Abstract

We investigate the localized photonic states and dynamic process in one-dimensional nonreciprocal coupled Su-Schrieffer-Heeger chain. Through numerical calculation of energy eigenvalue spectrum and state distributions of the system, we find that different localized photonic states with special energy eigenvalues can be induced by the nonreciprocal coupling, such as zero-energy edge states, interface states and bound states with pure imaginary energy eigenvalues. Moreover, we analyze the dynamic process of photonic states in such non-Hermitian system. Interestingly, it is shown that the nonreciprocal coupling has an evident gathering effect on the photons, which also break the trapping effect of topologically protected edge states. In addition, we consider the impacts of on-site defect potentials on the dynamic process of photonic states for the system. It is indicated that the photons go around the defect lattice site and still present the gathering effect, and different forms of laser pulses can be induced with the on-site defect potentials in different lattice sites. Furthermore, we present the method for the quantum simulation of current model based on the circuit quantum electrodynamic lattice.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

After the concept of topology is introduced into the field of condensed matter physics, it rapidly promotes the development of fundamental study of physics. Many novel topological materials have also emerged. Su-Schrieffer-Heeger (SSH) model is one of the most simplest one-dimensional (1D) topological insulators, which is originally used to describe the polyacetylene [1]. The most important and interesting feature of the SSH model is the topologically nontrivial edge states protected by time-reversal symmetry. Based on this model, many efforts have been devoted to investigating topological phase [24] and topological states [59]. Recently, many attentions have been paid to two-coupled SSH model, which present more intriguing physical phenomena in the investigation of the topological nodal points [10], topological states [11], and topological phase transition [12,13]. Moreover, the investigation of topological states are also connected with periodically driven system [14,15]. Moreover, the non-Hermitian extension of topological system is becoming one of the research focuses. A series of novel phenomena have been found in non-Hermitian topological system, such as bulk-boundary correspondence breaking [1619], non-Hermitian skin effect [2022], and fractional topological number [2325]. Most recently, Weidemann et al. demonstrated a highly efficient funnel for light by utilizing the non-Hermitian skin effect [26]. Moreover, many schemes have been proposed to investigate the SSH model with parity-time (${\mathcal {P}T}$) symmetry [2730] and nonreciprocal coupling [16,31]. It benefits from advantage of topologically protected edge state with robustness against perturbations and defects, many schemes have been proposed to realize the topological lasers based on the theory and experiments [3235].

On the other hand, with the rapid development of micro- and nanofabrication technologies, circuit QED system has become one of the most prospective platforms for the realization of quantum information processing [3643], quantum computing [44,45], and quantum simulation [46,47]. Circuit QED system describes the interaction between the nonlinear superconducting circuit and the photon storing in the transmission line resonator. Similar to the energy level structure of the atom, superconducting qubit also has ground state and excited state. In the study of quantum information processing, just the lowest two energy levels can be considered, and the other higher energy levels are often negligible. When a superconducting qubit is coupled to a resonator, the quantum state of superconducting circuit can be manipulated and read out by the detector [4850]. Moreover, circuit QED lattice system can be constructed when the highly coherent superconducting quantum qubits and microwave resonators are arranged in a periodic array, which can map to different one dimensional or even high dimensional topological systems [47,49,5153]. Furthermore, resorting to the bosonic statistical properties of the circuit QED lattice, it is easy to realize the detection of the topological edge states and topological invariants [47,49]. Experimentally, the simulation of quantum spin has been realized in the circuit QED system, which provides a method for controlling and simulating spin-lattice dynamics [54].

Inspired by above, we investigate the different localized states in one-dimensional (1D) nonreciprocal coupled SSH chain, where the nonreciprocal coupling depends on the direction of photons tunneling. For the nonreciprocal coupling strength $\Delta =0$, the current system is decoupled into two independent chains with SSH model structure. To investigate the influence of the nonreciprocal coupling on the system, we numerically calculate the energy eigenvalue spectrum of the system. It is shown that the nonreciprocal coupling can induce different localized photonic states with special energy eigenvalue both in the topologically nontrivial regime and topologically trivial regime, such as the zero-energy edge states, the interface states and bound states with pure imaginary energy eigenvalues. We also analyze these localized photonic states from the perspective of state distributions. Moreover, we investigate the dynamic process of photonic states in current non-Hermitian system, which reveals that the nonreciprocal coupling has a gathering effect on the photons and break the trapping effect of the topologically protected edge states. In addition, we consider the influence of defect potential on the dynamic process of the system, which can induce different forms of laser pulses. Particularly, we find that the gathering effect caused by the nonreciprocal coupling is immune to the on-site defect potential. Further, we propose the method for the quantum simulation of current model based on the circuit QED lattice.

The rest of this paper is structured as follows. In Sec. 2, we show the physical model and the corresponding Hamiltonian of the system and investigate the energy eigenvalue spectrum and the state distributions of the system. In Sec. 3, we investigate the dynamic process of the photonic states of the system. In Sec. 4, we present the quantum simulation of current model by using the circuit QED lattice. Finally, we give the conclusion in Sec. 5.

2. Energy eigenvalue spectrum and state distributions for 1D nonreciprocal coupled SSH chain

As shown in Fig. 1, consider the nonreciprocal coupling model, which consists of two identical SSH chains. The corresponding Hamilton of the system can be written as

$$\begin{aligned} H=&\sum_{j=1}^{L-2}\left(t_{1}a^{\dagger}_{j}a_{j+1}+t_{2}a^{\dagger}_{j+1}a_{j+2}+b^{\dagger}_{j}b_{j+1}+t_{2}b^{\dagger}_{j+1}b_{j+2}+\rm{H.c.}\right)\cr &+\Delta a_{L}b^{\dagger}_{1}-\Delta b_{1}a^{\dagger}_{L}, \end{aligned}$$
where $t_{1}=t\left (1-\delta \cos \theta \right )$ and $t_{2}=t\left (1+\delta \cos \theta \right )$ are the intracell and intercell nearest neighbor hopping amplitudes, $\theta$ is a periodic parameter varying from $-\pi$ to $\pi$, and $\delta$ is the periodically modulated amplitude. The nonreciprocal coupling strengths $\pm \Delta$ between chain $\textrm {I}$ and chain $\textrm {II}$ depend on the direction of particles hopping. The hopping from the site $a_{j}$ to $b_{1}$ corresponds to the coupling strength $\Delta$. Instead, the hopping from the site $b_{1}$ to $a_{j}$ corresponds to the coupling strength $-\Delta$.Obviously, in the case of $\Delta =0$, the 1D non-Hermitian system is decoupled into two independent chains which possess SSH model structure with qubit-assisted on-site potential $\xi$. Under the open boundary condition, this model has topologically nontrivial phase in the regime of $-\pi /2< \theta < \pi /2$ characterized by the gapless zero-energy edge states with even-numbered sites. While, in the regimes $-\pi < \theta < -\pi /2$ and $\pi /2< \theta < \pi$, the system without zero-energy edge states belongs to the topologically trivial phase. For odd-numbered sites, this model holds a single zero-energy edge state in the whole parameter regions $\theta$. This is the typical even-odd effect of the SSH model.

