Quantum Simulation of (Non-)Abelian Gauge Theories

Quantum Simulation of (non-)abelian gauge theories

Enrique Rico Ortega 13 February 2013

miércoles 13 de febrero de 13 1 The team • Albert Einstein Center - Bern University

D. Banerjee M. Bögli P. Stebler P. Widmer U.-J. Wiese • Complutense University • Technical University Madrid Vienna

M. Müller P. Rabl

• IQOQI - Innsbruck University

M. Dalmonte D. Marcos P. Zoller miércoles 13 de febrero de 13 2 The references

Atomic Quantum Simulation of Dynamical Gauge Fields Coupled to Fermionic Matter: From String breaking to Evolution after a Quench Phys. Rev. Lett. 109, 175302 (2012)

Atomic Quantum Simulation of U(N) and SU(N) non-abelian Gauge Theories arXiv:1211.2242 (2012)

Quantum Simulation of Dynamical Lattice Gauge Field Theories with Superconducting Q-bits

miércoles 13 de febrero de 13 3 Recent related works on dynamical gauge fields

H. Büchler, M. Hermele, S. Huber, M.P.A. Fisher, P. Zoller, Atomic quantum simulator for lattice gauge theories and ring exchange models, Phys. Rev. Lett. 95, 40402 (2005).

H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, H.P. Büchler, A [digital] open system Rydberg quantum simulator, Nat. Phys. 6, 382 (2010).

J.I. Cirac, P. Maraner, J.K. Pachos, Cold Atom Simulation of Interacting Relativistic Quantum Field Theories, Phys. Rev. Lett. 105, 190403 (2010).

E. Kapit, E. Mueller, Optical-lattice Hamiltonians for relativistic quantum electrodynamics, Phys. Rev. A 83, (2011).

E. Zohar, J.I. Cirac, B. Reznik, Simulating Compact Quantum Electrodynamics with ultracold atoms: Probing confinement and nonperturbative effects, Phys. Rev. Lett. 109, 125302 (2012).

L. Tagliacozzo, A. Celi, A. Zamora, M. Lewenstein, Optical Abelian Lattice Gauge Theories, Ann. Phys. 330, 160-191 (2013). Barcelona, Bern, Innsbruck, Leeds, Madrid, Munich, New York, Stuttgart, Tel Aviv, ... miércoles 13 de febrero de 13 4 … and on dynamical non-abelian gauge fields

E. Zohar, J.I. Cirac, B. Reznik, A cold-atom quantum simulator for SU(2) Yang-Mills lattice gauge theory, arXiv:1211.2241 (2012).

L. Tagliacozzo, A. Celi, P. Orland, M. Lewenstein, Simulations of non- Abelian gauge theories with optical lattices, arXiv:1211.2704 (2012).

Barcelona, Bern, Innsbruck, Leeds, Madrid, Munich, New York, Stuttgart, Tel Aviv, ... miércoles 13 de febrero de 13 5 Why quantum simulate Gauge theories?

The study of Gauge theories is the study of Nature.

miércoles 13 de febrero de 13 6 Why quantum simulate Gauge theories?

The study of Gauge theories is the study of Nature.

Gauge symmetry as a fundamental principle

miércoles 13 de febrero de 13 6 Why quantum simulate Gauge theories?

The study of Gauge theories is the study of Nature.

Gauge symmetry as a fundamental principle

Gauge symmetry as an emergent phenomenon

miércoles 13 de febrero de 13 6 Why quantum simulate Gauge theories?

The study of Gauge theories is the study of Nature.

Gauge symmetry as a fundamental principle

Gauge symmetry as an emergent phenomenon

Gauge symmetry as a resource

miércoles 13 de febrero de 13 6 Gauge symmetry as a fundamental principle Standard model: for every force there is a gauge boson,

miércoles 13 de febrero de 13 7 Gauge symmetry as a fundamental principle atoms Standard model: for every force there is a gauge boson,

• The photon is the “carrier” of the electromagnetic force.

miércoles 13 de febrero de 13 7 Gauge symmetry as a fundamental principle atoms Standard model: for every force there is a gauge boson,

• The photon is the “carrier” of the electromagnetic force. nucleons

• The W+, W- and Z0 are the “carriers” of the weak force.

miércoles 13 de febrero de 13 7 Gauge symmetry as a fundamental principle atoms Standard model: for every force there is a gauge boson,

• The photon is the “carrier” of the electromagnetic force. nucleons

• The W+, W- and Z0 are the “carriers” of the weak force.

• The gluons are the quarks “carriers” of the strong force.

miércoles 13 de febrero de 13 7 Gauge symmetry as a fundamental principle Gauge theories on a Non-perturbative approach to fundamental theories of discrete lattice structure. matter, e.g. Q.C.D.

K. Wilson, Phys. Rev. D (1974)

miércoles 13 de febrero de 13 8 Gauge symmetry as a fundamental principle Gauge theories on a Non-perturbative approach to fundamental theories of discrete lattice structure. matter, e.g. Q.C.D.

1 S[ ,U] O = [ ,U] e O [ ,U] K. Wilson, Phys. Rev. D h i Z D (1974) Z N 1 S[ ,U ] e n n O [ ,U ] ⇠ N n n n=1 X 1 O [ ,U ] ⇠ N n n S[ ,U ] P [Un] e n n Monte Carlo simulation = Classical/ XStatistical Mechanics miércoles 13 de febrero de 13 8 Gauge symmetry as a fundamental principle Achievements by classical Monte-Carlo simulations: • first evidence of quark-gluon plasma • ab-initio estimate of the entire hadronic spectrum

S. Dürr, et al., Science (2008)

miércoles 13 de febrero de 13 9 Gauge symmetry as an emergent phenomenon Gauge bosons can appear as collective fluctuations of a strongly correlated many-body quantum system F. Wegner, J. Math. Phys. (1971) J.B. Kogut, Rev. Mod. Phys. (1979) A. Kitaev, Ann. Phys. (2003) P.A. Lee, N. Nagaosa, X.G. Wen, Rev. Mod. Phys. (2006)

miércoles 13 de febrero de 13 10 Gauge symmetry as an emergent phenomenon Gauge bosons can appear as collective fluctuations of a strongly correlated many-body quantum system F. Wegner, J. Math. Phys. (1971) J.B. Kogut, Rev. Mod. Phys. (1979) A. Kitaev, Ann. Phys. (2003) Ex.- Kitaev model (Z2 gauge theory) P.A. Lee, N. Nagaosa, X.G. Wen, Rev. Mod. Phys. (2006)

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miércoles 13 de febrero de 13 11 Volume 94B, number 2 PHYSICS LETTERS 28 July 1980

DYNAMICAL STABILITY OF LOCAL GAUGE SYMMETRY Volume 94B, number 2 PHYSICS LETTERS 28 July 1980 Creation of LightGauge From Chaos symmetry as an

D. FOERSTERemergent phenomenon Service de Physique Thdorique, CEN Sac&y, [:-91190 Gif-sur-Yvette, France VolumeSome 94B, number questions: 2 PHYSICS LETTERS 28 July 1980 DYNAMICAL STABILITY OF LOCAL GAUGE SYMMETRY H.B.(i) NIELSEN fractionalization, The Niels Bohr Institute, University of Copenhagen, and NORDITA,Creation of Light From Chaos DK-2100(ii) Copenhagenconfinement-deconfinement Volume~, Denmark 94B, number 2 quantumPHYSICS LETTERS phase transition, 28 July 1980 (iii) spin-liquid physics...D. FOERSTER and Service de Physique Thdorique, CEN Sac&y, [:-91190 Gif-sur-Yvette, France DYNAMICAL STABILITY OF LOCAL GAUGE SYMMETRY M.Volume NINOMIYA 94B, number 2 CreationPHYSICS of LETTERSLight FromH.B. ChaosNIELSEN 28 July 1980 The Niels Bohr Institute, University of Copenhagen, DK-21 O0 CopenhagenThe Niels ~, Bohr Denmark Institute, University of Copenhagen, and NORDITA, DYNAMICAL STABILITY OF LOCAL GAUGE SYMMETRY D. FOERSTER DK-2100 Copenhagen ~, Denmark Service de Physique Thdorique, CEN Sac&y, [:-91190 Gif-sur-Yvette, France Received 14 May 1980 Creation of Light From Chaos and

