Simulating systems of itinerant spin-carrying particles using arrays of superconducting qubits and resonators

We propose possible approaches for the quantum simulation of itinerant spin-carrying particles in a superconducting qubit-resonator array. The standard Jaynes-Cummings-Hubbard setup considered in several recent studies can readily be used as a quantum simulator for a number of relevant phenomena, including the interaction with external magnetic fields and spin-orbit coupling. A more complex setup where multiple qubits and multiple resonator modes are utilized in the simulation gives a higher level of complexity, including the simulation of particles with high spin values and allowing more direct control on processes related to spin-orbit coupling. This proposal could be implemented in state-of-the-art superconducting circuits in the near future.

The most commonly studied setup for the JCH model comprises an array of cavities between which photons can hop and each of which contains a two-level atom. The Hamiltonian describing the setup is given bŷ where ω a is the atomic resonance frequency, ω 0 is the cavity resonance frequency, g is the atom-cavity coupling strength, and J is the inter-cavity coupling strength (or in other words the inter-cavity hopping strength). The operatorsσ (i) α (with α = x, y or z) are the Pauli matrices operating on the state of the two-level atom, whileâ i andâ † i are, respectively, the annihilation and creation operators of the cavity field; the indices i and j denote the location of the atom or cavity in the lattice, and the notation i, j indicates nearest-neighbour hopping. In writing Eq. (1), we have assumed that the different parameters are uniform across the entire lattice, and we have assumed that there is no direct coupling between the atoms [26].
One interesting property of the JCH model is the fact that in typical setups the number of excitations is not conserved; in most proposals excitations are to be created by coherently driving the cavities, thus creating quantum superpositions of states with different total particle numbers. As a result, one typically needs to consider the balance between the injection of excitations into the system by the externally applied fields (which can be pulsed or continuous) and the loss of excitations through various radiative or dissipative processes [11]. As such, one deals not with an equilibrium system, but rather a driven dissipative system. This situation differs drastically from commonly studied condensed-matterphysics problems (where particle number is assumed to be fixed throughout an experiment, and losses are typically treated as a deficiency that limits the power of the setup), and it could be seen as a disadvantage of the proposed JCH setups when viewed as alternative quantum simulators to replace, e.g., atomic gases in optical lattices [7,27]. On the other hand, the difference between the JCH model and other, well-established paradigms with conserved particle number offers an additional element of novelty, where new theories need to be developed in order to understand and possibly harness potential applications of the new paradigm. Furthermore, there is no fundamental difficulty in creating states with well-defined particle numbers in the JCH model: Rabi-oscillation dynamics in the atoms in principle allows the controllable creation of any desired total particle number, and the suppression of radiative losses could allow for motional thermalization of the fluid on a timescale that is short compared to the lifetime of the particles.
There have been proposals for implementing the JCH model in a variety of physical systems, including opticalfrequency cavities with implanted or electromagnetically trapped atoms [3], superconducting resonators and qubits fabricated on a chip [12,[14][15][16][17]19], nanomechanical resonators [28] and phonon cavities with impurities in silicon [29]. The superconducting platform is particularly promising from an experimental point of view. Superconducting qubits, which serve as artificial atoms in the model, and superconducting resonators are making rapid advances in terms of their quantum coherence and the level of control and addressability that they allow [30]. Coherent coupling between qubits and resonators has been achieved in various settings and has led to demonstrations of various phenomena predicted from the theory of quantum optics [31]. Furthermore, large arrays of superconducting resonators can readily be fabricated on an electronic chip with present-day technology [5,24,25]. An implementation of the JCH model in such circuits can be expected in the coming few years. We shall therefore focus on this implementation of the model.
Studies on the JCH model have mainly focused on the case where only one resonator mode and one qubit per site play an active role in the representation of the particles in the lattice. The result is that one ends up dealing with a system of effectively spinless particles. Even in this case, several interesting phenomena can be obtained. Examples include the fractional quantum Hall effect [10], unusual propagation behaviour [13], novel disorder effects [22], a rich phase diagram in the case of ultrastrong atomcavity coupling [18], topological models [20] and Dirac points [21].
