPlasma.– Confinement/deconfinement ofdynamicalcharges

–High-Tcsuperconductivity?

LATTICEGAUGETHEORIES

LATTICEGAUGETHEORIESHAMILTONIANFORMULATION

LATTICEGAUGETHEORY

Generators:

Gaugetransformation:

Gaugegroupelements:

Ur isanelementofthegaugegroup(intherepresentationr),oneachlink

Leftandrightgenerators:

LATTICEGAUGETHEORIESHAMILTONIANFORMULATION

Gaugetransformation:

Matter:

LATTICEGAUGETHEORIESHAMILTONIANFORMULATION

Matterdynamics:

Gaugefielddynamics (Kogut-SusskindHamiltonian):

Strongcouplinglimit:g>>1Weakcouplinglimit:g<<1

LATTICEGAUGETHEORIESHAMILTONIANFORMULATION

TOYEXAMPLE:U(1)

𝜓K 𝜓KL:

TOYEXAMPLE:U(1)

Hisinvariantunderglobal transformations:

TOYEXAMPLE:U(1)

Promotethetransformationtobelocal:

Addanewfield onthelinks:

𝜓K 𝜓KL:

TOYEXAMPLE:U(1)

Invarianceunderalocal gaugetransformations:

TOYEXAMPLE:U(1)

Gaugefield dynamics:

KS:“U(1)Rigidrotatator”

Gaugefielddynamics:PLAQUETTES

Inthecontinuumlimit,thisREDUCESto 𝛻×𝑨 ? : themagneticenergydensity.

TOYEXAMPLE:U(1)

COMPACTQED(cQED)

+

Electricenergy+Magneticenergy+Gauge-Matterinteraction

+

QUANTUMSIMULATIONS

REQUIREMENTS:HEPmodels

FieldsFermion MatterfieldsBosonic gaugefields

LocalgaugeinvarianceExact,orlowenergy,effective

RelativisticinvarianceCausalstructure,inthecontinuumlimit

QUANTUMSIMULATIONCOLDATOMS

Fermion matterfieldsBosonic gaugefields

Superlattices:

Atominternallevels

1)EFFECTIVEGAUGEINVARIANCE

Gauss’slaw isaddedasaconstraint.LeavingthegaugeinvariantsectorofHilbertspacecoststoomuchEnergy.

LowenergysectorwithaeffectivegaugeinvariantHamiltonian.

E.Zohar,B.Reznik,Phys.Rev.Lett.107,275301(2011)

Δ ≫ 𝛿𝐸

…..

𝛿𝐸

Gaugeinvariantsector

NotGaugeinvariant

2)EXACTGAUGEINVARIANCE

• AtomicSymmetries« LocalGaugeInvariance

ABELIANCASE:E.Zohar,J.I.Cirac,B.Reznik,Phys.Rev.A88 023617(2013)

NON-ABELIANCASE:E.Zohar,J.I.Cirac,B.Reznik,Rep.Prog.Phys.79,014401(2016)

LINKS

cQED LINK

F-BScattering

𝜓D

ΦQ,ΦE

FFermion

𝜓R

cQED LINK

FFermion

𝜓D

ΦQ,ΦE

𝜓R

LZ LZ - 1

cQED LINK

FFermion

𝜓D

ΦQ,ΦE

𝜓R

cQED LINK

FFermion

𝜓D

ΦQ,ΦE

𝜓R

LZ LZ +1

cQED LINK

𝜓RS ΦQS ΦE𝜓D + 𝜓DS ΦE

S ΦQ𝜓R

mF (c)

mF (d)

mF (a)

mF (b)

𝜓D

ΦQ,ΦE

𝜓R

cQED LINK

mF (c)

mF (d)

mF (a)

mF (b)

𝜓D

ΦQ,ΦE

𝜓R

𝜓RS ΦQS ΦE𝜓D + 𝜓DS ΦE

S ΦQ𝜓R

cQED LINK

mF (c)

mF (d)

mF (a)

mF (b)

