PHYSICAL REVIEW X 3, 041018 (2013)

Quantum Simulation of a Lattice Schwinger Model in a Chain of Trapped Ions

P. Hauke,1,* D. Marcos,1 M. Dalmonte,1,2 and P. Zoller1,2 1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria 2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria (Received 31 May 2013; published 22 November 2013) We discuss how a lattice Schwinger model can be realized in a linear ion trap, allowing a detailed study of the physics of Abelian lattice gauge theories related to one-dimensional quantum electrodynamics. Relying on the rich quantum-simulation toolbox available in state-of-the-art trapped-ion experiments, we show how one can engineer an effectively gauge-invariant dynamics by imposing energetic constraints, provided by strong Ising-like interactions. Applying exact diagonalization to ground-state and time- dependent properties, we study the underlying microscopic model and discuss undesired interaction terms and other imperfections. As our analysis shows, the proposed scheme allows for the observation in realistic setups of spontaneous parity- and charge-symmetry breaking, as well as false-vacuum decay. Besides an implementation aimed at larger ion chains, we also discuss a minimal setting, consisting of only four ions in a simpler experimental setup, which enables us to probe basic physical phenomena related to the full many-body problem. The proposal opens a new route for analog quantum simulation of high-energy and condensed-matter models where gauge symmetries play a prominent role.

DOI: 10.1103/PhysRevX.3.041018 Subject Areas: Atomic and Molecular Physics, Particles and Fields, Quantum Information

I. INTRODUCTION unexplored [42]. The aim of this article is to show that the excellent control of the microscopic dynamics reached At present, trapped ions are one of the physical systems in ion traps may offer interesting perspectives in this in quantum-information science with the largest number of direction. Concretely, we propose—relying on existing achievements [1,2]. By coupling pseudospins represented trapped-ion technology—a quantum simulation of the lattice by internal atomic states to collective lattice vibrations, Schwinger model [44], which is a one-dimensional (1D) high-fidelity entangling gates are now routinely performed version of quantum electrodynamics. in the lab [3–6]. A natural step forward taken in recent Currently, quantum simulation of gauge theories is receiv- years was to exploit these technological possibilities for ing an increasing degree of interest, as these theories represent the quantum simulation of spin models [7–9], in both one of the most solid and elegant theoretical frameworks able digital [5,10] and analog [11–18] protocols. The under- to capture a variety of physical phenomena. For example, in lying idea of such quantum simulations is to take advan- condensed-matter physics, gauge theories play a prominent tage of accurate control of a physical quantum system role in frustrated spin systems, where the identification of and to thus engineer an effective dynamics that mimics a emergent degrees of freedom in terms of gauge fields has quantum many-body model of interest [19–24]. First ex- provided a deeper insight into the physics of quantum-spin tensions into the domain of high-energy physics have also liquids and exotic insulators [45–47]. At a microscopic been undertaken, e.g., by simulating the Dirac equation level, in the standard model of particle physics, gauge theories [25], or with proposals for simulation of coupled quantum provide a natural description of interactions between fields [26] or the Majorana equation [27]. However, it fundamental constituents of matter [28–31]. Despite their remains an outstanding challenge in this field to extend apparent simplicity, solving gauge theories is generally chal- these ideas to lattice gauge theories (LGTs), whose many lenging, a long-standing example being the theory of strong facets, from static to dynamical properties, present several interactions—quantum chromodynamics [28–31]. One of the technical difficulties for classical computations [28–31]. In main reasons for the difficulties in solving gauge theories is the context of ultracold neutral atoms, several proposals for that they may present nonperturbative effects that are hard to the quantum simulation of lattice gauge theories have been capture within diagrammatic expansions, preventing the ap- made recently [32–41], but the realization of gauge sym- plication of unbiased analytical approaches to the many-body metries in alternative atomic or optical systems is currently problem. Some of these limitations can be circumvented in the *[email protected] framework of LGTs [28–31,48]. Here, by means of Monte Carlo simulation of the corresponding lattice action, Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- a broad regime of interaction parameters becomes accessible, bution of this work must maintain attribution to the author(s) and thus allowing the investigation of nonperturbative effects the published article’s title, journal citation, and DOI. with controlled numerical techniques [31,47]. Nevertheless,

2160-3308=13=3(4)=041018(18) 041018-1 Published by the American Physical Society P. HAUKE et al. PHYS. REV. X 3, 041018 (2013) classical simulations are severely limited by the sign prob- [57–61], which represent a generic class of models that lem, which prevents an accurate description of finite-density capture gauge invariance and, at the same time, facilitate regimes (as relevant, e.g., for the core of dense neutron stars) quantum simulations. In this framework, Abelian gauge fields and out-of-equilibrium dynamics (which is realized in can be represented by spin-S quantum operators, where the heavy-ion collider experiments). Given these difficulties, it limit S !1recovers the corresponding continuous-variable becomes particularly attractive to develop a quantum simu- gauge theory, such as quantum electrodynamics or quantum lator of LGTs. chromodynamics. For lower spin representations, QLMs re- The basic elements of a LGT are (typically fermionic) tain exactly the featured gauge symmetry and still share many matter fields c i occupying lattice sites i, coupled to bo- features with the corresponding continuum gauge theories, sonic degrees of freedom, the gauge fields Uij that live on such as the physics of confinement, string breaking, or false- the links between neighboring sites [see Fig. 1(a)]. In the vacuum decay [61]. This last feature is the example that we context of a quantum simulator consisting of ultracold will analyze in the present article. The goals of this article are neutral atoms in an optical lattice, the fermionic fields (i) to propose a realistic architecture, suitable for large ion can be naturally implemented with fermionic atoms. In chains, to realize such a QLM, and (ii) to illustrate how contrast, the basic degrees of freedom in a trapped-ion minimal instances of LGTs can be accessed even in basic quantum simulator are spins and bosons, constituted by setups. These points thus track a possible experimental internal states and collective ion vibrations, respectively. roadmap, where steps of increasing difficulty demonstrate To make fermionic matter fields accessible to an ion setup, significant signatures of many-body physics. we therefore consider one-dimensional ion chains, allow- The present article is organized as follows. First, we in- ing one to use the Jordan-Wigner transformation to map troduce a QLM version of the Schwinger model (Sec. II), fermionic degrees of freedom to pseudospins. Linear focusing especially on the spontaneous charge-conjugation chains have the additional advantage that they are the and parity-symmetry breaking encountered in this model. natural geometry implemented in linear Paul traps. Then, in Sec. III, we explain how engineered spin-spin inter- Regarding the gauge fields, recent ideas to simulate clas- actions can be used to enforce gauge invariance and to con- sical gauge fields have emerged in the context of trapped ions struct a microscopic Hamiltonian that—to second-order [49–51], following the optical-lattice counterpart [52–56]. perturbation theory—generates the dynamics of the ideal In that case, the classical field has no dynamics of its own QLM. In Sec. IV, we show the validity of the effective model and may be described by a simple phase picked up during a by studying static and dynamic properties of the microscopic iij Hamiltonian, using exact diagonalizations of finite chains. tunneling process, Uij ¼ e , with 2 R. Here, in contrast, Since experiments will likely start with very few ions, we we are interested in dynamical gauge fields, where Uij be- comes a quantum field. In the Wilson formulation of LGTs present in Sec. V a simpler (although, in contrast to Sec. III, [48], the U are characterized by continuous degrees of not scalable to larger chains) implementation with four ions. ij We show that this simple system already exhibits features freedom. In a trapped-ion setup with discrete degrees of related to the full many-body problem. In Sec. VI, we outline freedom, a natural formalism to implement the gauge fields a possible experimental sequence and discuss the most promi- is provided by the so-called quantum link models (QLMs) nent error sources. Finally, in Sec. VII,wepresentourcon- clusions and an outlook on future directions.

