Quantum simulation of the Abelian-Higgs lattice gauge theory with ultracold atoms

We present a quantum simulation scheme for the Abelian-Higgs lattice gauge theory using ultracold bosonic atoms in optical lattices. The model contains both gauge and Higgs scalar fields, and exhibits interesting phases related to confinement and the Higgs mechanism. The model can be simulated by an atomic Hamiltonian, by first mapping the local gauge symmetry to an internal symmetry of the atomic system, the conservation of hyperfine angular momentum in atomic collisions. By including auxiliary bosons in the simulation, we show how the Abelian-Higgs Hamiltonian emerges effectively. We analyze the accuracy of our method in terms of different experimental parameters, as well as the effect of the finite number of bosons on the quantum simulator. Finally, we propose possible experiments for studying the ground state of the system in different regimes of the theory, and measuring interesting high energy physics phenomena in real time.


Introduction
The study of natural systems governed by the laws of quantum mechanics is a complicated task. Most of the usual theoretical and numerical techniques employed to tackle the behavior of quantum systems become rapidly overwhelmed once their size starts increasing. In particular, the computational resources required to simulate a quantum mechanical system using a classical computer scale exponentially with the number of its constituents. This situation led Richard Feynman to think of the concept of a quantum simulator [1]. According to his idea, such a device would be governed itself by the laws of quantum mechanics and, exploiting this fact, could be used to study other quantum mechanical systems more efficiently. Feynman's idea was formalized as an algorithm that could be processed by a universal quantum computer [2]. This is referred to as a digital quantum simulation, since it is based on approximating the simulated system's dynamics by applying a discrete set of operations (quantum gates) on a highly-controllable quantum device. Quantum computers are very interesting from a theoretical point of view, since, in principle, they could simulate any other physical system using only polynomial resources, provided that certain locality conditions are fulfilled [2]. From the experimental side, however-and in spite of the great advances over the last decades regarding the manipulation of microscopic quantum systems-a practical implementation of a fully operational quantum computer is still a longterm goal.
Notwithstanding, having these controllable devices so well studied and reachable makes quantum simulation a reality, by means of simpler quantum devices that can perform calculations beyond the scope of classical machines. In particular, the so-called analog quantum simulators, although much more limited than an universal simulator, can be used to study a broad range of quantum systems within the reach of today's technological progress [3,4]. Analog simulators act by mapping the degrees of freedom of the simulated system to those of the simulating one. The latter can be controlled in the laboratory and its dynamics can be tailored (in particular, the corresponding Hamiltonian) to be equivalent to those of the system we are trying to study. This

