Witnessing edge modes in trimerized circuit quantum electrodynamic lattice

We propose a scheme to investigate and witness edge modes of general one-dimensional photonic trimers in a circuit quantum electrodynamic lattice. These in-gap edge modes are strictly and analytically solved and the criteria for their emergence are indicated respectively. Moreover, the energy spectrum of the system shows two different regimes characterized by a discrepancy in the number of edge modes. Specifically, while there are always a couple of edge modes only present at one single boundary in both the regimes, one of the regimes also shows another pair of edge modes localized on the opposite boundary. Furthermore, these edge modes are witnessed with the aid of continuous-time quantum walks and average photon number measurements. Our scheme provides a comprehensive method for studying the edge modes of matter.


Introduction
Inspired by the quantum Hall effect [1,2], the investigation for the topological phases of matter have attracted considerable attention in recent decades.One such celebrated example is described by topological insulators, which have been extensively explored in the realms of optics [3,4], acoustics [5,6], and condensed-matter physics [7,8].Different from conventional insulators only showing an insulating bulk gap, under an open boundary geometry, topological insulators simultaneously support the edge or surface states in the bulk gap, which leads to a conducting boundary of the system.These edge or surface states are protected by three discrete symmetries [9][10][11][12][13] and are robust against local perturbations and disorder [7,8,14,15], such as thermal fluctuations and defects.More importantly, the most extraordinary feature of topological insulators is the principle of bulk-boundary correspondence [16][17][18][19], where the emergence of these edge or surface states is inherently related to the nontrivial topological properties of the bulk and characterized by a bulk invariant [9,[20][21][22][23][24][25][26][27].Moreover, despite the simplicity of low-dimensional topological insulators, fruitful physical phenomena can be found in them.As an example, the Su-Schrieffer-Heeger model [28] of polyacetylene with spontaneous dimerization presents topological soliton excitation [29,30], fractional charge [31], and nontrivial edge states [32].Extending the standard Su-Schrieffer-Heeger model to more general situations, in particular to the trimer lattice model, many unique physical properties [33,34] and very rich phase diagrams [35,36] can be further addressed, e.g.exotic chiral edge states [37] and piecewise continuous rather than discrete Berry phase [38], to name but a few, which have made the trimer lattice model a focus of research interest in both theory [33,34,39] and experiment [40,41] and also stimulate us to study it in depth on a realizable platform.
On the other hand, quantum walks, the quantum counterpart of classical random walks, can be understood by the random dynamics of quantum particles on a discrete lattice, which are divided into continuous-time quantum walks and discrete-time quantum walks [42][43][44].Compared with the classical random walks, quantum walks display prominently different propagation behavior due to the coherent superposition of time-dependent wave packet and the presence of interference in evolution and open up avenues for potentially exploiting applications in quantum algorithms [45,46], quantum computation [47,48], quantum information processing [49], etc.Based on the superposition mechanism of eigenfunction, quantum walks also provide a powerful tool to indirectly reflect some fundamental properties of eigenstates of a system, e.g.localization.Additionally, quantum walks have been theoretically investigated in the frameworks combining topological phases [50], defects [51,52], or interactions [53,54] and experimentally implemented in various platforms, such as, trapped ions [55], optical resonators [56], coupled waveguide arrays [57], nuclear magnetic resonance [58], and fiber optics [59].
Furthermore, circuit-quantum electrodynamic (QED) systems, which can be deemed that superconducting artificial atoms couple to quantized microwave fields in transmission line resonators, generally consist of highly coherent superconducting quantum qubits and microwave resonators along the same direction and feature flexibility in structural designability and parameter adjustability on a single-site level [60][61][62][63][64][65][66][67][68][69].With the rapid developments and significant advances in micronano manufacturing and material processing technology, circuit-QED systems have become one of the promising candidates for studying quantum optics [70,71], quantum computation [72,73], and quantum information processing [74,75], and easy construction and outstanding controllability make them an ideal device to implement quantum simulation [76][77][78].For instance, benefiting from the Bose statistical properties of the system, circuit-QED lattices allow one to realize photonic topological insulators [79,80], where nontrivial edge modes emerging in topological phases and topological invariants of the system can be detected directly via the resonantly driving process and the input-output formalism [81], respectively.
Motivated by the ability of circuit-QED systems to implement quantum simulation, in this paper, we resort to a circuit-QED lattice to construct and investigate general one-dimensional (1D) photonic trimerized models.The energy spectrum of the system is exhibited and edge modes emerge in the bulk gap as the intercell hopping strength increases.Despite its extremely low spatial dimension, an interesting phenomenon is found in the energy spectrum of the system, where there are two different regimes characterized by a discrepancy in the number of edge modes.Specifically, both the regimes holds a couple of edge modes only present at one single boundary all the time, but one of the regimes also sustains another pair of edge modes localized on the other boundary.Moreover, the exact solutions of these edge modes are analytically given and the criteria for their emergence are explicitly provided.Furthermore, we witness these edge modes with the aid of continuous-time quantum walks and average photon number measurements.In the scenario of continuous-time quantum walks, the ballistic diffusion or dynamical localization of a quantum walker setting out from a boundary depends on the absence or presence of these corresponding edge modes, respectively, which can thus be distinguished indirectly.In the scenario of average photon number measurements, attributed to the Bose statistical properties of the system, based on the the resonantly driving process, we can measure the average photon number at the corresponding boundary in the steady state to directly detect these edge modes.
The remainder of this paper is organized as follows.In section 2, we present the Hamiltonian of the circuit-QED lattice and attain the desired general 1D photonic trimerized models.In section 3, we analyze the energy spectrum of the system in detail.Moreover, the exact solution of the edge modes in the band gap of the system is derived.In section 4, we introduce two approaches, continuous-time quantum walks and average photon number measurements, to witness these edge modes and the details of how to verify their presence are discussed.Finally, a conclusion is given in section 5.