II. ONE-DIMENSIONAL U(1) LATTICE GAUGE THEORIES AS QUANTUM LINK MODELS In this article, we focus on a conceptually simple instance of a gauge theory, a one-dimensional U(1) QLM with stag- gered fermions. This model is chosen as a good compromise between feasibility and fundamental interest, allowing one to demonstrate basic phenomena, such as string breaking [62], shared with other, more complex LGTs [34]. It can be under- stood as a QLM version of the Schwinger model, which represents quantum electrodynamics in one dimension. The FIG. 1. (a) Lattice gauge theory. Fermions with annihilation Hamiltonian describing the dynamics of this QLM is operators c i, living on lattice sites i, couple to gauge fields, XNm XNm represented by the operators Ui;iþ1,livingonlinksi, i þ 1.(b)A J ~ H~ ¼ ðc yU c þ H:c:Þþm ð1Þi c y c fermion tunnels from site i to i 1 and flips the gauge field Si1;i on 2 i i;iþ1 iþ1 i i the link in between. (c) A filled (empty) bullet denotes an occupied i¼1 i¼1 (empty) site, which are mapped by the Jordan-Wigner transforma- 2 XNm g 2 tion to the pseudospin up (down) state. (d) The tunneling process of þ ðEi;iþ1Þ (1) panel (b), expressed in spin language. 2 i¼1

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~ ~ ~ ~ ¼ HJ þ Hm þ Hg: (2) Gijphysi¼0 8 i: (5) c y c The field operators i and i at the lattice sites i are creation In the continuum limit, this condition reduces to the usual and annihilation operators, respectively, of ‘‘staggered- Gauss law of quantum electrodynamics, r~ E~ ¼ , where fermion’’ particles. Staggered fermions are characterized by is the density of charged matter. We call states fulfilling ~ amasstermHm with alternating sign, and they provide an the Gauss law the ‘‘physical’’ states of our problem. The elegant way of simultaneously incorporating matter and an- challenge for any quantum simulation of a gauge theory is timatter fields by using a single degree of freedom on bipartite to engineer the dynamics given by H, which connects only lattices. One may picture staggered fermions as quarks, where physical states, while avoiding gauge-variant perturbations a fermion on even sites corresponds to a quark (q)andthe ~ to states jc i with Gijc i 0. (Such perturbations corre- absence of a fermion on odd sites to an antiquark (q); the spond to, e.g., the spontaneous creation of an imbalance opposite configurations denote the absence of quarks or anti- between quarks and antiquarks and are therefore ‘‘unphys- quarks. In order to keep the symmetry between matter and ical.’’) Before explaining how this challenge can be met in antimatter, the number of lattice sites Nm has to be even. the trapped-ion chain, we first discuss the basic physics of ~ ~ In Hamiltonian H,thetermHg describes the energy of the the QLM of Eq. (1). gauge field, with Ei;iþ1 being an electric-field operator asso- þ ciated with the link i, i 1. While in the Wilson formulation A. Charge-conjugation and parity-symmetry breaking of LGTs, the link variables Ui;iþ1 are parallel transporters spanning an infinite-dimensional Hilbert space, in the QLM Despite its simplicity, the model of Eq. (1) captures the formulation they may be defined as spin operators of arbitrary physics of interesting phenomena associated with gauge ~þ ~z theories. When the quantum links are represented by spins representation S, by setting Ui;iþ1 ¼ Si;iþ1 and Ei;iþ1 / Si;iþ1. S 1, the competition between the electric-field term H~g This choice of ‘‘quantum links’’ retains the commutation 2 and the mass term H~m leads, for J m, g , to a crossover relations ½Ei;iþ1;Ui;iþ1¼Ui;iþ1 that are required to assure P between a ‘‘string’’ state characterized by hS~z i 0 gauge covariance of the gauge fields. The term H~J, finally, i;iþ1 2 ~z couples the matter and gauge fields via an assisted tunneling (for m g ) and a ‘‘meson’’ state featuring hSi;iþ1i¼0 [see Figs. 1(b) and 1(c)] and represents a key ingredient of any (for m g2)[34,43]. gauge theory in the presence of matter fields [28–31,45]. In the S ¼ 1=2 case that we are going to discuss in the The most distinctive feature of gauge theories, such as rest of this article [63], the electric-field term H~g is con- the QLM of Eq. (1), is the presence of local (gauge) stant, and the system dynamics is described by the com- symmetries. petition between the kinetic and the mass term. This These symmetries imply that, given a set of local sym- competition leads to the spontaneous breaking of the parity ~ metry generators fGig, the system Hamiltonian is invariant (P) and charge-conjugation (C) symmetries, defined as under gauge transformations of the form P P y y c c N c c Nm ¼ m ; Nm ¼ Nm ; (6a) Y Y þi 2 i þi i ~0 ~ ~ ~ 2 2 2 H ¼ expfiiGigH expfiiGig; (3) P y UNm Nm ¼ UNm Nm ; (6b) i i 2 þi; 2 þiþ1 2 i1; 2 i P N N ENm Nm ¼ E mi1; mi; (6c) where fig is an arbitrary set of parameters. For the gauge 2 þi; 2 þiþ1 2 2 theory we are interested in, the generators read C c iþ1 c y C c y iþ1 c i ¼ð1Þ iþ1; i ¼ð1Þ iþ1; ð1Þi 1 (6d) G~ ¼ E E þ c y c þ ; (4) i i1;i i;iþ1 i i 2 C y C Ui;iþ1 ¼ Uiþ1;iþ2; Ei;iþ1 ¼Eiþ1;iþ2: (6e) where the constant term is due to the use of staggered For sufficiently largeP positive values of m=J, the gauge fermions. ~z fields are polarized, ihSi;iþ1i 0 [Fig. 2(a), right side]. The gauge invariance expressed in Eq. (3) is equivalent In the corresponding state, the parity and charge- ~ ~ to the condition ½H; Gi¼0 8 i, implying that the conjugation symmetries are spontaneously broken. For Hamiltonian does not mix eigenstates of G~ with different i large negative values ofPm=J, parity and charge conjugation eigenvalues G~ . Thus, gauge invariance can be imposed ~z i are restored, yielding ihSi;iþ1i¼0 [Fig. 2(a), left side]. by restricting the Hilbert space to a sector with fixed ~ . Gi To illustrate the physical meaning of this symmetry In the case of the present U(1) gauge theory, the sector we breaking, consider the mapping to the associated quark are interested in is the one of balanced matter, i.e., equal picture [Fig. 2(b)]. In this picture, C transforms a quark number of quarks and antiquarks, and we will restrict our on site i to an antiquark on site i þ 1, etc. Now, for a considerations to this subspace. This subspace is defined pictorial visualization, consider a simplified scenario by ~ ¼ 0 8 i; i.e., it consists of all states satisfying the Gi where J ¼ 0, under periodic boundary conditions (for a so-called Gauss law [29,61] discussion of various boundary conditions, see Ref. [34]).