The simulated model
The spontaneous breaking of a symmetry refers to a situation characterized by a ground state which is not invariant under the same set of transformations as the theory's Lagrangian or Hamiltonian. The Higgs mechanism [103][104][105][106] describes this phenomenon in the presence of a (local) gauge symmetry. It can be applied, for instance, to the( ) ( ) SU U 2 1 gauge group, associated to the electroweak interaction in the Standard Model of particle physics, explaining how the corresponding gauge bosons,  W and Z, acquire a non-zero mass [107]. The basic features of the mechanism can be studied, however, in a much simpler case, the Abelian-Higgs model [46,108], involving complex scalar fields coupled to abelian gauge fields (appendix A.1).
The Abelian-Higgs model can be formulated as a lattice gauge theory [109][110][111][112][113][114][115], this is, as a field theory on a discretized space-time, invariant under U(1) local transformations. The phase diagram of the theory shows very interesting phenomena, related to the Higgs mechanism and the confinement phenomenon (appendix A.2).
For quantum simulation purposes, it is more convenient to express the theory in a Hamiltonian formulation. This makes the mapping between the simulating and the simulated system-in this case, ultracold atoms in optical lattices-more straightforward, since the latter is described usually in terms of a secondquantized many-body Hamiltonian. Here, we shall introduce the Abelian-Higgs model directly in the Hamiltonian formulation. We obtained the former by applying the transfer matrix method to the standard lattice gauge action used in the high energy physics literature. The details of these calculations can be found in appendix B.
The Abelian-Higgs theory involves two types of fields: gauge and Higgs fields. We will formulate it on a d dimensional spatial square lattice, with continuous time, and will fix the lattice spacing a = 1.
The gauge fields reside on the links of the lattice. They are represented by the unitary operators º q -( )Û e k q q n, i k n, , residing on the links (joining the vertices n and +k n , withk a unit vector in one of the d orthogonal directions). q is an integer denoting the U(1) representation; we will omit it from now on and focus on the case q = 1. On each link we introduce another operator,Ê k n, , with the commutation relations d d = where the sums in n and k run over the vertices and links of a spatial lattice in d dimensions, and the coupling constants R and g are introduced to recover the continuous Abelian-Higgs model in the limit  a 0. The Hamiltonian (6) is composed of a non-interacting part with local terms (charge and electric field) for the vertices and links of the lattice, and an interacting part that includes a plaquette-type term (figure 1) that creates electric field excitations along the four links of a plaquette, and nearest-neighbor vertex couplings, mediated by an excitation on the link that joins them. The second row in equation (6) corresponds, in particular, to the puregauge Kogut-Susskind Hamiltonian [116].
In the Hamiltonian formulation of lattice gauge theories, the gauge is fixed in the temporal direction, q = 0 n,0 (appendix B). However, it is not fixed in the spatial directions. Therefore, the Hamiltonian is still gauge invariant under a restricted subset of local transformations applied to the vertices of the spatial lattice, It is easy to check that such operator commutes with the Hamiltonian for every vertex n. The eigenvalues of the operatorsĜ n are called static charges q n , taking, as well, integer values. The Hilbert space  of the system is divided into sectors, each one associated to a different static charge configuration, Since the Hamiltonian commutes with the generators of the gauge transformations, the dynamics cannot mix different sectors. If the initial state belongs to a certain sector, the system will remain in the same sector through its time evolution. The above equation is known as the Gauss's law, and it is crucial for the structure of the physical states of the system.
In general, the total Hilbert space of the system would be the product over the vertices and links of the lattice of each infinite dimensional local Hilbert space. However, Gauss law (9) imposes a constraint on the allowed states where the Hamiltonian (6) acts. Let us first consider the pure-gauge Kogut-Susskind Hamiltonian, for the case R = 0, within the zero-static charge sector ( = q 0 n " n). In the strong coupling limit (  g 1), the ground state of the system is the empty state. This is, every link of the lattice is an eigenstate of the electric field operator with eigenvalue equal to zero. As the value of the coupling constant g decreases, the empty state is no longer the ground state of the system. The plaquette-interaction term can create excitations composed of closed electric flux lines. The ground state, and the rest of excited states, will be superposition of such loops, with an increasing number of them when the energy of the system grows ( figure 2).
Consider now a gauge-invariant sector with non-zero static charges in some vertices (figure 3). In this case, the empty state is not the ground state of the system even in the strong coupling limit. The reason for this is that Gauss law (9) requires an electric flux line to exist between every pair of static charges, being the ground state in the strong coupling limit characterized by the shortest of those paths. If the value of the coupling constant g decreases, closed loops will be formed all around the lattice, deforming the flux line between the static charges if they share a common link with the latter (figure 3).
If matter is present in the theory, the flux lines will not only fluctuate but can even break. For ¹ R 0, the gauge-matter interaction term can create charge excitations in two nearest-neighbor vertices. We can find, then, states characterized by separated pairs of dynamical charges joined via fluctuating flux lines, which we will call mesons. The difference with respect to the pure-gauge case is that, now, the flux lines can break, resulting in two new-and shorter-meson-like states (figure 4). This phenomenon resembles the behavior of QCD, where, due to the confinement mechanism, the flux lines between a quark-antiquark pair can break through the creation of a new pair from the vacuum (string breaking) [47].
Here, we have described some basic features of this lattice model. The complete phase diagram of the theory is described in appendix A.2 for different dimensions and different representations of the symmetry group.  (6). Since the plaquette term is gauge-invariant, these states are gauge invariant as well. This is clear observing that the Gauss law (9) is fulfilled on each vertex, forcing the closed loop structure. In the figure, positive values of the electric field are represented with arrows pointing right or upwards.

Quantum simulation
We shall now describe a quantum simulation scheme for the lattice Abelian-Higgs theory using ultracold bosonic atoms trapped in optical lattices. The goal is to impose certain conditions on this highly controllable atomic system, such that the Hamiltonian that describes it maps, approximately, the Hamiltonian under study (6). Both the parameters appearing in the latter, as well as the degree of approximation, can be controlled experimentally. This allows to study, for instance, the ground state of this many-body quantum theory in the interacting regime, which is a hard task to perform using standard analytical or numerical calculations.
The simulation proposal involves three main steps. First, starting from the most general multi-species Hamiltonian that describes ultracold bosons, the experimental requirements that are necessary for the simulation will be presented. After that, the transformations required for mapping the degrees of freedom of the atomic system to those of the Abelian-Higgs theory will be introduced. Finally, we will see how the desired Hamiltonian is effectively obtained, apart from correction terms, after increasing one of the energy scales of the system compared to the rest. Most of these corrections can be made as small as required by changing the experimental parameters of the atomic system, keeping the freedom to move through the phase space of the simulated theory, and by increasing the number of atoms used in the simulation.

Atomic Hamiltonian
Ultracold atomic gases consist of neutral atoms trapped and cooled down to almost absolute zero temperature, where quantum effects play a significant role. They represent one of the most relevant platforms for the study of collective quantum phenomena in regimes that are either hard to access experimentally, or where numerical simulations do not provide sufficiently good results. In particular, the so-called optical lattices-structures made of counter-propagating lasers-can be prepared in a way that emulates a periodic lattice structure with atoms bound to the vertices [5,7,9]. Within these lattices, interactions among the atoms can be tuned in order to recreate Hamiltonian models describing many-body systems, both in condensed matter and in high energy physics. In this work, we will consider the most general multi-species bosonic Hamiltonian [9], where the latin indices denote the different sites of an optical lattice, and the greek indices represent different bosonic species (either different kinds of atoms or atoms with different internal levels). The operators â b i, and â † b i, are bosonic creation and annihilation operators, respectively, fulfilling the commutation relations The parameters a b t i j can be fine-tuned using experimental techniques such as (optical) Feshbach resonances [7,[117][118][119]. The first term in (12) corresponds to a hopping process of individual atoms along the lattice sites, while the second is a two-body collision process. For the latter, the conservation of the total angular momentum F implies, in particular, that the following condition must be fulfilled in each collision, represents the hyperfine angular momentum state for the atomic species α.

