Abstract
Gauge theory plays the central role in modern physics. Here we propose a scheme of implementing artificial Abelian gauge fields via the parametric conversion method in a necklace of superconducting transmission line resonators (TLRs) coupled by superconducting quantum interference devices (SQUIDs). The motivation is to synthesize an extremely strong effective magnetic field for chargeneutral bosons which can hardly be achieved in conventional solidstate systems. The dynamic modulations of the SQUIDs can induce effective magnetic fields for the microwave photons in the TLR necklace through the generation of the nontrivial hopping phases of the photon hopping between neighboring TLRs. To demonstrate the synthetic magnetic field, we study the realization and detection of the chiral photon flow dynamics in this architecture under the influence of decoherence. Taking the advantages of its simplicity and flexibility, this parametric scheme is feasible with stateoftheart technology and may pave an alternative way for investigating the gauge theories with superconducting quantum circuits. We further propose a quantitative measure for the chiral property of the photon flow. Beyond the level of qualitative description, the dependence of the chiral flow on external pumping parameters and cavity decay is characterized.
Introduction
Circuit quantum electrodynamics (QED)^{1,2,3,4} is the realization of cavity QED^{5,6} in superconducting quantum circuits. It employs the superconducting coplanar transmission line resonators (TLRs)^{1,2} to substitute the standingwave optical cavities and the superconducting qubits^{7,8,9,10} to replace the atoms. Due to its flexibility and scalability, this onchip architecture has been regarded as a promising platform for quantum computation^{4} and quantum simulation^{15,16}. Recently, several theoretical schemes have been proposed to generate artificial gauge fields for microwave photons and polaritons^{11,12,13,14} in circuit QED lattices^{15,16}. While the idea of synthesizing gauge fields was first proposed and realized in the context of ultracold atoms^{17,18,19} and magnetooptical systems^{20}, circuit QED takes the advantages of individual addressing and in situ tunability of circuit parameters^{4,15,16}. Moreover, the effective strong photonphoton repulsion can be induced through a variety of mechanisms, including electromagnetically induced transparency (EIT)^{21,22,23}, JaynesCummingsHubbard (JCH) nonlinearity^{24,25}, and nonlinear Josephson coupling^{26,27,28}. Combining the strong photon correlation with the synthetic gauge fields, the circuit QED system is showing prospective potential in the investigation of bosonic fractional quantum Hall liquids^{29,30} and nontrivial topological edge states for microwave photons^{31}.
In the pioneering work^{13} and its generalization^{14}, a method of generating effective magnetic fields for polaritons in a two dimensional cavity lattice has been proposed. For each site on the lattice, the mixing phase between the atomic and photonic components of the polariton is controlled by an EIT type modulation of the atom trapped in the cavity, and the intersite polariton hopping is induced by the untunable evanescent coupling between neighboring cavities. When hopping from one particular site to its neighbor, the polariton acquires a hopping phase which is the difference of the mixing phases subjected to the two neighboring sites. To obtain nontrivial gauge fields, the hopping along the horizontal and vertical directions should be controlled independently, and twomode cavities (TMCs) and double EIT processes are consequently required. The construction and pumping of the complicated multilevel artificial atoms are still challenging in current experimental setup. There is also another scheme proposed in which a passive circulator is used to induce nontrivial hopping phases between TLRs capacitively connected to it^{12} through a virtualresonant process. Due to its dispersive nature, this scheme is suitable to construct Kagomé and honeycomb lattices with coordination number three because the circulator induces effective photon hopping between any two of the TLRs connected to it. When applied to other lattice configurations with coordination number larger than three (e. g. square lattice with coordination number four), this scheme will result in unwanted crosstalk.