D. FOERSTER H.B. NIELSEN M. NINOMIYA DYNAMICAL STABILITY OF LOCAL GAUGE SYMMETRY Service de Physique Thdorique,The Niels CEN Bohr Sac&y, Institute, [:-91190The University Niels Gif-sur-Yvette, Bohr of Institute, Copenhagen, France University and NORDITA, of Copenhagen, DK-21 O0 Copenhagen ~, Denmark And GodCreation said "Let of Light there From be Chaoslight", and thereDK-2100 was lightCopenhagen - Genesis ~, Denmark 1-3 H.B. NIELSEN Received 14 May 1980 and D. FOERSTER The Niels Bohr Institute, University of Copenhagen, and NORDITA, We show that the large distanceDK-2100 behavior Copenhagen of gauge~, Denmark theories is stable, within certain limits, with respect to addition of gauge noninvariantService de Physique interactions Thdorique, at small CEN distances. Sac&y, [:-91190M. NINOMIYA Gif-sur-Yvette, France The Niels Bohr Institute, University of Copenhagen, DK-21 O0 Copenhagen ~, Denmark and And God said "Let there be light", and there was light - Genesis 1-3 H.B. NIELSEN PHYSICAL REVIEW B 68, 115413 ͑2003͒ The Niels Bohr Institute,M. University NINOMIYA of Copenhagen, Received and 14 NORDITA, May 1980 We show that the large distance behavior of gauge theories is stable, within certain limits, with respect to addition of DK-2100 Copenhagen ~,The Denmark Niels Bohr Institute, University ofance Copenhagen, has a highDK-21 chance O0 Copenhagen of arising ~, Denmark spontaneously even 1. One of us (H.B.N.)Artificial has suggested light and [1 ] quantumearlier that order ingauge systems noninvariant of screened interactions at dipoles small distances. symmetries and physical laws should arise naturally if nature is not gauge invariant at the fundamental and Received 14 May 1980 from some essentially random dynamics rather thanAnd God saidscale. "Let there be light", and there was light - Genesis 1-3 M. NINOMIYA Xiao-Gang Wen* being postulatedDepartment to be exact of Physics,or adjusted Massachusetts by hand Institute+1. of Technology,1.Iliopoulos One of Cambridge, us and(H.B.N.) Nanopoulos Massachusetts has suggested [7] 02139, [1have ] earlier USAinformed that us ance has a high chance of arising spontaneously even miércoles 13 deThe febrero Niels Bohr de 13 Institute, University of Copenhagen, DK-21 O0 Copenhagen ~, Denmark 11 The idea of postulating͑Received 25 dynamical November 2002;stability revised ,2 manuscriptin theWe show received thatthatsymmetries the 24 theylarge January distance are and calculating 2003;physical behavior published laws of thegauge should 16renormalization theories September arise is naturally stable, 2003 within ͒ group/3- certainif limits,nature with is not respect gauge to invariantaddition of at the fundamental sense that coupling parameters Andshall God not saidbe contrived"Let theregauge benoninvariant light",functionfrom and interactions somethere for essentiallywas variousat lightsmall - distances. random gaugeGenesis breakingdynamics 1-3 ratherterms thanand hope forscale. ReceivedThe origin 14 May of light 1980 is an unsolved mystery in nature. Recently, it was suggested that light may originate from has though been spread for a long time. In earlier approximatebeing postulated gauge to be invariance exact or adjusted to appear by hand towards +1. the Iliopoulos and Nanopoulos [7] have informed us a new kind of order, quantumWe show order. that Tothe test large this distance idea inbehavior experiments, of gauge wetheories study is systemsstable, within of screened certain limits, magnetic/ with respect tothat addition they ofare calculating the renormalization group/3- papers some of the present authors (H.B.N. and M.N.) infrared.The idea of postulating dynamical stability ,2 in the electric dipoles in two-dimensionalgauge noninvariant͑2D interactions͒ and 3D lattices. at smallsense We distances. show that thatcoupling our models parameters contain shall an not artificial beance contrived light–ahas a high chancefunction of arising for various spontaneously gauge breaking even terms and hope for [5], and Chadha, have obtained Lorentz invariance1. One of us (H.B.N.)This has attemptsuggested is [1 similar ] earlier to that what Br6zin and Zinn- photonlikeAnd God said collective "Let there excitation. be light", Wesymmetries discuss and there how wasand to designlightphysicalhas - realisticthoughGenesis laws shouldbeen devices 1-3 spread arise that naturally realizefor a long our time. models. Inif natureearlier We show is not gaugeapproximate invariant at gauge the fundamental invariance to appear towards the in non-Lorentz covariant Yang-Mills theories and elec- Justin [8] did for an asymmetric ~.i/kl~iq)/(Pk~l theory. that the ‘‘speed of light’’ and the ‘‘fine-structurefrom some constant’’essentiallypapers of random the some artificial dynamics of the light present rather can be authorsthan tuned in (H.B.N. our models.scale. and M.N.) The infrared. They noticed that this theory becomes automatically trodynamics propertiesin Wethe show infrared of that artificial the limit1. large One atoms asdistance of compared us͑bound (H.B.N.) behaviorbeing states to hasof postulated a ofgauge suggested pairs theories of to artificial [1[5],be is ] stable,exactearlier and charges Chadha,orwithin that adjusted ͒ certainare haveance alsoby limits,obtained hand discussed.has with a+1. high Lorentzrespect chance The to existenceinvariance additionofIliopoulos arising ofof spontaneously and NanopoulosThis attempteven [7] have is similar informed to what us Br6zin and Zinn- fundamental artificialgaugescale, noninvariant e.g. light the͑assymmetries Planck wellinteractions as length. artificial and at smallphysical In electronThe distances.these idealaws͒ deri- in of should condensed-matterpostulating ariseO(n)in non-Lorentznaturally dynamical invariant systems stabilitycovariant in suggests theif nature ,2infrared Yang-Mills in that isthe not elementary under gauge theories that suitableinvariant particles, they and are elec-condi-at thecalculating fundamental Justin the [8] renormalization did for an asymmetric group/3- ~.i/kl~iq)/(Pk~l theory. vations we putsuch in as gauge light andinvariancefrom electron, some ,3 mayessentially as not an besenseassump- elementary.random that couplingdynamics They tions.trodynamics mayparameters rather be than collective shall in the not excitations infrared scale.be contrived limit of quantum as comparedfunction order into ourafor variousThey gauge noticed breaking that terms this andtheory hope becomes for automatically Iliopoulos and Nanopoulos [7] have informed us tion. It wouldvacuum. also be In nice ourbeing model, to show postulated light that is realized togauge behas exact asthoughinvari- a fluctuationor beenadjusted spread offundamental by string-netsThese handfor a +1.longare scale, and thetime. charges samee.g. In theearlier type as Planck the of ends length.arguments ofapproximate open In these stringsas thederi- gauge invarianceO(n) invariant to appear in the towardsinfrared theunder suitable condi- The idea of postulating dynamical stability ,2 in the that they are calculating the renormalizationtions. group/3- 1. ͑Oneor nodes of us of(H.B.N.) string netshas͒ .suggested [1papers ] earlier some that of the Lorentzvationspresentance has weauthors invariancea puthigh in (H.B.N.chance gauge derivation invarianceofand arising M.N.) by spontaneously ,3 twoas aninfrared. ofassump- us (H.B.N.even sense that coupling parameters shall not tion.be contrived It would also befunction nice to forshow various that gauge invari-breaking terms andThese hope are for the same type of arguments as the symmetries and physical laws should arise[5], naturally and Chadha, haveandif natureobtained M.N.) is not [3]Lorentz gaugeand invarianceChadha. invariant at the fundamentalThis attempt is similar to what Br6zin and Zinn- DOI: 10.1103/PhysRevB.68.115413has though been spread for a long time. In earlier PACSapproximate number͑s͒: 73.22.gauge Ϫinvariancef, 11.15.