 

Fig. 1. Schematic diagram of 1D nonreciprocal coupled SSH chain with alternating coupling strength.

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2.1 Energy eigenvalue spectrum analysis

In this section, we focus on the influence of nonreciprocal coupling strength on the system. We numerically calculate the energy eigenvalue spectrum of the system and present them as a function of $\theta$ with $L_\textrm {I}=50$ and $L_\textrm {II}=50$, as shown in Fig. 2. Here, $t$ is taken as the unit of energy, and the periodically modulated amplitude $\delta$ is set to be 0.5 throughout this paper.

 

Fig. 2. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\theta$ for 1D nonreciprocal coupled SSH chain, where we set $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the nonreciprocal coupling strengths (a) $\Delta =0.1$, (b) $\Delta =0.5$, (c) $\Delta =0.8$, (d) $\Delta =1$, (e) $\Delta =1.5$, (f) $\Delta =2$, (g) $\Delta =2.5$, and (h) $\Delta =3$, respectively.

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We first consider the topologically nontrivial regime $-\pi /2< \theta < \pi /2$. In the case of $\Delta =0.1$, the real part of the energy eigenvalue spectrum has a similar structure comparing with the standard SSH model, as shown in Fig. 2(a). The system possesses midgap modes in this regime. After checking the imaginary part of the energy eigenvalue spectrum, we find that the system holds two degenerate zero-energy edge states, and the energy eigenvalues of the rest ($L_\textrm {I}+L_\textrm {II}-2$) eigenstates are complex value. Notably, there also exists two interface states with pure imaginary energy eigenvalues (the real parts are degenerate at $E=0$), and the maximum and minimum absolute values of imaginary part occur at $\theta =0$ and $\theta =\pm \pi /2$, respectively. With the increase of $\Delta$, these two interface states gradually disappear and finally become two bound states with pure imaginary energy eigenvalues. While, the two degenerate zero-energy edge states can still exist. It is indicated that these zero-energy edge states are immune to the variation of the $\Delta$, as shown in Figs. 2(b)–2(h).

Next, we investigate the energy eigenvalue spectrum of the system in the topologically trivial regimes, i.e., $-\pi < \theta < -\pi /2$ and $\pi /2< \theta < \pi$. For $\Delta ~\le ~0.5$, the energy eigenvalue spectrum of the system possesses ($L_\textrm {I}+L_\textrm {II}$) complex energy eigenvalues with the form of $\pm a \pm ib$, as shown in Figs. 2(a)–2(b). In Figs. 2(c) and 2(d), we depict the energy eigenvalue spectrum of the system with $\Delta =0.8$ and $\Delta =1$, respectively. There exists four bound states with complex energy eigenvalue. The maximum absolute values of imaginary part appear at $\theta =\pm \pi$. However, with the increase of $\Delta$, the minimum absolute values of imaginary part gradually spread to the phase boundary points $\theta =-\pi /2$ and $\theta =\pi /2$, and finally reach to them for $\Delta =1$. When $\Delta =1.5$, one can see that the imaginary parts of energy eigenvalue spectrum split into two branches nearby the phase boundary points $\theta =\pm \pi /2$, as shown in Fig. 2(e). To continue to increase $\Delta$, these branches with pure imaginary energy eigenvalue gradually spread to $\theta =-\pi$ and $\theta =\pi$ and finally form two new interface states and two bound states with pure imaginary energy eigenvalues (the real parts are degenerate at $E=0$), as shown in Figs. 2(f)–2(h).

To see the change of energy eigenvalue spectrum of the system more clearly, we plot the real and imaginary parts of them as a function of the nonreciprocal coupling strength $\Delta$ for the system in different topological phases, as shown in Fig. 3. In Fig. 3(a), we show the energy eigenvalue spectrum of the system with $\theta =0$, and the system is in the topologically nontrivial phase. It is found that the system holds two degenerate zero-energy edge states in the whole parameter regions $\Delta$. Notably, one pair of imaginary part separates from the bulk states as long as $\Delta \not =0$, and the absolute value of energy eigenvalue presents an upward trend with the increase of $\Delta$, which corresponds to two interface states of the system (the real parts are degenerate at $E=0$). For $\theta =\pi /2$, this phenomenon begins to emerge at the phase boundary point when $\Delta > \Delta _{c,\pi /2}~=~1$, as shown in Fig. 3(b). For $\theta =\pi$, i.e., in the topologically trivial regime, one can see that two pairs of imaginary parts separate from the bulk states at $\Delta _{c_{1},\pi }=0.52$, which correspond to two pairs of degenerate bound states with complex energy eigenvalues. When the nonreciprocal coupling strength $\Delta _{c_{2},\pi }> 3$, these degenerate bound states with complex energy eigenvalues split into two branches and become two new interface states and two bound states with pure imaginary energy eigenvalues (the real parts are degenerate at $E=0$).

 

Fig. 3. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\Delta$ for 1D nonreciprocal coupled SSH chain. Here, we set $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the periodic parameters (a) $\theta =0$, (b) $\theta =\pi /2$, and (c) $\theta =\pi$, respectively.

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Furthermore, we also introduce the degree of complex energy levels, which is defined as the number of complex energy levels over the total number of energy levels. It can be used to indirectly determine the number of the zero energy existing in multiple degenerate levels. As shown in Fig. 4(a), we plot the degree of complex energy levels of the system as a function of nonreciprocal coupling strength $\Delta$ for choosing $\theta =0$. We note that the degree of complex energy levels is equal to 0.98 in the whole parameters regions $\Delta$. It is indicated that the system exists 98 complex energy levels, and the remaining two energy levels correspond to the two degenerate zero-energy edge states in the topologically nontrivial regime. Then, we show the degree of complex energy levels with $\theta =\pi /2$ and $\theta =\pi$, as shown in Figs. 4(b) and 4(c). We find that the degree of complex energy levels are both equal to 1. It is found that the corresponding energy eigenvalues both in the phase boundary and the topologically trivial phase are entirely complex energy eigenvalue. These results are consistent with previous analyses.