The addition of a spin degree of freedom to the particles can be expected to result in a variety of new phenomena that are absent in the spinless case. In this context it is worth mentioning the large number of phenomena encountered in the study of Bose-Einstein condensates of spin-carrying particles [32]. In particular, novel quantum states have been predicted in relation to the motion of spin-carrying atoms in optical lattices [33] and spin-orbit coupling in atomic gases [34][35][36]. In this paper we consider possible routes towards the addition of a spin degree of freedom to the effective particles in the JCH model.
There have been proposals to simulate spins in a JCHlike setup in the literature [9]. However, the simulated systems described stationary spins, as opposed to itinerant spin-carrying particles. The possible use of polariton spin, photon polarization or different modes in optical cavities in order to introduce additional degrees of freedom has also been discussed in the literature [4,6,37], and reference [38] proposed the simulation of a twocomponent JCH model. However, in these proposals the numbers of particles in the different components were conserved, and the addition of a spin degree of freedom was therefore not truly realized. A related setup for the simulation of Luttinger liquids and spin-charge separation corresponding to hard-core repulsion in a continuous medium was proposed in Ref. [39]. It should also be noted that these proposals generally rely on atoms with specific energy-level structures, making them less suited for implementation in a superconducting architecture.
Here we analyze the possibility of constructing a superconducting quantum simulator for itinerant particles that are free to hop between neighbouring sites in a regular lattice and that possess an internal (spin or pseudospin) degree of freedom. As will be discussed in detail in Sec. III, different approaches allow for the simulation of different possible combinations of spin-related phenom-ena.

II. THE DESIRED HAMILTONIAN
In studies on the JCH model, it is common to assume a large detuning between the qubits and the resonators, such that each qubit-resonator unit is in the so-called dispersive regime [40]. In this regime, there are two well-separated types of excitations with mostly atomic and mostly photonic natures. It is also common to assume that the hopping strength is small compared to the qubit-resonator detuning (or the qubit-resonator coupling strength in the case where the qubit and resonator are on resonance with each other), such that the itinerant excitations naturally split into two groups, the so-called lower-and upper-branch polaritons. These two branches (or excitation modes) correspond to the two types of excitations mentioned above. If one considers only one of the two branches, the Hamiltonian in Eq. (1) reduces to an effective Hamiltonian that corresponds to the standard Bose-Hubbard model: where t is the hopping strength, U is the on-site interaction strength, andb andb † are, respectively, the annihilation and creation operators of the bosonic particles in the lattice. Thus constructing the basic Hubbard-model Hamiltonian should be rather straightforward in the superconducting architecture. Our aim is to design a JCH setup where the resulting effective particles possess a spin degree of freedom. We would therefore like to engineer an effective Bose-Hubbard Hamiltonian with an additional, internal degree of freedom. A Hamiltonian that contains only hopping and on-site interaction terms would read where t α,β is a matrix of strengths that describe the process in which a particle with spin value β hops to a neighbouring site and spin value α, and U α,β,γ,δ is a tensor that describes the strengths of various local interaction processes that transform a pair of particles with spin values γ and δ to spin values α and β. As before, we have assumed that the different parameters are uniform across the entire lattice. The inclusion of nonuniform parameters, such as trapping potentials, would be quite straightforward in a superconducting architecture. A Hamiltonian describing a system of spin-carrying particles would in general contain a Zeeman term, which in the Hubbard model takes the form where g is the Landé g factor, S α,β is the (vector) spin operator for the spin value being simulated, and B is the magnetic field. It should also be noted that the hopping term in Eq. (3) also contains spin-changing hopping terms that are relevant to spin-orbit coupling, as we shall discuss below. The interaction term in Eq. (3) contains a tensor with several parameters, which means that designing a system where all these parameters are tunable can be a challenging task, and we do not attempt to do that here. We focus on the basic Hamiltonian that contains hopping and magnetic terms: This Hamiltonian would already exhibit several features that do not exist in the spinless case, and it can be seen as a natural first step towards a more complete model that would include interactions.

