Cold Atom Quantum Simulator for String and Hadron Dynamics in Non-Abelian Lattice Gauge Theory
UUMD-PP-020-8
Cold Atom Quantum Simulator for String and Hadron Dynamics in Non-AbelianLattice Gauge Theory
Raka Dasgupta and Indrakshi Raychowdhury Dept. of Physics, University of Calcutta, 92 A. P. C. Road, Kolkata-700009, India ∗ Maryland Center for Fundamental Physics and Department of Physics,University of Maryland, College Park, MD 20742, USA † (Dated: September 30, 2020)We propose an analog quantum simulator for simulating real time dynamics of (1 + 1)-d non-Abelian gauge theory well within the existing capacity of ultracold atom experiments. The schemecalls for the realization of a two-state ultracold fermionic system in a 1-dimensional bipartite lattice,and the observation of subsequent tunneling dynamics. Being based on novel loop string hadronformalism of SU(2) lattice gauge theory, this simulation technique is completely SU(2) invariant andsimulates accurate dynamics of physical phenomena such as string breaking and/or pair production.The scheme is scalable, and particularly effective in simulating the theory in weak coupling regime,and also bulk limit of the theory in strong coupling regime up to certain approximations. Thispaper also presents a numerical benchmark comparison of exact spectrum and real time dynamicsof lattice gauge theory to that of the atomic Hamiltonian with experimentally realizable range ofparameters. I. INTRODUCTION
Gauge field theories constitute an exceptionally pow-erful theoretical framework that describes three of thefour fundamental interactions of nature. Non-Abeliangauge symmetry lies at the heart of standard model ofparticle physics. Quantum chromodynamics (QCD) isa SU(3) gauge theory that accurately describes interac-tions between quarks via gluons as mediators. However,the phenomenon of quark confinement is yet to be es-tablished analytically from the fundamental theories inphysics. In 1974, Wilson proposed a regularization ofthe gauge theory on space-time lattices [1] that exhibitsthe phenomenon of confinement in the strong couplinglimit. Wilson’s lattice gauge theory (LGT) is an ex-tremely successful non-perturbative technique that hasbeen used extensively over past four-five decades. Oneof the reasons behind this versatility and success storyof LGT is that one can perform lattice QCD calcula-tions by Monte Carlo simulations [2] and world’s largestsuper-computing resources are now being employed forthe same [3].Although the lattice QCD program based on numeri-cal calculations is really successful, there is the infamous‘sign problem’ that forbids lattice QCD calculations [4]to work at certain regimes such as to address systemswith finite and non-zero density or calculating real timedynamics. The natural framework to study dynamics ofgauge theory is Hamiltonian framework. However, theexponential growth of Hilbert space dimension with sys-tem size for any quantum system claims that classicalcomputer may not be the best computational tool [5]for Hamiltonian simulation. Feynman’s visionary idea ∗ [email protected] † [email protected] [6] combined with recent technological advancement inquantum information science and technology leads to aperfect alternative in the form of quantum computationand/or simulation, and offers hope for settling these is-sues.The concept of analog quantum simulation involvesmimicking a quantum system described by a Hamilto-nian (simulated Hamiltonian) by another completely dif-ferent quantum system described by some other Hamilto-nian (simulating Hamiltonian). The idea is, if it is hardto analyze the first Hamiltonian mathematically or nu-merically, it can be mapped to the second Hamiltonian,and that Hamiltonian can be studied in an experimentby suitably tuning the parameters. Systems of ultracoldatoms [7] or ions [8] trapped in optical lattices serve asexcellent quantum simulators, as the relevant parameterscan be precisely measured and controlled. In the past fewyears, there has been a steady progress in this directionin the context of gauge theory. [9–22].The experimental realization of Bose-Einstein conden-sate in 1995 [23–25] and the subsequent exploration oftrapped ultracold fermions [26–29] in the degenerateregime paved the way towards newer directions of atomicand molecular physics. In the initial years, theoreticaland experimental works in this field mostly focused onthe emergence of macroscopic quantum coherent phe-nomena in a many-body system, and the formation ofmolecular condensates [30–32] . However, it was soondiscovered that ultracold atomic systems can serve aswonderful testing grounds for other branches of physicsas well, by virtue of the sheer tunability of the param-eters. The atom-atom scattering length (and thus, theinteraction strength ) in ultracold gases can be variedacross a wide range by methods of Feshbach resonances.The creation of optical lattices [33] by using two counter-propagating coherent laser beams took this tunability astep further : as the size, shape and dimensionality ofthe lattice could be easily controlled. Advanced cooling a r X i v : . [ h e p - l a t ] S e p and trapping methods has now led to temperatures aslow as in nanoKelvin and picoKelvin ranges, resulting inquantum engineering at an unparalleled precision level.This allows for each individual atom to be monitored,and one can have a perfect quantum simulator. In thepast, cold atomic systems have successfully emulated arich variety of systems and addressed problems in dis-ordered systems, spin liquids, superconductivity, nuclearpairing, artificial gauge fields and topology [34–36].Over the last few years, there has been a continuouspursuit towards analog quantum simulating lattice gaugetheories using cold atom systems. During the first halfof the past decade, these proposals mostly focused onconstructing quantum simulators for both Abelian andnon-Abelian gauge theories in Kogut-Susskind formal-ism [9–12] as well as Quantum Link Model formulation[13–15] and also Abelian Higgs Model in 2+1 dimensions[16]. All these schemes involved a careful designing of theset-up so that the system remains in the gauge invari-ant Hilbert space throughout the dynamics. There wasalso one generic experimental proposal for quantum sim-ulating non-Abelian gauge theories using Rydberg atomgates [17]. A general feature of all of these proposals are :the lattice sites (for matter field) and the links (for gaugefield) of the original lattice gauge theory are simulated bybosonic and/or fermionic atoms trapped in arrays of po-tential wells of optical lattices; there being a one-to-onecorrespondence between the LGT sites and the sites ofthe spatial optical lattice. Additionally, auxiliary atomswere also considered to effectively create the plaquetteterm of the gauge theory Hamiltonian [10, 12, 16]. How-ever, none of these proposals has yet been experimentallyimplemented or numerically benchmarked. We note thatthe proposals involving quantum link models are moresuited for practical realization. The amount of infor-mation of QED actually captured in finite dimensionalHilbert space of Quantum Link Model and its real timedynamics was studied in [18].Following the first experimental demonstration of adigital quantum simulation of a lattice Schwinger model[37], the first analog quantum simulation of Z gaugetheory on a two staggered site lattice by cold atom quan-tum simulator was reported in [19]. However, general-ization of this scheme either to make it scalable or tosimulate theory with continuous gauge groups has notyet been reported. The first experiment demonstratinga scalable quantum simulation of continuous gauge the-ory was reported very recently [20] that demonstrates theengineering of an elementary building block of the U(1)quantum link model in 1 spatial dimension using a mix-ture of bosonic atoms. This scheme is quite unique, asunlike the past proposals, a single potential well here doesnot merely hosts a site or link of gauge theory. Instead,a physical site here contains both the matter and thegauge states in the form of two different atoms, and alsothe matter-gauge field interactions : thus acting as thefundamental building block of the gauge theory Hamil-tonian. Exact Implementation of gauge invariance is an- other challenging task in any of these quantum simula-tors. Recently U(1) gauge invariance in the simulatedsystem has been observed experimentally for a long lat-tice [21]. In spite of these significant advances, a prac-tically realizable analog quantum simulation scheme forsimulating dynamics of a gauge theory even with the sim-plest non-Abelian, continuous gauge group such as SU(2)is absent in past literature.In this paper, we present a scalable and immediatelyrealizable quantum simulation proposal for quantum sim-ulating SU(2) gauge theory coupled to fermionic stag-gered matter in (1 + 1)-d. This work also provides nu-merical study of the spectrum of the simulating and simu-lated Hamiltonian and compare their real time dynamicsfor