In this manuscript, we consider an alternative mechanism of implementing artificial Abelian gauge fields and propose a minimum circuit to demonstrate this method. Our work is inspired by the laser assisted tunneling technique used in ultracold atoms^{17} and recent works of Josephsonembedded circuit QED systems^{28,33,34,35}. We consider a necklace consisting of three TLRs coupled by superconducting quantum interference devices (SQUIDs)^{26,28,32,33} which can be harmonically modulated. With appropriate modulating pulses, effective parametric photon conversion between eigenmodes of the necklace can be induced^{34}, which manifests itself as photon hopping between neighboring TLR sites. This modulation in turn leads to an accumulated hopping phase during the hopping process, which can be regarded as an effective magnetic field imposed on the photons. As our method endows hopping phases directly to the links between TLRs, we expect that the experimental setup of our scheme is much simpler because the complicated double EIT pumping is not necessary. In addition, since our scheme does not rely on the dispersive mechanism, the abovementioned crosstalk difficulty can also be circumvented. Moreover, the effective hopping strength in our scheme can be controlled by the modulating pulses. Such advantage may offer potential facilities in the future study of the competition between synthetic gauge fields, photon hopping, and Hubbard repulsion.
We further study the chiral photon flow dynamics of the necklace, which is a direct evidence of the breaking of time reversal symmetry (TRS) in the proposed circuit. We numerically simulate the photon flow dynamics in the presence of decoherence based on reported experimental data^{34,36,37}. Our results imply that the unidirectional character of the photon flow survives in the cavity decay. The feasibility of detecting such phenomena with the recentlydeveloped photon detection technique^{34,38,40} is also discussed. Moreover, to quantitatively describe the chiral flow, we introduce the concepts of photon position vector and chiral area. We show that the direction and the strength of the chiral photon flow can be represented by the chiral area which is the directed area swept by the photon position vector in a given time. With the proposed quantitative measure, we quantify the chiral flow character in general cases and investigate its detailed dependence on external pumping and decoherence processes.
Results
Implementing artificial gauge field with dynamic modulation method: the theoretical model
We start from a photon hopping process between three cavities described by the following Hamiltonianwhere are the annihilation/creation operators of the ith site for i = 1, 2, 3, g_{ij} are the i ↔ j hopping rates, and θ_{ij} are the corresponding hopping phases. We can imagine that there is a photon initially prepared in the cavity 1 and hopping on the cavity necklace. When finishing the 1 → 3 → 2 → 1 circulation, the photon accumulates a phase θ_{Σ} = θ_{12} + θ_{23} + θ_{31}, which is similar to the AharonovBohm phase of an electron circulating in an external magnetic field. Consequently, θ_{Σ} can be regarded as an artificial magnetic field imposed on the chargeneutral photon, and the TRS of this circuit keeps intact if and only if ^{11,12}. To break the TRS, we synthesize nontrivial θ_{12}, θ_{23} and θ_{31} by the dynamic modulation method^{39}: we consider a threecavity Hamiltonianwithwhere ω_{i} is the eigenfrequency of the ith cavity and Ω_{ij}(t) is the coupling constant between the ith and jth cavities. Here we assume that Ω_{ij}(t) can be tuned harmonically and in situ (we will discuss the physical realization in the next subsection). Moreover, we assume that the parameters in satisfy the far offresonance condition:
In the first step we consider the 1 ↔ 2 hopping. If Ω_{12}(t) is static, the 1 ↔ 2 photon hopping can hardly be induced because the two cavities are far offresonant. Meanwhile, we can implement the effective 1 ↔ 2 photon hopping by modulating Ω_{12}(t) dynamically asPhysically, Ω_{12}(t) carries energy quanta filling the gap between the two cavity modes. For a photon initially placed in the 1st cavity, it can absorb an energy quantum ω_{2} − ω_{1} from the 1 ↔ 2 link, convert its frequency to ω_{2}, and hop finally into the 2nd cavity. We can further describe this process in a more rigorous way: in the rotating frame with respect to , becomesbecause the other terms are fast oscillating and thus are safely neglected. From Eq. (7), we notice that both the effective hopping strength and the hopping phase can be controlled by the modulating pulse Ω_{12}(t). Similarly, we can induce the effective 2 ↔ 3 and 3 ↔ 1 hopping process by modulating Ω_{23}(t) and Ω_{31}(t) as
Summarizing the three pulses up, we getwhich directly reproduces the model (1).