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FIG. 2. Ground states of the QLM in several equivalent representations, for m=J !1(left) and m=J !þ1(right; only one of two degenerate solutions is drawn). Sketches are for vanishing quantum fluctuations (J ¼ 0þ), periodic boundary conditions, and gauge fields represented by spins 1=2. (a) Staggered-fermion QLM. Physical states are invariant under gauge transformations ~ with generators Gi, which span one matter and two gauge fields. (b) Associated quark picture. Fermions on even sites correspond to quarks (q, light green) and vacancies on odd sites to antiquarks (q, dark red); the opposite configurations denote the absence of (anti)quarks. For m=J !1, alternating sources and drains of flux completely cover the chain, and the ground state is charge and parity invariant. For m=J !þ1, the system is empty of matter and has an unbroken flux string, breaking C and P symmetry. In this case, the Gauss law (4) requires the same orientation for all gauge fields, allowing a double energy degeneracy, corresponding to the two directions of electric flux (only one of which is visualized here). (c) Equivalent spin model H~QLM. Fermions are mapped to spins via the Jordan-Wigner transformation. For S ¼ 1=2 quantum links, right- (left-) flowing flux translates to an up (down) gauge-fieldP spin. (d) Rotated spinP model HQLM for experimental implementation. At m=J !1,the z z order parametersP are characterizedP by ihSi;iþ1i¼1 and ihi i¼1.IntheC-andP-breaking state at m=J !1,theytake z z the values ihSi;iþ1i¼0 and ihi i¼1.

In that picture, at large negative m the ground state consists B. Mapping to a spin model of alternating sources and drains of flux and preserves C To make the QLM of Eq. (1) accessible to a trapped-ion and P. The ground state at large positive m, however, setup, we first map the fermionic fields to spin-1=2 degrees displays a gauge-field string that threads across the system. of freedom, using a Jordan-Wigner transformation, This state is doubly degenerate, corresponding to the two P i ð~z þ1Þ=2 polarization directions, and it breaks C and P. y k c ¼ e k

041018-4 QUANTUM SIMULATION OF A LATTICE SCHWINGER ... PHYS. REV. X 3, 041018 (2013) X accompanied by a flip of the additional spin degree of H ¼ V ðzSz þ zSz þ Sz Sz þ z ~ G i i1;i i i;iþ1 i1;i i;iþ1 i freedom, S. The second part in H~QLM is simply a staggered, i transverse magnetic field. The QLM model is now ex- z z þ Si1;i þ Si;iþ1 þ 2Þ: (15) pressed purely by spin operators, which can be represented in the ions by pseudospins consisting of two internal states. In view of the experimental implementation, the basis In the implementation discussed below, the two types of transformation (12) constitutes a major simplification, as spins are distinguished by choosing different internal levels it replaces interactions with alternating sign with uniform to represent and S spins, respectively. ones. As it becomes apparent in the first three terms of HG, The gauge-invariant subspace that we are interested in the way gauge invariance is recovered in this spin basis is fulfills the following constraint given by the spin version of reminiscent of the basic frustration induced by Ising inter- the Gauss law (4) and (5), actions in triangular (ladder) geometries at fixed magneti- zation [65]. The gauge-invariant subspace can therefore be 1 G~ jc i¼0; G~ ¼ ½S~z S~z þ ~z þð1Þi: seen as the ground-state manifold of an associated frus- i i 2 i1;i i;iþ1 i trated Ising model with a longitudinal field. Out of this (9) manifold (degenerate with respect to HG), the dynamics generated by HJ and Hm selects one state as the ground Here, we represent the electric-field operator by the Pauli z state of the QLM. ~z ~z matrix Si;iþ1, Ei;iþ1 Si;iþ1=2. The product states at m=J ¼1in this spin picture are sketched in Fig. 2(c). III. MICROSCOPIC MODEL In the microscopic model, where gauge invariance is not In this section, we discuss how HQLM and HG can be built in a priori, we will enforce gauge invariance by engineered in a trapped-ion experiment by employing ef- adding the term fective spin-spin interactions [66–68] mediated by the ion X vibrations. Our system consists of a linear chain of ions, H~ ¼ 2V ðG~ Þ2 (10) G i oriented along the z direction. For simplicity, we assume i equal spacings between ions, as can be achieved around the X center of a long ion chain or by employing individual z ~z z ~z ~z ~z ¼ V ½~i Si1;i ~i Si;iþ1 Si1;iSi;iþ1 microtraps [69,70] (see Sec. VI B for the consequences i of relaxing this conditions). As we will see later, to create i z ~z ~z H þð1Þ ð~i þ Si1;i Si;iþ1Þþ2: (11) the desired couplings G with correct weights, we need to distinguish the spins from the S spins. This distinction can For V jJj, jmj, this energetically suppresses unphysical, be made by encoding the two types of fields in pseudospins ~ gauge-variant transitions to states with Gijc i 0. with different resonance frequencies [Fig. 3(a)]. In one dimension, we can remove the various alternating Next, we describe how to engineer HG, as this will signs by the basis transformation (corresponding to a stag- enforce gauge invariance. The associated high-energy gered rotation about the x axis) scale will also be used to implement perturbatively the effective spin coupling J characteristic of HQLM. A dis- z i z y i y ~i !ð1Þ i ; ~i !ð1Þ i ; (12a) cussion of possible experimental errors will be given in ~z i z ~y i y Sec. VI B. Si1;i !ð1Þ Si1;i; Si1;i !ð1Þ Si1;i; (12b) while the x components remain unchanged. The product A. Effective spin-spin interactions states at m=J ¼1 are then rotated as sketched in Engineering HG in an ion setup requires the realization Fig. 2(d). In this basis, the model Hamiltonian becomes z of single-spin operators proportional to i and of spin-spin z X X interactions of zz type. The implementation of i operators J þ m z HQLM ¼ ði Si;iþ1iþ1 þ H:c:Þþ i is straightforward, e.g., by state-selective ac-Stark shifts 2 i 2 i due to off-resonant Raman transitions. The zz interactions can be engineered by a variety of schemes, e.g., by laser HJ þ Hm; (13) addressing of optical quadrupole qubits [4,68] or by driv- and the Gauss law reads ing rf transitions in ions in spatially varying magnetic fields [6,71]. The currently best-established technique for engi- 1 G c ¼ 0;G¼ð1Þi ðSz þ Sz þ z þ 1Þ: neering zz interactions [11,17] encodes the pseudospins in i i 2 i1;i i;iþ1 i two hyperfine states and uses Raman lasers in ‘‘moving- (14) standing-wave’’ configurations to generate interactions be- tween the spins with the help of state-dependent ac-Stark Therefore, the energetic restriction to the physical sub- shifts [66,67]. Here, we describe the main steps for obtain- space is achieved by the Hamiltonian ing effective spin-spin interactions with the last technique.

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FIG. 3. Proposed experimental scheme. (a) Two kinds of pseudospins encode matter and gauge fields in an alternating fashion, with associated spin matrices i and Si;iþ1. The corresponding internal states are labeled as j#i, j"i and j#iS, j"iS, respectively. (b) and (c) Spin interactions of zz type are transmitted by coupling the pseudospin levels (sketched for a single ion) to phonons. (b) For hyperfine qubits, one can engineer these couplings via a differential light shift induced by two pairs of Raman beams (with Rabi ";# frequencies 1;2 and detuning jj!; here ! is the qubit energy difference, with ¼ , S). The Raman transition is detuned by x from the phonon frequency !x. jni denotes the phonon-number Fock state. (c) For optical qubits, one can use two beams with Rabi 0 frequencies 1 ¼ 2 ¼ that are tuned (with a small detuning x) close to half the phonon frequency !x.