 

Fig. 4. The degree of complex energy levels of the system, where $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the periodic parameters (a) $\theta =0$, (b) $\theta =\pi /2$, and (c) $\theta =\pi$, respectively. The degrees of complex energy levels with blue line and pink line are equal 0.98 and 1.

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2.2 State distributions analysis

In above discussion, we have investigated the energy eigenvalue spectrum of the system. It is found that the nonreciprocal coupling can induce different localized photonic states with special energy eigenvalues, such as zero-energy edge states, interface states and bound states with pure imaginary energy eigenvalues. In the following, we analyze these localized photonic states of the system from the perspective of state distributions. In Fig. 5, we plot the state distributions of the system versus the lattice site $j$ and the nonreciprocal coupling strength $\Delta$ in different topological regimes. We first consider the state distributions with $\theta =0.35\pi$, as shown in Fig. 5(a). One can see that the interference fringes appear around the 1st and 100th lattice site and gradually disappear towards the center lattice site. Meanwhile, these interference fringes are not affected by the variation of nonreciprocal coupling strength. This phenomenon characterizes that the system holds two edge states steadily localized at two ends of system, which are immune to the effects of nonreciprocal coupling strength. On the other hand, for small $\Delta$, the same phenomena appear around the two middle lattice sites, which correspond to the left and right interface states of the system. With the increase of $\Delta$, we find that these interference fringes gradually disappear, and the maximum values of state distributions are in the 50th and 51st lattice sites. This process corresponds to the formation of two bound states. As shown in Fig. 5(b), we depict the state distributions of the system versus the lattice site $j$ and the nonreciprocal coupling strength $\Delta$ with $\theta =0.65\pi$. For $\Delta < 2$, the state distribution at each site are almost equal to 0, i.e., all the energy eigenstates are extended states. For $\Delta > 2$, we find that the interference fringes appear around the 49th and 51st lattice sites and gradually disappear towards the two boundaries of the system, which characterizes the formation of new left and right interface states of the system. In addition, the two bound states are always localized at the two middle lattice sites.

 

Fig. 5. (a) The state distributions $|\psi _{(50)}|^{2}$, $|\psi _{(51)}|^{2}$ (two red zero-energy edge states in Fig. 2) and $|\psi _{(49)}|^{2}$, $|\psi _{(52)}|^{2}$ (two red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically nontrivial regime $\theta =0.35\pi$. (b) The state distributions $|\psi _{(50)}|^{2}$, $|\psi _{(51)}|^{2}$ (two red bound states in Fig. 2) and $|\psi _{(49)}|^{2}$, $|\psi _{(52)}|^{2}$ (two new red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically trivial regime $\theta =0.65\pi$. The other parameters are selected as $L_\textrm {I}=50$, $L_\textrm {II}=50$, and $\delta =0.5$, respectively.

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In the following, we present the state distributions of localized photonic states in nonreciprocal coupled SSH chain. As shown in Fig. 6(a), we find that two degenerate zero-energy edge states are highly localized at both ends of the system with $\theta =0$ and $\Delta =0.1$. Meanwhile, the system possesses two interface states with pure imaginary energy eigenvalues $+0.0889i$ and $-0.0889i$ at two middle lattice sites. In Fig. 6(b) we plot the state distributions of the localized states with $\theta =0$ and $\Delta =3$. It is noted that two degenerate zero-energy edge states are always highly localized at both ends of the system. However, the original two interface states become the two bound states with pure imaginary energy eigenvalues $+2.932i$ and $-2.932i$, which are localized at two middle lattice sites. Next, we consider the state distributions of the bound states with $\theta =\pi$ and $\Delta =0.8$, as shown in Fig. 6(c). Two pairs of degenerate bound states are localized at 49th, 50th, 51st, and 52th lattice sites with complex energy eigenvalues $1.3930+0.2437i$, $-1.3930-0.2437i$, $1.3930+0.2437i$, and $-1.3930-0.2437i$, respectively. In Fig. 6(d), we show the state distributions of the bound states $\theta =\pi$ and $\Delta =3$. In Fig. 6(d), the original bound states become two bound states with pure imaginary energy eigenvalues $1.102i$ and $-1.102i$, which are localized at two middle lattice sites. In addition, two new interface states appear at the 49th and 52th lattice sites with pure imaginary energy eigenvalues $1.184i$ and $-1.184i$.

 

Fig. 6. The state distributions of localized photonic states for the system, where $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$. (a) $\theta =0$ and $\Delta =0.1$, (b) $\theta =0$ and $\Delta =3$, (c) $\theta =\pi$ and $\Delta =0.8$, (d) $\theta =\pi$ and $\Delta =3$. The red line, blue line, blue dotted line, green line, and green dotted line represent the state distributions of the zero-energy edge state, interface states with pure imaginary energy eigenvalues, new interface states with pure imaginary energy eigenvalues, bound states with complex energy eigenvalues, and bound states with pure imaginary energy eigenvalues, respectively.

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3. Dynamic process in 1D nonreciprocal coupled SSH chain

In this section, we investigate the dynamic process of the photonic states in nonreciprocal coupled SSH chain. The parameters are chosen as $\Delta =0.5$, $\theta =0$, $L_\textrm {I}=10$, and $L_\textrm {II}=10$, as shown in Fig. 7. In Fig. 7(a), when the 10th lattice site is excited, we find that the photons are localized on the two middle lattice sites with time evolution, which correspond to the site position of nonreciprocal coupling for the system. Then, when we excite the 5th and 15th lattice sites belonging to the bulk sites, the dynamic process of photonic states firstly presents the ballistic diffusion and finally shows the localization on the two middle lattice sites of the system, as shown in Figs. 7(b) and 7(c). In Figs. 7(d) and 7(e), for exciting the one and two lattice sites at the boundary, the photons are firstly trapped on the boundary with time evolution and finally localized on the two middle lattice sites of the system. When all the lattice sites are excited, one can see that all the photons are equally localized on the two middle lattice sites with time evolution. The above phenomena indicate that the nonreciprocal coupling of the system induces the gathering effect of the photons, which makes the dynamic process of the photons states towards the site position of nonreciprocal coupling.