Ϫ toq appear Lorentztowards invariancethe derivation by two of us (H.B.N. from some essentially random dynamics inrather non-Lorentz than covariantscale. Yang-Mills theories and elec- Justin [8] did for an asymmetric ~.i/kl~iq)/(Pk~l theory. papers some of the present authors (H.B.N.Instead and M.N.) we shallinfrared. introduce an exact gauge &varianceand M.N.) [3] and Chadha. +1 See alsobeing ref. postulated [2] for a reviewto be exactby Iliopoulos or adjusted and trodynamicsby a support-hand +1. in the infraredIliopoulos limit as and compared Nanopoulos to a [7] haveThey informed noticed us that this theory becomes automatically [5], and Chadha, have obtained Lorentzin invariance a formal way and thenThis attemptdeduce is the similar real to one what from Br6zin it inand Instead Zinn- we shall introduce an exact gauge &variance ing philosophyThe idea of by postulating Woo [3]. dynamical stabilityfundamental ,2 in the scale, e.g.+1that See the they also Planck ref.are [2] calculatinglength. for a reviewIn these the by renormalization deri-Iliopoulos O(n)and a support-invariantgroup/3- in the6 infrared under suitable condi- I. INTRODUCTIONin non-Lorentz covariant Yang-Mills theoriesstates and contain elec- a newJustin kind [8] of did order, for an a asymmetric topological ~.i/kl~iq)/(Pk~l order. Re- theory. t~ See e.g.sense Thome that [4].coupling parameters shall not vationsbe contrived we put in gaugethefunctioning infrared.invariance philosophy for variousWe ,3 by as shallWoo an gauge [3].assump-show breaking that aterms gaugetions. and theory hope forarises in a formal way and then deduce the real one from it in trodynamics in the infrared limit as comparedcently, to wea find thatThey even noticed gapless that this quantum theory becomes states inautomatically two, +3 For ahas review though of gauge been theories,spread for see a ref.long [6]. time. tion.In earlier It would also automaticallyt~beapproximate Seenice e.g. to Thome show gauge atthat[4]. large invariancegauge distances invari- to appear from towards aThese theory arethe that the samethe typeinfrared. of arguments We shall showas the that a gauge theory arises What is light? Where lightfundamental comes from? scale, Why e.g. lightthe Planck exists? length.+3three, InFor these a review or deri- other of gauge dimensionsO(n) theories, invariant see can ref. in contain [6].the infrared an order under that suitable is be- condi- papers some of the present authors (H.B.N. and M.N.) infrared. Lorentz invariance derivationautomatically by two at large of us distances (H.B.N. from a theory that Every one probably agreesvations that these we put are in fundamental gauge invariance ques- ,3 as anyond assump- Landau’s symmetry-breakingtions. theory.7,8 We call this or- [5], and Chadha, have obtained Lorentz invariance This attempt is similar to what Br6zinand and M.N.) Zinn- [3] and Chadha. tion. It would also be nice to show that gauge invari- These are the same type of arguments135 as the tions. Butin non-Lorentz one may wonder covariant if theyYang-Mills are scientific theories questions, and elec- Justinder quantum [8] did order.for an asymmetric A preliminary ~.i/kl~iq)/(Pk~l theoryInstead of theory. quantum we shall order introduce is an exact gauge &variance 135 +1 See also ref. [2] for a review by IliopoulosLorentz and a support- invariance derivation by two of us (H.B.N. philosophicaltrodynamics questions, in the or infrared even religiouslimit as compared question. to Beforea Theydeveloped. noticed We that find this sometheory quantum becomes ordersinautomatically a formal can way be character-and then deduce the real one from it in ing philosophy by Woo [3]. and M.N.) [3] and Chadha. answeringfundamental these questions scale, e.g. and the the Planck questions length. aboutt~ InSee these thee.g. Thome ques-deri- [4]. O(n)ized invariant by projective in the infrared symmetry under suitable groupthe infrared. condi-͑PSG ͒We justshall show like that a gauge theory arises Instead we shall introduce an exact gauge &variance tions, wevations would we like put toin gauge ask+1 threeSee invariance also more ref. [2],3 questions: foras +3 ana review Forassump- a What reviewby Iliopoulos isof gaugetions. symmetry-breakingand theories, a support- see ref. [6]. orders can be characterizedautomatically by at symmetry large distances from a theory that 46 in a formal way and then deduce the real one from it in phonon?tion. Where It would phonon also comes be niceing from? philosophy to show Why thatby phonon Woo gauge [3]. exists?invari- group.These Using are the quantum same type orders of arguments and PSG, as the we classified over t~ See e.g. Thome [4]. the infrared. We shall show that a gauge theory arises We know that these are scientific questions and we know Lorentz100 different invariance two-dimensional derivation by ͑two2D ͒ofspin us (H.B.N. liquids that have the 135 +3 For a review of gauge theories, see ref. [6]. 7automatically at large distances from a theory that their answers. Phonon is a vibration of a crystal. Phonon andsame M.N.) symmetry. [3] and Chadha.Intuitively, we can view quantum/ comes from a spontaneous translation symmetry breaking. topologicalInstead we order shall asintroduce a description an exact of gauge pattern &variance of quantum en- +1 See also ref. [2] for a review by Iliopoulos and a support- 8 135 Phonon existsing philosophy because by theWoo translation-symmetry-breaking[3]. intanglements a formal way in and a quantumthen deduce state. the realThe one pattern from it of in quantum phase actuallyt~ See e.g. exists Thome in [4]. nature. theentanglements infrared. We isshall much show richer that a thangauge pattern theory ofarises classical con- It is quite+3 For interestinga review of gauge to seetheories, that see our ref. understanding [6]. of a automaticallyfigurations. at large distances from a theory that gapless excitation phonon is rooted in our understanding of We know that the fluctuations of pattern of classical con- 1 phases of matter. According to Landau’s theory, phases of figurations ͑such as lattices͒ lead to low-energy135 collective matter are different because they have different broken sym- excitations ͑such as phonons͒. Similarly, the fluctuations of metries. The symmetry description of phases is very power- pattern of quantum entanglement also lead to low-energy ful. It allows us to classify all possible crystals. It also pro- collective excitations. However, we find that the collective vides the origin for gapless phonons and many other gapless excitations from quantum order can be gapless gauge excitations.2,3 bosons9–15 and/or gapless fermions. The fermions can even However, light, as a U͑1͒ gauge boson, cannot be a appear from pure bosonic models on lattice.7,11,12,14,16–18 Nambu-Goldstone mode from a broken symmetry. There- If we believe in quantum order, then the three questions fore, unlike phonon, light cannot originate from a symmetry- about light will be scientific questions. Their answers will be breaking state. This may be the reason why we treat light ͑A͒ light is a fluctuation of quantum entanglement, ͑B͒ light differently than phonon. We regard light as an elementary comes from the quantum order in our vacuum, and ͑C͒ light particle and phonon as a collective mode. exists because our vacuum contains a particular entangle- However, if we believe in the equality between phonon ment ͑i.e., a quantum order͒ that supports U͑1͒ gauge fluc- and light and if we believe that light is also a collective mode tuations. of a particular ‘‘order’’ in our vacuum, then the very exis- According to the picture of quantum order, elementary tence of light implies an order not found earlier in our particles ͑such as photon and electron͒ may not be elemen- vacuum. Thus, to understand the origin of light, we need to tary after all. They may be collective excitations of a bosonic deepen and expand our understanding of phases of matter. system. Without experiments at Planck scale, it is hard to We need to discover a new kind of order that can produce prove or disprove if photon and electron are elementary par- and protect light. ticles or not. However, one can make a point by showing that After the discovery of fractional quantum Hall ͑FQH͒ photon and electron can emerge as collective excitations in effect,4,5 it became clear that Landau’s symmetry-breaking certain lattice bosonic models. So photon and electron do not theory cannot describe different FQH states, since those have to be elementary particles. states all have the same symmetry. It was proposed that FQH The emergent gauge fluctuations from local bosonic mod-