As sketched in Fig. 3(b), in this scheme, two lasers with For concreteness, we consider the situation where the optical frequencies !1 and !2 are far off resonant with pseudospin is coupled to the radial phonon modes in the x respect to a dipole-allowed transition to an auxiliary ex- direction. To enable this coupling, the laser beat note !L cited state jei. To avoid spurious population of jei, the has to be tuned close to the corresponding phonon frequen- q detuning is much larger than the spontaneous decay rate cies !x . In a linear chain of ions in a Paul trap, the radial of the excited state and the Rabi frequencies i , where modes are ‘‘stiff,’’ meaning that the energy scale associated i ¼ 1, 2 numbers the laser beam and ¼# , " denotes the with a local ion vibration (given by the transverse trapping internal state which is acted on. Under these conditions, potential) outweighs the coupling of vibrations via the 2 2 3 one can eliminate the excited state, thus obtaining an Coulomb repulsion, i.e., x e =ðM!xd0Þ1, where effective Hamiltonian involving only the pseudospin states M is the ion mass, !x the trap frequency in the x direction, and collective vibrational modes. and d0 the inter-ion distance. In such a situation, the radial The resulting atom–light interaction can be written as an phonon modes are only slightly dispersed, and we can q effective Hamiltonian for the pseudospins [66,67,72], approximate !x !x. Thus, the lasers couple the internal states almost equally to all modes. The condition of XN @ tuning close to the phonon frequencies then becomes z n ik rni! t Hd ¼ sn e L L þ H:c: (16) ! ¼ ! , with ! , as sketched in Fig. 3(b). 2 L x x x x n¼1 Additionally, to address the vibrational modes in the x k direction, the difference of the laser wave vectors L has to Here, N is the number of ions, and the pseudospins are point along the x axis. Then, we can use the corresponding represented by operators sn, which denote or S spins, phonon operators to write depending on whether the nth ion encodes a matter field or r XN a gauge field. Further, n is the displacement of ion n k r y k k k L n ¼ nqaq þ H:c: (17) from its equilibrium position; L ¼ 1 2 is the differ- q¼1 ence of the laser wave vectors; !L ¼ !1 !2 is the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ? M q @ difference of the laser frequencies; and n ¼ð1;#2;# Here, nq ¼ nqkL= 2M!x = is the Lamb-Dicke pa- ? rameter, with k ¼jk j. The matrices M , obtained by 1;"2;"Þn=ð2Þ is the two-photon differential Rabi L L nq frequency. Here, we assume that the pseudospin energy diagonalizing the elasticity matrix of the ion crystal, trans- splitting ! is jj ( ¼ , S), meaning that the detun- form the localized vibrations of the nth ion into normal y ing to the excited level is approximately equal for the upper modes, with the creation operator aq . and the lower level. Additionally, in writing Eq. (16), we Typically, nq 1, so we can expand the exponential in exploited the fact that unwanted transitions between the Eq. (16) up to first order. Then, including the eigenenergies pseudospin levels are avoided as long as the condition of the collective vibrational modes, we obtain a spin- !L ! is ensured. phonon Hamiltonian of the form [67,72]

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XN XN @ XN Since and S spins have different resonance frequencies, @ q y z n y Hsp ¼ x aq aq þ i sn nqaq þ H:c: one can achieve different coupling strengths via global 2 q¼1 n¼1 q¼1 laser beams, where the beam that is close to resonance (18) for the S-spin transition has a larger intensity than the one for the -spin transition. This possibility of strengthening Here, we transformed the equation into an interaction z z the interaction Si1;iSi;iþ1 is the main reason why one needs picture with respect to the laser beat note ! , in which L two different types of pseudospins. Using n ¼ if n is a the eigenfrequencies of the phonon operators are shifted to q q spin and jnj¼jSj¼8jj if n denotes an S spin, we x !x !L. Moreover, we employed a rotating-wave obtain the Hamiltonian approximation, neglecting terms rotating at frequencies    q X 1 X 1 1 !x þ !L, which is valid for n !L. Since !L !x, z z z z z z HV ¼V S i Sj;jþ1 þ Si;iþ1Sj;jþ1 þ 6 i j ; for realistic trap frequencies lying in the range !x ¼ i;j Dij i

041018-7 P. HAUKE et al. PHYS. REV. X 3, 041018 (2013) the spin-phonon couplings with a larger number of laser J X H ¼ ðSþ þ : :Þ; frequencies, which allows one to generate only the desired J i i;iþ1 iþ1 H c (27) 2 i interaction pattern [77]. KB In Sec. IV, we will address the quantitative effects of the where we defined J V , assuming a homogeneous err gauge-invariant contributions given by HV . Before that, Ki;iþ1 ¼ K. Similar hopping terms, realized within four we will show how one can generate the desired dynamics hyperfine states of a single ion and for a single-mode HQLM within a perturbative framework, provided the sys- coupling, have been envisioned in Ref. [26]. Notably, tem dynamics is energetically constrained onto the gauge- Eq. (27) is a nearest-neighbor Hamiltonian; longer-range þ þ þ invariant subspace by means of a strong HG. interactions and terms such as i Si;iþ1iþ1 are suppressed by the energy penalty HG [79]. C. Engineering the system dynamics Summarizing, the microscopic model reads The desired dynamics of the QLM given by Eq. (13) err Hmicro ¼ HK þ HB þ Hm þ HG þ HV consists of the mass term Hm and the matter-field–gauge- X X X x x x m z field interaction HJ. The single-particle terms generating ¼ Kiji j þ BSi;iþ1 þ i i

X 2 j;xxj iqjq@!q Kij ¼ 2 2 : (26) q !q

x z To avoid cross terms of the type i j, it is convenient if the phonon modes addressed here are different from those used FIG. 4. Schematic-level diagram of a single ion summarizing the applied spin-phonon couplings (sketched for optical quadru- for the zz couplings, for example, by choosing vibrations in 40 þ a different direction. Alternatively, such cross terms will be pole qubits as in Ca ). The processes in Figs. 3(b) and 3(c) couple z and Sz to radial phonon modes. The correct weight energetically suppressed by the Gauss law. i i;iþ1 of zz interactions for the Gauss law requires effective coupling Taking H1 as a perturbation to HG, we can obtain an strengths jSj¼8jj. The xx interactions in HK may be effective Hamiltonian for the low-energy sector of HG, generated by Mølmer-Sørensen–type couplings of strength c c given by Ec h jHGj i¼0. This sector is equivalent . Coupling to undesired internal levels is avoided by c ;xx to the gauge-invariant subspace defined by Gij i¼0 8 i. selection rules and Zeeman splittings EZee;D=S. Whether the In second-order perturbation theory, we obtain the effec- ion represents an S or a spin is determined by its initialization tive interaction in the gauge-invariant subspace in the state j#iS or j#i.

041018-8 QUANTUM SIMULATION OF A LATTICE SCHWINGER ... PHYS. REV. X 3, 041018 (2013) effective Hamiltonian (27) arising in higher orders of J=V ¼ KB=V2. Such contributions can modify the system dynamics and compromise gauge invariance. We will show next that the parameter window left open by these two error sources allows us to retain approximate gauge invariance while, at the same time, observing the correct dynamics.