Experimental requirements
Our simulation proposal focuses on the 2 + 1 dimensional case, which is the first 'non-trivial' dimension, since it contains plaquette interactions. In particular, we need two superimposed optical lattices to perform a lattice gauge theory simulation, whose minima correspond to the vertices and links of the simulated lattice, respectively (figure 5(a)). Filling and combining two such lattices is possible in the two-dimensional case, however, it becomes more complicated in three-dimensions. All the theoretical calculations are valid, nevertheless, for To simulate the complete Abelian-Higgs Hamiltonian (6) we need six different bosonic species (figure 5(c)). Two of them, a and b, are placed on the links of the simulated lattice and represent the gauge degrees of freedom. Another pair of bosons simulates the scalar fields on the vertices (dynamical matter), one species on the odd vertices, c, and the other one on the even ones, d. The last two species, e and f, are distributed, again, on odd and even vertices, respectively, acting as auxiliary particles required to obtain the plaquette interactions in an effective way, as we will see. Each pair of bosonic species corresponds to the same type of atom, but their internal level differ in the value of the hyperfine angular momentum, m F . We will denote both types of dynamical bosons with η, and both auxiliary ones with χ, in those situations where a distinction between the hyperfine angular momentum level is not important.

Primitive Hamiltonian
For dimensions + d 1, with > d 1, the Hamiltonians that describe lattice gauge theories, such as the Abelian-Higgs one (6), include plaquette-type interactions made out of the product of four unitary operators (figure 1). Such four-body interactions are not found in the ultracold atomic Hamiltonian. In order to simulate them, we have to obtain these type of terms effectively [70,71], as we will explain in later sections. We , , Figure 5. (a) Two superimposed optical lattices, oriented with a  45 angle between their links, are required to simulate both the matter and gauge degrees of freedom. The vertices of one lattice (blue in the figure) represent the vertices of the simulated one, whereas the vertices of the rotated lattice (green) correspond with the links of the latter. The 'matter' lattice is filled with four types of bosons (c), two species per site, alternating on even and odd vertices. Two of these bosonic species represent dynamical matter (species c and d), and the other two are auxiliary bosons required for the simulation (species e and f ). The 'gauge' lattice, on the other hand, is filled with two types of bosons on each site (species a and b), with no distinction for even and odd links. The characteristics of these lattices must be configured such that there is no interactions between bosons from different sites of the 'gauge' lattice-using, for example, deep potentials-but allowing for interactions between them and the bosons from the nearest-neighbor vertices of the 'matter' lattice. This is the case if the corresponding Wannier functions have a non-vanishing intersection (b).
The first row contains the operators corresponding to auxiliary bosons living on the vertices of the lattice. ĉ † n and ĉ n are the bosonic creation and annihilation operators, respectively, and c c = ĉˆ † N n n n is the corresponding number operator. The second row contains all the terms of (6) except for the plaquette interaction. The primitive Hamiltonian is invariant as well under U(1) local transformations applied on the vertices of the lattice.
We will now focus on bringing the atomic Hamiltonian to the primitive (intermediate) form (15). Although only an approximate Hamiltonian can be obtained, the level of approximation can be controlled by using a different number of atoms in the simulation.

Operator transformations
The first step towards a quantum simulation is to construct a one-to-one mapping between the degrees of freedom of both the simulating and the simulated systems. Here, this is achieved by defining the operators that appear in the Abelian-Higgs Hamiltonian (6) in terms of bosonic creation and annihilation operators corresponding to the different atomic species. For a finite number of atoms trapped on the vertices and links of the optical lattice, the correspondence is not exact, in the sense that the defined operators will act only on a truncated Hilbert space. However, the approximation will improve when the number of atoms increases, as the connection between both systems becomes exact in the infinite number of atoms limit.
On the vertices of the lattice (both even and odds), we introduce the operatorsQ n , f n and f † n using the second-quantized operators ĥ n and ĥ † n for the dynamical bosons, where N v 0, is the density of η bosons. With these relations, both the canonical commutation relations for the bosonic operators and the commutation relations (4) are fulfilled. According to this definition, the operatorQ n is bounded from below by -N v 0, . For this reason, it is not exactly equivalent to the dynamical charge operator that appears in (6), however, we can think about it as the latter acting on a truncated Hilbert space-the one spanned by the basis ñ |m with -where it is equivalent to the dynamical charge operator. For this reason, we use the same notation for both, understanding, in the following, that we are referring to the operator acting on the truncated space.
Another problem concerning the finiteness of N l 0, is that, by imposing the commutation relations (4), ĥ n and ĥ † n cannot be unitary, which also means that they are not the exponential of a self-adjoint phase ĵ n , f = j -ê n i n . This is related to the quantum phase operator problem [120][121][122][123][124], and lies in the fact thatQ n is bounded from below. A unitary operator is well-defined only in the limit , such that the charge operator is no longer bounded.
We are working with a large, but finite, number of atoms N v 0, on each vertex. The reason it is still fine, from a physical point of view, to use these definitions-although they are not formally correct-is that in all the interaction terms of the Hamiltonian the bosonic operators appear in pairs, corresponding to nearest-neighbor vertices, h h + †k n n . Therefore, only phase differences between different vertices are relevant. The phase difference is a well-defined quantity, since it is canonically conjugate to a number operator, = -ˆN N N c d , which is not bounded from below. The operators associated to the gauge fields,Ê k n, andÛ k n, can be obtained from the bosonic operators on the links if we first represent them in a similar way as the atomic operators on the vertices, whereÛ k a n, andÛ k b n, are lowering operators, and Again, if the commutation relations (1) are fulfilled, the canonical relations for the bosonic operators will be satisfied.
The total number of bosons on each link, N l 0, is a conserved quantity, since, as we shall see, only products of bosonic operators of the formˆ † a b k k n n , , andˆ † b a k k n n , , will appear in the Hamiltonian. This implies that , . We use the latter as the electric field operator on the link ( ) k n, , which can also be expressed as We construct the unitary operators acting on the links as satisfying the correct commutation relations (1) with the electric field (21). Note that, in this case, the operator E k n, is not bounded from below.Û k n, andˆ † U k n, are, therefore, well-defined unitary operators, although they only act on a truncated Hilbert space that grows with N l 0, . Finally, the product of two bosonic operators on the links can be written as where we used the property In the following, we will assume this property-similarly for the operators on the vertices, In order to simplify the notation, we define the non-unitary operators , which become the desired unitary ones, f † , respectively. The connection with the atomic operators is summarized in the following relations,