Topological quantum computation: Deconfined phases of gauge models may have excitations with non-abelian statistics and degenerate ground states.

A. Kitaev, Ann. Phys. (2003) M.H. Freedman, A. Kitaev, M.J. Larsen, Z. Wang, Bull. Amer. Math. Soc. (2003) C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. (2008)

Some questions: (i) new materials, (ii) how to create and manipulate quasi-particles.

miércoles 13 de febrero de 13 12 Why quantum simulate Gauge theories?

Problems not solvable on a classical machines

Various flavors of sign problems in strongly correlated systems

miércoles 13 de febrero de 13 13 Why quantum Real time evolution: simulate Gauge Heavy ion experiments (collisions) theories?

miércoles 13 de febrero de 13 14 Why quantum Real time evolution: simulate Gauge Heavy ion experiments (collisions) theories?

QCD with finite density of fermions: Dense nuclear matter, color superconductivity (phase diagram of QCD) S. Hands, Contemp. Phys. (2001) M.G. Alford, A. Schmitt, K. Rajagopal, T. Schäfer, Rev. Mod. Phys. (2008) K. Fukushima, T. Hatsuda, Rep. Prog. Phys. (2011)

miércoles 13 de febrero de 13 14 Why quantum Real time evolution: simulate Gauge Heavy ion experiments (collisions) theories?

Frustrated spin models: Spin liquid physics, RVB states (High Tc superconductivity?) E. Dagotto, Science (2005) M.R. Norman, D. Pines, C. Kallinl, Adv. Phys. (2005) P. Wahl, Nat. Phys. (2012) miércoles 13 de febrero de 13 14 Why quantum simulate Gauge theories?

miércoles 13 de febrero de 13 15 Why quantum simulate Gauge theories? Feynman: “It is difficult to simulate quantum physics on a classical computer”

R.P. Feynman, Int. J. Theor. Phys. (1982)

Entanglement

= c + c + + c N | i 1| "" ···"i 2| "" ···#i ··· 2 | ## ···#i Huge miércoles 13 de febrero de 13 15 Why quantum simulate Gauge theories? Feynman’s universal quantum simulator: controlled quantum device which efficiently reproduces the dynamics of any other many-particle quantum system.

miércoles 13 de febrero de 13 16 Why quantum simulate Gauge theories? Feynman’s universal quantum simulator: controlled quantum device which efficiently reproduces the dynamics of any other many-particle quantum system. How?… cold atoms, ions, photons, superconducting circuit, etc.

J.I. Cirac, P. Zoller I. Bloch, J. Dalibard, S. Nascimbène R. Blatt, C.F. Roos, A. Aspuru-Guzik, P. Walther A.A. Hock, H.E. Türeci, J. Koch Nature Physics Insight - Quantum Simulation (2012) miércoles 13 de febrero de 13 16 Why quantum simulate Gauge theories?

NEED

Design a controlled microscopic quantum simulator for lattice gauge theories.

AIM

Investigate relevant phenomena, e.g., characterize the phase diagram and dynamics of strongly coupled lattice gauge models.

miércoles 13 de febrero de 13 17 Why quantum simulate Gauge theories?

simulators

Areas for quantum

miércoles 13 de febrero de 13 18 Lattice x : fermion gauge Contents theory Ux,x+1 : boson • Hamiltonian formulation of lattice gauge theories. [degrees of freedom, symmetry generators, dynamics]

String breaking miércoles 13 de febrero de 13 19 Lattice x : fermion gauge Contents theory Ux,x+1 : boson • Hamiltonian formulation of lattice gauge theories. [degrees of freedom, symmetry generators, dynamics]

• Wilson lattice gauge formulation vs. Quantum link models. [connections with quantum information and condensed matter]

• Phenomenology. [confinement, string-breaking, quark-gluon plasma]

• Wilson lattice gauge formulation vs. Quantum link models. [connections with quantum information and condensed matter]

• Phenomenology. [confinement, string-breaking, quark-gluon plasma]

• Implementation of quantum link models. [Bose-fermi mixture, fermionic alkaline earth atoms, superconducting q-bits]

• Wilson lattice gauge formulation vs. Quantum link models. [connections with quantum information and condensed matter]

• Phenomenology. [confinement, string-breaking, quark-gluon plasma]

• Implementation of quantum link models. [Bose-fermi mixture, fermionic alkaline earth atoms, superconducting q-bits]

• Observability of interesting phenomena

• Wilson lattice gauge formulation vs. Quantum link models. [connections with quantum information and condensed matter]

• Phenomenology. [confinement, string-breaking, quark-gluon plasma]

• Implementation of quantum link models. [Bose-fermi mixture, fermionic alkaline earth atoms, superconducting q-bits]

• Observability of interesting phenomena

• Conclusions & Outlook String breaking miércoles 13 de febrero de 13 19 Hamiltonian formulation of lattice gauge theories A gauge invariant model is defined by:

Set of local dynamical operators acting on the vertices (matter fields) and on the links (gauge fields)

miércoles 13 de febrero de 13 20 Hamiltonian formulation of lattice gauge theories A gauge invariant model is defined by:

Set of local dynamical operators acting on the vertices (matter fields) and on the links (gauge fields)

U3,4 4 3 i j x† y

U4,1 U2,3 ij Ux,y (operator acting on a Hilbert space)

1 U1,2 2 1:U(1) J.B. Kogut, L. Susskind, PRD (1975) i = : U(2) ref. Creutz and Montvay/Muenster books "# J.B. Kogut, Rev. Mod. Phys. (1979) (brg : U(3) miércoles 13 de febrero de 13 20 Hamiltonian formulation of lattice gauge theories

Set of local generators of gauge transformations

ki U~x,+ˆy U ij i ~x,+ˆx ~x mi U~x, xˆ il U~x, yˆ miércoles 13 de febrero de 13 21 Hamiltonian formulation of lattice gauge theories

Set of local generators of gauge transformations Generators of the local symmetry:

~ ~ ~ ~ i z ✓z Gz i i z ✓z Gz ij j e xe = ⌦x x P P j ~ ~ ~ ~ X i z ✓z Gz ij i z ✓z Gz ik kl jl e Ux,ye = ⌦x Ux,y⌦y ⇤ P PU ki ~x,+ˆy Xk,l U ij i ~x,+ˆx ~x mi U~x, xˆ il U~x, yˆ miércoles 13 de febrero de 13 21 Hamiltonian formulation of lattice gauge theories

Set of local generators of gauge transformations

0 Define the Hilbert space: 0" # 1 2 ~ ~ Gx = B C Gx physical =0 B 1 C | i x ⇣ ⌘ B " # C X B C B . C B ..C ki B C U~x,+ˆy @ A U ij Block-diagonal Hilbert i ~x,+ˆx space ~x mi U~x, xˆ il U~x, yˆ miércoles 13 de febrero de 13 22 Hamiltonian formulation of lattice gauge theories

Set of local generators of gauge transformations

Gx = x† x Ex,x+ˆµ Ex µ,xˆ matter Xµˆ electric field

⇢ ~ E~ =0 : Gauss’ law ex.- U(1) group r · phys

h i ki U~x,+ˆy U ij i ~x,+ˆx ~x mi U~x, xˆ il U~x, yˆ miércoles 13 de febrero de 13 23 Hamiltonian formulation of lattice gauge theories

Gauge invariant quantum Hamiltonian:

H, G~ =0 x Local conserved quantities 8 Gauge (local) symmetries h i

miércoles 13 de febrero de 13 24 Wilson formulation: continuum valued operator, infinite-dimensional local Hilbert space

ex.- U(1) group

@ ix,y Ux,y e Ex,y i ! ! @x,y

Implementation in AMO setup very challenging

E. Kapit, E. Mueller, Phys. Rev. A (2011) E. Zohar, B. Reznik, Phys. Rev. Lett. (2011) miércoles 13 de febrero de 13 25 Quantum link formulation: gauge fields span a finite- dimensional local Hilbert space

D. Horn, Phys. Lett. B (1981) P. Orland, D. Röhrlich, Nucl. Phys. B (1990) S. Chandrasekharan, U.-J. Wiese, Nucl. Phys. B (1997)

miércoles 13 de febrero de 13 26 Quantum link formulation: gauge fields span a finite- dimensional local Hilbert space

D. Horn, Phys. Lett. B (1981) P. Orland, D. Röhrlich, Nucl. Phys. B (1990) S. Chandrasekharan, U.-J. Wiese, Nucl. Phys. B (1997)

Q.C.D. can be formulated as a non-abelian quantum link model

R. Brower, S. Chandrasekharan, U.-J. Wiese, Phys. Rev. D (1999) R. Brower, S. Chandrasekharan, S. Riederer, U.-J. Wiese, Nucl. Phys. B (2004)

miércoles 13 de febrero de 13 26 Quantum Link models Connections with Quantum Information (Z2 gauge theory-Kitaev model) H. Weimar, M. Müller, I. Lesanovsky, P. Zoller, H.P. Büchler, Nat. Phys. (2010)

miércoles 13 de febrero de 13 27 Quantum Link models Connections with Quantum Information (Z2 gauge theory-Kitaev model) H. Weimar, M. Müller, I. Lesanovsky, P. Zoller, H.P. Büchler, Nat. Phys. (2010) Local degrees of freedom.-