IV. NUMERICAL ANALYSIS As explained in the previous section, at small J=V the microscopic model Hmicro, Eq. (28), reproduces the physics of the lattice gauge theory HQLM, Eq. (13). In this section, we test these perturbation-theoretical considerations via exact diagonalizations of finite chains. We consider six matter-field spins plus six gauge-field spins, under periodic boundary conditions to avoid boundary effects (see FIG. 5. Validity of the microscopicP model. (a) and (b) Gauge 2 2 Sec. VI B for a discussion of possible effects of nonhomo- invariance, quantified by G iGi =Nm, is only violated sub- geneous ion distances and open boundary conditions). For stantially when V=J is not a strong energy scale. (c) At large 2 x simplicity, we set B ¼ K, and we assume a dipolar decay PJ=V ¼ KB=V , the gauge fields acquire a finite S Sx =N of the interactions in HV and HK. In particular, we study i i;iþ1 m. The detrimental effects encountered in panels the influence of these error terms on the system when (a)–(c) are enhanced around m ¼ 0, i.e., close to the quantum changing the sign of m (with J>0). Below, we first phase transition of the QLM. Curves in panels (a) and (c) are for m=J 0, as given in the figures. The behavior for m=J < 0 is consider ground-state properties. Afterwards, we estimate c c the optimal parameter values to see the false-vacuum decay very similar. (d) The overlap F ¼h ðm; J; VÞj idealðm; JÞi be- tween the ground states of the microscopic model and the ideal in an adiabatic sweep from the C- and P-invariant to the QLM, Eq. (13), is largest at intermediate J=V and increases with C- and P-breaking situation. increasing jm=Jj.

A. Ground state of the microscopic model err the undesired dipolar terms HV . These terms form a The degree of gauge invariance achieved by the micro- relevant contribution when J=V & 1=8 or jmj=V & 1=8. scopic model is measured by violations toP the Gauss Being gauge invariant, they do not affect the Gauss law, but 2 2 law (14), which can be quantified by G ihGi i=Nm they do change the dynamics of the system. [Figs. 5(a) and 5(b)]. As expected, when KB=V2 is not a To analyze how much these deviations affect the ground- small energy scale, second-order perturbation theory state phaseP diagram, we now study the order parameters z z z breaks down, and HG is no longer efficient in energetically PS ihSi;iþ1i=Nm [Figs. 6(a) and 6(b)] and z suppressing gauge-variant terms. The onset of this effect ihi i=Nm [Fig. 6(c)]. As discussed in Fig. 2(d),atm=J ¼ is linear in J=V and quadratic when expressed in K=V,as 1, we expect z ¼ 1 and Sz ¼1 for the ideal model. expected from second-order perturbation theory. Up to the At m=J ¼1, we expect z ¼1 and Sz ¼ 0. The last range of J=V 0:3, however, the system remains approxi- value is realized by a superposition of the two parity- mately gauge invariant. breaking states with fluxes in the negative and positive As J=V is increased, because of the presence of HB directions [see Fig. 2(a)], i.e., by the two antiferromagnetic Sx z Pthe gauge fields acquire a finite polarization configurations of Si;iþ1 that are shifted by one lattice x ihSi;iþ1i=Nm [Fig. 5(c)]. Simultaneously, because of HK spacing. The expected behavior at large jmj is indeed the matter fields display an antiferromagnetic ordering in what is observed in the microscopic model. the x direction. Neither effect is foreseen in the ideal model Deviating from the QLM, around m ¼ 0, we find the (13). These effects are more pronounced close to m ¼ 0, precursor of a plateau where the order parameters remain which corresponds to a potential critical region in the QLM. constant at Sz z 1=3 over an entire range of m=J. A quantitative indicator for deviations resulting from This plateau becomes more pronounced at smaller J=V these perturbations is provided by the overlap of the exact (for the curve at J=V ¼ 0:125, it stretches beyond the microscopic ground state to the ideal one [Fig. 5(d)]. In plotted range). As a result, the corresponding curves general, this overlap is a very strict measure, and we can change the sign of their curvature twice. Since peaks in understand it as a loose lower bound to the quality of the @ z=@ðm=JÞ or @Sz=@ðm=JÞ are good indicators for quan- ground state. As expected from the analysis of Sx and the tum phase transitions in the thermodynamic limit, this gauge invariance, it decreases at large J=V and in a suggests two distinct transition points in the microscopic relatively broad region around m ¼ 0. Additionally, the model. This is in contrast to the ideal QLM, where we overlap is strongly suppressed at small J=V because of expect only a single quantum phase transition [64].

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The assumption of two transitions is confirmed by the reason, the extent of the intermediate region increases at fidelity susceptibility, which is a reliable tool to detect smaller J=V. However, because of the dipolar nature of err conventional quantum phase transitions [80,81]. It is de- HV , this region underlies appreciable finite-size effects at fined as the considered system sizes. Additionally, it presents a considerable asymmetry upon m !m. This asymmetry logjhc ðm; JÞjc ðm þ m; JÞij2 z z ðm; JÞ¼ lim ; (29) derives from the i Si;iþ1 interactions, which favor the m!0 m2 state achieved at m=J !1 over the one reached at m=J !þ1. where jc ðm; JÞi is the ground state of the microscopic To summarize this analysis, while there seems to be a model at fixed m and J. In analogy to other, long- novel phase because of Herr in the region around m ¼ 0, established susceptibilities, the fidelity susceptibility mea- V the transition from a parity-conserving state to a parity- sures the response of the ground state towards an external broken state can be clearly observed in the microscopic perturbation, in this case, a change of the Hamiltonian. model, even at small chain lengths. Further, by restricting Close to a quantum critical point, the wave function oneself to the regime of J=V 0:2, one can realize a changes its behavior drastically, yielding peaks in situation where J dominates over the error terms Herr— ðm; JÞ. For the microscopic model, we find two such V and thus induces the desired dynamics—while retaining peaks [see Fig. 6(d)]. As can be expected, at the position approximate gauge invariance. of these peaks, the Gauss law is more strongly violated [see Fig. 5(b)]. The presence of two peaks in the fidelity susceptibility B. Sweep across the transition indicates the appearance of an intermediate phase around We now want to check the possibility of observing the m ¼ 0, which is absent in the ideal QLM. In this central false-vacuum decay and the associated spontaneous C and P region, the physics of the microscopic model is governed breaking in experiments along the lines of Refs. [11,13–16]. err by a complex interplay between Hm and HV — while the For this end, we consider an adiabatic sweep from the mass term Hm favors the states sketched in Fig. 2(d), with a C- and P-invariant region to the C- and P-broken region. finite z polarization of the matter fields, the frustration We keep V and J positive and constant and apply a linear err arising from HV tends to suppress such polarizations. quench of the mass term, mðtÞ¼minit þ tm=t, with Because of this interplay, transitions away from this region minit mðt ¼ 0Þ < 0. We finish the sweep at mfin err occur roughly when Hm dominates over HV , i.e., when mðtfinÞ¼minit. The duration of the quench, tfin, depends jmj=V * 1=8 or, equivalently, jmj=J * V=ð8JÞ. For this on the quench range and speed. In our analysis, we consider a system of four matter-field spins plus four gauge-field spins for similar parameters as in the preceding section. As a strict measure for how well this sweep reproduces the false-vacuum decay, we study the quench fidelity. It is defined as the overlap of the microscopic state after c the sweep, fin, with the ground state of the ideal QLM c at the final parameter values, idealðmfin=JÞ, i.e., Fqu 2 jhc finjc idealðmfin=JÞij . As seen in Fig. 7(a), with increas- ing J=V, the errors committed in the perturbative treatment of Sec. III C reduce Fqu. More drastic, however, is the err effect of the dipolar error terms HV : In order to start and finish in the correct states of Fig. 2(d), we require jminit=JjV=ð8JÞ. Otherwise, the sweep happens within the central phase, which is not present in the ideal QLM, and the overlap with the ideal final state vanishes. Furthermore, as usual, a small m=t is desirable to FIG. 6. Ground-state behavior of the microscopic model. (a)– assure the adiabaticity of the process. The parameter region (c) Similar to the ideal QLM, the system reaches Sz 1 and that is most relevant for our purposes, 1=8 & J=V 1,is z z z 1 for m=J !1, and S 0 and 1 for m=J !1. especially sensitive to a change of sweep speeds. For a This denotes a transition from a parity- and charge-symmetric good fidelity, one will thus want to choose not only a large state to a parity- and charge-symmetry breaking state. Curves from jm =Jj but also a small m=t, both of which increase lighter to darker and shorter to longer dashes: J=V ¼ 0:125, 0.25, init 0.275. Solid line: Ideal QLM. The curves change their sign of the total quench time tfin. In experiment, where available curvature twice, indicating two phase transitions. (d) Similarly, time scales are restricted (e.g., by the lifetime of the upper the fidelity susceptibility ðm; JÞ has two peaks, suggesting two pseudospin level and spontaneous Raman scattering from phase transitions. This behavior is in contrast to the ideal model, the optical beams [82]), one will therefore have to find a where ðm; JÞ has a single peak at m ¼ 0. compromise.