III. POSSIBLE ROUTES FOR ADDING SPIN DEGREES OF FREEDOM TO THE JCH SETUP
In this section we discuss some possible approaches that can be used to introduce internal degrees of freedom to the particles in the JCH setup. In particular, the use of multiple polariton branches is rather straightforward and could be used to achieve some demonstrations of the desired behaviour, while the inclusion of multiple qubits coupled to each resonator could lead to simulations of more complex phenomena.

A. Polariton branches
Although most studies consider the spinless case and implicitly focus on only one excitation mode (or branch), the JCH setup inherently supports two excitation modes, namely the lower-and upper-polariton modes. These modes could be used to represent the |↑ and |↓ states of spin-1/2 particles. One can therefore achieve a quantum simulation of itinerant spin-1/2 (or more accurately pseudospin-1/2) particles using the basic JCH setup.
There are a few points that must be noted in this context. One of the main issues that require care is the following: if only the |↑ state or the |↓ state is occupied at any given lattice site (i.e. if only one of the two polariton branches is populated at a given site), the energy-level ladder allows multiple occupation of that site. A difficulty arises, however, when dealing with quantum states that contain particles in both states |↑ and |↓ at the same site. The mapping between polariton excitations and spin-carrying particles fails in this case, as can be seen by direct inspection. The mapping is therefore only applicable when dealing with systems where it suffices to consider at most a single particle per site, which generally corresponds to dilute systems.
We now consider the above point more quantitatively. We take two neighbouring lattice sites with one excitation in each. Regardless of which polariton branches are occupied in this state, the inter-cavity coupling term mixes the state with all four states that where the two excitations occupy the same lattice site (these four states corresponding to the combination of two sites and two polariton branches). Such states would be problematic for a number of reasons. For example, the mapping between the excited states in the JCH setup and the states describing spin-carrying particles is not clearly defined in this case. Furthermore, the tunneling allows all kinds of mixing between the different spin states. This difficulty is avoided if one considers only the regime of weak intercavity coupling (i.e. where J is much smaller than the energy splitting between the two polariton branches) and small average number of excitations/particles per site. In this case, double occupation is prohibited, and the particles behave as hard-core bosons that do not interact unless they are about to occupy the same site.
Another issue to be considered is that we also need to make sure that the Hamiltonian describes a non-trivial setup of spin-1/2 particles. As mentioned above we ignore inter-particle interactions (except for the hard-core condition) and focus on a Hamiltonian that contains only the hopping and Zeeman terms. In this work the hopping is assumed to take place through the resonators. When the qubit-resonator detuning is large (i.e. in the dispersive regime), the effective hopping strength for the photonic excitations will be much larger than that for the atomic excitations. In order to make the two hopping strengths comparable, or even equal, one could move away from the dispersive regime and consider the case of exact resonance between the qubits and resonators (where the two hopping strengths are exactly equal).

vac UP LP
FIG. 1: (Color online) Schematic diagram of a Raman process for converting excitations between the two polariton modes in the JC model. This process is used to simulate the Zeeman term in the Hamiltonian, where the two states of a spin 1/2 particle mix locally, i.e. at a single site in the JCH lattice. The states |LP and |UP are states with a single excitation in the lower-and upper-polariton branches, respectively. The state |vac is the vacuum state with no excitations. The solid lines represent the three relevant energy levels, and the dashed line represents a virtual energy level whose location is determined by the detuning between the driving ac field and the real energy levels.
In the system obtained thus far still, the numbers of particles in the two spin branches are conserved. In order to simulate the Zeeman term with magnetic fields pointing in arbitrary directions, we need to induce mixing be-tween the two spin states, or in other words conversion of excitations between the lower-and upper-polariton modes. First we consider the Zeeman term. Since the two modes have different energies, one might think of achieving the conversion by driving the system at the frequency separation between the two modes. However, the relevant matrix elements for the driving fields that can be applied easily [with driving operators (â +â † ) or σ x ] vanish (because these operators change the excitation number while the excitation number is conserved in the Zeeman term), precluding the conversion between modes using one of these candidates for the driving field. The conversion could be achieved, however, using a Raman process with simultaneous driving at two frequencies, as illustrated in Fig. 1. Depending on whether the driving fields are chosen such that the Raman process occurs with perfect energy matching or not, one can obtain an effective magnetic field that points in any desired direction in the xz plane. Note that this effective (artificial) magnetic field couples only to the spin degree of freedom and does not lead to phenomena related to the quantum Hall effect, which is related to the coupling between the orbital motion and external magnetic fields.
