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!