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frequency of which changes as a function of the coupling parameters J=V. Since we take a snapshot at the fixed time tfin, these oscillations are imprinted on the final state. To summarize this part, the fidelity of the process can reach large values of approximately 90% while retaining approximate gauge invariance. A reasonable choice of pa- 2 @ rameters could be minit=J ¼4 and m=t ¼ 0:2V = . For a realistic value of V=@ ¼ 21kHz [8,17] (see Sec. VI B), this choice corresponds to a sweep time of tfin ¼ 40@=V ¼ 6:5ms. This is a reasonable value compared to experimental time scales. The latter are limited by dephas- ing times, which typically lie in the range 5–10 ms and can reach up to 100 ms [17]. The quality of the final state could be further increased by optimizing the quench protocol, FIG. 7. Final state after a linear sweep, mðtÞ¼minit þ tm=t, for example, by choosing faster sweep velocities in regions with mfin ¼minit and fixed V and J. Solid lines from lighter to where the gap to the first excited state is large [83]. darker shades [bottom to top in panel (a)]: minit=J ¼2, 3, 4 2 @ for m=t ¼ 0:2V = . Short (long) dashed lines: minit=J ¼3 and m=t ¼ 0:4ð0:1ÞV2=@. The arrows show how the precision V. MINIMAL IMPLEMENTATION USING FOUR IONS of the quench increases for larger minit and smaller quench velocities m=t. The total times considered range from Up to now, we have considered an implementation of @ 2 @ @ tfin ¼ 15 =V (for minit=J ¼3, m=t ¼ 0:4V = )to60 =V the QLM (13) that is suitable also for large ion chains. ¼ ¼ @ ¼ (minit=J 3, m=t 0:1). For V= 21kHz [8,17], these However, current experiments are typically restricted to a correspond to 3.2–9.5 ms. (a)–(d) For a value of J=V 0:15–0:2, the final, C- and P-breaking state is reached with (a) high fidelity, few ions. In the following, we discuss a minimal experi- (b) small gauge variance, and the desired behavior for (c) the ment that can be realized with four ions and by only gauge field and (d) the matter field. addressing axial modes, but which already contains the physics of false-vacuum decay explained above. We focus In contrast to the quench fidelity, the gauge invariance is on the unrotated QLM (8) with S ¼ 1=2 and two sites plus not decreased by a possible nonadiabaticity of the sweep. two links, under periodic boundary conditions. This system Instead, it remains on the level of the ground-state values at is sketched in Fig. 8. minit=J, as can be appreciated by comparing Fig. 7(b) to A. Possible implementation Fig. 5(a). For this reason, a large jðm=JÞinitj appears to be more important than a small m=t. The implementation of this two-site QLM is straightfor- For the parameter regimes where the quench fidelity is ward: To induce the interactions for H~G, we exploit the fact large, the order parameters Sz [Fig. 7(c)] and z [Fig. 7(d)] reach the final value expected for the C- and P-breaking state (0 and 1, respectively). For J=V & 0:05, however, the quench happens entirely in the central region, and Sz and z practically remain at their initial values; for a system of four matter-field spins and four gauge-field spins, these are Sz ¼0:5 and z ¼ 0. In the range 0:05 & J=V & 0:1, the quench still ends in the central region, but now it starts close to the C- and P-retaining state sketched in Fig. 2(d), left side, where Sz ¼1 and z ¼ 1. The starting values of the order parameters now lie farther away from the ones of the C- and P-breaking state. Therefore, in the parameter region of 0:05 & J=V & 0:1, the final values of the order parameters actually seem worse FIG. 8. Setup for the minimal implementation using four ions. than for J=V & 0:05. Only when the sweep starts in a C- and Spin-spin interactions are created by pairs of Raman beams, as ¼ P-retaining state and ends in the C- and P-breaking region explained in Fig. 3, with strengths S 2. By tuning close to the vibrational mode q characterized by amplitudes M ¼ (or vice versa) do we obtain the expected final values of 0 nq0 ð Þ z z 1; 1; 1; 1 n=2, one obtains zz spin-spin couplings, as desired the order parameters S and . Depending on jðm=JÞinitj, H~ * for G [Eq. (10)] (green solid lines). An additional undesired, this result is achieved for J=V 0:2. weaker interaction (orange dashed line) does not affect the Besides this overall behavior, all considered quantities system dynamics. Mediated by a different vibrational mode, an display some oscillations as a function of J=V. These additional laser pair generates the xx interaction, as required for derive from oscillations during the time evolution, the HK (blue dotted line).