One can therefore achieve a system where the tunneling strength can be made spin independent and an effective magnetic field can be engineered freely using the basic JCH setup. This setup can therefore be used to demonstrate a basic quantum simulator for itinerant spin-carrying particles. We would ideally also like to simulate spin-orbit coupling, as well as spin values larger than 1/2. In Sec. III D we will give an alternative system where these goals could be achieved. But first, we digress and discuss two more setups that at first sight seem to be promising for the purpose of constructing the desired quantum simulator.

B. Higher qubit states
Superconducting qubits are in fact multi-state quantum systems where only two states are used when representing qubit states. A recent experiment [41] made use of the additional quantum states in order to simulate a single spin larger than 1/2. Constructing a resonator array with a single one of these multi-level circuits might therefore seem to be a promising approach to obtaining the desired system. However, the higher energy levels in the qubit are almost (but not exactly) equally spaced. A similar lack of tunability also arises when considering the matrix elements that describe the qubit-resonator coupling. Consequently the tunneling strengths in the resulting JCH model would be constrained to follow a certain, partially regular pattern, limiting the ability of the system to probe even the basic parameter regimes of main interest. Constructing the desired quantum simulator using this system is therefore not as straightforward as it may seem at first sight.

C. Multiple resonator modes
Owing to their extended structure, transmission-line resonators (TLRs) generally support a large number of modes, the so-called fundamental mode with frequency ω f and modes with frequencies that are close to integermultiples of the fundamental frequency (i.e. at frequencies close to n m ω f where the mode index n m = 2, 3, 4, ...). One therefore automatically obtains multiple (potentially usable) degrees of freedom in a TLR. Excitations in the different modes can be used to represent particles in the different internal states. In particular, 2s + 1 modes are needed in order to simulate spin-s particles. The recently demonstrated parametric coupling between two modes of a TLR [42] can be used to simulate spin-changing terms in the Hamiltonian.
There are, however, a number of difficulties associated with following this approach. The hopping strength in a system with capacitive coupling between the resonators is proportional to the capacitive energy between two resonators, which is proportional to the product of the charges accumulated across the capacitor. For a given mode n m , each one of the charges is proportional to √ n m × n p , where n p is the number of photons in mode n m . As a result the hopping strength is proportional to n m . In order to simulate particles whose hopping strength is independent of spin, one would like to have no such dependence, or at least one would like to have tunable coupling strengths with the ability to set all of them to a single value.
Another complication that arises with the use of multiple TLR modes is the simple relation between the frequencies of the different modes. For example, if one drives the system at the fundamental frequency, the drive signal would be resonant with multiple processes including single-mode and multi-mode processes (Note that some of these resonances can be multi-photon resonances occurring at integer multiples of the driving frequency). One possible method to circumvent the detrimental effects of this regular structure of frequencies is to use a combination of modes that are not integer-multiples of each other, e.g. use the modes n m = 2 and n m = 3 in order to simulate spin-1/2 particles or use the modes n m = 2, 3, 5 in order to simulate spin-1 particles. However, this solution becomes more demanding for larger spin values. Because of this and the previously mentioned difficulty, this approach is also not as straightforward as it may seem at first sight.

D. Multiple qubits coupled to each resonator
We now consider the incorporation of multiple qubits coupled to each resonator. Each qubit would then represent one spin state of the simulated particles, keeping in mind that the effective particles would actually be qubitresonator hybridized excitations. It is worth emphasizing here that this setup is not unrealistic since there have been experiments with large arrays of resonators as well as experiments with multiple qubits coupled to a single resonator.
FIG. 2: (Color online) Schematic diagram of a superconducting resonator lattice with multiple qubits coupled to each resonator. The cyan areas represent superconducting material, which is used to define the resonators, while the red dots represent the qubits. The resonators are arranged in a Kagome lattice, which is rather natural for the surperconducting architecture [15]. In this particular example, we include three qubits for each resonator, which would be the case when simulating spin-1 particles.