Gauge-invariant interactions
Local gauge invariance is not a fundamental symmetry in a system of ultracold atoms. It is essential, however, for the description of the interactions present in high energy physics, as we have seen. If one aims to correctly describe the latter, gauge symmetry must be imposed on the simulating system. This can be done either in a digital [91] or in an analog simulation framework. For the latter, there are two distinct ways to achieve this [71], one where gauge invariance is obtained effectively, as an emerging, low energy symmetry [73,74,76], and another where it is mapped, exactly, to a fundamental symmetry of the system-such as the conservation of hyperfine angular momentum in atomic collisions. In this case, as opposed to an effective gauge symmetry, gauge invariance is exact, making the simulation more robust against experimental imperfections. Using the second approach, quantum simulations of both abelian and non-abelian lattice gauge theories have been proposed [70,93]. Here, we will employ it to simulate the U(1) gauge invariance of the Abelian-Higgs model. In order to obtain the correct 'gauge-invariant' interactions, the hyperfine angular momenta of the six bosonic species must fulfill certain conditions (figure 6), Since only collision processes that conserve the total angular momentum are allowed, this condition selects the type of terms that appear in the atomic Hamiltonian. Consider, for instance, the interactions between a dynamical boson on a vertex and a gauge boson on one of its nearest-neighbor links. The conservation of hyperfine angular momentum allows both species-changing collisions (figure 7(a)), and non-species-changing ones, The sum of these terms becomes a constant, proportional to the number of bosons, when summed over all vertices, if the scattering lengths are properly tuned. The same kind of processes appears for the interaction between gauge and auxiliary bosons ( figure 7(b)). The species-changing collisions can be seen to be equivalent to the matter-gauge interactions in (6), after applying the canonical transformations k k x n n k k n n n n 1 2 Both c and d operators can be denoted now by η, and both e and f by χ. Using the transformations defined in the last section (25), the species-changing collisions for the dynamical bosons are just where a new coupling constant has been defined, For the auxiliary bosons, the species-changing collisions transform to Note that these kind of interactions are possible provided the Wannier functions of the corresponding bosons on the lattice overlap ( figure 5(b)).  The conservation of hyperfine angular momentum allows the two types of bosons to interact as well. This would result in non-desired terms in the simulated Hamiltonian. For this reason, the system must be prepared such that this kind of interaction does not occur, using, for example, Feshbach resonances to reduce the corresponding scattering length.
Consider now the non-interacting terms in (15). Both the charge and the electric field kinetic terms arise from on-site collisions. For the vertices we have Finally, the hard-core constraint for the auxiliary bosons can be obtained from on-site collision among them, as well as atomic mass terms, Collecting all the building blocks we see how the atomic Hamiltonian corresponds to , , which, in the limit  ¥ N N , results in (15). In an experiment, only a finite number of atoms is placed on each link and vertex. However, this Hamiltonian provides a good approximation, even for a small number of atoms [83]. We will discuss the effect that the number of atoms have on the simulation after obtaining the effective Hamiltonian.

Hard-core bosons
We shall next describe how the plaquette interactions can be obtained in an effective way from the primitive Hamiltonian (37). The main idea is to impose a strong energy penalty in the atomic Hamiltonian, in such a way that the Abelian-Higgs Hamiltonian (6) is obtained as a low energy effective approximation, including the plaquette interactions [70]. The advantage of this method is that the building blocks from which the plaquettes are constructed are already gauge-invariant. Therefore, even if the effective description contains undesired terms, these will not violate the gauge symmetry.
The method requires the use of auxiliary particles, which can be either fermions or bosons [71]. In this case, we make use of hardcore auxiliary bosons sitting on the vertices of the lattice, which we have already introduced in the primitive Hamiltonian (37). The energy penalty comes from the on-site interaction between auxiliary bosons, which is the desired plaquette interaction. Terms of the type  ln n 1 only contribute for even values of n. Therefore, the latter is, in practice, a second order correction. Apart from the plaquette term, many other terms appear in the effective expansion. Most of them arise as an effect of the non-unitary behavior of the operators (24) when the number of atoms is finite, vanishing in the large number of atoms limit.
The use of hard-core bosons has several advantages, compared to obtaining the effective plaquettes using auxiliary fermions [70]. First, the penalty term should be easier to implement experimentally, since it is homegeneous across the vertices of the lattice. Also, as we will see when we apply it to the complete lattice Abelian-Higgs theory, no undesired divergence terms or inhomogeneous renormalizations appear in the effective Hamiltonian, as opposed to in the fermionic case. Finally, the quantum simulation will only require bosonic atoms, rather than bosons and fermions, which simplifies the experiment as well.