a small lattice.We aim to quantum simulate the Kogut-Susskind (KS)Hamiltonian for LGT [38]. In a recent study [39], ithas been demonstrated that amongst many variants ofHamiltonian formulation of non-Abelian gauge theories[40–45], the loop string hadron (LSH) formalism [46] isthe most convenient and computationally least expen-sive one for (1 + 1)-d within the scope of classical com-putation. The reason is, being a fully gauge invariantformalism, the LSH Hamiltonian describes the dynam-ics of only relevant physical degrees of freedom. In 1dspatial lattice, that is precisely the dynamics of stringsand hadrons. It can be shown [39, 47] that in 1+1d,any gauge theory with open boundary condition can bemapped to a theory of only fermions, i.e., equivalent tothe XYZ model and hence much simpler to analyze. Thenovel LSH formalism shares many features of this purelyfermionic formalism but can actually be generalized toperiodic boundary conditions as well as to higher dimen-sions [46].The present paper exploits this versatility of LSH for-malism of SU(2) gauge theory. Here, different parame-ter regimes of SU(2) gauge theory are mapped to differ-ent parameter regimes of an atomic Hamiltonian: thatof an ionic Hubbard model with different total numberof fermions on the lattice. We consider the half filledHubbard model that is exactly equivalent to the gaugetheory Hilbert space containing strong coupling vacuum.We show that the spectrum, obtained with exact diago-nalization of both the simulating and simulated systemcompared remarkably in the weak coupling regime. Wealso provide a benchmark comparison of the dynamics ofatomic system directly mapped to the pair production-string breaking dynamics of the low energy sector ofSU(2) gauge theory. The numerical analysis employs pa-rameters and experimental set-ups already realized withultracold atom systems. We demonstrate two key points:i) the full gauge theory Hamiltonian can be reduced to anapproximated LSH Hamiltonian, which, in turn, can beperfectly mimicked by the atomic system to the low en-ergy dynamics in the weak coupling limit of gauge theory,and ii) for the strong and intermediate coupling regimes,the difference between the full gauge theory Hamilto-nian and the approximated Hamiltonian is slightly moreprominent. But one can still access dynamics of stringsand hadrons in presence of a background gauge field inthe bulk limit of the lattice by tuning the on-site in-teraction parameter of the Hubbard Hamiltonian. Fur-ther improvements of this scheme to include dynamicalgauge fields in higher dimension, and also generalizationto SU(3) gauge theory will take us close to quantum sim-ulating the full QCD.In this proposal, a system of ultracold fermionstrapped in optical lattices is considered as quantum simu-lating platform for non-Abelian gauge theories. The planof the paper is as follows: In section II we briefly discusslattice gauge theory Hamiltonian including LSH frame-work in general and also in the weak coupling limit. Insection III we discuss the atomic system to be used forthe quantum simulation scheme, a fermionic Hubbardmodel on a bipartite lattice. In section IV we map thegauge theory Hamiltonian to the Hubbard model Hamil-tonian introduced before for both weak and strong cou-pling regimes of gauge theory. In section V the proposedexperimental set-up is described. Section VI contains nu-merical study and comparison of the spectrum and realtime dynamics of both the simulating and simulated sys-tems using the parameters for the proposed experimentalscheme. Finally, in section VII we discuss our results andalso future prospects. II. SU(2) LATTICE GAUGE THEORY IN (1 + 1) -D Hamiltonian or canonical formulation of lattice gaugetheories was developed by Kogut and Susskind [38] rightafter Wilson introduced lattice gauge theory originally inEuclidean formalism [1]. While, classical computing forlattice gauge theory has explored the original Euclideanformulation, the Hamiltonian framework, being not muchuseful in classical computation era remain unexplored.However, the interest in exploring Hamiltonian descrip-tion of lattice gauge theories is renewed, as it turns out tobe the natural framework to work with in the upcomingquantum simulation/computation era. The mostly usedformalism in this context is Quantum Link Model repre-sentation of gauge theory as it provides a finite dimen-sional representation of the gauge fields. There is a draw-back though : in smaller dimensions, that are accessibleby present-day quantum technology, the quantum linkmodel does not have the desired spectrum as obtainedwith the original Kogut-Susskind Hamiltonian [39, 40].In this work we consider the original Kogut-SusskindHamiltonian for simplest non-Abelian gauge group, i.e.SU(2) and proceed to construct a quantum simulator forthe same in (1 + 1)-d.The Kogut-Susskind (KS) Hamiltonian describingSU(2) Yang Mills theory coupled to staggered fermionson (1 + 1)-d (1d spatial lattice and continuous time) [38]can be written as: H (KS) = H (KS) E + H (KS) M + H (KS) I . (1) Where, H (KS) E corresponds to electric part of the Hamil-tonian given by, H (KS) E = g a N − (cid:88) j =0 3 (cid:88) a =1 E a ( j ) E a ( j ) . (2)Here, (cid:88) a =1 E a ( j ) E a ( j ) = (cid:88) a =1 E aL ( j ) E aL ( j ) = (cid:88) a =1 E aR ( j ) E aR ( j )for left and right electric fields E L/R associated with alink connecting sites j and j + 1.The staggered fermionic matter ψ , in the fundamen-tal representation of SU (2) consisting of two components (cid:0) ψ ψ (cid:1) yields a staggered mass term: H (KS) M = m N (cid:88) j =0 ( − j (cid:2) ψ † ( j ) · ψ ( j ) (cid:3) . (3) H (KS) I denotes interaction between the fermionic andgauge fields and is given by: H (KS) I = 12 a N − (cid:88) j =0 (cid:2) ψ † ( j ) U ( j ) ψ ( j + 1) + h . c . (cid:3) . (4)The gauge link U ( j ) is a 2 × j and j + 1. A temporal gaugeis chosen to derive the above Hamiltonian which sets thegauge link along the temporal direction equal to unity.The color electric fields E aL/R are defined at the left L and right R sides of each link and they satisfy thefollowing commutation relations (su(2) algebra) at eachend: [ E aL ( j ) , E bL ( j (cid:48) )] = i(cid:15) abc δ jj (cid:48) E cL ( j ) , [ E aR ( j ) , E bR ( j (cid:48) )] = i(cid:15) abc δ jj (cid:48) E cR ( j (cid:48) ) , [ E aL ( j ) , E bR ( j (cid:48) )] = 0 , (5)where (cid:15) abc is the Levi-Civita symbol. The electric fieldsand the gauge link satisfy the following quantization con-ditions at each site,[ E aL ( j ) , U ( j (cid:48) )] = − σ a δ jj (cid:48) U ( j ) , [ E aR ( j ) , U ( j (cid:48) )] = U ( j ) σ a , (6)where σ a are the Pauli matrices. The Hamiltonian in (1)is gauge invariant as it commutes with the Gauss’ lawoperator, G a ( j ) = E aL ( j ) + E aR ( j −
1) + ψ † ( j ) σ a ψ ( j ) (7)at each site j . The physical sector of the Hilbert spacecorresponds to the space consisting of states annihilatedby (7).For LGT, the natural and most convenient basis isformed out of eigenstates of the electric-field operator.Tensor product of fermionic occupation number basis andelectric field basis constitute the full Hilbert space. Thisparticular basis, being eigenbasis of the diagonal Hamil-tonian ( H E + H M ) in the g → ∞ limit, is called thestrong-coupling basis of LGT.In the strong coupling regime, lattice gauge theoryshows desired physics such as quark confinement and fi-nite mass gap. In this limit, along with finite latticespacing a , the interaction terms in (4) that involves tran-sitions between different eigenstates of the electric-fieldoperator becomes insignificant, and hence in the Hamilto-nian, diagonal terms dominate over the off-diagonal onesin the strong coupling basis. As g → ∞ , only very smallelectric flux configurations on the lattice contribute tothe low energy sector of the theory. In this regime, lat-tice Hamiltonian matrices can be analyzed perturbativelywith the electric part as the unperturbed Hamiltonian.Order by order perturbation corrections yield a finite di-mensional Hilbert space, within a cut-off imposed on thebosonic quantum number corresponding to gauge flux.The computation cost rises exponentially with increas-ing Hilbert space dimension, that grows with system sizeas well as cut-off [39]. As a result, calculating Hamilto-nian dynamics for an arbitrary large system even withthe largest possible computer seems impossible. How-ever, the continuum limit of the LGT lies in the oppositeregime, where g → , a → N → ∞ . In this regime, the dynamicsbecomes too much cut-off sensitive, all possible electricflux states do contribute to the low energy spectrum ofthe theory with major contribution coming from strongcoupling states with electric flux values to be larger andlarger with g →
0. The Hamiltonian moves away fromdiagonal structure as (4) becomes dominant with a → A. Loop-String-Hadron (LSH) Hamiltonian