Quantum two level system living on the link

(3) , (1) { x,y x,y}

miércoles 13 de febrero de 13 27 Quantum Link models Connections with Quantum Information (Z2 gauge theory-Kitaev model)

Local generator of gauge transformations.-

Local unitary transformation around every vertex

(1) (1) (1) (1) Gvert = 1,22,33,44,1

G (3) G = (3) vert 1,2 vert 1,2

Z2 gauge transformation miércoles 13 de febrero de 13 28 Quantum Link models Connections with Quantum Information (Z2 gauge theory-Kitaev model)

Local generator of gauge transformations.-

“Physical” Hilbert space (Gauss’ law) G phys = phys vert| i | i

8-vertex model: equal parity subspace miércoles 13 de febrero de 13 29 Quantum Link models Connections with Quantum Information (Z2 gauge theory-Kitaev model)

Gauge invariant Hamiltonian.-

H = (3)(3)(3)(3) + (1) 1,2 2,3 3,4 4,1 x,y plaq x,y X hXi magnetic term electric term

[H, G ]=0, vertex vert 8

miércoles 13 de febrero de 13 30 Quantum Link models Connections with Condensed Matter (U(1) gauge theory-Quantum Spin Ice model)

Local degrees of freedom.-

Quantum two level system living on the link

(3) (+) ( ) , , { x,y x,y x,y }

L. Balents, Nature (2010) C. L. Henley, Ann. Rev. Cond. Matt. Phys. (2010) C. Castelnovo, R. Moessner, and S.L. Sondhi, Ann. Rev. Cond. Matt. Phys. (2012) miércoles 13 de febrero de 13 31 Quantum Link models Connections with Condensed Matter (U(1) gauge theory-Quantum Spin Ice model) Local generator of gauge transformations.-

Local generator around every vertex

✓vert (+) ✓vert (+) exp i G exp i G = ei✓vert 2 vert 1,2 2 vert 1,2  

(3) (3) (3) (3) Gvert = 1,2 + 2,3 + 3,4 + 4,1

U(1) gauge transformation

miércoles 13 de febrero de 13 32 Quantum Link models Connections with Condensed Matter (U(1) gauge theory-Quantum Spin Ice model) Local generator of gauge transformations.-

“Physical” Hilbert space (Gauss’ law) G phys =0 vert| i

6-vertex model: zero magnetization subspace miércoles 13 de febrero de 13 33 Quantum Link models Connections with Condensed Matter (U(1) gauge theory-Quantum Spin Ice model)

Gauge invariant Hamiltonian.-

+ + + + H = + 1,2 2,3 3,4 4,1 1,2 2,3 3,4 4,1 plaq X ⇥ magnetic term ⇤

[H, G ]=0, vertex vert 8

miércoles 13 de febrero de 13 34 Quantum Link models

Local degrees of freedom.-

Quantum link carrying an electric flux

+ Ux,y Sx,y Spin-½: Spin-1: ⌘ E=+1 E=1/2 E=0, no flux E S(3) E=-1/2 E=-1 x,y ⌘ x,y

miércoles 13 de febrero de 13 35 Quantum Link models Gauss’ law.- G phys =0 ~ E~ =0 vert| i , r ·

configuration obeying ice rules, except for defects (charge or monopoles) miércoles 13 de febrero de 13 36 Quantum Link models

Gauge invariant Hamiltonian.-

2 g 2 1 H = [E ] U † U U † U + U U † U U † 2 x,y 4g2 1,2 2,3 3,4 4,1 1,2 2,3 3,4 4,1 x,y plaq hXi X h i

Electric term Magnetic term

miércoles 13 de febrero de 13 37 Quantum Link models Rishon (Schwinger) representation

x y x y + Link operator U S = c† c x,y ⌘ x,y y x

Electric field (3) 1 E S = c† c c† c [U(1) generator] x,y ⌘ x,y 2 y y x x ⇥ ⇤ c ,c† = Schwinger fermions (rishons) { x y} x,y [cx,cy† ]=x,y Schwinger bosons miércoles 13 de febrero de 13 38 Quantum Link models Rishon (Schwinger) representation

Spin representation: 2 Nx,y Nx,y Nx,y = c† cy + c† cx S~ +1 y x x,y ⌘ 2 2  Spin-½: h i

E=1/2 E=-1/2

miércoles 13 de febrero de 13 39 Quantum Link models Rishon (Schwinger) representation

Spin representation: 2 Nx,y Nx,y Nx,y = c† cy + c† cx S~ +1 y x x,y ⌘ 2 2  Spin-1: h i

E=+1 E=0 E=-1

miércoles 13 de febrero de 13 40 Non-abelian quantum link models Rishon (Schwinger) representation with internal degrees of freedom

x y x y ij i j Ux,y cxcy†

miércoles 13 de febrero de 13 41 Non-abelian quantum link models Rishon (Schwinger) representation with internal degrees of freedom

1:U(1) i = : U(2) x y x y (b"#rg : U(3) ij i j Ux,y cxcy†

x y x y x y U(1) group U(2) group U(3) group miércoles 13 de febrero de 13 41 Non-abelian quantum link models Rishon (Schwinger) representation with internal degrees of freedom

Local degrees of freedom.-

ij i j Link operator U c c † x,y ⌘ x y

1 i i i i Electric field [U(1) generator] E c †c c †c x,y ⌘ 2 y y x x ⇥i i i i ⇤ Representation [occupation] Nx,y = cy†cy + cx†cx

miércoles 13 de febrero de 13 42 Non-abelian quantum link models Rishon (Schwinger) representation with internal degrees of freedom

Local degrees of freedom.-

Non-abelian electric fields [SU(N) generators]

Left generators Right generators

a i a i a i a i Lx,y = cx†ijcx Rx,y = cy†ijcy exp i✓aLa U exp i✓aLa =exp[ i✓aa]U x x,y x,y x x,y x x,y exp ⇥i✓aRa ⇤U exp ⇥ i✓aRa ⇤ = U exp i✓aa y x,y x,y y x,y x,y y miércoles 13 de febrero de 13 43 ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ Non-abelian quantum link models Rishon (Schwinger) representation with internal degrees of freedom

Local generators.-

Gx = Ex,x+kˆ Ex k,xˆ Xk ⇣ ⌘ a a a Gx = Lx,x+kˆ + Rx k,xˆ Xk ⇣ ⌘

miércoles 13 de febrero de 13 44 Non-abelian quantum link models Rishon (Schwinger) representation with internal degrees of freedom

Hamiltonian.-

g 2 g2 2 2 H = 0 (E )2 + L~ + R~ 2 x,y 2 x,y x,y x,y x,y  hXi hXi ⇣ ⌘ ⇣ ⌘ 1 U † U U † U + U U † U U † 4g2 1,2 2,3 3,4 4,1 1,2 2,3 3,4 4,1 plaqX h i miércoles 13 de febrero de 13 45 Non-abelian quantum link models with matter

i i j j i ij j † c c † x† Ux,y y x x y y

x y x y

i ij j i i j j H = t †U +h.c. + = t †c c † +h.c. + x x,y y ··· 2 x x 0 y y1 3 ··· x,y ,i,j x,y i ! j h Xi hXi X X 4 @ A 5

Matter - gauge interaction = hopping of fermions mediated by a quantum link = correlated hopping of fermions and rishons miércoles 13 de febrero de 13 46 Non-abelian quantum link models with matter