Effective expansion
Let us derive the complete effective Hamiltonian that acts on the sector of the energy constraint characterized by just one auxiliary boson per vertex. We assume that λ is large compared to the rest of the coupling parameters, This can be achieved experimentally using Feshbach resonances. In order to calculate the effective Hamiltonian, we use time-independent perturbation theory, following [125,126]. The effective Hamiltonian is perturbatively expanded in terms of l 1 . More details about the calculations can be found in appendix D. Here, we present the results up to fourth order. Notice that all the terms obtained in the effective expansion are gauge invariant, since the symmetry was exact in the primitive Hamiltonian (37).
The first order contributions to the effective Hamiltonian include all the terms in the primitive one (37) that commute with the penalty term (38), In the third and fourth order, we get, apart from different renormalizations of the first order terms (41), new contributions that were not present in the primitive Hamiltonian (37). The most important one is the already mentioned plaquette interaction (39). Due to the non-unitary character of the operators (24), the effective expansion provides, as well, a plethora of interaction terms that do not appear in the simulated Hamiltonian (6). Most of these corrections can be made negligible (see appendix D). The only extra terms that could affect the simulation in some regimes are the following fourth order contributions, where the last term involves an interaction between nearest-neighbor links. The importance of the correction terms is analized now for different situations.
3.6. Pure-gauge theory Before discussing the complete effective Hamiltonian, let us consider the pure-gauge theory. Most of the discussion generalizes when dynamical matter is present. Here, the situation corresponds to a quantum This Hamiltonian, although simplified, still contains some non-desired correction terms. We study now how we can control them experimentally.  . We will see that the last correction is the most relevant one. Note that, since no terms of order  ln n 1 , for n odd, appear in the effective expansion, it is enough to impose  l  ( ) 1 2 to make higher order correction negligible. This also applies to the rest of the situations we will consider. It is important to notice that, at the fourth order, the effective Hamiltonian in the large atom limit does not include any non-desired contribution, agreeing perfectly with the Kogut-Susskind Hamiltonian.

Large number of atoms
We write now the experimentally controllable parameters, ò and μ, in terms of the coupling constant g of the Kogut-Susskind Hamiltonian (44). In order to do so, since we are only interested in the ratio between the electric and the magnetic part of the Hamiltonian, we multiply everything by a constant α, whose effect is just to rescale the energy. This constant, and the coupling g, are defined as where the most important higher order corrections are written explicitly, allowing us to discuss their importance in the different regimes of the theory. By using the Feshbach resonance technique to fine-tune the ratios  l and  m , one can explore the phase diagram of the Kogut-Susskind Hamiltonian. Consider, in particular, two extreme relevant limits.
• Strong coupling limit (  g 1) To achieve this, we require In this limit, only the electric part of the Hamiltonian is relevant. All the corrections are negligible as well.
• Weak coupling limit (  g 1) In this case, we need Here, the electric part vanishes and the plaquette interaction becomes relevant. The first two corrections in (48) are small compared to the latter, since  m is very large. Therefore, the biggest correction in this limit is of order  l ( ) 2 compared to the plaquette interaction. From this, we conclude that both  l and m l must be small in order to make the higher order corrections in (48) negligible. However, we still have freedom to change the ratio  m to achieve different values of g.

Finite number of atoms
Consider now a more realistic case, where the number of atoms on each link is finite, and non-desired contributions appear in the effective Hamiltonian. At the same time, a different approach to generate the electric field operators is introduced. We set m = 0, and use the renormalization terms that were obtained in the effective expansion (at the second order) as the electric field part of the Hamiltonian. Proceeding this way, many correction terms disappear, and we are left with the following effective Hamiltonian, 3 0, 0, 2 "n.n." , , , Defining, as before, the coupling constant in terms of the system parameters in a proper way, h.c.
Since N l 0, is a fixed quantity, once the experiment starts, the coupling g is controlled with the ratio  l ( ) 2 . Consider, again, the weak and strong coupling limits.
• Strong coupling limit (  g 1) To access this regime, the experimental parameters must be chosen such that Higher order corrections will be negligible provided that  l  1. In this limit, the electric part of the Hamiltonian dominates. Therefore, even for a small number of atoms in each link, the atomic Hamiltonian accurately simulates the Kogut-Susskind one.
• Weak coupling limit (  g 1) This regime is more problematic, and the accuracy of the simulation depends strongly on the number of atoms used in the experiment. To access it, we have to impose where the second condition is required to make higher order corrections negligible. In this regime, the number of atoms (and correspondingly, the eigenvalues of the electric field operator) are not good quantum numbers, since the corresponding operators undergo large fluctuations. But as long as condition (56)  . Higher values of | | m are relevant when the plaquette interaction dominates over the electric part, however, their influence rapidly decreases with N l 0, . In particular, for a large number of atoms, the non-desired corrections vanish faster than  k n, converges tô U k n, ; the latter problem again being the most important one.
A final comment about the energy scale of the simulated Hamiltonian is in order. To be able to obtain a plaquette interaction term that dominates over the electric field one, the latter should be made smaller. Therefore, the energy scale of the simulated Hamiltonian in the weak limit is very small. As a consequence, one would have to wait a longer time to be able to observe relevant physical phenomena. In this situation, the coherence time of the experimental system plays an important role.
Apart from the two extreme limits, it is especially interesting to consider the intermediate case, withg 1. This corresponds to the condition This situation is not as hard to achieve in an experiment as the weak limit, but it is very relevant since most of the usual analytical approaches to study lattice gauge theories fail in this intermediate region [116].
Setting m = 0 has several advantages with respect to the more conventional approach [70]. As we have seen, it eliminates some of the non-desired correction terms. Also, the condition to enter the weak regime is easier to realize here, compared to (50). In this case, we can access the regime by making  l smaller than N l 0, . The ratio should still be larger than 1, but just enough to make higher order corrections small.