LSH formalism of lattice gauge theory is based on pre-potential framework, where, the original canonical con-jugate variables of the theory, i.e color electric field andlink operators are replaced by a set of harmonic oscillatordoublets, defined at each end of a link [50–58]. In pre-potential framework, the SU(2) gauge group is confinedto each lattice site allowing one to have local gauge in-variant operators and states at each site. For pure gaugetheory, these local gauge invariant operators and statescan be interpreted as local snapshots of Wilson loop op-erators of original gauge theory. One can now constructlocal loop Hilbert space by action of local loop operatorson strong coupling vacuum of the theory (no flux state)defined locally at each site. At this point, we must men-tion that, mapping the local loop picture to original loopdescription of gauge theory requires one extra constrainton each link, that states N L ( j ) = N R ( j ) (8)where, N L ( R ) is occupation number of prepoten-tials/Schwinger bosons at the left(right) end of a linkconnecting sites j and j + 1. This constraint is actuallya consequence of the constraint E L = E R mentioned insection II.Inclusion of staggered fermionic matter fields for SU(2)gauge theory at each lattice site, combines smoothly withlocal loop description obtained in prepotential frameworkas both the prepotential Schwinger bosons and matterfields transform as fundamental representation of the lo-cal SU(2) at that site. In addition to local gauge in-variant loop operators, one can now combine matter andprepotentials to construct local string operators, that de-notes start of a string from a particle and/or end of astring at an antiparticle. Matter fields combine into localgauge invariant configurations representing hadrons like-wise in the original formalism. This complete descriptionis named as LSH formalism as in [46]. We are not goinginto the details of the full LSH formalism here. Instead,we will focus on the application of LSH formulation toone spatial dimension only, and describe the appropriateframework.Within LSH framework, the gauge invariant and or-thonormal LSH basis is characterized by a set of threeintegers n l ( j ) , n i ( j ) , n o ( j ) that satisfies Gauss’ law con-straint: G a ( j ) | n l ( j ) , n i ( j ) , n o ( j ) (cid:105) = 0 , ∀ j, a. (9) FIG. 1. Two staggered sites in the LSH formulation on a1d spatial lattice. Each site carries three types of operatorsnamely incoming string, outgoing string and flux. The Hilbertspace is characterized by the corresponding quantum numbers n l , n i , n o respectively for each and every site of the lattice. These three quantum numbers signify loop, incomingstring and outgoing string at each site. The allowed val-ues of these integers are given by0 ≤ n l ( j ) ≤ ∞ (10)0 ≤ n i ( j ) ≤ ≤ n O ( j ) ≤ n l is bosonic excitation, whereas n i , n o arefermionic in nature. However, it is important to notethat, unlike fermionic matter field in the original theory,the fermionic operators building the ‘local string’ Hilbertspace are SU(2) invariant bilinears of one bosonic prepo-tential operator and one fermionic matter field, yieldingoverall fermionic statistics. Hence, the string states con-tain the information of both gauge field and matter con-tent.At this point, we define a set of LSH operators consist-ing of both diagonal and ladder operators locally at eachsite as following:ˆ n l | n l , n i , n o (cid:105) = n l | n l , n i , n o (cid:105) (13)ˆ n i | n l , n i , n o (cid:105) = n i | n l , n i , n o (cid:105) (14)ˆ n o | n l , n i , n o (cid:105) = n o | n l , n i , n o (cid:105) (15)ˆ λ ± | n l , n i , n o (cid:105) = | n l ± , n i , n o (cid:105) (16)ˆ χ + i | n l , n i , n o (cid:105) = (1 − δ n i , ) | n l , n i + 1 , n o (cid:105) (17)ˆ χ − i | n l , n i , n o (cid:105) = (1 − δ n i , ) | n l , n i − , n o (cid:105) (18)ˆ χ + o | n l , n i , n o (cid:105) = (1 − δ n o , ) | n l , n i , n o + 1 (cid:105) (19)ˆ χ − o | n l , n i , n o (cid:105) = (1 − δ n o , ) | n l , n i , n o − (cid:105) (20)In the above set of equations, we have not mentionedexplicit site index as these are considered to be definedat a particular site.One major benefit of using LSH formalism is that, oneno longer needs to solve/satisfy SU(2) Gauss’ law (7) ateach site as the basis states are SU(2) gauge invariant byconstruction. Note that, for non-Abelian gauge theoriesimposing Gauss’ law is a non-trivial task and that givesrise to a whole range of complications as discussed in [39].However, the LSH formalism still carries the constraint (8) that is necessary to glue SU(2) invariant states resid-ing at neighboring sites to yield original non-local gaugeinvariant Hilbert space of the theory. In terms of LSHoperators, this constraint (8) reads as:ˆ n l ( j ) + ˆ n o ( j )(1 − ˆ n i ( j ))= ˆ n l ( j + 1) + ˆ n i ( j + 1)(1 − ˆ n o ( j + 1)) (21)Comparing each side of (21) to that of (8) upon actingon LSH basis states, we get: N L ( j ) = n l ( j ) + n o ( j )(1 − n i ( j )) (22) N R ( j ) = n l ( j + 1) + n i ( j + 1)(1 − n o ( j + 1)) (23)where, N L ( j ) and N R ( j ) count bosonic occupation num-bers at each end of the link connecting site j and j + 1.As mentioned earlier, the bosonic occupation number ateach end of a link has contribution coming from fermionicexcitation n i and n o as well. Pictorially, left and rightside of (21) and/or (8) is represented by the numberof thick solid lines at left and right end of a link con-necting sites j and j + 1 in FIG. 1. As in [39, 46],definition of a hadronic state in LSH basis is given by | n l = 0 , n i = 1 , n o = 1 (cid:105) at one particular site.Hamiltonian of the theory, exactly equivalent to theoriginal Hamiltonian (1) in terms of LSH operators isgiven by: H (LSH) = H (LSH) E + H (LSH) M + H (LSH) I (24)where, H (LSH) E is the electric energy term, H (LSH) M is themass term and H (LSH) I is the matter-gauge interactionterm of the Hamiltonian. Explicitly, in terms of LSHoperators defined in (13-20), each part of the Hamiltonianis as below: H (LSH) E = g a (cid:88) n (cid:34) ˆ n l ( j ) + ˆ n o ( j )(1 − ˆ n i ( j ))2 , × (cid:18) ˆ n l ( j ) + ˆ n o ( j )(1 − ˆ n i ( j )2 ) + 1 (cid:19) (cid:35) (25) H (LSH) M = m (cid:88) n ( − j (ˆ n i ( j ) + ˆ n o ( j )) , (26) H (LSH) I = 12 a (cid:88) n (cid:112) ˆ n l ( j ) + ˆ n o ( j )(1 − ˆ n i ( j )) + 1 × (27) (cid:104) S ++ o ( j ) S + − i ( j + 1) + S −− o ( j ) S − + i ( j + 1)+ S + − o ( j ) S −− i ( j + 1) + S − + o ( j ) S ++ i ( j + 1) (cid:105) × (cid:112) ˆ n l ( j + 1) + ˆ n i ( j + 1)(1 − ˆ n o ( j + 1)) + 1 . Here (27) contains LSH ladder operators in the followingcombinations (suppressing the explicit site index), S ++ o = ˆ χ + o ( λ + ) ˆ n i (cid:112) ˆ n l + 2 − ˆ n i (28) S −− o = ˆ χ − o ( λ − ) ˆ n i (cid:112) ˆ n l + 2(1 − ˆ n i ) (29) S + − o = ˆ χ + i ( λ − ) − ˆ n o (cid:112) ˆ n l + 2ˆ n o (30) S − + o = ˆ χ − i ( λ + ) − ˆ n o (cid:112) ˆ n l + 1 + ˆ n o ) (31)and S + − i = ˆ χ − o ( λ + ) − ˆ n i (cid:112) ˆ n l + 1 + ˆ n i ) (32) S − + i = ˆ χ + o ( λ − ) − ˆ n i (cid:112) ˆ n l + 2ˆ n i (33) S −− i = ˆ χ − i ( λ − ) ˆ n o (cid:112) ˆ n l + 2(1 − ˆ n o ) (34) S ++ i = ˆ χ + i ( λ + ) ˆ n o (cid:112) ˆ n l + 2 − ˆ n o . (35)The strong coupling ( ga (cid:29) , ma =fixed) vacuum of theLSH Hamiltonian is given by: n l ( j ) = 0 ∀ xn i ( j ) = 1 , n o ( j ) = 1 for j odd (36) n i ( j ) = 0 , n o ( j ) = 0 for j evenIt is easy to check that (36) satisfies Abelian Gauss law(21). One should also consider a suitable boundary con-dition for one dimensional spatial lattice as discussed indetail in [39] as:Open Boundary Condition (OBC): N R (0) = l OBC i Periodic Boundary Condition (PBC): N R (0) = N L ( N − ≡ l PBC i . where, N L , N R are defined in (22) and (23) for thefirst (0) and last ( N −
1) site of a N site lattice. l i can be any positive semi definite integer. Now, onecan easily check that, for any gauge invariant state (cid:81) N − j =0 | n l ( j ) , n i ( j ) , n o ( j ) (cid:105) , the bosonic quantum numbers n l ( j ) for all values of j are completely determined by theboundary flux l i and constraint (21) imposed on each andevery link of the lattice starting from one end as: n l ( j ) = l i + j − (cid:88) y =0 ( n o ( y ) − n i ( y )) − n i ( j ) (1 − n o ( j )) . (37)For OBC, any physical state in LSH formalism is com-pletely determined by ( n i , n o ) quantum numbers at eachside. For PBC, the gauge invariant or LSH Hilbert spaceis characterized by many copies of the same fermionic( n i , n o ) configurations with different winding number ofclosed loops, that plays the exact role as the l i and fixesthe n l ’s throughout the lattice. We exploit this partic-ular feature in the analog quantum simulation proposaloutlined in the present work . Note that, n l being deter-mined does not mean that we describe a static gauge fieldtheory; rather, truly relevant or physical gauge degreesof freedom are contained into the ( n i , n o ) excitation ofany physical state. The numerical analysis performed in this work is for OBC ingauge theory as simulating the same in an experiment is easierthan that for PBC.