In order to clearly identify the different qubits with the different spin states, the qubits corresponding to the same spin state could all be set to a single frequency throughout the lattice. Each spin state therefore has a unique qubit frequency associated with it. This condition naturally leads to spin-conserving hopping processes; an excitation in one qubit can only hop to qubits that have the same frequency at neighbouring sites. If multiple qubits are near-resonant with a single resonator mode, it would be necessary to minimize their hybridization with the state of the resonator, a condition that is desirable in order to avoid complicating the coexistence of multiple particles. As a result, one would like to work in the dispersive regime. In this regime, the resonator serves mainly as a mediator of hopping. The effective hopping strength is then given by an expression of the form ∼ J × [g/(ω a − ω 0 )] 2 where the coupling strength g and detuning ω a −ω 0 are specific to the spin state in question. Since the different spin states correspond to different values of ω a , and the difference must be large enough to suppress spin-nonconserving hopping (and assuming that the values of g for the different qubits are comparable to each other), one obtains a situation where the effective hopping strengths for the different spin states are drastically different. This difficulty can, however, be avoided by setting the different qubit frequencies near resonance with different resonator modes. By adjusting the qubit-resonator detunings, one can now obtain an effective hopping strength ∼ n m ×J ×[g/(ω a −ω 0 )] 2 that is independent of spin. The hopping strength can of course be made spin dependent as well. As an added advantage of using multiple resonator modes, the large-detuning requirement can be relaxed.
Transitions between the different spin states, which are needed in order to simulate magnetic fields and the Zeeman term in the Hamiltonian, can be obtained by driving the qubits using any one of the recently realized microwave-driving-based techniques for implementing SWAP operations between superconducting qubits [43]. The resonators naturally provide an effective interqubit coupling that can be used in these conversion protocols. As discussed in Sec. III A, the simulated magnetic field can be designed to point in any direction depending on the amplitude, phase and detuning of the driving fields.
FIG. 3: (Color online) Schematic diagrams illustrating the processes involved in a spin-orbit coupled Hubbard model and how these processes could be induced in a JCH system. The red arrows describe spin-conserving hopping processes, while the purple arrows describe spin-changing hopping processes. In (b) the black solid lines describe the single-excitation qubit energy levels, the blue dashed lines describe the delocalized single-photon energy levels in the resonators, and the green dotted lines show virtual energy levels whose locations are determined by the detunings between the ac fields and the real energy levels. The virtual energy levels are somewhat detuned from the single-photon energy levels in order to avoided populating the resonators with real excitations.
We now consider the simulation of spin-orbit coupling. This goal can be achieved by inducing spin-changing hopping processes. The processes needed in the case of spin-1/2 particles are illustrated in Fig. 3a. These processes can be realized by driving Raman transitions, as shown in Fig. 3b. One point that requires some care here is that when the qubits are biased at their symmetry points, convenient operators [such as (â +â † ) andσ x ] have zero matrix elements for the desired transitions. Modulating the qubit frequencies [i.e. driving the qubits using the operatorσ z ], however, would produce these transitions. If one wishes to achieve the maximum possible coupling with this approach, one needs to work in the Landau-Zener regime, where the qubit frequencies are driven up and down past the resonator frequencies [44]. Clearly such large amplitudes are not realistic for the large frequencies needed to drive some of the transitions in this approach. This constraint would limit the magnitude of the effective spin-changing hopping matrix elements. Another possibility for driving the Raman transitions is using tunable couplers between the qubits and the resonators and modulating the coupling strength at the required frequencies [45,46]. The effective matrix elements for the spin-changing hopping processes obtained this way are only limited by the hopping strength J (which occurs when the driven qubit-resonator transitions have strengths comparable to J), meaning that with the appropriate driving strengths suitable values of the matrix elements can be obtained. In order to obtain a controllable form of spin-orbit coupling, one needs to have the ability to control the amplitude and phase of the matrix elements describing the spin-changing hopping processes. For example, in order to obtain Rashba spin-orbit coupling in a square lattice, the two spin-changing hopping matrix elements along one of the two spatial dimensions need to have the same amplitude but opposite signs [35]. Spin-orbit coupling can also be obtain in one dimension by designing the two spin-changing processes to have opposite signs [36]. This sign difference can be achieved by adjusting the phases of the ac fields that drive the Raman transitions. This goal can be achieved with the proper choice of parameters, as we show in the following derivation.