𝜓D

ΦQ,ΦE

𝜓R

𝜓RS ΦQS ΦE𝜓D + 𝜓DS ΦE

S ΦQ𝜓R

cQED LINK

𝐿L = ΦQS ΦE ;𝐿V = ΦE

S ΦQ

𝐿W =:?𝑁Q − 𝑁E ;𝑙 = :

?𝑁Q + 𝑁E ΦQ,ΦE

cQED LINK

andthuswhatwehaveis

𝜓RS𝐿L𝜓D ~𝜓R

S𝑒]^𝜓D

𝜓RSΦQ

S ΦE𝜓D + 𝜓DSΦE

S ΦQ𝜓R

𝐿L = ΦQS ΦE ;𝐿V = ΦE

S ΦQ

𝐿W =:?𝑁Q − 𝑁E ;𝑙 = :

?𝑁Q + 𝑁E

whereforlarge𝑙 ,𝑚 ≪ 𝑙𝐿L~𝑒] _`V_a ≡ 𝑒]^

ΦQ,ΦE

cQED LINK

DYNAMICALFERMIONS

StaggeredFermions”L.SusskindPhys.Rev.D16,3031(1977)

DYNAMICALFERMIONSSHWINGERMODEL

NON-ABELIANLINK

Ur =elementofthegaugegroup

Ur

NON-ABELIANLINKS

LR“NON-ABELIANCHARGE”

𝐻c]Kd = { 𝑗,𝑚,𝑚g } =⊕j [𝑗l ⊗ 𝑗B]oQp4

SU(2)EXACT

𝐻c]Kd = 0⊕ (12⊗

12)

SU(2)EFFECTIVE

Ancillary“constraint”Fermion

Oneachlink– a1,2 bosonsontheleft,b1,2 bosonsontheright

“color”fermions

PLAQUETTES

PLAQUETTES

1delementarylinkinteractions– alreadygaugeinvariantbuildingblocksofeffectiveplaquettes

Auxiliaryfermions:=

PLAQUETTES

1delementarylinkinteractions– alreadygaugeinvariantbuildingblocksofeffectiveplaquettes

Auxiliaryfermions:=

PLAQUETTES

1delementarylinkinteractions– alreadygaugeinvariantbuildingblocksofeffectiveplaquettes

Auxiliaryfermions– virtualprocesses

PLAQUETTES

1delementarylinkinteractions– alreadygaugeinvariantbuildingblocksofeffectiveplaquettes

Auxiliaryfermions– virtualprocesses

PLAQUETTES

1delementarylinkinteractions– alreadygaugeinvariantbuildingblocksofeffectiveplaquettes

Auxiliaryfermions– virtualprocesses- plaquettes.

PLAQUETTES

1delementarylinkinteractions– alreadygaugeinvariantbuildingblocksofeffectiveplaquettes

Auxiliaryfermions– virtualprocesses- plaquettes.

OKAYfor:discrete,abelian&non-abelian groups

DIGITALSIMULATION

2

C

1

Threeatomiclayersw.movablecontrolatoms

E.Zohar,A.Farace,B.Reznik,J.I.Cirac, PRA2017.E.Zohar,A.Farace,B.Reznik,J.I.Cirac,PRL2017.

LatticeGaugeTheorywithStators

ML

C

MatterFermionsLink(Gauge)degreesoffreedomControldegreesoffreedom

E.Zohar,A.Farace,B.Reznik,J.I.Cirac, Phys.Rev.A2017.E.Zohar,A.Farace,B.Reznik,J.I.Cirac,Phys.Rev.Lett.2017.