Circuit-QED lattice model and Hamiltonian
We consider a circuit-QED lattice comprised of a 1D array of transmission line resonators and superconducting qubits.The circuit-QED lattice contains three different resonators in each unit cell and two nearest-neighbor resonators are alternately coupled by three different two-level superconducting qubits, as shown in figure 1.The Hamiltonian of the system reads where )] , Here H f describes the free energy of the circuit-QED lattice.o † n and ε o (o = a, b, c) are the photonic creation operator and frequency of resonator o in unit cell n, respectively.ε j (j = 1, 2, 3) is the energy spacing of superconducting qubit Q j and can be modulated by the magnetic field provided by a flux-bias line [82,83], σ z n,j = |e⟩ n,j ⟨e| − |g⟩ n,j ⟨g| denotes the corresponding Pauli z operator with |g⟩ and |e⟩, respectively, being the ground and excited states of a superconducting qubit.H h represents the photonic hopping between nearest-neighbor resonators tuned by a superconducting qubit.g j is the coupling strength between resonator and superconducting qubit and σ n,j = |g⟩ n,j ⟨e|.Specifically, the open boundary condition is imposed so that the circuit-QED lattice terminates in the resonator a 1 at the left boundary and in the resonator c N at the right boundary, respectively, with N giving the number of unit cells.
Moreover, in the interaction picture with respect to the energy spacings ε 1 , ε 2 , and ε 3 and in the rotating frame with respect to an external driving frequency ε d , when all the superconducting qubits are prepared on the ground state, we can obtain the effective Hamiltonian of the system where , and respectively, being the detunings of resonator and superconducting qubit.Apparently, the present circuit-QED lattice formally realizes the well-known commensurable Aubry-André-Harper model with period 3.
For the sake of convenience, we set λ a = λ b = λ c = 0.The effective Hamiltonian of the system can thus be simplified as Subsequently, for a more compact expression, we assign − = ν, and − = ω, respectively, where µ, ν, and ω are all assumed to be nonnegative real parameters throughout the paper.The final total effective Hamiltonian can be rewritten as follows It is obvious that equation ( 5) indicates a nearest-neighbor tight-binding lattice and the present system describes general 1D photonic trimerized models with intracell hopping strengths µ, ν and intercell hopping strength ω.

Edge modes in the circuit-QED lattice
In this section, we mainly investigate the energy spectrum of the circuit-QED lattice and exactly solve the edge modes present in the bulk gap, including the criteria for their emergence.