Local generators.-

i i U(1) generator Gx = x† x Ex,x+kˆ Ex k,xˆ Xk ⇣ ⌘

SU(N) generator a i a j a a Gx = x†ij x + Lx,x+kˆ + Rx k,xˆ Xk ⇣ ⌘

“Physical” Hilbert space G~ x phys =0 x | i 8

miércoles 13 de febrero de 13 47 Non-abelian quantum link models with matter

Hamiltonian.-

[H, G ]=[H, Ga]=0 x x x 8

miércoles 13 de febrero de 13 48 Non-abelian quantum link models

Strong coupling Hamiltonian with staggered fermions

2 2 2 2 g0 2 g i ij j x i i H = (E ) + L~ + R~ t †U +h.c. + m ( 1) † 2 x,y 2 x,y x,y x x,y y x x x,y x,y  x,y ,i,j x,i hXi hXi ⇣ ⌘ ⇣ ⌘ h Xi X Staggered mass Non-abelian electric field Electric field Matter-gauge interaction

miércoles 13 de febrero de 13 49 Non-abelian quantum link models

Strong coupling Hamiltonian with staggered fermions

Staggered fermions: energy

empty mass gap mass gap filling = ½ filled 0 1 2 ... fermi sea miércoles 13 de febrero de 13 49 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model) ( 1)x 1 Gauss’ law Gx = x† x + (Ex,x+1 Ex 1,x) 2

Gx phys =0 ⇢ ~ E~ =0 Spin-1 representation | i , r · 1 0 +1 0 1 | i | i | i | i | i

miércoles 13 de febrero de 13 50 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model) ( 1)x 1 Gauss’ law Gx = x† x + (Ex,x+1 Ex 1,x) 2

Gx phys =0 ⇢ ~ E~ =0 Spin-1 representation | i , r · 1 0 +1 0 1 | i | i | i | i | i Even sites

miércoles 13 de febrero de 13 50 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model) ( 1)x 1 Gauss’ law Gx = x† x + (Ex,x+1 Ex 1,x) 2

Gx phys =0 ⇢ ~ E~ =0 Spin-1 representation | i , r · 1 0 +1 0 1 | i | i | i | i | i Even sites Odd sites

miércoles 13 de febrero de 13 50 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model)

Vacuum state

2 2 g (3) x H = S + m ( 1) † 2 x,y x x x,y x hXi ⇣ ⌘ X

2x 1 2x 2x +1 + Creating a quark - antiquark pair: 2†xS2x,2x+1 2x+1

miércoles 13 de febrero de 13 51 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model)

Vacuum state

2 2 g (3) x H = S + m ( 1) † 2 x,y x x x,y x hXi ⇣ ⌘ X

2x 1 2x 2x +1 + Creating a quark - antiquark pair: 2†xS2x,2x+1 2x+1

miércoles 13 de febrero de 13 51 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model)

Vacuum state

2 2 g (3) x H = S + m ( 1) † 2 x,y x x x,y x hXi ⇣ ⌘ X

2x 1 2x 2x +1 + Creating a quark - antiquark pair: 2†xS2x,2x+1 2x+1

miércoles 13 de febrero de 13 51 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model)

miércoles 13 de febrero de 13 52 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model)

meson meson

miércoles 13 de febrero de 13 52 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model)

meson meson

miércoles 13 de febrero de 13 52 Phenomenology Confinement and string breaking: QED in 1+1D (Schwinger model)

Microscopic picture:

g2 Lm E = (L 1) string 2 2

(L 2) m E = g2 meson 2 2m L =2+ c g2 miércoles 13 de febrero de 13 53 Quantum Chromodynamics: Confinement under normal conditions

i Quarks and gluons carry a color charge x i =

Quarks are confined into color-neutral (color singlet) bound states (hadrons)

qqq baryons: proton, neutron, ... qq- mesons: pions (lightest), kaon, rho, ...

Quarks interact by exchanging gluons i ij j x†Ux,y y miércoles 13 de febrero de 13 54 QCD under extreme conditions

Compress or Hadrons Confinement heat baryons overlap is “lost”

Expect interesting/unsual behaviour

Thermal excitation of pions

Increased baryon density Temperature (T)

pressure or chemical potential (µ B ) miércoles 13 de febrero de 13 55 QCD under extreme conditions

Compress or Hadrons Confinement heat baryons overlap is “lost”

Expect interesting/unsual behaviour Early universe Thermal excitation of pions Heavy-ion collisions

Neutron Increased stars baryon density Temperature (T)

pressure or chemical potential (µ B ) miércoles 13 de febrero de 13 55 Implementation of (non-)abelian quantum link models Strong coupling Hamiltonian with staggered fermions

miércoles 13 de febrero de 13 56 Implementation of (non-)abelian quantum link models Strong coupling Hamiltonian with staggered fermions

x y x y

i ij j i i j j H = t †U +h.c. + = t †c c † +h.c. + x x,y y ··· 2 x x 0 y y1 3 ··· x,y ,i,j x,y i ! j h Xi hXi X X 4 @ A 5 Matter - gauge interaction = hopping of fermions mediated by a quantum link = correlated hopping of fermions and rishons miércoles 13 de febrero de 13 56 Implementation of (non-)abelian quantum link models

i i H = t˜ †c +h.c. hop x x i X(Color singlet) hopping fermion-rishon

miércoles 13 de febrero de 13 57 Implementation of (non-)abelian quantum link models

i i H = t˜ †c +h.c. hop x x i X(Color singlet) hopping fermion-rishon

miércoles 13 de febrero de 13 57 Implementation of (non-)abelian quantum link models

i i H = t˜ †c +h.c. hop x x i X(Color singlet) hopping 1 fermion-rishon ( c c ) p2 | " # i| # " i

The building block is already gauge invariant (summation over internal degrees of freedom)

Action of the hopping fermion-rishon swaps the local singlet to nearest-neighbor ones miércoles 13 de febrero de 13 57 Implementation of (non-)abelian quantum link models x 1 x x +1

x 1 x x +1

t˜ U t˜ U t˜

(Color-singlet interaction/constraint) number of rishon per link

2 2 i i i i H = U [N n] = U c †c + c †c n U x,y x x y y " i # X Constraint: on-site SU(2N) interaction miércoles 13 de febrero de 13 58 Implementation of (non-)abelian Alkaline-earth(-like) atoms quantum link models

Implementation: fermionic alkaline earth atoms

Strontium / Ytterbium

87Sr(I =9/2)

173 i) fermionic alkaline earths have nuclear spin I>0 Yb(I =5/2) • fermionic alkaline-earthsii) scattering independent have of the nuclear nuclear spin spin I > 0 miércoles 13 de febrero de 13 59

2 Implementation of (non-)abelian quantum link models Ground state hyperfine Zeeman levels encode the color degrees of freedom

1 P1 = 32MHz

3 P2 3 P1 3 P0 = 461nm = 698nm

1 =1mHz S0

miércoles 13 de febrero de 13 60 Implementation of (non-)abelian quantum link models 11/2 9/2 7/2 5/2 3 3/2 1/2 P1 1/2 3/2 5/2 7/2 9/2 11/2 0.02 0.05 0.11 + 0.18 0.27 1 9/2 7/2 0.38 0.51 S0 5/2 3/2 0.65 0.82 1/2 1/2 1 1 2 3/2 5/2 | i | i 7/2 9/2 1 2 | i | i

Different polarizations are used to trapped different internal levels

miércoles 13 de febrero de 13 61 Implementation of (non-)abelian quantum link models i i j j i ij j † c c † x† Ux,y y x x y y

x y x y At second order in perturbation theory (t/U):

2 i i i i i i i i H = U c †c + c †c n t˜ †c + c † +h.c. micro x x y y x x y y " i # i X X

i i j j H = t †c c † +h.c. + e↵ 2 x x 0 y y1 3 ··· x,y i ! j hXi X X miércoles 13 de febrero de 13 4 @ A 5 62 Implementation of (non-)abelian quantum link models i i j j i ij j † c c † x† Ux,y y x x y y

x y x y At second order in perturbation theory (t/U):

2 i i i i i i i i H = U c †c + c †c n t˜ †c + c † +h.c. micro x x y y x x y y " i # i X X

i i j j H = t †c c † +h.c. + e↵ 2 x x 0 y y1 3 ··· x,y i ! j hXi X X miércoles 13 de febrero de 13 4 @ A 5 62 Observability of phenomena

Preparation of many body states (Mott phase)