Complete Abelian-Higgs theory
Let us consider now the complete Abelian-Higgs theory. Following the discussion of the pure-gauge case, we choose, again, m = 0. This automatically removes many of the correction terms obtained in the last section.
Again, we multiply the effective Hamiltonian by a properly chosen constant α, and define g as before (52). In order to get the correct coupling constant in front of the charge term, and the interaction term between the dynamical charges and the electric field, we define R as Once g is fixed through  l , R can be fine-tuned by changing the ratio   ¢ | | . Note that  ¢ must be negative, since the charge term and the gauge-matter interaction have opposite signs in the simulated Hamiltonian (6).
The  For small values of R and large values of g, the atomic system maps exactly to the Abelian-Higgs theory. When R grows and g decreases, the effect of the non-unitary character of the operators  k n, and F n becomes important, as well as the relevance of the non-desired electric field terms, as it was discussed for the pure-gauge case. Making N l 0, and N v 0, larger will allow us to perform better simulations, going deeper into the large R and small g regime. The discussion for the strong regime in R is analogous to the weak regime in g, and vice versa.

Possible experiments
In the last section, we derived the experimental conditions under which the Hamiltonian describing a system of ultracold bosons is well approximated by the Hamiltonian associated to the Abelian-Higgs lattice gauge theory (6). Once this is achieved, several experiments can be performed on such a system, providing useful information about different high energy physics phenomena.

Initialization: static charge sectors
As a first step, the atomic system is initialized in the ground state of the non-interacting part of the Hamiltonian (6). Therefore, we choose the experimental parameters such that  g 1 (52) and R = 0 (58) and consider the vertices of the lattice. Any dynamical charge configuration maps directly to the number of dynamical bosons of species c and d (17). The ground state corresponds, in particular, to an initial atomic system with N v 0, dynamical bosons on each vertex (of the corresponding species, with the same number for each case), such that there are no dynamical charges present at the beginning of the simulation ( figure 9(a)). In addition, the vertices of the lattice are also filled with one auxiliary boson per site, alternating the two species (e and f ) on odd and even vertices, as we described in section 3.5.
For the links, there is also a mapping between the eigenstates of the electric part of the Hamiltonian and the difference in the number of gauge bosons of species a and b (21), noting that their roles interchange (29). The difference here is that the ground state of the Hamiltonian does not correspond, in general, to the product of the ground states of each link (zero electric field configuration). As a consequence of gauge invariance, the Hilbert space of the system is divided into different sectors (10)-characterized by different static charge configurationsand the states that belong to them are constrained by the Gauss law (9). Consider, for example, the sector characterized by two static charges of opposite sign ( figure 9(a)). The states that belong to such a sector present an electric flux line-or a superposition of them-that joins the two vertices where the static charges are placed. Consequently, some of the links will be in excited states of the electric part of the Hamiltonian, whereas others remain in the ground state-since, altogether, everything must satisfy all the local Gauss laws (9). The ground state corresponds to the shortest path of electric field excitations between the charges, in this case, a straight line. This is the situation, in general, for any gauge-invariant sector. Therefore, any configuration of static charges can be achieved by initializing the atomic system with the proper number of atoms of type a and b. Since this number will not be the same on every link for a non-zero static charge configuration (as opposed to the vertices), single-atom addressing techniques will be required [127].

Turning on the interactions
Once the system is initialized in the ground state of the non-interacting part of the Hamiltonian, the interactions can be introduced by modifying the value of the coupling constants R and g, tuning the corresponding experimental parameters (58). This can be done either adiabatically or suddenly (quench). If the change is adiabatic, the system will adapt to a new state that corresponds to the ground state of the new Hamiltonian. Measuring the new state allows us to explore the phase diagram of the system. In the case of a quench, the system will not have enough time to adapt to the change in the Hamiltonian. Instead, it will remain in the initial state, which will correspond now to an excited state of the new Hamiltonian. Here, measuring the state of the system will provide us with information about the non-equilibrium and thermalization properties of the theory.

Real-time dynamics
We propose two different dynamical effects that can be measured in this setup.