DigitalLatticeGaugeTheories

ML

C

TheZ2 example:

- Plaquette interactions

- Linkinteractions

Plaquettes:Four-bodyInteractions

M1

C 2

3

4

Stators:two-bodyinteractionsà four-bodyinteractions

Plaquettes:Four-bodyInteractions

M1

C 2

3

4

Stators:two-bodyinteractionsà four-bodyinteractions

Plaquettes:Four-bodyInteractions

M1

C 2

3

4

Stators:two-bodyinteractionsà four-bodyinteractions

Plaquettes:Four-bodyInteractions

M1

C 2

3

4

Stators:two-bodyinteractionsà four-bodyinteractions

Plaquettes:Four-bodyInteractions

M1

C 2

3

4

Stators:two-bodyinteractionsà four-bodyinteractions

Plaquettes:Four-bodyInteractions

M1

C 2

3

4

Stators:two-bodyinteractionsà four-bodyinteractions

DIGITALSIMULATION

AbipartitesingletimestepTrotterized timeevolution,ofalreadygaugeinvariantselements

QUANTUMSIMULATIONSCOLDATOMS– EXPERIMENTS

QUANTUMSIMULATIONSCOLDATOMS– EXPERIMENTS

QUANTUMSIMULATIONSCOLDATOMS– EXPERIMENTS

QUANTUMSIMULATIONSIONS– EXPERIMENTS

QUANTUMSIMULATIONSCOLDATOMS– EXPERIMENTS

Oberthaler group

QUANTUMSIMULATIONSCOLDATOMS– EXPERIMENTS

CONFINEMENTTOYMODELS

• 1+1D:Schwinger’smodel.

• cQED:2+1D:nophasetransitionInstantons giverisetoconfinementat𝑔 < 1(Polyakov).(ForT>0:thereisaphasetransitionalsoin2+1D.)

• cQED:3+1D: phasetransitionbetweena strongcouplingconfiningphase,anda weakcouplingcoulombphase.

• Z(N):forN ≥ 𝑁R: Threephases:electricconfinement,magneticconfinement,andnonconfinement.

References

• E.Zohar,B.Reznik,ConfinementandlatticeQEDelectricflux-tubessimulatedwithultracold atoms,Phys.Rev.Lett.107,275301(2011).

• E.Zohar,J.I.Cirac,B.Reznik,SimulatingCompactQuantumElectrodynamicswithultracold atoms:Probingconfinementandnonperturbativeeffects.Phys.Rev.Lett.109,125302(2012).

• E.Zohar,J.I.Cirac,B.Reznik,Acold-atomquantumsimulatorforSU(2)Yang-MillslatticegaugetheoryPhys.Rev.Lett.110,125304(2013).

• E.Zohar,J.I.Cirac,B.Reznik,Simulating2+1dLatticeQEDwithdynamicalmatterusingultracold atoms..Phys.Rev.Lett.110,055302(2013).

• E.Zohar,J.I.Cirac,B.Reznik,Quantumsimulationsofgaugetheorieswithultracold atoms:localgaugeinvariancefromangularmomentumconservation,Phys.Rev.A88 023617(2013).

• E.Zohar,J.I.Cirac,B.Reznik,QuantumSimulationsofLatticeGaugeTheoriesusingUltracold AtomsinOpticalLattices.,Rep.Prog.Phys.79,014401(2016).

• E.Zohar,A.Farace,B.Reznik,J.I.Cirac ,Digitallatticegaugetheories., Phys.Rev.A(2017).

• E.Zohar,A.Farace,B.Reznik,J.I.Cirac ,Digitalquantumsimulationofℤ2latticegaugetheorieswithdynamicalfermionicmatter.Phys.Rev.Lett.(2017).

Hep simulations/GROUPS

– ICFO,Barcelona(Lewenstein)– Innsbruck(Zoller,Blatt)– UniversityofBern(Wiese)– Heidelberg(Oberthaler,Berges)– UGENT(Verstraete)– Caltech,UMD(Preskill,Jordan)– …– TensorNetworkswithLGI(cQED in1+1,2+1),MPQ,UGENT,Ulm,Mainz

Hep simulations/GROUPS

– ICFO,Barcelona(Lewenstein)– Innsbruck(Zoller,Blatt)– UniversityofBern(Wiese)– Heidelberg(Oberthaler,Berges)– UGENT(Verstraete)– …

ThankYou!