The superconducting circuit implementation: a SQUIDcoupled threeTLR necklace
Here we show explicitly the implementation of Eq. (1) in a circuit QED necklace. We propose a circuit consisting of three TLRs with different lengths L_{n} but the same capacitance c and inductance l per unit length, coupled by three grounding SQUIDs with capacitance C_{Jα} and maximal Josephson coupling energy E_{α} for n = 1, 2, 3 and α = a, b, c, as shown in Fig. 1(a) (this architecture has also been exploited to study the entanglement generation through dynamic Casimir effect recently^{35}). For each of the SQUID loops, an external static flux bias is added to modulate the effective Josephson coupling energy as with Φ_{0} = h/2e the flux quantum. We assume that the inductance/capacitance of the resonator is much bigger than the inductance/capacitance of the SQUIDs such that the following inequalities hold:where is the effective inductance of the αth SQUID for α = a, b, c, L = L_{1} + L_{2} + L_{3} is the total length of the TLR necklace, and ΔL = min{L_{i} − L_{j}, i ≠ j} characterizes the length difference between the TLRs. Focusing only on the lowest three modes, this necklace can be described by the Hamiltonianwhere ω_{m} is the eigenfrequency of the mth eigenmode for m = 1, 2, 3, and are the corresponding annihilation/creation operators. While Eq. (13) is derived in detail in Methods, we can explain the mode structure of the necklace in an intuitive way. The presence of the grounding SQUIDs can modify the eigenmodes of the individual TLRs and induce the TLRTLR coupling. From the point of view of TLR 1, the SQUID a plays the role of a shortcut of TLR 2, because the current coming from TLR 2 will largely flow through SQUID a directly to the ground, without crossing TLR 1. This allows us to define separated and localized modes for the TLR necklace: due to the small inductances of the grounding SQUIDs, the edges of the TLRs can be regarded as grounding nodes, and the lowest three eigenmodes can be approximated by the three individual λ/2 modes of the TLRs. For the mth eigenmode, we calculate its normalized node flux distribution function f_{n}_{,m}(x) in the nth TLR based on data from recent experiments^{34,36,37} and study its localization property versus the Josephson coupling energies of the grounding SQUIDs. For the TLRs, the circuit parameters are chosen as c = 1.6 × 10^{−10} F · m^{−1}, l = 4.08 × 10^{−7} H·m^{−1}, L_{1} = 6 mm, L_{2} = 7 mm and L_{3} = 8 mm. For the grounding SQUIDs, we choose the effective critical currents I_{α} = 2πE_{Jα}/Φ_{0} of the three grounding SQUIDs on the level of I_{α} ∈ [0.5 μA, 4 μA] for α = a, b, c. As shown in Fig. 1(b) and 1(c), larger critical currents lead to better localization, this is consistent with our previous description of the roles played by the grounding SQUIDs. With proper choice of the parameters, the eigenmode amplitudes f_{n}_{,m}(x)^{2} become sufficiently large only for n = m while negligible for n ≠ m.
Moreover, since the currents of two neighboring TLRs flow to the ground through the same grounding SQUID, by the modulation of the grounding SQUIDs we can establish the effective interTLR parametric hopping. For the 1 ↔ 2 coupling, we add an extra a. c. flux driving δΦ_{a}(t) = ΔΦ_{a} cos[(ω_{1} − ω_{2})t − θ_{a}] to the static which induces the parametric coupling Hamiltonian (see Methods)with g_{a} the coupling strength between modes 1 and 2. In the rotating frame with respect to in Eq. (13), such modulation results in the effective 1 ↔ 2 hopping which can be described byIn addition, we can add similar pumping pulses on the SQUIDs b and c to induce the 2 ↔ 3 and 3 ↔ 1 hoppings, respectively. Summing up the three modulations, we get the effective Hamiltonianwhich is identical with Eq. (1) through the mappings g_{a} → g_{12}, g_{b} → g_{23} and g_{c} → g_{31}. We should emphasize that g_{a}, g_{b}, and g_{c} can be modulated independently by the amplitudes of the a. c. pulses, and the three phases θ_{a}, θ_{b}, and θ_{c} are determined by the initial phases of the corresponding pulses. The range of the effective coupling strength g_{α} can be estimated base on reported experimental data: we set I_{α} ∈ [1, 4] μA, , and ΔΦ_{α}/Φ_{0} ∈ [0.01, 0.02] for α = a, b, c. The resulted coupling strengths are in the range g_{α}/2π ∈ [10, 30] MHz. For simplicity, in the following we consider the homogenous hopping situation g_{T} = g_{a} = g_{b} = g_{c}.