Energy spectrum of the system
Energy spectrum is a straightforward manner to elucidate the intrinsic nature of a system and we plot in figure 2(a) the energy spectrum of the circuit-QED lattice as a function of the intercell hopping strength ω under open boundary condition with intracell hopping strengths µ = 2 and ν = 1.Visibly, there are three major bulk bands in the energy spectrum of the system, which are separated by two bulk gaps.Furthermore, it is clear that four energies emerge in the two bulk gaps as ω increases, two are pinned at E = ±2 and the other two are pinned at E = ±1.The in-gap energies E = ±2 correspond to a pair of edge modes localized on the left boundary, as shown in figures 2(b1) and (b2) with ω = 1.5 and figures 2(c3) and (c4) with ω = 2.5, whereas a pair of edge modes localized on the right boundary can be captured for the in-gap energies E = ±1, as shown in figures 2(c1) and (c2) with ω = 2.5.Interestingly, we find that the critical value ω c for the emergence of these in-gap energies is different.To be specific, the in-gap energies E = ±2 emerge when ω > ν with ω c = 1, if ω > µ, the in-gap energies E = ±1 emerge with ω c = 2, which leads to three different regimes in the energy spectrum of the system.In regime I, there is no in-gap energy in the range of 0 < ω < 1, in regime II, only a pair of in-gap energies E = ±2 emerge in the range of 1 < ω < 2, and four in-gap energies E = ±2 and ±1 exist in regime III in the range of ω > 2. As a result, depending on the value of ω, one can observe a discrepancy in the number of edge modes in the system: there are always a couple of edge modes present at the left boundary in regimes II and III, whereas regime III also shows a pair of edge modes localized on the right boundary.
It is well known that the topologically-protected edge mode can show an excellent immunity to moderate disorder.On the one hand, according to the conventional bulk-edge correspondence nontrivial topological invariants indicated the presence of edge modes.However, the general 1D photonic trimerized model does not seem to support a winding number due to the absence of the chiral symmetry [84], but there are still edge modes in the energy spectrum of the system, which is attributed to an inherent connection with the celebrated commensurate off-diagonal Aubry-André-Harber lattice [85,86].On the other hand, disorder is always unavoidable in the practical experimental implementation.These motivate us to examine whether these edge modes are also robust against disorder even without the protection of any symmetry.To this end, we introduce random disorder in the nearest-neighbor hopping so that µ n = µ + δχ 1  n , ν n = ν + δχ 2 n , and ω n = ω + δχ 3  n , where δ is the disorder strength and χ 1,2,3 n are randomly and uniformly distributed between −1 and 1 with n being the unit cell index.As an example, we plot in figure 3 the energy spectrum of the system as a function of the disorder strength δ when µ = 2, ν = 1, and ω = 5, averaged over 1000 disorder realizations.The maximum of the disorder strength δ is chosen as 1 to make each nearest-neighbor hopping strength nonnegative.It is clear that as the disorder strength δ increases, despite the energies E = ±2 of the couple of the left edge modes (red dashed line) and the energies E = ±1 of the couple of the right edge modes (blue dashed-dotted line) being all slightly shifted, they do not merge into the adjacent bulk bands.On the contrary, they still persist in the two bulk gaps and remain respectively localized on the left and right boundaries.Therefore, these edge modes are insensitive to disorder.
In the next subsection, we will provide the exact solutions of these edge modes and demonstrate the critical value ω c given above for their emergence.

Solutions of the edge modes
In order to solve these edge modes emerging in the two bulk gaps, we first write out the energy eigenequation of the system where µ, ν, and ω are the parameters defined in section 2 and ψ n,o (o = a, b, c) is the amplitude of wave function at resonator o in unit cell n.
According to equation ( 6), the two edge modes localized on the right boundary with ψ n,a = 0 satisfy the energy eigenequation  ψn,c = µ ω , respectively, and the normalized coefficient is also Similarly, the energy eigenequation for the two edge modes localized on the left boundary with ψ n,c = 0 reads The energies of the two left edge modes are E = ±µ, also conforming to E = ±2 in figure 2(a) with µ = 2.
For energy E = µ, the amplitudes of the wave function at resonators a n and b n are ψ n,a = ψ n,b and ψ n,b = − ν ω , respectively, and the normalized coefficient is given by ψ n,b = ν ω , respectively, and the normalized coefficient is also as shown in figures 2(b2) and (c4).The two left edge modes emerge when ω > ν, also in accordance with ω c = 1 in figure 2(a) with ν = 1.