 

Fig. 7. The dynamic process of the photonic states in 1D nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, and $\Delta =0.5$, respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively.

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Next, we show the dynamic process of photonic states with strong nonreciprocal coupling strength $\Delta =3$. The selected initial states and other parameters are same as that in Fig. 7. It is found that the gathering effect of the photons still appear in the system when we excite the lattice sites at the middle site, the bulk site, and the boundary site, respectively, as shown in Fig. 8. Apparently, compared with Fig. 7, the evolution time needed for the localization of dynamic process is reduced significantly with the increase of nonreciprocal coupling strength. In other words, the strong nonreciprocal coupling strength enhances the gathering effect of the photons.

 

Fig. 8. The dynamic process of the photonic states in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, and $\Delta =3$, respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively.

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Furthermore, we show the dynamic process for exciting the lattice sites at the boundary in Hermitian SSH chain, correspondingly, the interchain coupling term can be described by $H_\textrm {I-II}=\Delta a_{L}b^{\dagger }_{1}+\Delta b_{1}a^{\dagger }_{L}$. For $\Delta =0.5$, one can see that the photons are always localized at the boundary sites with time evolution. Even in the strong coupling condition with $\Delta =3$, the dynamic process of photonic states also presents the localization on two boundary sites of the system, as shown in Fig. 9(b). This is because that, in such Hermitian system, the localization properties own to the topologically protected edge states, which have trapping effect on the photons. Obviously, compared with Figs. 7(d)–7(e) and 8(d)–8(e), we find that the results of dynamic process of Hermitian system are different from that in non-Hermitian system. It indicates that the nonreciprocal coupling breaks the trapping effect of the edge states.

 

Fig. 9. The dynamic process of the photonic states in Hermitian SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, (a) $\Delta =0.5$, and (b) $\Delta =3$. The excited positions are both the 1st and 20th lattice sites for Figs. (a)-(b).

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In the following, we investigate the influence of on-site defect potentials on the dynamic process of photonic states in nonreciprocal coupled SSH chain. The parameters are chosen as $\Delta =0.5$, $\theta =0$, $L_\textrm {I}=10$, $L_\textrm {II}=10$, and the initial state of the system is located at the 1st lattice site. In Fig. 10(a), consider that the defect is at the 1st lattice site, i.e., the left boundary site, one can see that the evolution time for the localization of dynamic process is longer than that without defect potential in Fig. 7(d). When the on-site defect potential is at the 2th lattice site, we find that the photons go around the defect lattice site, and they are finally localized on the two middle lattice sites corresponding to the site position of nonreciprocal coupling, as shown in Fig. 10(b). The same phenomena also occur for the defect potentials are in the other bulk sites ($j=3$ and $j=4$), as shown in Fig. 10(c)–10(d). It indicates that the gathering effect of the photons caused by the nonreciprocal coupling is not affected by the on-site defect potentials. Notably, we find that different forms of laser pulses can be induced around the left boundary site with the on-site defect potentials in different positions. However, the photons are finally localized on the two middle lattice sites with time evolution because of the nonreciprocal coupling.

 

Fig. 10. The dynamic process of the photonic states with on-site defect potentials in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $t=1$, $\theta =0$, and $\Delta =0.5$, respectively.The excited positions are the 1st, 2nd, 3rd, and 4th lattice sites, corresponding to Figs. (a)-(d), respectively.

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4. Realization of quantum simulation for 1D nonreciprocal coupled SSH chain

In this section, we briefly present the quantum simulation of the nonreciprocal SSH chain. We consider an array of transmission line resonators including chain I and chain II, in which these two chains have a nonreciprocal coupling with the strengths $\pm \Delta$. Moreover, we consider that each transmission line resonator is coupled to a two-level superconducting flux qubit represented by excited state $|e\rangle$ and ground state $|g\rangle$. The corresponding Hamiltonian of the system can be written as ($\hbar =1$)

$$H=H_\textrm{I}+H_\textrm{II}+H_\textrm{I-II},$$
with
$$\begin{aligned} H_\textrm{I}=&\sum_{j=1}^{L}\left(\frac{\omega_{1}}{2}\sigma_{z, j}^{a}+\omega_{a}a_{j}^{\dagger}a_{j}+g_{a,j}\sigma_{a,j}^{+}a_{j}+g_{a,j}\sigma_{a,j}^{-}a_{j}^{\dagger}\right)\cr\cr &+\sum_{j=1}^{L-2}\left(t_{1}a_{j}^{\dagger}a_{j+1}+t_{2}a_{j+1}^{\dagger}a_{j+2}+\rm{H.c.}\right),\cr\cr H_\textrm{II}=&\sum_{j=1}^{L}\left(\frac{\omega_{2}}{2}\sigma_{z, j}^{b}+\omega_{b}b_{j}^{\dagger}b_{j}+g_{b,j}\sigma_{b,j}^{+}b_{j}+g_{b,j}\sigma_{b,j}^{-}b_{j}^{\dagger}\right)\cr\cr &+\sum_{j=1}^{L-2}\left(t_{1}b_{j}^{\dagger}b_{j+1}+t_{2}b_{j+1}^{\dagger}b_{j+2}+\rm{H.c.}\right),\cr\cr H_\textrm{I-II}=&\Delta a_{L}b_{1}^{\dagger}-\Delta b_{1}a_{L}^{\dagger}, \end{aligned}$$
where $a_{j}$ ($a_{j}^{\dagger }$) and $b_{j}$ ($b_{j}^{\dagger }$) are the annihilation (creation) operators, $\sigma _{a,j}^{+}$ ($\sigma _{a,j}^{-}$) and $\sigma _{b,j}^{+}$ ($\sigma _{b,j}^{-}$) are the raising (lowering) Pauli operators, $\sigma _{z, j}^{a}$ and $\sigma _{z, j}^{b}$ are the Pauli $z$ operators, $\omega _{1}$ and $\omega _{2}$ represent the transition frequencies between the excited state $|e\rangle$ and the ground state $|g\rangle$ of the resonators $Q_{j}^{a}$ and $Q_{j}^{b}$, $\omega _{a}$ and $\omega _{b}$ are the frequencies of the resonator $a_{j}$ and $b_{j}$, $g_{a,j}$ and $g_{b,j}$ denote the qubit-resonator coupling strengths, $t_{1}$ and $t_{2}$ are the nearest neighbor hopping amplitudes between the resonators. The nonreciprocal coupling device in Fig. 11 can be implemented by using a long Josephson junction operates in the flux-flow regime [55]. More specifically, the preferred direction of the electromagnetic wave can be created by combining the polarities of the bias current with magnetic field. The propagation is facilitated when the external microwave propagates along the direction of flux flow. On the contrary, the propagation is damped in opposite direction in a long Josephson junction.