The realization and detection of the chiral photon flow
To demonstrate the presence of the synthetic gauge field we study the chiral photon flow in this necklace which is the analog of electron circulation in an external magnetic field. We initialize the necklace such that there is initially a photon in the 1st mode and numerically simulate its subsequent time evolution in the presence of decoherence using the master equationwhere ρ is the density matrix of the necklace and κ_{j} is the decay rate of the jth eigenmode. The circuit parameters are chosen as the same as those used in the calculation of Figs. 1(b) and 1(c), with the pumping strength g_{T}/2π = 20 MHz and the homogenous decay rate κ/2π = 250 kHz. In the three situations θ_{Σ} = π/2, π and 3π/2, we calculate the energy stored in the TLRs and plot our results in Fig. 2. As shown in the first panel which corresponds to θ_{Σ} = π/2, the energy population in the three TLRs exhibits clear temporal phase delay which implies that the photon is flowing unidirectionally, first from TLR 1 to TLR 2 and then from TLR 2 to TLR 3. Such chiral character is a significant demonstration of the breaking of TRS in this system. Similarly, the third panel which corresponds to θ_{Σ} = 3π/2 describes the chiral photon flow with the opposite direction. Meanwhile, the second panel implies that, in the trivial case θ_{Σ} = π, the energy initially stored in TLR 1 transfers symmetrically to its left and right. It should be emphasized that, although the cavity decay rate κ we choose is stronger compared with reported experimental data^{38}, the chiral character of the photon flow in the first and third panel still survives. The environment causes severe photon damping but influences little on the unidirectional character of the photon dynamics. Therefore, we expect that the chiral photon flow pattern in this necklace can be realized and measured by the photon number detection technique developed in recent experiments^{34,38,40}. For the three measurement devices shown in Fig. 1(a), we can use three phase qubits capacitively coupled to the corresponding TLRs^{34,38}. Since the frequencies of the qubits can be adjusted by their d. c. bias currents, the initialization and the measurement of the chiral flow dynamics can be proceeded by the following steps: in the first step, we tune the frequencies of the qubits to be large offresonance with the TLRs, and prepare the qubit 1 in its excited state. In this step, the coupling between the qubits and TLRs are effectively turned off. In the second step, we tune on the qubitTLR coupling by adiabatically tuning the qubit 1 in resonance with the TLR 1 for a duration T_{1} = π/2g_{q}_{1} such that the excited qubit 1 emit a photon to the TLR 1 (here g_{qn} denote the coupling strengths between TLR n and qubit n for n = 1, 2, 3). Through this manipulation, the singlephoton initial state is prepared. After the initialization, we turn off the qubitTLR coupling and turn on the external a. c. flux pumping on the grounding SQUIDs for a duration T_{0} during which the TLR necklace experiences the chiral photon flow. To measure the photon flow dynamics, we could prepare the three qubits all in their ground state, turn off the external a. c. flux pumping, and then turn on the TLRqubit coupling by tuning the frequencies of the qubits in resonance with the corresponding TLR modes. In this way we can load the TLR photons into the corresponding qubits and extract the evolution of photon distribution on the necklace through the measurement of the three qubits^{8}.