B. Weak coupling approximation
The strong coupling vacuum of the theory is defined byzero gauge flux i.e n l = 0 at all lattice sites. Howeveras one approaches the weak coupling regime, the statescontaining large amount of bosonic flux do contribute tothe low energy spectrum of the theory. In [48], a weakcoupling vacuum ansatz was proposed and justified forthe 2 + 1 dimensional pure SU(2) gauge theory withinprepotential framework. In that proposal, each latticesite contains a large but mean value for the local loopquantum numbers. The (1 + 1)-d version of that ansatzwithin LSH framework (i.e prepotential + staggered mat-ter) would be equivalent to each site containing more andmore gauge fluxes, i.e n l (cid:29)
0, for all sites as one ap-proaches the weak coupling limit g →
0. As discussedbefore, the incoming flux or boundary flux l i fixes thebosonic loop quantum numbers n l ’s at each of the latticesite for any configurations of n i , n o throughout the lat-tice as per (37). Hence, choosing for l i (cid:29) n l ( j ) = l i ≡ n l ∀ j. (38)We derive the following approximate Hamiltonian thatacting on the LSH states on the 1d spatial lattice with theboundary flux l i (cid:29) H (approx) E = g a (cid:88) j (cid:34) ˆ n l ( j )2 ˆ n l ( j )2 (cid:35) (39) H (approx) M = m (cid:88) j ( − j (ˆ n i ( j ) + ˆ n o ( j )) (40) H (approx) I = 12 a (cid:88) j (cid:104) ˆ χ + o ( j ) ˆ χ − o ( j + 1) + ˆ χ − o ( j ) ˆ χ + o ( j + 1)+ ˆ χ + i ( j ) ˆ χ − i ( j + 1) + ˆ χ − i ( j ) ˆ χ + i ( j + 1) (cid:105) (41)The derivation of the approximated Hamiltonian givenabove is detailed in Appendix A. For n l (cid:39) l i (cid:29) g → , a →
0, the weak coupling LSH Hamiltonian givesan accurate description of low energy spectrum of thecontinuum. In this limit, the Abelian Gauss law con-straint (8) is automatically satisfied as (22) and (23) ef-fectively become equal.We present the details of an atomic quantum simula-tion scheme to simulate this approximated weak couplingHamiltonian in next section.The systematic correction to recover the full Hamilto-nian is also discussed in Appendix A that is to be takeninto account in order to improve upon this particularquantum simulation proposal. Systematic generalizationof this proposal in a series of future works would finallylead to a complete and scalable quantum simulator fornon-Abelian gauge theories.
III. ATOMIC HAMILTONIAN : HUBBARDMODEL ON A BIPARTITE LATTICE
We consider a Fermi-Hubbard model in a one-dimensional lattice. The lattice is a superposition of aprimary lattice and a secondary lattice. The primarylattice has the form V ( x ) = − (cid:88) j V L δ ( x − x j ) , (42)and can also act as the trapping potential. Here V L marksthe primary lattice depth. In the secondary lattice, thelattice depth is V A and V B on alternating sites. The sec-ondary lattice thus is a Kronig-Penney type two-colorlattice: the potential consisting of δ -combs of two differ-ent strengths: V ( x ) = (cid:88) j =odd V A δ ( x − x j ) + (cid:88) j =even V B δ ( x − x j ) (43)The structure of the secondary lattice is shown in Fig. 2.We introduce two new parameters : V = ( V A + V B ) / V (cid:48) = ( V A − V B ) / | ↑(cid:105) and | ↓(cid:105) respectively; ψ ↑ ( x ) and ψ ↓ ( x ) being thecorresponding field operators. The Hamiltonian can bewritten as H = H + H int . (44)Here H is the non-interacting part of the Hamiltonian,and H int is the fermion-fermion interaction. H = (cid:90) (cid:0) ψ †↑ ( x ) H ( x ) ψ ↑ ( x ) + ψ †↓ ( x ) H ( x ) ψ ↓ ( x ) (cid:1) dx, (45)where H ( x ) = H ( x ) + H ( x ) (46) H ( x ) = − (cid:126) m ∂ ∂x + V ( x ) (47)is the contribution from the kinetic energy and the pri-mary lattice potential, same for all the sites. The contri-bution from the secondary lattice is H ( x ) = V ( x ) = V + V (cid:48) (48)for odd sites. And H ( x ) = V ( x ) = V − V (cid:48) (49)for even sites.In the low-energy scattering regime, the atoms usuallyinteract via s-wave scattering. The corresponding cou-pling constant is given by g = 4 π (cid:126) a s m a s being the scattering length. The interacting part ofthe Hamiltonian is given by: H int = g (cid:90) ψ †↑ ( x ) ψ †↓ ( x ) ψ ↓ ( x ) ψ ↑ ( x ) dx. (50)Now, if the lattice potentials are sufficiently deep, thefield operators can be expanded in terms of single-particleWannier functions, localized to each lattice site: ψ σ ( x ) = (cid:88) j c σ ( j ) W ( x − x j ); σ = ↑ , ↓ , (51)where c σ ( j ) is the fermionic annihilation operator for spinindex σ and site j . The single-particle Hamiltonian canbe written as: H = H (cid:15) + H V + H V (cid:48) + H hopping . (52)Here, each term can be expressed using the fermioniccreation and annihilation operators c † σ ( j ) and c σ ( j ); andalso the number operators c † σ ( j ) c σ ( j ) = N jσ .The first term is : H (cid:15) = (cid:88) j (cid:15) j N ( j ) (53)where the parameter (cid:15) j is given by (cid:15) j = (cid:90) W ( x − x j ) H ( x ) W ( x − x j ) dx (54)and N ( j ) = N ↑ ( j )+ N ↓ ( j ) is the total number of fermionsin site j . In the subsequent part of the paper, we neglectthe H (cid:15) term because if simply gives a constant energyshift.As for the terms arising from the secondary lattice : H V = (cid:88) j V N ( j ) (55)and H V (cid:48) = V (cid:48) (cid:88) j =odd N ( j ) − V (cid:48) (cid:88) j =even N ( j ) (56) H V is a constant, too, but we keep this, in order to makea direct correspondence with the reduced LSH Hamilto-nian.The hopping term, which represents the tunneling be-tween sites is given by : H hopping = − (cid:88) j t i,j (cid:0) c †↑ ( j ) c ↑ ( i ) + c †↓ ( j ) c ↓ ( i ) (cid:1) . (57)Tunneling to next-nearest neighbors is usually suppressedby one order of magnitude, in comparison with the near-est neighbor tunnelling. So we consider hopping betweenadjacent sites only. The tunneling rate from site j to( j + 1) is given by the matrix element t j, ( j +1) = − (cid:90) W ( x − x j ) H W ( x − x j +1 ) dx. (58) FIG. 2. Structure of the two-color lattice
As for the interaction between up-spin and down-spinfermions sharing the same site H int = u (cid:88) j N ↑ ( j ) N ↓ ( j ) . (59)Here the on-site interaction matrix element is given by u = g (cid:90) |W ( x − x j ) | dx. (60)The total Hamiltonian thus translates to H = − (cid:88) j t j, ( j +1) (cid:0) c †↑ ( j ) c ↑ ( j + 1) + c †↓ ( j ) c ↓ ( j + 1) (cid:1) + u (cid:88) j N ↑ ( j ) N ↓ ( j ) + (cid:88) j V N ( j )+ V (cid:48) (cid:88) j =odd N ( j ) − V (cid:48) (cid:88) j =even N ( j ) . (61)If the hopping − t is a constant throughout the lattice,this model essentially is a 1D Hubbard model with al-ternating potential, often termed as the “ ionic Hubbardmodel”, defined on a bipartite lattice. Here, in addi-tion to a site-independent hopping − t and the on-siteinteraction u , there is a difference in the energy offset2 V (cid:48) between sublattice A and sublattice B . This modelwas originally proposed to study transitions in organiccrystals [59], and later, found application in the studiesof ferroelectric transitions [60]. In the recent past, thismodel has been experimentally realized [61] in a systemof ultracold atoms. So we consider this to be a very suit-able candidate to simulate lattice gauge theories.At half-filling, the ionic Hubbard model is capable ofdescribing a band-insulator [62]. However, this modelhas a rich phase diagram, and at higher inter-atomic in-teraction strengths, can support transitions to differentstates, including Mott insulator [62], correlated insulator[63, 64], AFM insulator and half-metal [64] phases; andcertain combinations of u and V (cid:48) can even lead to super-fluidity [65]. As we will see in the later part of this work,we will have to carefully choose our parameters such that the entire dynamics remains confined to a single param-agnetic phase in order to mimic the dynamics of gaugetheory. IV. SIMULATING AND SIMULATEDHAMILTONIAN AND THEIR PARAMETERS
We are now in a position to compare the weak couplingLSH Hamiltonian and the atomic Hamiltonian. For aparticular site j , we make the following identification : n i ( j ) = N ↑ ( j ); n o ( j ) = N ↓ ( j ) (62) χ + i ( j ) = c †↑ ( j ); χ − i ( j ) = c ↑ ( j ) (63) χ + o ( j ) = c †↓ ( j ); χ − o ( j ) = c ↓ ( j ) (64) m = V (cid:48) (65)Also, the magnitude of V has to be chosen to be mappedto electric part of the gauge theory Hamiltonian for aparticular n l , fixed by the open boundary condition.The electric term of approximated LSH Hamiltonianis mapped to: H (approx) E → (cid:88) j V N ( j ) . (66)Similarly, the potential V (cid:48) is fixed by the mapping: H (approx) M → V (cid:48) (cid:88) j =odd N ( j ) − V (cid:48) (cid:88) j =even N ( j ) (67)and the hopping terms are identically related as, H (approx) I → − t (cid:88) j (cid:0) c †↑ ( j ) c ↑ ( j + 1) + c †↓ ( j ) c ↓ ( j + 1) (cid:1) +h.c . (68)Note that, there is no term in the weak coupling LSHHamiltonian that corresponds to the on site interactionterm (59). So, in the limit u →