Fluctuation of electric flux lines
An initial state with well-defined electric flux lines is an eigenstate of the non-interacting part of the Hamiltonian. Reducing the value of g increases the probability of a plaquette-type electric flux loop to appear ( figure 3). If we change this parameter adiabatically, more of these flux loops will appear, being the state of the system a superposition of many different electric field configurations. The initial straight flux line can be deformed if one plaquette excitation coincides with one of its links ( figure 3). This effect can be detected dynamically by measuring the state of the atoms on the links of the lattice.

Breaking of electric flux lines
Another effect that could be measured dynamically is the breaking of electric flux lines (figure 9). By increasing the value of R, the probability of new pairs of dynamical charges of opposite sign to appear grows. If such excitations coincide with an existing electric field line between other pair of charges, it will cause the breaking of the latter, resulting in two new pairs joined by shorter lines ( figure 9).
This phenomenon, relevant in the understanding of confinement of matter in gauge theories, is hard to study using analytical or numerical techniques, since real-time dynamics are difficult to simulate using standard Monte Carlo methods. Using this quantum simulation scheme, the dynamical breaking of electric fluxes could be accurately measured.

Measuring the state of the system
We showed that there is a mapping between the state of the simulated system in the basis of the eigenstate of the electric field and charge operator, and the number of atoms of each species on the vertices and links of the lattice. Therefore, in order to know the state of the system at any point of the simulation, it is enough to measure the number of atoms locally, using single-atom detection techniques [128]. By doing that we could obtain a detailed picture of the flux lines and dynamical charge configuration that characterizes the state of the system.

Summary
In this paper, we proposed a quantum simulation scheme for the lattice version of the Abelian-Higgs theory, using ultracold bosonic atoms trapped in an optical lattice. The interest in performing such simulation is twofold. On the one hand, and despite its simplicity, this model shows very interesting high energy physics phenomena. It allows the possibility to study both the Brout-Englert-Higgs mechanism and the confinement of dynamical Higgs matter. On the other hand, it can serve as a benchmark to study the validity of several quantum simulation techniques, since the phase diagram of this theory is well known from conventional theoretical and numerical calculations, and can be compared to the one obtained in a quantum simulation.
Starting from the lattice action of the Abelian-Higgs theory [109], the corresponding quantum Hamiltonian was obtained using the transfer matrix method (6). Such a Hamiltonian is a generalization of the pure-gauge Hamiltonian obtained by Kogut and Susskind [116], including, as well, the degrees of freedom corresponding to the scalar matter fields.
The quantum simulation of this model using ultracold atoms is achieved by utilizing two superimposed optical lattices, filled with six different bosonic species. These correspond to three different types of atoms, each one possessing two accessible hyperfine angular momentum levels. One type of atom simulates the gauge degrees of freedom, and another one the dynamical matter. The last type of atom serves as auxiliary particles, required for obtaining the plaquette interactions. An intermediate 'primitive Hamiltonian' is first obtained by selecting the correct hyperfine angular momentum levels for the six species, and tuning some atomic scattering lengths using Feshbach resonances. After transforming the atomic degrees of freedom and expressing them in terms of the lattice Abelian-Higgs operators, the atomic Hamiltonian results in a gauge-invariant Hamiltonian. The corresponding operators are approximated with the bosonic ones, improving the approximation when the number of bosons increases. The primitive Hamiltonian contains all the terms of the Abelian-Higgs Hamiltonian except for the plaquette interactions. From the former, and with the help of the 'hard-core' auxiliary bosons, the Abelian-Higgs Hamiltonian can be approximated effectively, provided that the energy scales of the system are tuned correctly. The degree of approximation can be controlled by increasing the number of atoms used in the simulation, making the non-desired correction terms in the effective expansion negligible.
Finally, we described an experimental scheme for measuring, in real time, interesting phenomena such as the fluctuation and breaking of electric flux lines, and for adiabatically preparing the system in the ground state, which could help to explore the phase diagram of this high energy physics theory.
Next steps toward a complete understanding of the Abelian-Higgs theory might include higher dimensional cases, as well as non-fundamental representations of the gauge group ( > q 1), allowing a richer phase diagram to be accessed in a quantum simulation experiment. In a more general context, scalar Higgs matter could also be studied for non-abelian theories.
, however, the minimum is found for f á ñ = 0 0 . To take into account the effect of quantum fluctuations, one can expand the complex field f ( ) x around the semiclassical result, using two real scalar fields, s ( ) x and p ( ) x , such that A.6 x R i Plugging this expression into (A.1), the Lagrangian can be expressed in terms of the new fields. In particular, the terms containing the gauge field A μ alone are now the following, Thus, the non-zero expectation value of the scalar field f generates a mass term for the gauge field A μ or, in other words, the photon becomes massive, which shows how the Higgs mechanism takes place in this simple model.
In the limit l  ¥ m, , with l m 2 fixed, the radial field σ decouples, and the effective Lagrangian for the remaining fields is just