A quantitative measure of chiral photon flow
We further consider how to describe the chiral pattern of the photon flow. In Fig. 2 as well as in Refs. 11 and 12, the dynamics of perfect chiral flow and perfect nonchiral flow have been investigated. Meanwhile, in more general cases the chiral character which is not perfect but does exist becomes fogged. To characterize how “chiral” the photon flow is, in the following we introduce a quantitative method. As shown in Fig. 3, we assign three unit vectors to the three TLRs, for the TLR 1, for the TLR 2, and for the TLR 3. We then represent the photon distribution on the necklace by the photon position vector where is the photon number population in the TLR j for j = 1, 2, 3. The initial condition used in Fig. 2 corresponds to the initial position , and the state evolution can be expressed by the motion of on the two dimensional plane. The traces of for some typical values of θ_{Σ} and κ are shown in Figs. 4(a) and 4(b). While the 6th panel of Fig. 4(a) indicates the perfect chiral flow in the situation θ_{Σ} = π/2 and κ = 0, the other traces become chaotic and irrational as θ_{Σ} departs from π/2 and the influence of decoherence is taken into account.
To grasp the chiral character from the complicated traces of the photon position vector, our idea is to sum up the area swept by in a given time, as shown in Fig. 3. We define the chiral area aswhere T is a time scale sufficiently longer than 1/g_{T} but significantly smaller than 1/κ. S has the following properties which make it a suitable measure of the chiral character:
S is directed. A clockwise trace and its counterclockwise correspondence (e. g. the θ_{Σ} = π/2 and 3π/2 situations shown in Fig. 2) result in chiral areas with opposite signs. Therefore, the sign of S can be used to represent the direction of the chiral flow.
In the perfect nonchiral case θ_{Σ} = π shown in the middle panel of Fig. 2, the photon transfers symmetrically to the TLR 2 and TLR 3. Therefore, the trace of is always along the direction of and zero chiral area is obtained, i. e. S = 0 for θ_{Σ} = π.
Obviously, S achieves its maximal value when the photon flow is perfect chiral. Moreover, a faster rotation of (which means larger g_{T}) and a longer (which means more photons involved) lead certainly to a bigger S. From this point of view, S can also be used to demonstrate the influence of driving and dissipation on the chiral photon flow.
The chiral area concept can help us to go beyond the qualitative description of the state evolution in special cases and move into a quantitative and general level of investigation. Despite the complicated evolution might undergo, the chiral area presents an intuitive and physical description of the chiral character. With this definition we calculate the chiral area versus the θ_{Σ} and κ, and show our results in Fig. 4(c). It can be seen that the chiral area S decreases rapidly as the total phase θ_{Σ} departs from π/2. This is in agreement with our observation of the vector traces shown in Figs. 4(a) and 4(b). Our calculation indicates that the θ_{Σ} window suitable for the observation of chiral photon flow is not wide. Meanwhile, the width of this window is barely influenced by the decay rate κ.
Discussion
The implementation of the Abelian gauge field in the threeTLR necklace can be regarded as a minimal model. Through a variety of generalizations, the proposed dynamic approach can be used to construct a scalable and flexible quantum simulator of gauge theories. First of all, we consider the scalability of this method, i. e. how to synthesize a gauge field on a circuit QED lattice using this dynamic approach. We take the construction of a square lattice as an example. As shown in Fig. 1(d), we can build a twodimensional square TLR lattice by four kinds of TLRs with eigenfrequencies ω_{1}/2π = 8 GHz, ω_{2}/2π = 9 GHz, ω_{3}/2π = 10 GHz and ω_{4}/2π = 11 GHz placed in an interlaced form and connected to the ground by the grounding SQUIDs. We assume that the small capacitance/inductance conditions in Eqs. (11) and (12) still hold. To introduce effective passive photon hopping with nontrivial hopping phases, we add extra twotone a. c. flux driving with frequencies 1 GHz and 3 GHz and appropriate initial phases to the loops of the grounding SQUIDs. In this situation, the crosstalk between nextnearestneighbor TLRs can be neglected because the corresponding frequency (2 GHz) is far offresonant with the a. c. flux pumping. Moreover, we can introduce offdiagonal disorder^{41,42} (i. e. random magnetic fields) into the lattice through randomization of the initial phases of the a. c. driving pulses. While the simulation of diagonal disorder has been realized in ultracold atoms^{43} and in optical fiber systems^{45}, the simulation of offdiagonal disorder of bosonic particles has also attracted research interest in recent years^{44}. We can also introduce the effective photonphoton interaction to the noninteracting system described in this manuscript by the JCH method, i. e. we couple the TLRs to superconducting qubits resonantly. Though the resonant coupling, the photons are dressed by the qubits and inherit the nonlinearity of the qubits^{24,25}. From the above points of view, we can expect that this dynamic modulation approach may pave a new way of investigating the novel physics of the competition between artificial gauge fields, diagonal and offdiagonal disorder, and Hubbard repulsion in the circuit QED system.