0, one would have a com-plete mapping between atomic system and weak couplinglimit of gauge theory.We would like to point out that although the LSHHamiltonian contains explicit bosonic modes n l ( j ), theseare actually non-dynamical in the weak coupling approx-imation as discussed before, and hence we do not keepactual bosons in the atomic system. Instead, we incor-porate the effect of these bosons in the potential itself,in the form of a constant energy shift. This enables us toi) keep n l uniform for each site, and ii) ensure that thebosonic and the fermionic modes are completely decou-pled: as there remains no chance of any boson-fermionscattering.Moving away from weak coupling approximation, theelectric part becomes dominant and hence correction tothe same becomes most important. The complete correc-tion to the electric part of Hamiltonian is given in (A6)within mean field ansatz. The on-site interaction in theatomic Hamiltonian, which does not have an equivalentin the approximate LSH Hamiltonian, can be tuned torecover the exact contribution of (25) within the meanfield ansatz of LSH Hamiltonian given in (A6).As discussed in Appendix A, the correction term to beadded to the weak coupling approximated electric Hamil-tonian (39) to yield full electric Hamiltonian (25) in thebulk limit is given by:∆ H (LSH) E = g a N (cid:18) n l (cid:19) (69)Now, for a Hubbard model at half-filling, all the fouraccessible states | (cid:105) , | ↑(cid:105) , | ↓(cid:105) and | ↑↓(cid:105) are equally likelyas long as the system remains in the paramagnetic phase.So, N |↓(cid:105) (the number of sites belonging to state | ↓(cid:105) ) ≈ N/ N being the total number of lattice sites. Similarly N |↑↓(cid:105) , the number of sites with doublons, would be ≈ N/
4, too and those many configurations contribute to(59). Hence, one can utilize the on-site interaction termto recover the exact correction term, u (cid:88) j N ↑ ( j ) N ↓ ( j ) = u N −→ ∆ H (LSH) E (70)Note that, (70) is exact, only in the bulk limit, i.e N (cid:29) H (mLSH) E −→ H V + H int (71) H (LSH) M −→ H V (cid:48) (72) H (approx) I −→ H hopping (73)provided we fix V and u such that the system, staying inthe desired phase, mimics the dynamics of gauge theoryas shown in Fig. 3.The correction to the approximate interaction Hamil-tonian is negligible in weak coupling regime, and alsoinsignificant in the strong coupling limit. In this work,we do not consider any correction to the interaction term. We now, explicitly calculate the parameters of theatomic Hamiltonian to simulate desired gauge theory dy-namics, that has to be tuned in the experiment. First, wescale the gauge theory Hamiltonian to be dimensionlessin order to make comparison with that of atomic system. A. Scaling of the gauge theory Hamiltonian
It is convenient to scale the Hamiltonian H (KS) givenin (1) as per [66], so as to make it dimensionless:˜ H = 2 g a H ( KS ) = (cid:88) j E ( j ) (cid:124) (cid:123)(cid:122) (cid:125) ˜ H E + µ (cid:88) j ( − j (cid:2) ψ † ( j ) · ψ ( j ) (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) ˜ H M + x (cid:88) j (cid:2) ψ † ( j ) U ( j ) ψ ( j + 1) + h . c . (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) ˜ H I . (74)Here, x = g a and µ = 2 √ x mg are dimensionlesscoupling constants of the theory. Evolving this ˜ H withscaled time (from zero to ˜ τ )˜ τ = τ gauge x (75)is due to the unitary operator: U (˜ τ ) = exp (cid:16) − i ˜ H ˜ τ (cid:17) = exp (cid:18) − i g a H (KS) g a τ gauge (cid:19) = exp (cid:16) − i aH (KS) τ gauge (cid:17) . (76)Here, 2 aH (KS) is another scaled Hamiltonian with dimen-sionless parameters given by,2 aH (K S ) = 1 x (cid:88) x E ( x )+2 mg √ x (cid:88) j ( − j (cid:2) ψ † ( j ) · ψ ( j ) (cid:3) + (cid:88) j (cid:2) ψ † ( j ) U ( j ) ψ ( j + 1) + h . c . (cid:3) . (77)The strong coupling limit is defined for x →
0, wherethe interaction part of Hamiltonian become less dom-inant as evident from both (74) and (77), whereas inthe weak coupling limit defined at x → ∞ , interac-tion part of the Hamiltonian becomes the most impor-tant term that cannot be treated perturbatively. Thesescaling rules work equivalently on the LSH Hamiltoniandefined in (25,26,27) as the LSH Hamiltonian is exactlyequivalent to the original Kogut-Susskind Hamiltonian.0 FIG. 3. Cartoon representation of dynamics of 1d ionic Hubbard model mimicking that of SU(2) lattice gauge theory in onespatial dimension. (a) Initial state: fully filled odd sites and empty even sites mimicking the strong coupling vacuum consistingof no particles ( n i = 0 , n o = 0 on even sites) and no antiparticles ( n i = 1 , n o = 1 on odd sites). Under Hamiltonian evolution,one atom hops from an odd site to neighboring even site in Hubbard model, that mimics creation of a particle antiparticle pairat two neighboring staggered site of the gauge theory, connected by one unit of flux to form a gauge singlet string configuration.One further hopping as shown in the figure mimics the dynamics in gauge theory as elongation of the string and creation abaryon on one site. In these three states, the total number of particle (antiparticle) for the gauge theory are respectively 0 , , B. Weak coupling regime of gauge theory:
For the Hamiltonian given in (74), we consider thebosonic loop quantum number to take the average value n l ≈ O (10 p ) ⇒ ˜ H (approx) E ≈ O (10 p ) . (78)For a comparative mass and interaction contribution ofthe Hamiltonian, i.e.˜ H LSHM ≈ O (10 p ) (79)& ˜ H (approx) I ≈ O (10 p + p (cid:48) ) ) , (80) is obtained for the following scaling of the parameters: mg ≈ O (10 p − p (cid:48) ) √ x ≈ O (10 p + p (cid:48) ) , ∀ p (cid:48) ∈ Z + . (81)The exact values of the dimensionless parameters ofgauge theory can be taken as: n l = ˜ n l × p (82) µ = ˜ µ × p (83) x = ˜ x × p + p (cid:48) ) , ∀ integer p, p (cid:48) (84)= 10 p for the choice, p (cid:48) = 0 , ˜ x = 1 . (85)1Now, the dynamics of this scaled Hamiltonian ˜ H in(74), is to be simulated by the simulating Fermi-HubbardHamiltonian given in (61) in the time scale ˜ τ as definedin (75), such thatexp( − i ˜ H ˜ τ ) −→ exp( − iHτ ) (86)where, H is the atomic Hamiltonian given in (61) withthe parameters: V (cid:48) = ˜ µ (87) V = 14 (cid:0) ˜ n l + 2˜ n l (cid:1) (88) u = 0 (89) t = − . (90)Here, all the parameters are fixed in units of ‘ t ’. Theonly choice that we have made in setting the parametersis p (cid:48) = 0 in (84). Gauge theory with a nonzero p (cid:48) can beequivalently simulated by the same atomic system withtuning V (cid:48) to smaller values ˜ µ × − p (cid:48) in an experiment.This will access all mass values of gauge theory in thequantum simulation protocol. C. Strong coupling regime of gauge theory:
We consider the scaled Hamiltonian in (77) in strongcoupling regime x <
1. As discussed earlier, the bulklimit of the Fermi-Hubbard Hamiltonian in the param-agnetic phase will correspond to the exact mean fieldelectric term (A6) and mass term (26). Although theinteraction term is approximated, will not make majordifference in spectrum and/or dynamics as x → l i to be a fixed integer, but is of O (1). We map the gaugetheory Hamiltonian to Fermi-Hubbard Hamiltonian withparameters V (cid:48) = 2 1 √ x · mg (91) V = 1 x · l i l i + 2) (92) u = 1 x ·
14 (2 l i + 3) (93) t = − . (94)Here also, all the parameters are fixed in units of ‘ t ’. Itis clear from the above relations, for a fixed value of l i ,smaller values of x require larger V /t and u/t for theatomic system. However, we will have to be careful to re-main in the same paramagnetic phase such that our anal-ysis of compensating errors in electric Hamiltonian fromthe uniform potential are well compensated by the self in-teraction term. For this purpose, i.e in order to keep u/t below the critical point for paramagnetic-ferromagneticphase transition one can not really expect to simulate x → x < x = 1 besides accurately sim-ulating intermediate coupling range x ≈ −
100 as willbe demonstrated in the numerical analysis.Likewise the weak coupling case, the simulating andsimulated dynamics are comparable up to a factor τ atomic = 2 a × τ gauge . (95)where, a is small but finite in strong coupling limit.In the next section, we propose the precise experimen-tal set-up that is close to already performed experimentsfor Ionic-Hubbard model following the above mentionedscheme, where strong coupling regime of lattice gaugetheory dynamics is mapped to ionic Hubbard model with u/t >
1, whereas the weak coupling regime is mapped tothe same with u/t ≈ V. EXPERIMENTAL REALIZATION
The experimental scheme calls for the realization of1-dimensional Fermi-Hubbard model in a bipartite lat-tice. In the recent past, the ionic Fermi-Hubbard modelwas experimentally realized in a honeycomb lattice [61],and its bosonic counterpart was implemented on a bi-partite chequerboard lattice [67]. Also, a 1-dimensionalFermi-Hubbard model was implemented in a experimentby Scherg et al.[68]. Both [61] and [68] used a degener-ate gas of fermionic K of numbers ≈ and 10 re-spectively. We propose that a combination of these twomethods can successfully yield a 1-dimensional Hubbardmodel with alternating lattice potentials. A. Proposed set-up