A.2. Lattice gauge action
The Abelian-Higgs model can be formulated as a lattice gauge theory [109][110][111][112][113][114][115], this is, as a field theory on a discretized space-time.  i n, reside on the links that joins the vertices n and m +n a , with m a unit vector in one of the + d 1 orthogonal directions. The coupling constants R and g are introduced to obtain the correct expression in the continuum limit. Finally, q is an integer number denoting the representation of the U(1) group under which the Higgs field is transforming, where the fundamental one corresponds to q = 1.
The first term in (A.9) is an interaction between nearest-neighbor vertices, mediated by the link that connects them. The second one is a plaquette interaction, which involves the fields on the four links of a plaquette.
The phase diagram of this lattice gauge theory depends on the specific group representation under which the Higgs field transforms. In particular, one finds a phase diagram with two phases if the representation is the fundamental one (q = 1), and another one with three phases if any other representation is used [109,114]. We shall describe now the properties of the possible phases one can find in the theory for  + d 1 4, based on the results from [109]. If the Higgs field transforms according to a representation of U(1) with > q 1, the theory presents three different phases ( figure A1(a)): • Higgs phase (large R, small g). In this phase the photon is massive (Higgs mechanism), the force between different particles is short-ranged, and there is no confinement of charges.
• Coulomb phase (small R, small g). Here, the photon is massless, and the charges are free, as well.
• Confinement phase (large g). In this regime there are no free charges (confinement) and the photon has a finite mass.
In the fundamental representation (q = 1), it can be shown that there is no phase boundary between the Higgs and confining regimes [109]. The theory presents only two different phases ( figure A1(b)), a Coulomb phase as in the previous case, and a Higgs-confinement phase, characterized by massive photons, short-ranged forces, and the absence of free charges.
In the 2 + 1 dimensional case there is no Coulomb phase for finite coupling constants. In the fundamental representation, there is only one phase, the Higgs-confinement one, whereas for > q 1, there is a phase boundary separating the Higgs and confinement regimes [109].

Appendix B. Hamiltonian formulation
We compute the quantum Hamiltonian corresponding to the Abelian-Higgs action (A.9) by applying the transfer matrix method. The application of the method to the pure-gauge theory leads to the standard Kogut-Susskind Hamiltonian [52]. Here, we extend it to the matter part of the action for the fundamental representation of U(1) (q = 1), First, we separate the temporal and the spatial directions. We introduce the notation m = ( ) k 0, , with k = 1,K, d, and = ( ) n n n , 0 , where the first component corresponds to the temporal direction and the second to the d spatial ones. Both n 0 and n take integer values from 0 to N. We also consider the temporal gauge, The action is expressed, then, as  where a τ and a are the lattice spacings in the temporal and spatial directions, respectively. To obtain the correct continuum limit, the fields j n should depend on a τ as j j -µ t + a n n n n , 1 , 0 0 . Therefore, we can expand the cosine in terms of this parameter. We get, disregarding irrelevant terms (in the limit    T , e S , , and can be thought as the matrix elements of a transfer operatorT over a complete and orthonormal product basis j j The partition function is expressed in terms of the transfer operator as Motivated by the latter, a quantum Hamiltonian is defined through the transfer operator, Na fixed. In order to get an expression for the transfer operator, we introduced the second-quantized operators ĵ n and Q n , following the commutation relations: B . 1 4 n n It can be shown that the following property holds, where q k n, is a quantum operator corresponding to the gauge field q k n, , obtained after applying the similar procedure to the gauge degrees of freedom on the links [52]. Taking the continuum limit in the time direction, a τ , one can read the Hamiltonian (up to irrelevant constants) from the transfer operator, using equation (B.12), In order to avoid the quantum phase operator problem [120][121][122][123][124], the operatorQ n must be unbounded from below. Only in that case ĵ e i n is a well-defined unitary operator. The situation is equivalent for the conjugate momentum of q k n, ,Ê k n, , defined in [52]   In that case,Ĉ is the identity, and the property (C.2) reduces to Since this operator is acting on the ground state sector, P 0 can be removed. The first one involves, again, the virtual hopping of auxiliary bosons to the nearest-neighbor vertices. The difference is that, in this case, the the Hamiltonian ( ) H eff 1 (which does not change the sector) is applied before the auxiliary boson comes back to its original vertex. This gives rise to non-local contributions, however, they will cancel with the second part of (D.11), which is itself non-local. The latter is composed of the first and second order corrections. Performing similar computations as in the second order case, we obtain the following local terms,   Most of these terms are composed of first, second and third order processes. As in the third order case, many non-local contributions arise, but they all cancel out in the end. Let us consider first the last term in (D.13). It involves the four virtual hopping steps of an auxiliary boson, returning to its original position. There are two main processes attached to this. The first one, already explained, related to the hopping of an auxiliary boson around the fourth links of a plaquette (8) The second non-trivial contribution comes from the virtual process that involves hopping twice to a nextnearest-neighbor vertex, and then hopping back to the original position. This can be done in many different ways ( figure D2). The total contribution, after canceling out the non-local terms, involve interactions between the electric fields of nearest-neighbor links,

H P H K H K H P H P H K H K H P H P H K H P P H K H P P H K H P H P H K H K H K H P P H K H K H K H P
The rest of the terms in (D.13) provide many other contributions. Apart from renormalizations to the electric part of the Hamiltonian, we obtain terms that also depend on different powers of the electric field operator, and it is written in terms of the commutator between them.

D.2. Relevant corrections
We study the importance of the correction terms (for the case m = 0), obtained in the effective expansion of the Hamiltonian, by expressing them, first, in terms of the simulated parameters, R (58) and g (52), and multiplying everything by α (53). Let us consider the terms proportional to  l ( ) 4 . They are proportional to the following