In summary, we have investigated the implementation of artificial gauge fields in the circuit QED system by the method of dynamic modulation. We have numerically simulated the photon chiral flow in a threeTLR necklace, discussed the feasibility of observing this phenomenon, and proposed a quantitative measure of the chiral character. Our work may offer new perspectives to future studies of quantum simulation and parametric quantum optical physics in SQUIDembedded circuit QED systems.
Methods
Quantization of the TLR necklace
The Lagrangian of the system can be written as:where V_{n} (x, t) describes the voltage distribution on the nth TLR for n = 1, 2, 3, is the corresponding node flux distribution, and V_{α}/φ_{α} are the voltage/node flux at locations of the SQUID α for α = a, b, c. In deriving Eq. (19), we have linearized the grounding SQUIDs as . This assumption is valid because φ_{α}(t) ≈ 0. Following the EulerLagrangian equation, we get the equation of motion of the node flux in the bulk of the TLRswith . At the edges of TLRs, based on Kirchhoff's law we get the following boundary conditionswhere x_{a} = L_{1}, x_{b} = L_{1} + L_{2}, x_{c} = 0 and are the locations of the grounding SQUIDs (the cyclic boundary condition is applied as shown in Eq. (21)). Here without loss of generality we assume the three SQUIDs have the same effective inductance L_{J} and capacitance C_{J}. The eigenmodes of the necklace can be obtained by the method of separation of variables. We set φ_{n}(x, t) = f_{n}(x)g(t) withwhere A_{n}/θ_{n} are the normalized amplitude/phase of the eigenmode in the nth TLR, and k is the wave vector. Substituting Eq. (27) to Eqs. (21)–(26) we get the following transcendental equations
When the phases and the wave vector are obtained, the amplitude distribution of the eigenmode in the three TLRs can be further determined byand the mode normalization conditions. Then we can write the flux distribution of the necklace as a superposition of the eigenmodes where f_{n}_{,m}(x) corresponds to the mth solution of Eqs. (28)–(31).
Due to the of the orthonormality of f_{n}_{,m}(x), the Lagrangian can be further simplified aswhere ω_{m} is the eigenfrequency of the mth eigenmode determined by Eqs. (28)–(31). The corresponding Hamiltonian iswith annihilation and creation operatorswhere is the canonical momentum of g_{m}. When only the lowest three eigenmodes are taken into consideration, Eq. (13) is reproduced from Eq. (35).
The parametric coupling Hamiltonian
For the grounding SQUID α, the introduction of an extra a. c. flux driving δΦ_{a}(t) resulting an additional intermode coupling which can be described by Ref. 35with for m = 1, 2, 3. Since the 3rd eigenmode is highly localized in the TLR 3, we have , and consequently neglect all terms in . Based on Eq. (6), we set δΦ_{a}(t) = ΔΦ_{a} cos[(ω_{1} − ω_{2})t − θ_{a}] to induce the photon conversion between modes 1 and 2. Omitting the counterrotating terms, we simplify Eq. (38) aswith . Notice that when the driving frequency ω_{1} − ω_{2} is comparable with the SQUID plasma frequency , the device can not be considered as a passive element because complex quasiparticle excitation behavior will emerge. Meanwhile, with parameters chosen in this manuscript, we can verify that is satisfied.
References
 1.
Blais, A., Huang, R. S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).
 2.
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
 3.
Chiorescu, I. et al. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature 431, 159–162 (2004).
 4.
You, J. Q. & Nori, F. Atomic physics and quantum optics using superconducting circuits. Nature 474, 589–597 (2011).
 5.
Raimond, J. M., Brune, M. & Haroche, S. Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys. 73, 565 (2001).
 6.