We now present a quantitative calculation showing how the the different processes are arise in an effective Hamiltonian with the proper choice of parameters and driving conditions. For the simulation of spin-1/2 particles, where we only need to consider two resonator modes, the Hamiltonian of the entire system can be expressed as (note that we shall not explicitly include the hat symbol for the operators, and we seth = 1) Here we have introduced the annihilation ad creation operators for (mostly) qubit excitations c and c † with subscripts ↑ and ↓ as qubit labels. As above we use a and a † as the resonator operators with subscripts 1 and 2 as resonator labels. The different parameters in the Hamiltonian as self explanatory. For this calculation we consider only two neighbouring lattice sites, and we focus on the single-excitation subspace. We also assume that the coupling strengths are modulated with sinusoidal time dependence, i.e. g(t) = g+f cos(ωt+φ). Ignoring terms that do not have any significant effect on the system (e.g. the dc component of the coupling between between highly detuned subsystems), the Hamiltonian can be expressed as ,j a 1,j + ω r,2 a † 2,i a 2,i + a † 2,j a 2,j +g ↓,1,i c † ↓,i a 1,i + c ↓,i a † 1,i + g ↓,1,j c † ↓,j a 1,j + c ↓,j a † 1,j + g ↑,2,i c † ↑,i a 2,i + c ↑,i a † 2,i + g ↑,2,j c † ↑,j a 2,j + c ↑,j a † 2,j +f ↓,1,i cos( We now split the problem of obtaining an effective Hamiltonian for the qubit excitations into two separate problems, one where we consider only the effects of the time-independent qubit-resonator coupling terms (with coefficients denoted by the symbol g) and one where we consider only the effects of the ac terms (with coefficients denoted by the symbol f ), such that we assume no interference between these two types of terms in producing the effective Hamiltonian. We shall discuss at the end why this approximation is justified.
We first consider the Hamiltonian without the ac terms: We now perform an adiabatic elimination of the resonator modes via the transformationH =Ũ HŨ † , wherẽ where ∆ 1 = ω r,1 − ω ↓ and ∆ 2 = ω r,2 − ω ↑ . After truncation to terms that are at least of order (g/∆) 2 and ignoring non-resonant terms, this transformation results in the effective Hamiltoniañ All but the last two terms in the above Hamiltonian correspond to terms in the original Hamiltonian with some small shifts in the effective parameters from the original values (Note that the shifts can be set to equal values, such that they do not detune previously resonant energy levels). The last two terms in the new Hamiltonian describe processes that were not explicitly present in the original Hamiltonian, namely the hopping of c ↑ and c ↓ excitations between neighbouring sites. Note that these processes preserve the spin of the hopping particles. It is also important to emphasize here that the appearance of these terms was the result of applying a transformation that eliminated (to lowest order) the qubit-resonator coupling terms from the Hamiltonian. Next we consider the Hamiltonian with the ac terms (and without the time-independent terms, whose effect has already been calculated). We shall set ω 1 + ω 2 = ω 4 − ω 3 = ω ↑ − ω ↓ in order to drive the desired spin-changing processes. We also set ω ↓ = 0 as an energy reference in order to simplify the expressions below. We now perform a rotating-frame transformation H ′ = U HU † , where After applying the rotating-wave approximation, i.e. ignoring terms that oscillate with frequencies that are on the order of the driving ac field frequencies, we obtain the Hamiltonian: where δ 1 = ω r,1 − ω 1 , δ 2 = ω r,2 − ω 4 . We now perform an adiabatic elimination of the resonator modes via the transformationH ′ =Ũ ′ H ′Ũ ′ † , wherẽ After truncation to terms that are at least of order (f /δ) 2 and ignoring non-resonant terms, we obtain the effective Hamiltoniañ Once again, all but the last two terms describe small renormalization effects to the Hamiltonian parameters.