Witnessing the edge modes
In this section, we witness these edge modes with two approaches, viz.continuous-time quantum walks and average photon number measurements.In the former, the ballistic propagation or dynamical localization of a quantum walker setting out from a boundary depends on the absence or presence of these corresponding edge modes, respectively, which can thus be reflected indirectly.In the latter, attributed to the Bose statistical properties of the system, based on the input-output formalism, we can also detect these edge modes by directly measuring the average photon number at the corresponding boundary in the steady state.

Continuous-time quantum walks
According to the superposition principle of eigenfunction, when an edge mode is presented and a quantum walker sets out from a boundary resonator populated by the edge mode, the initialization of the quantum walker will contain a superposition of the edge mode and one thus expects a localized component to emerge at the boundary resonator in the whole evolution so that the presence of the edge mode in the circuit-QED lattice can be verified intuitively.Here, the quantum walker is considered to be initialized at the three leftmost resonators and the three rightmost resonators, respectively, i.e. resonators a 1 , b 1 , c 1 , a N , b N , and c N .
The dynamics of the quantum walker is governed by the time-dependent Schrödinger equation with denotes the vacuum state and C n,o (t) is the time-dependent probability amplitude on each resonator.Hence, by solving equation ( 9) numerically, we can obtain the probability distribution of the quantum walker in the circuit-QED lattice at time t We first investigate the quantum walks in the circuit-QED lattice with ω = 0.5, namely, the system is in regime I, as shown in figures 4(a)-(f).One can observe that a ballistic propagation is displayed for all the six initializations, which implies the absence of these edge modes.However, when the system is in regime II, e.g.ω = 1.5, the dynamical behavior of the quantum walker setting out from the two leftmost resonators is dramatically different from figure 4, as shown in figures 5(a) and (b).It is found that a localized component emerges at the left boundary whether the quantum walker is initially located at resonators a 1 or b 1 , which arises from the presence of a pair of left edge modes in regime II, whereas there is still only a ballistic propagation for the rest of the initializations, as shown in figures 5(c)-(f).Furthermore, for the system in regime III, if the quantum walker is initially injected into resonators b N or c N , compared with figures 4 and 5, a localized component is exhibited at the right boundary, as shown in figures 6(e) and (f), since a pair of right edge modes are presented now.Moreover, in addition to a ballistic propagation in figures 6(c) and (d), we can find in figures 6(a) and (b) that the localized component at the left boundary is enhanced, which ascribes to that the localization of the two left edge modes on resonators a 1 and b 1 is enhanced with the increase of ω, as shown in figures 2(c3) and (c4), and their weight in the two initializations thus increases.Accordingly, the presence or absence of these edge modes can be indicated indirectly in terms of whether a dynamical localization can be observed at the corresponding boundary or only a ballistic propagation emerges in the whole evolution, respectively.Based on these differences of quantum walks in the six initializations, we can further probe the configurations of these left and right edge modes and distinguish the three regimes.