Greiner et al. (2002) Joerdens et al. (2008) Schneider et al. (2008) miércoles 13 de febrero de 13 63 NATURE|Vol 453|19 June 2008 INSIGHT REVIEW can create a highly entangled multiparticle state20,21, known as a ‘cluster possibility is to exploit the symmetry of the underlying two-particle state’36, which can be used as a resource state for quantum information wavefunctions to create the desired gate operations, even in the case processing. The superposition principle of quantum mechanics allows of completely spin-independent interactions between atoms. This this to be achieved in a highly parallel way, using a state-dependent optical principle lies at the heart of two experiments to control the spin–spin lattice, in which different atomic spin states experience different periodic interactions between two particles using exchange symmetry23,39,49, and potentials20,21. Starting from a lattice where each site is filled with a single builds on original ideas and experiments involving double quantum- atom, the atoms are first brought into a superposition of two internal dot systems25,26. spin states. The spin-dependent lattice is then moved in such a way that Research teams at the National Institute of Standards and Technol- an atom in two different spin states splits up and moves to the left and ogy (NIST) at Gaithersburg, Maryland, and the University of Mainz, right simultaneously so that it collides with its two neighbours. In a single Germany, have demonstrated such interactions for two atoms in a operation, a whole string of atoms can thereby be entangled. However, if double-well potential. How do these exchange interactions arise, and the initial string of atoms contained defects, an atom moving to the side how can they be used to develop primitives (or building blocks) for may have no partner to collide with, so the length of the entangled cluster quantum information processing? As one of the fundamental principles would be limited to the average length between two defects. The sorted of quantum mechanics, the total quantum state of two particles (used in arrays of atoms produced by an ‘atomic sorting machine’ could proveObservability to two experiments) has to be either of symmetrical phenomena in the case of bosons or be an ideal starting point for such collisional quantum gates, as the initial antisymmetrical for fermions, with respect to exchange of the two par- arrays are defect free. In addition, defects could be efficiently removed by ticles. When trapped on a single lattice site, a two-particle bosonic wave- further active cooling of the quantum gases in the lattice. Indeed, such function can be factored intoEvolution a spatial component, which describes the cooling is necessary to enhance the regularity of the filling achieved with positions of the two particles,(Super-exchange) and a spin component, which describes the current large-scale ensembles. Several concepts related to ‘dark state’ cooling methods from quantum optics and laser cooling could help in a this case. The atoms could be actively cooled into the desired many-body quantum state, which is tailored to be non-interacting (that is, dark) with 37,38 the applied cooling laser field . 2∆ When constructing such entangled states, the particles’ many degrees of freedom can couple to the environment, leading to decoherence, J J which will destroy the complex quantum superpositions of the atoms. U To avoid such decoherence processes, which affect the system more the larger it becomes, it is desirable to construct many-particle states, which are highly insensitive to external perturbations. Unfortunately, when using the outlined controlled-collisions scheme to create an atomic J cluster state, the atomic qubits must be encoded in states that undergo ex maximal decoherence with respect to magnetic field fluctuations. Two recent experiments have shown how decoherence could be avoided, by implementing controlled exchange interactions between atoms23,39; this b 0.5 could lead to new ways of creating robust entangled states (discussed in Anderlini et al. (2007) the next section). Another way to avoid the problem of decoherence is to 0 Trotzky et al. (2008) apply faster quantum gates, so more gate operations couldmiércoles be carried 13 de febreroout de 13 64 within a fixed decoherence time. For the atoms of ultracold gases in optical –0.5 lattices, Feshbach resonances40,41 can be used to increase the collisional interactions and thereby speed up gate operations. However, the ‘unitarity 0123 limit’ in scattering theory does not allow the collisional interaction energy 0.5 to be increased beyond the on-site vibrational oscillation frequency, so the lower timescale for a gate operation is typically a few tens of microseconds. 0 Much larger interaction energies, and hence faster gate times, could be achieved by using the electric dipole–dipole interactions between polar –0.5 42 43,44 molecules , for example, or Rydberg atoms ; in the latter case, gate 02468 times well below the microsecond range are possible. For Rydberg atoms, a and spin imbalance Population 0.5 phase gate between two atoms could be implemented by a dipole-blockade mechanism, which inhibits the simultaneous excitation of two atoms and 0 thereby induces a phase shift in the two-particle state only when both atoms are initially placed in the same quantum state. The first signs of such a Rydberg dipole-blockade mechanism have been observed in mesoscopic –0.5 cold and ultracold atom clouds45–48, but it remains to be seen how well they 0 50 100 150 200 can be used to implement quantum gates between two individual atoms. Time (ms) Rydberg atoms offer an important advantage for the entanglement of neu- Figure 5 | Superexchange coupling between atoms on neighbouring lattice tral atoms: they can interact over longer distances, and addressing single sites. a, Virtual hopping processes (left to right, and right to left) mediate atoms in the lattice to turn the interactions between these two atoms on an effective spin–spin interaction with strength Jex between the atoms, and off avoids the need for the atoms to move. In addition, the lattice does which can be controlled in both magnitude and sign by using a potential not have to be perfectly filled for two atoms to be entangled if their initial bias ∆ between the wells. U is the on-site interaction energy between the position is known before applying the Rydberg interaction. atoms on a single lattice site, and J is the single-particle tunnel coupling. b, The effective spin–spin interaction emerges when increasing the Novel quantum gates via exchange interactions interaction U between the particles relative to their kinetic energy J (top to bottom). It can be observed in the time evolution of the magnetization Entangling neutral atoms requires state-dependent interactions. A nat- dynamics in the double well. Blue circles indicate spin imbalance, and ural way to achieve this is to tune the collisional interactions between brown circles indicate population imbalance. The curves denote a fit to a atoms to different strengths for different spin states, or to allow explicitly theoretical model taking into account the full dynamics observed within the only specific spin states into contact for controlled collisions. Another Hubbard model. (Reproduced, with permission, from ref. 39.) 1019 NATURE | Vol 462 | 5 November 2009 LETTERS

CCD d a 1.0 b

0.8 e 0.6 g f 0.4 Intensity (a.u.) 0.2 b

25 μm 0.0 a y –1.0 –0.5 0.0 0.5 1.0 0 1,000 2,000 Position ( m) Counts μ

c Figure 2 | Imaging single atoms. a, Field of view with sparse site occupation. z b, Response of a single atom, derived from sparse images: shown are Figure 1 | Diagram of the quantum gas microscope. The two-dimensional horizontal (filled circles) and vertical (open circles) profiles through the atom sample (a) is located a few micrometres below the lower surface of a centre of the image generated by a single atom. The black line shows the hemispherical lens inside the vacuum chamber. This lens serves to increase expected Airy function for a perfect imaging system with a numerical the numerical aperture (NA) of the objective lens outside the vacuum (b)by aperture of 0.8. The blue dashed line denotes the profile expected from a the index of refraction, from NA 5 0.55 to NA 5 0.8. The atoms are single atom, taking into account only the finite width of the CCD pixels and illuminated from the side by the molasses beams (c) and the scattered the finite extension of the probability distribution of the atom’s location. fluorescence light is collected by the objective lens and projected onto a CCD The data are from the responses of 20 atoms in different locations within the camera (d). A 2D optical lattice is generated by projecting a periodic mask field of view which have been precisely superimposed by subpixel shifting (e) onto the atoms through the same objective lens via a beam splitter before averaging. (f). The mask is a periodic phase hologram, and a beam stop (g) blocks the residual zeroth order, leaving only the first orders to form a sinusoidal pixels and the expected extent of the atom’s on-site probability dis- potential. tribution within the lattice site during the imaging process. As the same high-resolution optics are used to generate both the lattice and of a phase hologram. This is in contrast to conventional optical lattice the image of the atoms on the CCD camera, the imaging system is experiments in which lattice potentials are created by superimposing very stable with respect to the lattice, which is important for single- separate laser beams to create optical standing waves. The advantage site addressing26. The observed drifts in the 2D plane are very low, less of the new method is that the geometry of the lattice is directly given by than 10% of the lattice spacing in one hour with shot to shot fluctua- the pattern on the mask. The imaged light pattern, and hence the tions of less than 15% r.m.s. potential landscape, can be arbitrary within the limits set by the avail- Pair densities within multiply occupied lattice sites are very high able imaging aperture and by polarization effects that can arise due to due to the strong confinement in the lattice. When resonantly illu- the large aperture imaging beyond the paraxial limit. Here, we create minated, such pairs undergo light assisted collisions and leave the blue detuned square lattice potentials with a periodicity a 5 640 nm trap within a time of the order of 100 ms, long before they emit and an overall Gaussian envelope. A major additional advantage is the sufficient photons to be detected27. Therefore the remaining number fact that the lattice geometryObservability is not dependent on the wavelength20, of phenomena apart from diffraction limits and chromatic aberrations in the lens for large wavelength changes. This allows us to use spectrally wide ‘white’ light with a short coherence length to reduce unwanted disorder from stray light interference. With a light source centred around 758 nm, we generate a conservative lattice potential with a lattice depth of up to 35 2 2 Erec, where Erec 5 h /8ma is the recoilDetection energy of the effective lattice wavelength, with m(Single-sitethe mass of 87Rb. fluorescence) The projection method also enables us to dynamically change the wavelength of the lattice light without changing the lattice geometry. This is important, as we strongly increase the lattice depth for site- resolved imaging in order to suppress diffusion of the atoms between sites due to recoil heating by the imaging light13. For this, we switch ARTICLE RESEARCH the light in the 2D lattice and the vertical standing wave to near- resonant narrow band light, increasing the lattice depth to 5,500 5 μm Erec (to 380 mK). The main use of the microscope set-up is the col- lection of fluorescence light and high-resolution imaging of the atoms. With thea atoms pinned to the deep lattice, we illuminateb c d the sample with red detuned near-resonant light in an optical 640 nm molasses configuration, which simultaneously provides sub- Doppler cooling24,25. Figure 2 shows a typical image obtained by loading the lattice with a very dilute cloud, showing the response of individual atoms. The spot function of a single atom can be Figure 3 | Site-resolved imaging of single atoms on a 640-nm-period directly obtained from such images. We measure a typical single optical lattice, loaded with a high density Bose–Einstein condensate. Inset, atom emission FWHM size as 570 nm and 630 nm along the x and magnified view of the central section of the picture.Bakr The lattice et al. structure (2010) and y direction, respectively, which is close to the theoretical minimum the discrete atoms are clearly visible. OwingWeitenberg to light-assisted collisions et al. (2011) and value of ,520 nm (Fig. 3). This minimum is given by the diffractiony molecule formation on multiply occupied sites during imaging, only empty limit from the objective combined with the finite size of the camera and singly occupied sites can be seen in the image. 2 μm 75 ©2009 Macmillan Publishers xLimited. All rights reserved