041018-11 P. HAUKE et al. PHYS. REV. X 3, 041018 (2013) that in a chain of four ions, there is a vibrational mode q0 behavior. In the following, we focus on the cleaner case M ¼ð Þ with amplitudes nq0 1; 1; 1; 1 n=2. The index n described in Fig. 8. labels the ions from 1 to 4, where n ¼ 1, 3 denotes the matter fields 1 and 2, and n ¼ 2, 4 denotes the gauge B. Numerical analysis of the ground-state behavior fields S and S , respectively. The frequency spacing of 12 21 pffiffiffiffiffiffiffi We now illustrate that the minimal implementation of the phonon modes is / ! , where, as before, ! Fig. 8 already contains much of the physics of the full- denotes the trap frequency in direction , and ¼ fledged many-body problem. Using again exact diagonal- e2=ðM!2 d3Þ is the associated stiffness parameter. For the 0 izations, we analyze in this section the ground-state axial direction, this energy spacing is sufficiently large to behavior. In the next section, we study the sweep dynamics. tune the beat note ! of a laser pair close to mode q , with L 0 For better comparison with the implementation discussed detuning ¼ ! ! , while negligibly driving the q0 q0 L in Sec. IV, we interpret our results, presented in Fig. 10, other modes. in the rotated basis (12). For convenience, we reproduce in To obtain the correct weights of the interactions, we Fig. 9 the product states at m=J 1for the present case of assume that the effective Rabi frequencies n coupling two sites under periodic boundary conditions. the S spins to the phonons is twice as large as for the As seen in Fig. 10(a), for this small system, gauge ¼ ions, S 2. Applying Eq. (21) to this case gives invariance is even better retained than what we saw in V Reð? Þ@2k2 M M =ð8M! Þ.We mn m n L mq0 nq0 q0 q0 Sec. IV. Again, deviations are stronger at large J=V and thus obtain the interaction strength between ion m and n, around m ¼ 0, i.e., close to the expected phase transition 0 1 of the ideal QLM (13). The order parameters Sz and z 0 1 1 1 B 2 C behave very similarly to an ideal QLM of the same size B 10 12 C [Figs. 10(b) and 10(c)]. In particular, at large jmj, the V B C mn ¼VB C ; (30) polarization of the product states sketched in Fig. 9 is @ 1 101 A 2 attained. The behavior of the order parameters is practi- 1 2 10mn cally independent of J=V. This finding suggests that in the

¼j j2@2 2 ð Þ with V kL= 16M!q0 q0 . For negative detun- ing, these interactions constitute exactly the zz interaction terms of the Gauss-law constraint (10) [i.e., the one before the rotation (12) in spin space]. Additionally, we obtain the z z undesired, but weaker interaction 12V=2. This interac- tion is gauge invariant, and we find that it does not alter the FIG. 9. Product states at m=J ¼1for the minimal imple- system dynamics (see Sec. VB). mentation of four ions. (a) State at m=J !1, conserving C x x The 12 interactions that generate HK can be mediated and P invariance. (b) State at m=J !þ1, breaking C and P via a different phonon mode, with lasers that address only invariance. This state is degenerate with the opposite configura- the pseudospins. Notably, because of the large mode tion of gauge-field spins (i.e., jS12i¼j"i, jS21i¼j#i). spacing, all interactions in this minimal implementation can be generated by using only axial modes. For the architecture presented in Secs. III and IV, this was not err possible since a sufficiently fast decay of HV requires the use of radial modes. An alternative four-ion implementation that avoids the use of two types of pseudospins can be obtained by group- ing the gauge-field spins in the center. Then, the order of the ions in the trap does not correspond to their position in the lattice, but an adequate addressing of the vibrational modes can engineer the desired interactions. For example, the correct weight of the interactions in HG can be gen- erated simply by off resonantly driving the center-of-mass mode with a laser that is focused more strongly on the FIG. 10. Observables in the minimal implementation. (a) The gauge invariance is only violated substantially where V=J is not a center of the chain. The xx interactions between matter- z z z strong energy scale. The order parameters (b) S ðS12 þ S21Þ=2 field spins 1 and 2 (now at positions 1 and 4 of the ion z z z and (c) ð1 þ 2Þ=2 behave similarly to the ideal QLM chain) can be transmitted by a mode that has amplitudes (solid line). The transition from a C- and P-invariant M nq ð0:66; 0:25; 0:25; 0:66Þn. We checked numeri- state to a C- and P-breaking state can be clearly seen. The blue cally that the additional, unwanted xx interactions in- curves from lighter to darker and shorter to longer dashes denote duced by that mode do not compromise the ground-state J=V ¼ 0:125, 0.25, 0.275, but practically coincide.

041018-12 QUANTUM SIMULATION OF A LATTICE SCHWINGER ... PHYS. REV. X 3, 041018 (2013) considered parameter range, the errors that are introduced by generating the system dynamics perturbatively are negligible (see Sec. III C). Moreover, the interactions z z V12=2 do not alter the ground state. Such interactions z would favor an antiferromagnetic correlation between 1 z and 2, a behavior that violates the Gauss law (14). The influence of this interaction is therefore canceled by the stronger HG. As a result, the region at small m=V that is not contained in the ideal QLM, and that appeared in the large- err chain scheme because of the dipolar interactions HV ,is absent in this minimal model. Consequently, the fidelity susceptibility (not shown) has only a single peak at m ¼ 0, in accordance with the ideal QLM.

FIG. 11. Final state after a linear sweep for the minimal C. Numerical analysis of adiabatic sweeps implementation. We change mðtÞ¼minit þ tm=t, with mfin ¼ In order to provide a better connection to experimental minit, and keep V and J fixed. Solid lines from lighter to darker 2 @ realizations, we now study adiabatic sweeps through the shades: minit=J ¼2, 3, 4 for m=t ¼ 0:2V = . Short 2 @ C- and P-breaking transition. For ease of comparison, we (long) dashed lines: minit=J ¼3 and m=t ¼ 0:4ð0:1ÞV = . consider the same linear quench protocol as in Sec. IV B, The arrows show how the precision of the quench increases for namely, a sweep from m=J 0 to m=J 0, with the smaller quench velocities m=t. The total times considered ¼ @ ¼ ¼ 2 @ same sweep speeds and ranges. range from tfin 15 =V (for minit=J 3, m=t 0:4V = ) to 60@=V (minit=J ¼3, m=t ¼ 0:1). For V=@ ¼ 2 1 kHz As the quench fidelity shows [Fig. 11(a)], for the con- [8,17], these correspond to 3.2–9.5 ms. (a)–(d) For a value of sidered parameter range, the accuracy of the sweep does J=V * 0:1, the final C- and P-breaking state is reached (a) with practically not depend on minit=J but only on the chosen high fidelity, (b) good gauge invariance, and the desired behavior quench speed. As in the scheme of Sec. IV, we remark that for (c) the gauge field and (d) the matter field. there is a considerable drop of fidelity at low J=V. In this case, however, the drop has its origin not in dipolar error terms but in the fact that the intrinsic energy scale of the VI. EXPERIMENTAL CONSIDERATIONS system dynamics decreases with J=V. This results in a smaller gap, thus requiring slower evolutions to retain In this section, we discuss some experimental issues, adiabaticity. Therefore, slower sweeps particularly improve including an outline of a meaningful experimental run, as the quench fidelity at small J=V. well as possible error sources. This behavior is reflected in the order parameters Sz [Fig. 11(c)] and z [Fig. 11(d)]. For the considered quench A. Experimental sequence velocities, the sweep is nonadiabatic at low J=V, and the A meaningful experimental sequence should typically order parameters almost retain their initial values Sz ¼1 contain (i) initialization in a state that can be prepared and z ¼ 1 [cf. Fig. 9(a)]. Only when J=V * 0:1 do the with high fidelity, (ii) time evolution under a desired final values of the order parameters approach the values Hamiltonian, and (iii) measurement of a relevant observ- expected for the ideal QLM at large m, namely, Sz ¼ 0 and able. All of these points can be implemented with high z ¼1 [cf. Fig. 9(b)]. We observe, again, an oscillatory accuracy in state-of-the-art trapped-ion experiments. behavior as a function of J=V. Similar to Sec. IV B, these For the initialization, one can choose the ground states oscillations appear because we take a snapshot of the at m=J ¼1. Since these are product states [as sketched evolution at a fixed final time tfin. in Fig. 2(d)], they can be prepared with high fidelity by In contrast to quench fidelity and order parameters, the optically pumping each ion into the appropriate internal overall breaking of gauge invariance rises with J=V and state (this step includes preparing the ions as or S spins, does not depend much on the quench speed [Fig. 11(b)]. as appropriate). Then, to protect gauge invariance, one It does, however, acquire stronger oscillations for faster needs to induce the Gauss-law interactions HG by turning sweeps. Thus, while adiabaticity requires a sufficiently on the corresponding lasers. Afterwards, one can turn on large J=V, to retain approximate gauge invariance, large the system dynamics given by HK þ HB þ Hm,inan J=V should be avoided—again, there is an optimal pa- adiabatic or an abrupt manner, depending on whether one rameter window. For the considered parameters, it lies wants to study ground states or quantum dynamics after a around J=V 0:1–0:2. Staying within this window, the sudden quench. Notably, applying only HK þ HB, without minimal model retains approximate gauge invariance generating Hm, corresponds to a quench from m=J ¼1 while displaying the C- and P-breaking physics character- to m ¼ 0. This quench ends in the critical region, which is istic of the full many-body problem. a highly interesting situation [84].