 

Fig. 11. Schematic diagram of 1D non-Hermitian circuit QED lattice system consisting of an array of transmission line resonators denoted by $a_{j}$ and $b_{j}$, each of them coupled to a two-level superconducting flux qubits $Q_{j}^{a}$ and $Q_{j}^{b}$. The nonreciprocal coupling between chain $\textrm {I}$ and chain $\textrm {II}$ can be implemented by using a long Josephson junction operates in the flux-flow regime.

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In the rotating frame with respect to the driving field frequency $\omega _\textrm {d}$, and all the superconducting flux qubits are prepared in their ground states, the effective Hamiltonian of the system is given by

$$\begin{aligned} H^{'}=&\sum_{j=1}^{L-2}\left(t_{1}a_{j}^{\dagger}a_{j+1}+t_{2}a_{j+1}^{\dagger}a_{j+2}+t_{1}b_{j}^{\dagger}b_{j+1}+t_{2}b_{j+1}^{\dagger}b_{j+2}+\rm{H.c.}\right)\cr\cr &+\sum_{j=1}^{L}\left(\Delta_{a}-\frac{g_{a,j}^{2}}{\Delta_{1}}\right)a_{j}^{\dagger}a_{j}+\sum_{j=1}^{L}\left(\Delta_{b}-\frac{g_{b,j}^{2}}{\Delta_{2}}\right)b_{j}^{\dagger}b_{j}\cr\cr &+\Delta a_{L}b_{1}^{\dagger}-\Delta b_{1}a_{L}^{\dagger}, \end{aligned}$$
where $\Delta _{a}=\omega _{a}-\omega _{d}$, $\Delta _{b}=\omega _{b}-\omega _{d}$, $\Delta _{1}=\omega _{1}-\omega _{d}$, and $\Delta _{2}=\omega _{2}-\omega _{d}$. Setting the parameters as follows: $\Delta _{a}-\frac {g_{a,j}^{2}}{\Delta _{1}}=\Delta _{b}-\frac {g_{b,j}^{2}}{\Delta _{2}}=\xi$, $t_{1}=t\left (1-\delta \cos \theta \right )$, and $t_{2}=t\left (1+\delta \cos \theta \right )$. In this parameter regime, the Hamilton in Eq. (4) becomes
$$\begin{aligned} H^{\prime\prime}=&\sum_{j=1}^{L-2}\Big[\left(1-\delta\cos\theta\right)a_{j}^{\dagger}a_{j+1}+\left(1+\delta\cos\theta\right)a_{j+1}^{\dagger}a_{j+2}\cr\cr &+\left(1-\delta\cos\theta\right)b_{j}^{\dagger}b_{j+1}+\left(1+\delta\cos\theta\right)b_{j+1}^{\dagger}b_{j+2}+\rm{H.c.}\Big]\cr\cr &+\sum_{j=1}^{L}\xi a_{j}^{\dagger}a_{j}+\sum_{j=1}^{L}\xi b_{j}^{\dagger}b_{j}+\Delta a_{L}b_{1}^{\dagger}-\Delta b_{1}a_{L}^{\dagger}. \end{aligned}$$
Obviously, the above Hamiltonian corresponds the implementation of nonreciprocal coupled SSH chain based on the circuit QED lattice.

5. Conclusions

In conclusion, we have investigated the localized photonic states induced by the nonreciprocal coupling and dynamic process of photonic states in 1D nonreciprocal coupled SSH chain. Through analyzing the energy eigenvalue spectrum of the system, we find that different localized photonic states can be induced by the nonreciprocal coupling. In the topologically nontrivial regime, the interface states with pure imaginary energy eigenvalues gradually disappear and finally become the bound states with pure imaginary energy eigenvalues with the increase of the coupling strength. However, the zero-energy edge states are robust to the variation of nonreciprocal coupling, which are highly localized on two ends of the system. Meanwhile, two pairs of degenerate bound states with complex energy eigenvalues become two bound states and two new interface states with pure imaginary energy eigenvalues, respectively, in the topologically trivial regime. Further, we analyze the dynamic process of photonic states in such non-Hermitian system. Interestingly, it is shown that the nonreciprocal coupling has an evident gathering effect on the photons, which breaks the trapping effect of topologically protected edge states. In addition, we consider the impacts of on-site defect potentials on the dynamic process of the system. It is shown that the photons go around the on-site defect lattice site and still present the gathering effect. Particularly, different forms of laser pulses can be induced with the on-site defect potentials in different lattice sites. Moreover, we have presented a simple method to realize the quantum simulation of nonreciprocal coupled SSH chain. Our scheme provide a new approach to investigate the localized photonic states and the dynamic process in non-Hermitian system.

Funding

National Natural Science Foundation of China (11874132, 61575055, 61822114).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979). [CrossRef]  

2. L. Li, Z. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014). [CrossRef]  

3. X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017). [CrossRef]  

4. X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020). [CrossRef]  

5. E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016). [CrossRef]  

6. F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018). [CrossRef]  

7. F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018). [CrossRef]  

8. D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019). [CrossRef]  

9. C. Liu, M. V. Gurudev Dutt, and D. Pekker, “Robust manipulation of light using topologically protected plasmonic modes,” Opt. Express 26(3), 2857–2872 (2018). [CrossRef]  

10. C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017). [CrossRef]  

11. C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A 97(4), 042118 (2018). [CrossRef]  

12. K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018). [CrossRef]  

13. Y. Kuno, “Phase structure of the interacting Su-Schrieffer-Heeger model and the relationship with the Gross-Neveu model on lattice,” Phys. Rev. B 99(6), 064105 (2019). [CrossRef]  

14. H. L. Wang, L. W. Zhou, and J. B. Gong, “Interband coherence induced correction to adiabatic pumping in periodically driven systems,” Phys. Rev. B 91(8), 085420 (2015). [CrossRef]  