Average photon number measurements
Attributed to the Bose statistical properties of the circuit-QED lattice, photons in the system can occupy the same eigenmode simultaneously.When the resonators are externally driven by pulses with the driving frequencies tuned as the eigenenergy of the system, the corresponding eigenmode will be occupied by some weights.Therefore, we can detect the desired eigenmode by directly measuring average photon number distribution in the circuit-QED lattice in the steady state.
The driving Hamiltonian in the rotating frame with respect to the external driving frequency ε d can be written as where Ω n,o (o = a, b, c) is the driving amplitude of resonator o in unit cell n.Taking dissipation into account, we can obtain the mean value of optical field o n in the steady state from the Lindblad master equation where is the Lindblad term and Γ is the decay rate of resonator.In the bases T with T being the transpose of matrix, when the condition of steady-state solution ⟨ȯ n ⟩ = 0 is imposed, the mean values all optical fields can be integrated into a compact form here the 3N × 3N matrix Ξ takes the form For the system in regime II, a pair of left edge modes mainly concentrated on resonators a 1 and b 1 are presented.We can thus excite either of them by externally driving resonators a 1 or b 1 with the driving pulses chosen as ⃗ η = (Ω 1,a , 0, 0, . . ., 0, 0, 0) T or (0, Ω 1,b , 0, . . ., 0, 0, 0) T .Meanwhile, the frequencies of the driving pulses need to be tuned to make the resonator detuning ∆ o matches the energy of one of the left edge modes.For example, the average photon number distributions in the circuit-QED lattice are exhibited in figures 7(a) and (b) when we respectively drive resonators a 1 and b 1 with ∆ o = 2.One can observe that in both the cases, photons mostly populate at resonators a 1 and b 1 and the average photon number distributions are consistent with the amplitude distribution of the left edge mode in the upper bulk gap, as shown in figure 2(b1).However, when the rest of boundary resonators are driven with the same pulse, the left edge mode will be nearly not excited and there is not a large photonic population at the left boundary, as shown in figures 7(c)-(f), and all photons will eventually decay into the vacuum.
Similarly, for the system in regime III, figures 8 and 9 also display the average photon number distributions in the circuit-QED lattice when these boundary resonators are respectively driven with ∆ o = 2 and 1.We can find in figures 8(a) and (b) that only resonators a 1 and b 1 are almost occupied now compared with figures 7(a) and (b), which reflects the enhanced localization of the left edge mode as ω increases, as shown in figure 2(c3).Furthermore, it is obvious that a large photonic population at resonators b N and c N can be found in figures 9(e) and (f) due to the excited right edge mode in the upper bulk gap.Moreover, the manner of photons occupying these right boundary resonators still follows the amplitude distribution of the right edge mode, as shown in figure 2(c1).The average photon number distributions for the rest of boundary resonators in figures 8 and 9 are also the same as figure 7.In a word, we can excite these edge modes at each corresponding boundary resonator via a resonant drive and detect them by directly measuring the average photon number distribution in the circuit-QED lattice.

Conclusions
In conclusion, we have proposed a scheme to construct and investigate general 1D photonic trimerized models in a circuit-QED lattice.It turns out that the energy spectrum of the system shows the edge modes emerging in the bulk gap with the increase of the intercell hopping strength and there exist two different regimes indicating a discrepancy in the number of edge modes.In other words, both the regimes support a couple of edge modes only present at one single boundary all the time, and another pair of edge modes localized on the opposite boundary are also manifested in one of the regimes.We have also exactly calculated these edge modes and explicitly provided the criteria for their emergence.Furthermore, two approaches have been introduced to witness these edge modes, i.e. continuous-time quantum walks and average photon number measurements.In the scenario of continuous-time quantum walks, when we consider a quantum walker initially injected into boundary, its ballistic diffusion or dynamical localization in evolution depends on whether these corresponding edge modes are absent or not, respectively, which can thus verify their presence indirectly.In the scenario of average photon number measurements, if an edge mode is presented and we further use an external pulse tuned as the energy of the edge mode to drive the corresponding boundary, we can measure a large average photon number occupying the corresponding boundary in the steady state, which allows us to detect these edge modes directly.

Figure 1 .
Figure 1.Schematic diagram of the circuit-QED lattice consisting of a 1D array of transmission line resonators and superconducting qubits.In each unit cell, there are three different resonators a, b, and c.While two nearest-neighbor resonators in a unit cell are coupled by superconducting qubits Q1 and Q2, respectively, two nearest-neighbor resonators in adjacent unit cells are coupled by another superconducting qubit Q3.The magnetic field provided by a flux-bias line (FBL) can be applied to modulate the energy spacing of each superconducting qubit.

Figure 2 .
Figure 2. (a) Energy spectrum of the circuit-QED lattice versus the intercell hopping strength ω with the number of unit cells N = 40.The intracell hopping strengths are µ = 2 and ν = 1, respectively.The bulk bands are depicted by the solid line, the dashed (dashed-dotted) line delineates the left (right) edge modes.The blank, purple, and cyan regions correspond to regimes I, II, and III, respectively.The amplitude distributions of the two left edge modes when ω = 1.5 with energies (b1) E = 2 and (b2) E = −2.The amplitude distributions of the two right edge modes when ω = 2.5 with energies (c1) E = 1 and (c2) E = −1.The amplitude distributions of the two left edge modes when ω = 2.5 with energies (c3) E = 2 and (c4) E = −2.The yellow, green, and orange bars label the amplitudes of the wave function at resonators a, b, and c, respectively.