Wu, Y. & Yang, X. Algebraic method for solving a class of coupledchannel cavity QED models. Phys. Rev. A 63, 043816 (2001).
 7.
Mooij, J. E. et al. Josephson PersistentCurrent Qubit. Science 285, 1036–1039 (1999).
 8.
Martinis, J. M., Nam, S., Aumentado, J. & Urbina, C. Rabi Oscillations in a Large JosephsonJunction Qubit. Phys. Rev. Lett. 89, 117901 (2002).
 9.
Koch, J. et al. Chargeinsensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).
 10.
Schreier, J. A. et al. Suppressing charge noise decoherence in superconducting charge qubits. Phys. Rev. B 77, 180502 (2008).
 11.
Nunnenkamp, A., Koch, J. & Girvin, S. M. Synthetic gauge fields and homodyne transmission in JaynesCummings lattices. New. J. Phys 13, 095008 (2011).
 12.
Koch, J., Houck, A. A., Hur, K. L. & Girvin, S. M. Timereversalsymmetry breaking in circuitQEDbased photon lattices. Phys. Rev. A 82, 043811 (2010).
 13.
Cho, J., Angelakis, D. G. & Bose, S. Fractional Quantum Hall State in Coupled Cavities. Phys. Rev. Lett. 101, 246809 (2008).
 14.
Yang, W. L. et al. Quantum simulation of an artificial Abelian gauge field using nitrogenvacancycenter ensembles coupled to superconducting resonators. Phys. Rev. A 86, 012307 (2012).
 15.
Schmidt, S. & Koch, J. Circuit QED lattices: Towards quantum simulation with superconducting circuits. Annalen der Physik 525, 395 (2013).
 16.
Houck, A. A., Türeci, H. E. & Koch, J. Onchip quantum simulation with superconducting circuits. Nature Phys. 8, 292–299 (2012).
 17.
Jaksch, D. & Zoller, P. Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. New J. Phys. 5, 56 (2003).
 18.
Galitski, V. & Spielman, I. B. Spinorbit coupling in quantum gases. Nature 494, 49–54 (2013).
 19.
Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523 (2011).
 20.
Wang, Z., Chong, Y. D., Joannopoulos, J. D. & Soljačic̀, M. Observation of unidirectional backscatteringimmune topological electromagnetic states. Nature 461, 772–775 (2009).
 21.
Hartmann, M. J., Brandão, F. G. S. L. & Plenio, M. B. Strongly interacting polaritons in coupled arrays of cavities. Nature Physics 2, 849–855 (2006).
 22.
Hu, Y., Ge, G. Q., Chen, S., Yang, X. F. & Chen, Y. L. CrossKerreffect induced by coupled Josephson qubits in circuit quantum electrodynamics. Phys. Rev. A 84, 012329 (2011).
 23.
Rebić, S., Twamley, J. & Milburn, G. J. Giant Kerr Nonlinearities in Circuit Quantum Electrodynamics. Phys. Rev. Lett. 103, 150503 (2009).
 24.
Lang, C. et al. Observation of Resonant Photon Blockade at Microwave Frequencies Using Correlation Function Measurements. Phys. Rev. Lett. 106, 243601 (2011).
 25.
Greentree, A. D., Tahan, C., Cole, J. H. & Hollenberg, L. C. L. Quantum phase transitions of light. Nature Physics 2, 856–861 (2006).
 26.
Leib, M., Deppe, F., Marx, A., Gross, R. & Hartmann, M. J. Networks of nonlinear superconducting transmission line resonators. New J. Phys. 14, 075024 (2012).
 27.
Jin, J. S., Rossini, D., Fazio, R., Leib, M. & Hartmann, M. J. Photon Solid Phases in Driven Arrays of Nonlinearly Coupled Cavities. Phys. Rev. Lett. 110, 163605 (2013).
 28.
Bourassa, J., Beaudoin, F., Gambetta, J. M. & Blais, A. Josephsonjunctionembedded transmissionline resonators: From Kerr medium to inline transmon. Phys. Rev. A 86, 013814 (2012).
 29.