The interference pattern of two counter propagatinglasers is used to create an optical lattice. The latticedepth is proportional to the intensity of the laser beamand is measured in units of the recoil energy E R .In the experiment by Messer et al.[61], first a regularhoneycomb lattice was created, and that fixed the hop-ping parameter t on each bond. Next a staggered energyoffset of ∆ was independently applied between sites of A and B sublattices. In our 1-dimensional structure, anequivalent would be to set up the primary lattice withlattice depth V L : V ( x ) = − V L cos (x)and superpose that with V ( x ) = V on each site. This fixes the hopping parameter t . Then,on top of it, energy offsets V (cid:48) and − V (cid:48) can be inde-pendently applied on the odd sites and even sites re-spectively, so that V ( x ) = V + V (cid:48) for odd sites, and V ( x ) = V − V (cid:48) for even sites.2Just like the hopping t , the on-site interaction u , toodepends on the lattice depth. However, u can be inde-pendently controlled as well, by means of Feshbach reso-nance. As for the two fermionic states, any two hyperfinestates of a particular atom can be employed. In [68], thehyperfine states | ↑(cid:105) = | F = − / m F = − / (cid:105)| ↓(cid:105) = | F = − / m F = − / (cid:105) of ultracold K atoms were used. In [61], in addition tothe above, the combination | ↑(cid:105) = | F = − / m F = − / (cid:105)| ↓(cid:105) = | F = − / m F = − / (cid:105) was also employed in order to obtain desired range of u .In [61], the ionic Hubbard model was studied on ahoneycomb lattice. In contrast, our model requires theimplementation of the ionic Hubbard model in a simpleone-dimensional geometry. Regarding the dimensional-ity of the system, it may be recalled that in the recentpast, ultracold atom experiments have successfully con-fined bosonic and fermionic atoms to one dimension (1D).The basic idea is to tightly confine the particles in twotransverse directions, and make them weakly confined inthe axial direction. Thus, their motion in the transversedirections are completely frozen. So effectively, these arequasi-1D systems.For example, in our proposed set up, suppose both V y and V z , the potentials in the transverse directions, arekept fixed at a large value (Like, 33 E R as in [69], or42 E R as in [70]). V ( x ), The lattice depth in the axialdirection is governed by both V L and V , and the finaldepth is kept in a range of 5 E R − E R . We note thatin Hubbard model experiments, the potentials are to bedeep enough ( V ≥ E R ) so that the single-band descrip-tion of Hubbard model remains valid. On the other hand, V ( x ) cannot be as deep as the potentials in the transversedirection, so as to restrict the dynamics in 1-dimensiononly. The hopping parameter t is a function of the latticedepth, and can be estimated using the Wannier functions[71].The actual lattice depth is given by | V L − V | , so dif-ferent combinations of V L and V can result in the samelattice depth. This offers a tremendous advantage in theexperimental pursuit, as the same optical lattice can beassumed to be split in different pairs of V L and V : al-lowing one to explore a wide range of V values (that, inturn, enables one to access a wide range of x and/or l i as per (92)). It is to be noted that both V L and V aretheoretical parameters in the model that leads to con-stant shifts in the energy only : bearing no effect on thedynamics of the fermions.Accordingly, we consider two configurations :(i) V L = 6 E R and V = 0 . E R and(ii) V L = 6 . E R and V = 1 E R . In both the cases, the resultant uniform lattice depthis 5 . E R for all the sites. This results in a hopping t =0 . E R . The combinations we have mentioned translateto (i) V ≈ . t and (ii) V ≈ . t respectively. In addition, an offset of V (cid:48) and − V (cid:48) is in-dependently applied on the odd and even sites. In ourscheme, we choose V (cid:48) = 1 . t and stick to this value in allour numerical simulations. The on-site interaction u canbe controlled by applying a Feshbach field.To simulate the weak coupling limit, we restrict our-selves to the weakly interacting atomic limit : u/t << u = 0 . t . On the other hand, simulationof the strong coupling limit calls for the realization ofthe strongly interacting atomic limit : u/t (cid:29)
1, and wechoose u ≈ . t . We note that these V (cid:48) /t and u/t val-ues comfortably fall in the parameter regimes accessed inrecent experiments [61, 68, 72]. B. Initial state preparation:
The initial state has to be prepared in a Charge-density-wave (CDW) configuration where all the oddsites are occupied by the fermionic particles and the evensites are completely empty. This can be done using somesort of filtering sequence in the experiment. For example,in [72, 73], this was achieved by superposing the primarylattice (with wavelength λ ) with an additional long lat-tice (with wavelength 2 λ ) in the following way: V ( x ) = − V l (cos ( kx/ φ )) − V s cos ( kx ) (96)with k = 2 π/λ .The lattice depths V l and V s and the relative phase φ can be adjusted independently. Here V s stands forthe depth of the original (and short) lattice with V s = | V L − V | ; and V l is the depth of the additional long lat-tice. This long lattice is utilized during the preparationof the initial CDW state. Initially, the long lattice ismade quite deep (like, 20 E R , as in [72]), and the shortlattice is ramped up to that depth at a non-zero relativephase φ to create a tilted lattice of double wells. Now itis so arranged that the odd sites host lower energy wellsthan the even sites, and it is possible to load all atoms inthe odd sites only. The tilt offsets are made sufficientlylarge so that the particles cannot escape from the oddsites and tunnel to the even sites. After loading all theatoms, the longer lattice is switched off, and the shortlattice is ramped down to its desired final value (In ourcase, 5 . E R ). The offset V (cid:48) and − V (cid:48) is added to the oddsites and even sites respectively, to create the bipartitestructure. Now tunneling is possible between adjacentsites, and the the dynamics begins.3 C. Observing the dynamics
The observable can be defined as the population im-balance P between the even sites and odd sites, definedas P = N e − N o N e + N o . (97)Here N e is the total number of atoms in the even sites,and N o is the total number of atoms in the odd sites.The time evolution of the parameter P is to be stud-ied in order to visualize the particle number dynamics ofgauge theory. A site-resolved technique is thus neededto determine the number of atoms on even and odd lat-tice sites separately. In [72, 73], a band-mapping schemewas successfully employed using the long lattice. Oncethe desired time evolution in the primary (short) latticeis over, the long lattice is introduced again to create thetilted lattice, and tunneling stops. The phase φ is chosensuch a way that the odd sites constitute the lower wellsin the array of double wells. Now the population distri-bution across the odd and even sites gets sealed. Next,the depth of the long lattice is ramped to a much highervalue : and the atoms in the even sites get transferred tothe third Bloch band of the superlattice. Atoms in theodd sites remain in the first band . The density profile inthe different bands can be obtained using Time-of-flight(TOF) images and absorption imaging [72]. VI. SIMULATED DYNAMICS ANDOBSERVABLES
We present numerical analysis of our proposal anddemonstrate the comparison between the simulating andsimulated spectrum as well as dynamics.
A. Spectrum comparison:
In weak coupling regime:
We aim to quantumsimulate gauge theory Hamiltonian, with the values ofdimensionless parameters given by: x = 10 − & m/g = 1 . × − acting on the LSH Hilbert space characterized by n l = 5 × at all sites and correspond to to p = 5 , p (cid:48) = 0 in (82-84).The fermionic (string) configurations remain completelydynamical as n i , n o can take all possible values at sites0 , , , ... N . Following (87-89) we obtain the parametersof the atomic Hamiltonian to be fixed at: V = 8 . t , V (cid:48) = 1 . t , u = 0 . t. (98)Note that, we have chosen a feasible but small value of theparameter u . Smaller and smaller values of u will enable to mimic the dynamics of gauge theory more accuratelyas we take p (cid:29)
1. We perform exact diagonalization forboth the Hamiltonians with a small number of sites, thatis doable on a PC. Our scheme, being completely scalable,the agreement in spectrum as in Fig. 4 holds true for anysize of lattice as per experimental capabilities.