miércoles 13 de febrero de 13 65 e f

Figure 2 | Single-site addressing. a, Top, experimentally obtained and | 1æ, such that only the addressed atoms were observed. Bottom, the fluorescence image of a Mott insulator with unity filling in which the spin of reconstructed atom number distribution shows 14 atoms on neighbouring selected atoms was flipped from | 0æ to | 1æ using our single-site addressing sites. c–f, As for b, but omitting the atom number distribution. The images scheme. Atoms in state | 1æ were removed by a resonant laser pulse before contain 29 (c), 35 (d), 18 (e) and 23 (f) atoms. The single isolated atoms in detection. Bottom, the reconstructed atom number distribution on the lattice. b, e and f were placed intentionally to allow for the correct determination of the Each filled circle indicates a single atom; the points mark the lattice sites. b, Top, lattice phase for the feedback on the addressing beam position. as for a except that a global microwave sweep exchanged the population in | 0æ atoms, which ideally should remain unaffected. For this purpose, we within 200 ms. At the same time, the other lattices were lowered to monitored the probability of finding a hole at the sites next to the Vy 5 30 Er and Vz 5 23 Er, which reduced the external confinement addressed ones (dark blue regions in Fig. 3a, b and points in Fig. 3c). In along the x direction, but still suppressed tunnelling in the y and z order to distinguish accidentally flipped neighbouring atoms from directions. After a varying hold time t, allowing the atoms to tunnel holes that originate from thermal excitations of the initial Mott insu- along x, the atomic distribution was frozen by a rapid 100 ms ramp of all 28 lator , we also monitored the probability of finding a hole at the lattice axes to 56–90 Er. By averaging the resulting atomic distribution second next neighbours (light blue regions and points in Fig. 3). As along the y direction and repeating the experiment several times, we both yielded the same hole probability of 6(2)%, we attribute all holes obtained the probability distribution of finding an atom at the different to thermal excitations and conclude that the probability of addressing lattice sites (Fig. 4, bottom row). a neighbouring atom is indiscernibly small. We fitted the hole prob- This probability distribution samples the single-atom wave- ability p0(dx) of the addressed site with a flat-top model function (see function after a coherent tunnelling evolution. We observed how Methods), keeping the offset fixed at the thermal contribution of 6%. the wavefunction expands in the lattice and how the interference of From the fit, we derived a spin-flip fidelity of 95(2)%, an FWHM of different paths leads to distinct maxima and minima in the distri- sa 5 330(10) nm and an edge sharpness of ss 5 50(10) nm (Fig. 3c). bution, leaving for example almost no atoms at the initial position These values correspond to 60% and 10% of the addressing beam after a single tunnelling time (Fig. 4c). This behaviour differs mark- diameter, demonstrating that our method reaches sub-diffraction- edly from the evolution in free space, where a Gaussian wave packet limited resolution, well below the lattice spacing. disperses without changing its shape, always preserving a maximum The observed maximum spin-flip fidelity is currently limited by the probability in the centre. For longer hold times, an asymmetry in the population transfer efficiency of our microwave sweep. The edge spatial distribution becomes apparent (Fig. 4d), which originates from sharpness ss originates from the beam pointing error of = 0.1 alat an offset between the bottom of the external harmonic confinement and from variations in the magnetic bias field. The latter causes fre- and the initial position of the atoms. quency fluctuations of ,5 kHz, which translate into an effective We describe the observed tunnelling dynamics by a simple (0) pointing error of 0.05 alat at the maximum slope of the addressing Hamiltonian including the tunnel coupling J between two neighbour- beam profile. The resolution sa could in principle be further reduced ing sites and an external harmonic confinement, parameterized by the by a narrower microwave sweep, at the cost of a larger sensitivity to trap frequency vtrap, and the position offset xoffs (Methods). A single fit the magnetic field fluctuations. A larger addressing beam power to all probability distributions recorded at different hold times yields (0) would reduce this sensitivity, but we observed that this deformed J /B 5 940(20) Hz, vtrap/(2p) 5 103(4) Hz and xoffs 526.3(6) alat. 2 the lattice potential, owing to the imperfect s -polarization, allowing This is in agreement with the trap frequency vtrap/(2p) 5 107(2) Hz neighbouring atoms to tunnel to the addressed sites. obtained from an independent measurement via excitation of the dipole mode without the x lattice, whose contribution to the external confine- Coherent tunnelling dynamics ment is negligible compared to the other two axes. From J(0), we calcu- The preparation of an arbitrary atom distribution opens up new pos- lated a lattice depth of Vx 5 4.6(1) Er, which agrees with an independent sibilities for exploring coherent quantum dynamics at the single-atom calibration via parametric heating. The expansion of the wave packet level. As an example, we studied the tunnelling dynamics in a one- can also be understood by writing the initial localized wavefunc- dimensional lattice (Fig. 4) which allowed us to determine how much tion as a superposition of all Bloch waves of quasi-momentum Bq, our addressing scheme affects the vibrational state of the atoms. We with 2p/alat # q # p/alat. To each quasi-momentum Bq, one can 1 LE started by preparing a single line of up to 18 atoms along the y direction assign a velocity vq~ q, determined by the dispersion relation (0) B L before we lowered the lattice along the x direction to Vx 5 5.0(5) Er E(q) 522J cos(qalat) of the lowest band. The edges of the wave

Simpler atomic/molecular/solid state implementations (not in the talk: QLM with magnetic atoms/polar molecules!)?

Superconducting Qubits (D. Marcos,...), Dipoles and Rydberg (A. Glaetzle, ...)

Connection with gauge magnets and spin liquids (in principle accessible within this toolbox)

Finite-temperature confinement/deconfinement phase transition, deconfined criticality in ‘feasible quantum link’?

Still very far away from QCD (even with the SU(3)): next steps? miércoles 13 de febrero de 13 66 The team • Albert Einstein Center - Bern University

D. Banerjee M. Bögli P. Stebler P. Widmer U.-J. Wiese • Complutense University • Technical University Madrid Vienna

M. Müller P. Rabl

• IQOQI - Innsbruck University

M. Dalmonte D. Marcos P. Zoller miércoles 13 de febrero de 13 67