15. R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020). [CrossRef]  

16. S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018). [CrossRef]  

17. L. Jin and Z. Song, “Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry,” Phys. Rev. B 99(8), 081103 (2019). [CrossRef]  

18. T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020). [CrossRef]  

19. L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020). [CrossRef]  

20. F. Song, S. Yao, and Z. Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” Phys. Rev. Lett. 123(17), 170401 (2019). [CrossRef]  

21. H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019). [CrossRef]  

22. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020). [CrossRef]  

23. D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017). [CrossRef]  

24. C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018). [CrossRef]  

25. T. Yoshida, K. Kudo, and Y. Hatsugai, “Non-Hermitian fractional quantum hall states,” Sci. Rep. 9(1), 16895 (2019). [CrossRef]  

26. S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020). [CrossRef]  

27. M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017). [CrossRef]  

28. Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017). [CrossRef]  

29. C. Yuce and Z. Oztas, “Pt symmetry protected non-Hermitian topological systems,” Sci. Rep. 8(1), 17416 (2018). [CrossRef]  

30. X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$ symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020). [CrossRef]  

31. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019). [CrossRef]  

32. M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018). [CrossRef]  

33. G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018). [CrossRef]  

34. M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018). [CrossRef]  

35. A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020). [CrossRef]  

36. N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013). [CrossRef]  

37. P. M. Billangeon, J. S. Tsai, and Y. Nakamura, “Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits,” Phys. Rev. B 91(9), 094517 (2015). [CrossRef]  

38. Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017). [CrossRef]  

39. T. Liu, B. Q. Guo, C. S. Yu, and W. N. Zhang, “One-step implementation of a hybrid Fredkin gate with quantum memories and single superconducting qubit in circuit QED and its applications,” Opt. Express 26(4), 4498–4511 (2018). [CrossRef]  

40. L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019). [CrossRef]  

41. A. J. X. Han, J. L. Wu, Y. Wang, Y. Y. Jiang, Y. Xia, and J. Song, “Multi-qubit phase gate on multiple resonators mediated by a superconducting bus,” Opt. Express 28(2), 1954–1969 (2020). [CrossRef]  

42. J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020). [CrossRef]  

43. S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020). [CrossRef]  

44. Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017). [CrossRef]  

45. M. D. Kim and J. Kim, “Scalable quantum computing model in the circuit-QED lattice with circulator function,” Quantum Inf. Process. 16(8), 192 (2017). [CrossRef]  

46. J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013). [CrossRef]  

47. L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018). [CrossRef]  

48. L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010). [CrossRef]  

49. F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015). [CrossRef]  

50. C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019). [CrossRef]  

51. J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010). [CrossRef]  

52. J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with purcell protection and tunable coupling,” Phys. Rev. Lett. 106(3), 030502 (2011). [CrossRef]  

53. Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014). [CrossRef]  

54. Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015). [CrossRef]  

55. K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012). [CrossRef]  

References

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  1. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
    [Crossref]
  2. L. Li, Z. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
    [Crossref]
  3. X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017).
    [Crossref]
  4. X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
    [Crossref]
  5. E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
    [Crossref]
  6. F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
    [Crossref]
  7. F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018).
    [Crossref]
  8. D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019).
    [Crossref]
  9. C. Liu, M. V. Gurudev Dutt, and D. Pekker, “Robust manipulation of light using topologically protected plasmonic modes,” Opt. Express 26(3), 2857–2872 (2018).
    [Crossref]
  10. C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017).
    [Crossref]
  11. C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A 97(4), 042118 (2018).
    [Crossref]
  12. K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018).
    [Crossref]
  13. Y. Kuno, “Phase structure of the interacting Su-Schrieffer-Heeger model and the relationship with the Gross-Neveu model on lattice,” Phys. Rev. B 99(6), 064105 (2019).
    [Crossref]
  14. H. L. Wang, L. W. Zhou, and J. B. Gong, “Interband coherence induced correction to adiabatic pumping in periodically driven systems,” Phys. Rev. B 91(8), 085420 (2015).
    [Crossref]
  15. R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020).
    [Crossref]
  16. S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
    [Crossref]
  17. L. Jin and Z. Song, “Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry,” Phys. Rev. B 99(8), 081103 (2019).
    [Crossref]
  18. T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
    [Crossref]
  19. L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
    [Crossref]
  20. F. Song, S. Yao, and Z. Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” Phys. Rev. Lett. 123(17), 170401 (2019).
    [Crossref]
  21. H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
    [Crossref]
  22. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020).
    [Crossref]
  23. D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
    [Crossref]
  24. C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
    [Crossref]
  25. T. Yoshida, K. Kudo, and Y. Hatsugai, “Non-Hermitian fractional quantum hall states,” Sci. Rep. 9(1), 16895 (2019).
    [Crossref]
  26. S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
    [Crossref]
  27. M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
    [Crossref]
  28. Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
    [Crossref]
  29. C. Yuce and Z. Oztas, “Pt symmetry protected non-Hermitian topological systems,” Sci. Rep. 8(1), 17416 (2018).
    [Crossref]
  30. X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$PT symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020).
    [Crossref]
  31. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
    [Crossref]
  32. M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
    [Crossref]
  33. G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
    [Crossref]
  34. M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
    [Crossref]
  35. A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
    [Crossref]
  36. N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013).
    [Crossref]
  37. P. M. Billangeon, J. S. Tsai, and Y. Nakamura, “Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits,” Phys. Rev. B 91(9), 094517 (2015).
    [Crossref]
  38. Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
    [Crossref]
  39. T. Liu, B. Q. Guo, C. S. Yu, and W. N. Zhang, “One-step implementation of a hybrid Fredkin gate with quantum memories and single superconducting qubit in circuit QED and its applications,” Opt. Express 26(4), 4498–4511 (2018).
    [Crossref]
  40. L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
    [Crossref]
  41. A. J. X. Han, J. L. Wu, Y. Wang, Y. Y. Jiang, Y. Xia, and J. Song, “Multi-qubit phase gate on multiple resonators mediated by a superconducting bus,” Opt. Express 28(2), 1954–1969 (2020).
    [Crossref]
  42. J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
    [Crossref]
  43. S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
    [Crossref]
  44. Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
    [Crossref]
  45. M. D. Kim and J. Kim, “Scalable quantum computing model in the circuit-QED lattice with circulator function,” Quantum Inf. Process. 16(8), 192 (2017).
    [Crossref]
  46. J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013).
    [Crossref]
  47. L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
    [Crossref]
  48. L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
    [Crossref]
  49. F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
    [Crossref]
  50. C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019).
    [Crossref]
  51. J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
    [Crossref]
  52. J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with purcell protection and tunable coupling,” Phys. Rev. Lett. 106(3), 030502 (2011).
    [Crossref]
  53. Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
    [Crossref]
  54. Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
    [Crossref]
  55. K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012).
    [Crossref]