Umucalilar, R. O. & Carusotto, I. Fractional Quantum Hall States of Photons in an Array of Dissipative Coupled Cavities. Phys. Rev. Lett. 108, 206809 (2012).
 30.
Hayward, A. L. C., Martin, A. M. & Greentree, A. D. Fractional Quantum Hall Physics in JaynesCummingsHubbard Lattices. Phys. Rev. Lett. 108, 223602 (2012).
 31.
Bardyn, C. E. & Imamoğlu, A. Majoranalike Modes of Light in a OneDimensional Array of Nonlinear Cavities. Phys. Rev. Lett. 109, 253606 (2012).
 32.
Leib, M. & Hartmann, M. J. Many body physics with coupled transmission line resonators. Physica Scripta T153, 014042 (2013).
 33.
Peropadre, B. et al. Tunable coupling engineering between superconducting resonators: From sidebands to effective gauge fields. Phys. Rev. B. 87, 134504 (2013).
 34.
ZBajjani, E. et al. Quantum superposition of a single microwave photon in two different ‘colour’ states. Nature Physics 7, 599–603 (2011).
 35.
Felicetti, S. et al. Dynamical Casimir Effect Entangles Artificial Atoms. Phys. Rev. Lett. 113, 093602 (2014).
 36.
Frunzio, L., Wallraff, A., Schuster, D. I., Majer, J. & Schoelkopf, R. J. Fabrication and characterization of superconducting circuit QED devices for quantum computation. IEEE Trans. Appl. Supercond. 15, 860 (2005).
 37.
Sillanpää, M. A., Park, J. I. & Simmonds, R. W. Coherent quantum state storage and transfer between two phase qubits via a resonant cavity. Nature 449, 438–442 (2007).
 38.
Wang, H. et al. Deterministic Entanglement of Photons in Two Superconducting Microwave Resonators. Phys. Rev. Lett. 106, 060401 (2011).
 39.
Fang, K. J., Yu, Z. F. & Fan, S. H. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature photonics 6, 782–787 (2012).
 40.
Mariantoni, M. et al. Planck Spectroscopy and Quantum Noise of Microwave Beam Splitters. Phys. Rev. Lett. 105, 133601 (2010).
 41.
Altland, A. & Simons, B. D. Field theory of the random flux model. Nucl Phys B 562, 445–476 (1999).
 42.
Xie, X. C., Wang, X. R. & Liu, D. Z. KosterlitzThoulessType MetalInsulator Transition of a 2D Electron Gas in a Random Magnetic Field. Phys. Rev. Lett. 80, 3563 (1998).
 43.
SanchezPalencia, L. & Lewenstein, M. Disordered quantum gases under control. Nature Physics 6, 87–95 (2010).
 44.
Edmonds, M. J. et al. From Anderson to anomalous localization in cold atomic gases with effective spinorbit coupling. New J. Phys. 14, 073056 (2012).
 45.
Lahini, Y. et al. Anderson Localization and Nonlinearity in OneDimensional Disordered Photonic Lattices. Phys. Rev. Lett. 100, 013906 (2008).
Acknowledgements
Y.P.W. would like to acknowledge useful discussions with XinYou Lü. The work was supported in part by the National Fundamental Research Program of China (Grants No. 2012CB922103 and No. 2013CB921804), the National Nature Science Foundation of China (Grants No. 11104096, No. 11374117, No. 11274351, No. 11375067 and No. 11275074), and the PCSIRT (No. IRT1243). Y.H. is supported by the fellowship of Hong Kong Scholars Program (Grant No. 201280).
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Affiliations
Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, China
 YanPu Wang
 , Wei Wang
 , Yong Hu
 & Ying Wu
Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
 ZhengYuan Xue
 & Yong Hu
Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
 ZhengYuan Xue
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
 WanLi Yang
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Contributions
Y.H. proposed the idea. Y.P.W. carried out all calculations under the guidance of Y.H. W.W., Z.Y.X., W.L.Y. and Y.W. participated in the discussions. Y.P.W., Y.H. and Z.Y.X. contributed to the interpretation of the work and the writing of the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to ZhengYuan Xue or Yong Hu.
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