The last two terms describe spin-changing hopping processes. The new terms appearing in Eqs. (11) and (16) along with their tunable parameters allow us to engineer an effective spin-orbit coupling. By taking the loop (↓, i) → (↑, j) → (↑, i) → (↓, j) → (↓, i), a particle picks up a phase of φ 1 + φ 2 + φ 3 − φ 4 . In the example of the Rashba spin-orbit coupling mentioned above, one can obtain the necessary minus signs by setting the phases to appropriate values, in particular having the combination φ 1 + φ 2 + φ 3 − φ 4 equal to zero for one direction and π for the other direction in the two dimensional square lattice. Changing the phases of the driving fields would lead to a continuum of different types of spin-orbit coupling. One might worry that there can be interference between processes in the two problems that we treated separately above. In particular, one might think that the eight different transitions contributing to the four Raman processes might produce additional combinations. If the inter-site hopping strengths J are much smaller than the detunings ∆ and δ, however, all of these additional combinations will describe non-resonant Raman processes that can be ignored. One must, however, be careful about such interference when generalizing the above construction to an array of lattice sites. For example, if the same frequency combinations are used to drive the transitions between sites i and i+1 and between sites i+1 and i+2, then undesirable transitions will be driven as well. This problem would be avoided if one uses different frequencies (or in other words different virtual energy levels) for the different pairs of neighbouring lattice sites.
State preparation and readout are rather straightforward when encoding the different spin states in different qubits. Microwave-controlled quantum operations driven via local antennas can be used to initialize individual qubits in their excited states, thus allowing the preparation of well defined numbers of particles in the different spin states. Similarly the state readout of individual qubits can be readily achieved in state-of-the-art setups, and this readout would yield the positions and spin states of the simulated particles.

IV. EXPERIMENTAL PARAMETERS
Superconducting qubits and resonators have typical frequencies in the range 1-20 GHz. One could therefore think of a resonator that has a fundamental frequency of about 4 GHz, with a few additional modes at multiples of this frequency. Qubits with tunable frequencies in the vicinity of these resonator frequencies can be fabricated in present-day state-of-the-art experiments. Qubitresonator coupling strengths can reach hundreds of megahertz. One can therefore take 100 MHz as a typical, realistic value for the coupling strength. Resonator arrays with coupling strengths of 30 MHz have been fabricated [24], and values in the 50-100 MHz range should be possible. Effective spin-conserving hopping matrix elements on the order of 10 MHz or higher should therefore be achievable.
The single-transition matrix elements in the Raman processes used in driving the spin-changing hopping processes are limited by the fact that one needs to keep these transitions virtual. In other words, one needs to keep the matrix elements at maximum driving strength smaller than the detuning between the resonator frequency and the virtual energy level used in the Raman process. This detuning can be 100 MHz or more. With the singletransition matrix elements tuned to around 50 MHz, the spin-changing hopping processes can have effective matrix elements on the order of 10MHz. The fact that both spin-conserving and spin-changing hopping matrix elements can be engineered in similar ranges allows one to design any desired combination of hopping processes and therefore any desired type of spin-orbit coupling.
Decay rates of superconducting qubits and resonators are steadily improving (i.e. decreasing). Decay rates of 10-100 kHz, implying excitation lifetimes of 10-100 µs (long compared to the hopping matrix elements), are now quite realistic. An implementation of the proposed setup could therefore be realized in the coming few years.

V. CONCLUSION
We have considered the possibility of simulating itinerant spin-carrying particles using lattices of superconducting qubits and resonators. The basic JCH setup could be used for initial demonstrations of such a quantum simulator, while a more complex system employing multiple qubits coupled to each resonator offers more flexibility and could lead to more sophisticated simulations in the future. In particular, such a system would allow the design of various combinations of spin-conserving and spinchanging hopping processes.
Experiments on the use of superconducting circuits for implementing JCH systems are in the early stages of development but are progressing at a fast pace. They hold promise of great controllability and measurability, two properties that are highly desirable in a quantum simulator. We expect that the ability to add internal degrees of freedom to the simulated particles, along with the ability to engineer various spin-related physical processes, will add to the power of this platform for quantum simulation.