In strong coupling regime:
We aim to quantumsimulate gauge theory Hamiltonian (77), with the valuesof dimensionless parameters given by: x = 0 .
69 & m/g = 1 . . This Hamiltonian acts on LSH Hilbert space character-ized by l i = 6 as in (38) , while n i , n o can take all possiblevalues at sites 0 , , , ... N . Following (91-93), the mim-icking atomic system is defined by parameters: V = 17 . t , V (cid:48) = 1 . t , u = 5 . t. (99)Likewise weak coupling case, we also perform exact diag-onalization for this case to compare and obtain the spec-trum as in Fig. 4. Note that, from our analysis we onlyexpect exact match of spectrum in N → ∞ limit, that isbeyond scope of exact diagonalization and is not reportedhere. Performing numerical calculations for a longer lat-tice is beyond scope of exact diagonalization, but can beperformed using state of the art tensor network techniqueand that study would establish proper benchmark for thescheme in strong coupling regime. However, tensor net-work can only calculate a particular (low energy) sectorof the theory with accuracy and quantum simulation isexpected to outperform the same.However, even with limited computational resources,we make the following observations: • It appears from (91-93) that, by increasing V /t in the atomic system, one would be able to accesssmaller and smaller values of the gauge theory pa-rameter x . However, the consequence is that, inorder to mimic exact strong coupling dynamics, u/t has to be increased as well. • With an increasing u/t (even for a fixed value of V /t ), gaps are introduced in the atomic spectrumas the atomic system experiences a quantum phasetransition (see Fig. 6) and enters into Mott In-sulator phase [74]. Then the system can no longermimic dynamics of gauge theory as there is no suchquantum phase transition in the gauge theory spec-trum. Hence, this quantum simulation scheme isnot suitable for x → • Instead, if one can arrange the experimental set-upto fix V /t at a smaller value, the atomic systemsimulates the intermediate coupling regime of thefull gauge theory reliably. We illustrate such anagreement for V /t = 0 . x = 120) in Fig 4.4 FIG. 4. Spectrum of the Ionic Hubbard model, Full SU(2) gauge theory (KS or LSH) Hamiltonian (without any approximation)and the Weak coupling approximated LSH calculated by exact diagonalization for 6 site system and scaled to fit between 0to 1. The left panel shows the spectrum in weak coupling regime of gauge theory with parameters as per (98) for differentvalues of p as discussed in subsection IV B. The right panel shows spectrum obtained for the string coupling analysis, for V = 17 . , .
75 & 0 . V becomes experimentally feasible. We propose to quantum simulate strongcoupling spectrum within a mean field approximation and at bulk limit, whereas the plots are only for small lattices and henceshowing magnified deviation of the mean field spectrum from that of the full gauge theory. The approximated LSH is onlyvalid in weak coupling regime and matches with full gauge theory for p (cid:29) B. Simulated Dynamics:
One important dynamical phenomenon to observe inreal time dynamics in gauge theory is the dynamics of pair production and string breaking as illustrated via car-toon in Fig. 3. We consider preparing the system in astate in which all even sites are completely empty (noparticle) and all odd sites are completely filled (no an-tiparticle). The real time Hamiltonian evolution of the5
FIG. 5. Simulated particle density dynamics, corresponding to the cartoon of Fig. 3(a) is plotted against a scaled time τ . Theparameters are identical to that used for spectrum analysis in Fig. 4. The simulated dynamics is almost exact to that of thefull gauge theory for weak coupling limit. The mismatch between full gauge theory dynamics and Hubbard model dynamicsin strong coupling regime is expected to get minimized at bulk limit. The approximated LSH is only valid in weak couplingregime and matches with full gauge theory for p (cid:29) atomic system involves atoms hopping from one site toanother, simulating the event of pair creation and particlenumber dynamics of gauge theory. Within LSH frame-work, for the no particle-no antiparticle state | Ψ (cid:105) on a1d lattice of N staggered sites, we define the following quantity to describe particle density, ρ ( τ ) = 1 + 1 N (cid:104) Ψ | ˆ U † ( τ ) ˆ O ˆ U ( τ ) | Ψ (cid:105) (100)where, ˆ O = (cid:80) j (cid:0) ( − j (ˆ n i ( j ) + ˆ n o ( j )) (cid:1) and U ( τ gauge ) isdefined in (76).6 FIG. 6. A Quantum phase transition is observed with theionic Hubbard model at a particular value of u/t , beyondwhich the spectrum becomes gapped and hence the Hubbardmodel can no longer mimic the dynamics of gauge theory.This particular plot is obtained with the parameters of Hub-bard model given in (99) except varying u/t . Choosing alarger value of V /t corresponds to smaller value of x via(92), but following (93) it will always be in the Mott Insulat-ing phase. The simulated dynamics in Hubbard model is mea-sured by the observable P , as defined in (97). Its con-nection with the particle number dynamics of gauge the-ory can be obtained by looking at the parameter 1 + P .In Fig. 5 we plot the quantities against a scaled time τ = τ atomic = 2 aτ gauge following (75).As done in the spectrum analysis, we consider thesame parameter values for calculating pair-productionand string breaking dynamics as well. From the simu-lated dynamics we can conclude the following : • The proposed simulation scheme simulates the dy-namics of weak coupling gauge theory perfectly andthat is evident even from the numerical analysisusing a small system. Here the particle densitydynamics resulting from i) full gauge theory, ii)the approximated LSH theory and iii) the atomicHamiltonian all agree very well. • The difference between the actual dynamics dueto the original Hamiltonian and the dynamics dueto the approximated Hamiltonian is quite pro-nounced in the intermediate coupling/ strong cou-pling regimes. However, by adjusting the on-siteinteraction parameter, it was possible to recoverthe correction in the electric energy term (69) sub-stantially, and hence the ionic-Hubbard dynamics isnow closer to the dynamics of the full gauge theory,when compared to the same with the approximatedLSH formulation. • The discrepancy that still exists in the intermedi-ate/ strong coupling regimes will surely get reducedif one can simulate using a long enough lattice, such that in the statistical limit, one can really recoverthe correction in electric energy term (69) in fullby choosing the atomic self-interaction accordingly.Considering that we used a small lattice (6 site sys-tem) for our numerical simulation and yet man-aged to observe a good agreement, it is extremelylikely that in an actual experiment (or, tensor net-work calculation) involving a large number of lat-tice sites, the error will be insignificant.It is discussed in Sec. V, how one can measure the dy-namics in an actual experiment. However, the actualtime measured in ms during the experiment is related tothe scaled times as: τ exp = (cid:126) τ atomic t ≡ τ atomic1 . ⇒ ≡ aτ gauge1 . . (102)Thus, for different values of lattice spacing, the sameexperiment would simulate real time dynamics of gaugetheory happening in different smaller time scale. VII. DISCUSSIONS AND FUTUREDIRECTIONS
This paper presents the very first practically imple-mentable quantum simulation proposal for simulatingSU(2) lattice gauge theory in (1 + 1)-d, that specificallysimulates the spectrum and dynamics of gauge theoryin weak coupling regime as well as intermediate couplingregime for a large lattice with good accuracy. Experimen-tal implementation of this particular scheme will demon-strate why quantum simulators can be a very effectivetool to study different aspects of gauge theories.The proposal is completely scalable that accesses dif-ferent regimes of gauge theory (with a varying degree ofaccuracy) and quantum simulate different symmetry sec-tors. A suitable scaling scheme presented in this paperenables one to model different regimes of LGT with asingle experimental set up, just by tuning the control-lable experimental parameters. For example, the weakand strong coupling limits of gauge theory is accessed bytaking u/t to 0 and u < u c respectively in the atomic sys-tem, where u c is a quantum critical point beyond whichthe atomic system enters into a Mott insulating phaseas observed in this particular study with small lattice(see Fig. 6). The only requirement here is that the sys-tem requires to remain in the same paramagnetic phasethroughout the course of its dynamics, so that in a bulklimit, all the allowed states are equally probable at half-filling.Future works will address the issue of going beyondmean field approximation by simulating dynamical gaugefields that exists beyond 1 spatial dimension. The LSHformalism for gauge theories in higher dimensions shouldbe equally useful in constructing atomic quantum sim-ulators for the same. Specifically, within LSH frame-7work, the matter gauge coupling remains the same asin 1d in any higher dimension, including the featureof non-dynamic loop degrees of freedom at matter sites[46, 48, 58]. Hence we expect the present proposal toremain as a useful building block for higher dimensionalquantum simulators as well. Work is in progress in thesedirections and will be reported elsewhere. The presentscheme can also be generalized for gauge group SU(3)upon generalization of LSH formalism for SU(3) gaugetheory and that will build a concrete step towards quan-tum simulating QCD. ACKNOWLEDGEMENT
We would like to thank Zohreh Davoudi and RudranilBasu for useful discussions and also for careful reading ofthe manuscript and helpful comments. R.D. would like toacknowledge support from the Department of Science andTechnology, Government of India in the form of an In-spire Faculty Award (Grant No. 04/2014/002342). I. R.is supported by the U.S. Department of Energy (DOE),Office of Science, Office of Advanced Scientific Comput-ing Research (ASCR) Quantum Computing ApplicationTeams (QCAT) program, under fieldwork Proposal No.ERKJ347. [1] Kenneth G. Wilson, “Confinement of quarks,” Phys. 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Appendix A: Approximate LSH Hamiltonian in theweak coupling limit
In this appendix we derive the approximate Hamilto-nian given in (39,40,41) starting from the Hamiltoniangiven in (25,26,27).