2020 (11)

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020).
[Crossref]

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020).
[Crossref]

X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$PT symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020).
[Crossref]

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

A. J. X. Han, J. L. Wu, Y. Wang, Y. Y. Jiang, Y. Xia, and J. Song, “Multi-qubit phase gate on multiple resonators mediated by a superconducting bus,” Opt. Express 28(2), 1954–1969 (2020).
[Crossref]

J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
[Crossref]

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

2019 (9)

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

T. Yoshida, K. Kudo, and Y. Hatsugai, “Non-Hermitian fractional quantum hall states,” Sci. Rep. 9(1), 16895 (2019).
[Crossref]

F. Song, S. Yao, and Z. Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” Phys. Rev. Lett. 123(17), 170401 (2019).
[Crossref]

H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
[Crossref]

L. Jin and Z. Song, “Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry,” Phys. Rev. B 99(8), 081103 (2019).
[Crossref]

Y. Kuno, “Phase structure of the interacting Su-Schrieffer-Heeger model and the relationship with the Gross-Neveu model on lattice,” Phys. Rev. B 99(6), 064105 (2019).
[Crossref]

D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019).
[Crossref]

2018 (13)

C. Liu, M. V. Gurudev Dutt, and D. Pekker, “Robust manipulation of light using topologically protected plasmonic modes,” Opt. Express 26(3), 2857–2872 (2018).
[Crossref]

F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
[Crossref]

F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018).
[Crossref]

C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A 97(4), 042118 (2018).
[Crossref]

K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018).
[Crossref]

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

C. Yuce and Z. Oztas, “Pt symmetry protected non-Hermitian topological systems,” Sci. Rep. 8(1), 17416 (2018).
[Crossref]

T. Liu, B. Q. Guo, C. S. Yu, and W. N. Zhang, “One-step implementation of a hybrid Fredkin gate with quantum memories and single superconducting qubit in circuit QED and its applications,” Opt. Express 26(4), 4498–4511 (2018).
[Crossref]

2017 (8)

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
[Crossref]

M. D. Kim and J. Kim, “Scalable quantum computing model in the circuit-QED lattice with circulator function,” Quantum Inf. Process. 16(8), 192 (2017).
[Crossref]

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref]

X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017).
[Crossref]

C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017).
[Crossref]

2016 (1)

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
[Crossref]

2015 (4)

H. L. Wang, L. W. Zhou, and J. B. Gong, “Interband coherence induced correction to adiabatic pumping in periodically driven systems,” Phys. Rev. B 91(8), 085420 (2015).
[Crossref]

P. M. Billangeon, J. S. Tsai, and Y. Nakamura, “Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits,” Phys. Rev. B 91(9), 094517 (2015).
[Crossref]

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

2014 (2)

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

L. Li, Z. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

2013 (2)

N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013).
[Crossref]

J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013).
[Crossref]

2012 (1)

K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012).
[Crossref]

2011 (1)

J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with purcell protection and tunable coupling,” Phys. Rev. Lett. 106(3), 030502 (2011).
[Crossref]

2010 (2)

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[Crossref]

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

1979 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Abdelghany, M.

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Amo, A.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

An, C. S.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

An, F. A.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
[Crossref]

Bai, C. H.

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Ballester, D.

J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013).
[Crossref]

Bandres, M. A.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

Barends, R.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Billangeon, P. M.

P. M. Billangeon, J. S. Tsai, and Y. Nakamura, “Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits,” Phys. Rev. B 91(9), 094517 (2015).
[Crossref]

Blais, A.

J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with purcell protection and tunable coupling,” Phys. Rev. Lett. 106(3), 030502 (2011).
[Crossref]

Bliokh, K. Y.

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref]

Campbell, B.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Cao, J.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Cartarius, H.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
[Crossref]

Carusotto, I.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Chen, G.

F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
[Crossref]

Chen, S.

H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
[Crossref]

C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
[Crossref]

L. Li, Z. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

Chen, Y.

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Chen, Z.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Chiaro, B.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Chong, Y. D.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017).
[Crossref]

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref]

Chow, J. M.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Christodoulides, D. N.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

Cleland, A. N.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Daniilidis, N.

N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013).
[Crossref]

Dast, D.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
[Crossref]

DeGottardi, W.

K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018).
[Crossref]

Deng, T.

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

Devoret, M. H.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Di Candia, R.

J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013).
[Crossref]

DiCarlo, L.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Dunsworth, A.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Dutt, A.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Fan, S.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Fang, M.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Fazio, R.

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

Fedorov, K. G.

K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012).
[Crossref]

Feng, M.

J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
[Crossref]

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

Feng, W.

C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019).
[Crossref]

Filipp, S.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Frunzio, L.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Gadway, B.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
[Crossref]

Gambetta, J. M.

J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with purcell protection and tunable coupling,” Phys. Rev. Lett. 106(3), 030502 (2011).
[Crossref]

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Geller, M. R.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Girvin, S. M.

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[Crossref]

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Goldman, N.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Gong, J. B.

H. L. Wang, L. W. Zhou, and J. B. Gong, “Interband coherence induced correction to adiabatic pumping in periodically driven systems,” Phys. Rev. B 91(8), 085420 (2015).
[Crossref]

Gong, W. J.

X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$PT symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020).
[Crossref]

Gorman, D. J.

N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013).
[Crossref]

Greiter, M.

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Guo, B. Q.

Guo, C.

Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
[Crossref]

Guo, F. Q.

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

Gurudev Dutt, M. V.

Hafezi, M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Häffner, H.

N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013).
[Crossref]

Han, A. J. X.

Harari, G.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

Hatsugai, Y.

T. Yoshida, K. Kudo, and Y. Hatsugai, “Non-Hermitian fractional quantum hall states,” Sci. Rep. 9(1), 16895 (2019).
[Crossref]

Heeger, A. J.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Hegde, S. S.