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The most important information can be read out easily presence of small gauge-variant terms (see Refs. [37,86] by measuring the local polarization, i.e., the occupation for a general treatment and a discussion in the context of of the pseudospin levels. From this simple measurement, quantum simulators). one can extract the signature of the parity-breaking In our analysis for the longer chains (Sec. IV), we z z phase transition through hSi;iþ1i or hi i. From the corre- assumed equidistant ions, but in a realistic linear Paul 2 sponding two-point correlations, one can compute hGi i trap, this is typically not truly fulfilled, resulting in inho- and thus quantify the breaking of gauge invariance. mogeneous coupling strengths. To achieve equal distances, Complementary observables such as the x spin compo- one could employ unharmonic trap potentials, study only nents can be measured by rotating the spins prior to the central part of a large ion chain, or resort to individual detection. ion traps [69,70]. However, even if the interactions are not homogeneous, the preservation of gauge invariance en- B. Error sources forced by HG will remain valid, although with a position- dependent strength (the position dependence of V may An important restriction of the proposed scheme is the possibly even be compensated by the one of K, so that condition J=V 1. A small J signifies a slow dynamics, J ¼ KB=V remains homogeneous). so one may ask whether physical phenomena associated Moreover, realistic ion chains are governed by open with the QLM (13) are observable within realistic experi- boundary conditions, while in our numerical calculations mental time scales. For approximately realizing the correct of Sec. IV, we considered periodic boundary conditions gauge-invariant dynamics, we require values of J V=5 because these better reproduce the physics of larger sys- (see Sec. IV). State-of-the-art experiments reach interac- tems. Nevertheless, since all involved phases are gapped, @ tion strengths as large as V= 21kHz [8,17,85]. we can expect a strong influence of the boundary condi- The corresponding time scale for the system dynamics is tions only close to quantum phase transitions. The physics @ =J 0:8ms. This time scale is well below the dephasing of the false-vacuum decay should depend only weakly on time, which typically is 5–10 ms and can reach up to the choice of boundary conditions. 100 ms [17]. It is also far below realistic relaxation times, Finally, let us note that the interactions used in our which can reach up to 1 s [8,17]. As we have seen in Fig. 7, scheme have, in a different context, all been successfully an adiabatic sweep with good fidelity requires only a few implemented in experiments. Namely, quantum simula- ms, so the C- and P-breaking transition should be observ- tions using hyperfine qubits have been realized by employ- able in realistic setups. ing both the zz couplings depicted in Figs. 3(b) and 3(c) For small J=V, another type of error might arise in the [11,17] and xx-type spin-spin couplings generated by derivation of the effective spin-spin interactions. Namely, Mølmer-Sørensen gates [12–16,18]. For optical qubits, the step from the spin-phonon coupling (18) to the spin-spin high-fidelity gates of xx type [3,5] and zz type have been interaction (20) involves truncating the canonical transfor- q demonstrated [4]. An alternative implementation, based on mation (19) to leading order in jnjnq=ð2x Þ. One rf coupling of hyperfine qubits in spatially varying mag- could think that higher-order terms neglected when deriving netic fields, has also been realized recently [6]. Rotations the strong interactions V may compete with the first-order of the coordinate system allow one to choose from these terms of weaker interactions such as H1 [Eq. (25)]. coupling schemes the most practical one for a given setup. However, the Lamb-Dicke parameter is nq 1 in typical For example, it may be simpler to implement yy interac- trapped-ion quantum simulations, so can easily be on the tions than zz interactions and to redefine the coordinate order of 1=8 [15]. Consequently, higher-order terms in the system accordingly. canonical transformation are of order 2 ¼ 0:01 [66]. We found in Sec. IV that reasonable parameters for H1 are KB VII. CONCLUSION AND OUTLOOK J ¼ V V=5 or, equivalently, K=V, B=V 0:45 0:01. Generally speaking, the inherent errors discussed in In conclusion, we have presented a scheme to realize, err Sec. IV, namely, the dipolar terms HV and gauge-variant with existing technology, a lattice gauge theory in a chain contributions to H1 that are not sufficiently suppressed by of trapped ions. Studying ground-state and dynamical be- HV, will by far dominate over other error sources, such havior via exact diagonalizations, we identified the optimal as experimental inaccuracies. For example, for typical parameter regime for the observation of false-vacuum experimental parameters, errors due to finite phonon oc- decay. An interesting roadmap for experiments would be cupation, phonon heating, and micromotion can be esti- to demonstrate the basic building block with two or three mated to be at most a few percent [66,72]. Additional ions. Then, one could proceed to the proof-of-principle errors, appearing if noncommuting spin-spin interactions realization of the parity-breaking transition with the use are induced simultaneously, are strongly suppressed if the of four ions and finally aim at a full-fledged quantum used phonon frequencies are sufficiently different [66]. simulation on larger ion chains. Let us also remark that the low-energy, many-body prop- Our scheme leaves way for adjustment to the concrete erties of the system are expected to be robust in the experimental situation. For example, the distinction

041018-14 QUANTUM SIMULATION OF A LATTICE SCHWINGER ... PHYS. REV. X 3, 041018 (2013) between matter-field and gauge-field spins could alterna- Mu¨ller, E. Rico, Ch. Roos, and U.-J. Wiese for useful tively be achieved via staggered fields that shift the reso- and fruitful discussions. M. D. acknowledges support by nance frequencies for every other spin, or via individual the European Commission via the integrated project laser addressing, or by separating the ions spatially, e.g., by AQUTE. We further acknowledge support by SIQS, ERC employing a narrow zigzag chain [87–90]. Moreover, if a Synergy Grant UQUAM, and the Austrian Science Fund large number of laser frequencies is available, some of the FWF (SFB FOQUS F4015-N16). most prominent errors due to dipolar interactions could be removed by appropriately engineering the spin-phonon couplings [77]. An additional advantage of our scheme is that imperfections in the realization of the Gauss law can [1] H. Ha¨ffner, C. F. Roos, and R. 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Freericks, and C. Monroe, ACKNOWLEDGMENTS Quantum Simulation of the Transverse Ising Model with Trapped Ions, New J. Phys. 13, 105003 (2011). We thank D. Banerjee, R. Blatt, R. Gerritsma, C. [15] R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Hempel, P. Jurcevic, B. Lanyon, S. Montangero, M. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang,

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