Electric Hamiltonian:
The electric part of the LSHHamiltonian as given in (25) can be written as: H (LSH) E = g a (cid:88) j h E ( j ) (A1)At each site j , depending upon the fermionic quantumnumbers n i , n o , the local contribution to electric energyis given by, n i n o h E n l (cid:0) n l + 1 (cid:1) n l +12 (cid:0) n l +12 + 1 (cid:1) n l (cid:0) n l + 1 (cid:1) n l (cid:0) n l + 1 (cid:1) (A2)The site index ( j ) is omitted in the above equation as itis on one particular site. Within the average electric fieldansatz, i.e for n l ( j ) = n l ⇒ h E ( j ) = h E for all sites j ,resulting, H (approx) E = g a N h E (A3)where, N is total number of staggered sites on the latticeand h E = n l (cid:0) n l + 1 (cid:1) . Note that, for n l (cid:29)
0, one canactually consider h E = h E ≡ n l .At any site j , the onsite electric energy h E ( j ) differfrom h E iff n i ( j ) = 0 , n o ( j ) = 1, and that difference,that is relevant in strong coupling regime (for n l >
0) isgiven by:∆ h E = n l + 12 (cid:18) n l + 12 + 1 (cid:19) − h E = n l . (A4)This correction term to H (approx) E is particularly impor-tant for strong as well as intermediate coupling regime,0where we consider mean value of gauge flux, that is notvery large compared to that considered in weak couplingregime. Within the mean field ansatz the total electricpart of the LSH Hamiltonian Hamiltonian is given by: H (LSH) E = g a N h E + (cid:88) { j (cid:48) } (cid:18) n l (cid:19) (A5)where, { j (cid:48) } denotes the sites with fermionic configuration n i ( j (cid:48) ) = 0 , n o ( j (cid:48) ) = 1. In the bulk limit of the lattice, theoccurrence of j (cid:48) will be N/ H (mLSH) E = g a (cid:20) N n l (cid:16) n l (cid:17) + N (cid:18) n l (cid:19)(cid:21) (A6) Mass Hamiltonian:
The mass term (26), being inde-pendent of gauge field configuration remain the same inthe mean field ansatz, also for both the strong and weakcoupling regime. H (approx) M = m (cid:88) j ( − j (ˆ n i ( j ) + ˆ n o ( j )) (A7) Interaction Hamiltonian:
The matter-gauge fieldinteraction term is the most complicated within LSH framework as detailed in (27). In the strong couplinglimit of the theory, this particular term gives small con-tribution to the Hamiltonian (see subsection IV A) andcan be treated perturbatively. However, in the weak cou-pling regime, this term becomes significant. The purposeof the present approximation scheme is to bring the in-teraction Hamiltonian into simple form, yet describingmatter gauge dynamics in the weak coupling regime.The approximation scheme that we follow is replacingthe local loop quantum numbers n l ( j ) by a constant n l (cid:29) H LSH I = 12 a N − (cid:88) j =0 h I ( j, j + 1) (A8)where, h I ( j, j + 1) = h I ( j, j + 1) + h I ( j, j + 1)+ h I ( j, j + 1) + h I ( j, j + 1) (A9)Each of these terms, can be further decoupled into left( L ) and right ( R ) parts located at site j and site j + 1respectively, h [ s ] I ( j, j + 1) = h [ s ] I ( L ) h [ s ] I ( R ) , [ s ] = 1 , , , . (A10)Now, considering each term separately, one would obtainthe following: h [1] I ( L ) = 1 (cid:112) ˆ n l + ˆ n o ( j )(1 − ˆ n i ( j )) + 1 ˆ χ + o ( λ + ) ˆ n i ( j ) (cid:112) ˆ n l + 2 − ˆ n i ( j ) = ˆ χ + o ( λ + ) ˆ n i ( j ) ˆ C ( L ) (A11) h [2] I ( L ) = 1 (cid:112) ˆ n l + ˆ n o ( j )(1 − ˆ n i ( j )) + 1 ˆ χ − o ( λ − ) ˆ n i ( j ) (cid:112) ˆ n l + 2(1 − ˆ n i ( j )) = ˆ χ − o ( λ − ) ˆ n i ( j ) ˆ C ( L ) (A12) h [3] I ( L ) = 1 (cid:112) ˆ n l + ˆ n o ( j )(1 − ˆ n i ( j )) + 1 ˆ χ + i ( λ − ) − ˆ n o ( j ) (cid:112) ˆ n l + 2ˆ n o ( j ) = ˆ χ + i ( λ − ) − ˆ n o ( j ) ˆ C ( L ) (A13) h [4] I ( L ) = 1 (cid:112) ˆ n l + ˆ n o ( j )(1 − ˆ n i ( j )) + 1 ˆ χ − i ( λ + ) − ˆ n o ( j ) (cid:112) ˆ n l + 1 + ˆ n o ( j )) = ˆ χ − i ( λ + ) − ˆ n o ( j ) ˆ C ( L ) (A14)and h [1] I ( R ) = ˆ χ − o ( λ + ) − ˆ n i ( j +1) (cid:112) ˆ n l + 1 + ˆ n i ( j + 1)) (cid:112) ˆ n l + ˆ n i ( j + 1)(1 − ˆ n o ( j + 1)) + 1 = ˆ χ − o ( λ + ) − ˆ n i ( j +1) ˆ C ( R ) (A15) h [2] I ( R ) = ˆ χ + o ( λ − ) − ˆ n i ( j +1) √ ˆ n l + 2ˆ n i (cid:112) ˆ n l + ˆ n i ( j + 1)(1 − ˆ n o ( j + 1)) + 1 = ˆ χ + o ( λ − ) − ˆ n i ( j +1) ˆ C ( R ) (A16) h [3] I ( R ) = ˆ χ − i ( λ − ) ˆ n o ( j +1) (cid:112) ˆ n l + 2(1 − ˆ n o ( j + 1)) (cid:112) ˆ n l + ˆ n i ( j + 1)(1 − ˆ n o ( j + 1)) + 1 = ˆ χ − i ( λ − ) ˆ n o ( j +1) ˆ C ( R ) (A17) h [4] I ( R ) = ˆ χ + i ( λ + ) ˆ n o ( j +1) (cid:112) ˆ n l + 2 − ˆ n o ( j + 1) (cid:112) ˆ n l + ˆ n i ( j + 1)(1 − ˆ n o ( j + 1)) + 1 = ˆ χ + i ( λ + ) ˆ n o ( j +1) ˆ C ( R ) (A18)The only approximation made in the above set of equa-tions is n l ( j ) , n l ( j + 1) → n l , where n l is the mean field value. The explicit operator form of the coefficientsˆ C [ s ] ( L/R )’s are the following:1 n i n o ˆ C ( L ) ˆ C ( L ) ˆ C ( L ) ˆ C ( L ) ˆ C ( R ) ˆ C ( R ) ˆ C ( R ) ˆ C ( R )0 0 1 1 1 (cid:114) n l + 1 n l + 2 1 (cid:114) n l n l + 1 (cid:114) n l + 2 n l + 1 (cid:114) n l + 2 n l + 10 1 (cid:114) n l + 2 n l + 1 (cid:114) n l + 2 n l + 1 (cid:114) n l + 2 n l + 1 (cid:114) n l + 2 n l + 1 1 (cid:114) n l n l + 1 (cid:114) n l n l + 1 11 0 (cid:114) n l + 1 n l + 2 1 1 (cid:114) n l + 1 n l + 2 1 1 1 (cid:114) n l + 1 n l + 21 1 (cid:114) n l + 1 n l + 2 1 1 1 (cid:114) n l + 2 n l + 1 (cid:114) n l + 2 n l + 1 (cid:114) n l n l + 1 1 (A19)It is clear from the above set of coefficients that in thelimit n l (cid:29)
0, all of the coefficients can be approximatedto be equal to identity operators, that is their leadingorder contribution. One can expand the coefficients andadd corrections order by order. However, for this workas the very first step we keep ourselves confined to theleading order contribution. In this regime we also approximate λ ± as identity op-erator as per the approximation, n l + 1 ≈ n l . Hence, theapproximated interaction Hamiltonian is given by, H (approx) I = 12 a (cid:88) j (cid:104) χ + o ( j ) χ − o ( j + 1) + χ − o ( j ) χ + o ( j + 1)+ χ + i ( j ) χ − i ( j + 1) + χ − i ( j ) χ + i ( j + 1) (cid:105)(cid:105)