Figure 3 . 7 )
Figure 3. Energy spectrum of the circuit-QED lattice versus the disorder strength δ, averaged over 1000 disorder realizations.Here, µ = 2, ν = 1, and ω = 5.The bulk bands are depicted by the black solid line, the red dashed (blue dashed-dotted) line delineates the left (right) edge modes.
as shown in figures 2(b1) and (c3).For energy E = −µ, the amplitudes of the wave function at resonators a n and b n are ψ n,a = −ψ n,b and ψ n,b = (

Figure 7 .
Figure 7. Average photon number distributions in the steady state in the circuit-QED lattice with ω = 1.5 when we adjust the resonator detuning ∆o = 2 and drive resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) bN , and (f) cN , respectively.Here, Ωa1 = Ω b1 = Ωc1 = ΩaN = Ω bN = ΩcN = Γ = 0.1 and other parameters are the same as figure 2. The left edge mode in the upper bulk gap is excited, which allows us to measure a large average photon number occupying the left boundary for driven resonators a1 and b1.

Figure 8 .
Figure 8.Average photon number distributions in the steady state in the circuit-QED lattice with ω = 2.5 when we adjust the resonator detuning ∆o = 2 and drive resonators (a) a1, (b) b1, (c) c1, (d) aN, (e) bN, and (f) cN, respectively.Here, Ωa1 = Ω b1 = Ωc1 = ΩaN = Ω bN = ΩcN = Γ = 0.1 and other parameters are the same as figure 2. The left edge mode in the upper bulk gap is excited, which allows us to measure a large average photon number occupying the left boundary for driven resonators a1 and b1.

Figure 9 .
Figure 9. Average photon number distributions in the steady state in the circuit-QED lattice with ω = when we adjust the resonator detuning ∆o = 1 and drive resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) , and (f) cN , respectively.Here, Ωa1 = Ω b1 = Ωc1 = ΩaN = Ω bN = ΩcN = Γ = 0.1 and other parameters are the same as figure 2. The right edge mode in the upper bulk gap is excited, which allows us to measure a large average photon number occupying the right boundary for driven resonators bN and cN .

Figure 10 .
Figure 10.Continuous-time quantum walks in the circuit-QED lattice with the on-site disorder strength ξ = 0.5, averaged over 100 disorder realizations, where the quantum walker is initially positioned at resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) bN , and (f) cN , respectively.Other parameters are the same as figure 4.

Figure 11 .
Figure 11.Continuous-time quantum walks in the circuit-QED lattice with the on-site disorder strength ξ = 0.5, averaged over 100 disorder realizations, where the quantum walker is initially positioned at resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) bN , and (f) cN , respectively.Other are the same as figure 5.

Figure 12 .
Figure 12.Continuous-time quantum walks in the circuit-QED lattice with the on-site disorder strength ξ = 0.5, averaged over 100 disorder realizations, where the quantum walker is initially positioned at resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) bN , and (f) cN , respectively.Other parameters are the same as figure 6.

Figure 13 .
Figure 13.Continuous-time quantum walks in the circuit-QED lattice with the on-site disorder strength ξ = 0.1, averaged over 100 disorder realizations, where the quantum walker is initially positioned at resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) bN , and (f) cN , respectively.Other parameters are the same as figure 4.

Figure 14 .
Figure 14.Continuous-time quantum walks in the circuit-QED lattice with the on-site disorder strength ξ = 0.1, averaged over 100 disorder realizations, where the quantum walker is initially positioned at resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) bN , and (f) cN , Other parameters are the same as figure 5.

Figure 15 .
Figure 15.Continuous-time quantum walks in the circuit-QED lattice with the on-site disorder strength ξ = 0.1, averaged over 100 disorder realizations, where the quantum walker is initially positioned at resonators (a) a1, (b) b1, (c) c1, (d) aN , (e) bN , and (f) cN , respectively.Other parameters are the same as figure 6.

Figure 16 .
Figure 16.Energy spectrum of the circuit-QED lattice as a function of the on-site disorder strength ξ, averaged over 1000 disorder realizations.Here, µ = 2, ν = 1, and ω = 5.The bulk bands are depicted by the black solid line, the red dashed (blue dashed-dotted) line delineates the left (right) edge modes.