K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018).
[Crossref]

Heinsoo, J.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Helbig, T.

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

Hodaei, H.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

Hofmann, T.

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Houck, A. A.

J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with purcell protection and tunable coupling,” Phys. Rev. Lett. 106(3), 030502 (2011).
[Crossref]

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[Crossref]

Huang, B. H.

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

Huang, C.

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref]

Hur, K. L.

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[Crossref]

Imhof, S.

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Jeffrey, E.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Jia, S.

F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
[Crossref]

Jiang, H.

H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
[Crossref]

C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
[Crossref]

Jiang, L.

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

Jiang, X. X.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Jiang, Y. Y.

Jin, L.

L. Jin and Z. Song, “Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry,” Phys. Rev. B 99(8), 081103 (2019).
[Crossref]

Johnson, B. R.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Kang, Y. H.

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

Kawabata, K.

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020).
[Crossref]

Kelly, J.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Khajavikhan, M.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

Kiessling, T.

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Kim, J.

M. D. Kim and J. Kim, “Scalable quantum computing model in the circuit-QED lattice with circulator function,” Quantum Inf. Process. 16(8), 192 (2017).
[Crossref]

Kim, M. D.

M. D. Kim and J. Kim, “Scalable quantum computing model in the circuit-QED lattice with circulator function,” Quantum Inf. Process. 16(8), 192 (2017).
[Crossref]

Klett, M.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
[Crossref]

Koch, J.

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[Crossref]

Kremer, M.

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

Kudo, K.

T. Yoshida, K. Kudo, and Y. Hatsugai, “Non-Hermitian fractional quantum hall states,” Sci. Rep. 9(1), 16895 (2019).
[Crossref]

Kuno, Y.

Y. Kuno, “Phase structure of the interacting Su-Schrieffer-Heeger model and the relationship with the Gross-Neveu model on lattice,” Phys. Rev. B 99(6), 064105 (2019).
[Crossref]

Kurpiers, P.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Kwek, L. C.

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

Lamata, L.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Lang, L. J.

H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
[Crossref]

Las Heras, U.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Lee, C. H.

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Leung, N.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Leykam, D.

X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017).
[Crossref]

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref]

Li, C.

C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017).
[Crossref]

Li, L.

C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
[Crossref]

L. Li, Z. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

Li, R.

R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020).
[Crossref]

Li, X. Q.

C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019).
[Crossref]

Li, X. S.

X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$PT symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020).
[Crossref]

Li, Z. Z.

X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$PT symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020).
[Crossref]

Liang, E. J.

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

Lin, S.

C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017).
[Crossref]

Liu, C.

Liu, F.

D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019).
[Crossref]

Liu, J. X.

J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
[Crossref]

Liu, T.

Lu, L.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Lü, R.

C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
[Crossref]

Lumer, Y.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

Ma, L. H.

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

Main, J.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
[Crossref]

Martinis, J. M.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Megrant, A.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Mei, F.

F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
[Crossref]

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

Meier, E. J.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
[Crossref]

Mella, J.

F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018).
[Crossref]

Mezzacapo, A.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Minkov, M.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Molenkamp, L. W.

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Molina, M. I.

F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018).
[Crossref]

Mondal, M.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Munoz, F.

F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018).
[Crossref]

Mutus, J. Y.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Nakamura, Y.

P. M. Billangeon, J. S. Tsai, and Y. Nakamura, “Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits,” Phys. Rev. B 91(9), 094517 (2015).
[Crossref]

Neill, C.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Nie, W.

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

Nori, F.

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref]

O’Malley, P. J. J.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Obana, D.

D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019).
[Crossref]

Okuma, N.

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020).
[Crossref]

Oppliger, M.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Orenstein, M.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Ozawa, T.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Oztas, Z.

C. Yuce and Z. Oztas, “Pt symmetry protected non-Hermitian topological systems,” Sci. Rep. 8(1), 17416 (2018).
[Crossref]

Padavic, K.

K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018).
[Crossref]

Parto, M.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Pedernales, J. S.

J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013).
[Crossref]

Pekker, D.

Pinilla, F.

F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018).
[Crossref]

Potocnik, A.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Price, H. M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Qi, L.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Qian, J.

R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020).
[Crossref]

Quan, Y. M.

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

Quintana, C. M.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Rechtsman, M. C.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

Reed, M. D.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Ren, J.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

Rotzinger, H.

K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012).
[Crossref]

Roushan, P.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Salathé, Y.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Sank, D.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Sato, M.

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020).
[Crossref]

Schoelkopf, R. J.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Schrieffer, J. R.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Schuster, D.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Segev, M.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

Shi, Z. C.

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

Shiozaki, K.

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020).
[Crossref]

Shitov, S. V.

K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012).
[Crossref]

Simon, J.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Solano, E.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013).
[Crossref]

Song, A. Y.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Song, F.

F. Song, S. Yao, and Z. Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” Phys. Rev. Lett. 123(17), 170401 (2019).
[Crossref]

Song, J.

A. J. X. Han, J. L. Wu, Y. Wang, Y. Y. Jiang, Y. Xia, and J. Song, “Multi-qubit phase gate on multiple resonators mediated by a superconducting bus,” Opt. Express 28(2), 1954–1969 (2020).
[Crossref]

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

Song, Z.

L. Jin and Z. Song, “Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry,” Phys. Rev. B 99(8), 081103 (2019).
[Crossref]

C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017).
[Crossref]

Stegmaier, A.

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

Su, Q. P.

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

Su, S. L.

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
[Crossref]

R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020).
[Crossref]

Su, W. P.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Sun, L.

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Sun, X. Q.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Szameit, A.

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

Thomale, R.

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

Tian, L.

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
[Crossref]

N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013).
[Crossref]

Tsai, J. S.

P. M. Billangeon, J. S. Tsai, and Y. Nakamura, “Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits,” Phys. Rev. B 91(9), 094517 (2015).
[Crossref]

Ustinov, A. V.

K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012).
[Crossref]

Vainsencher, A.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Vishveshwara, S.

K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018).
[Crossref]

Wakabayashi, K.

D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019).
[Crossref]

Wallraff, A.

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Wang, D. Y.

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Wang, G.

Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
[Crossref]

Wang, H.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Wang, H. F.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Wang, H. L.

H. L. Wang, L. W. Zhou, and J. B. Gong, “Interband coherence induced correction to adiabatic pumping in periodically driven systems,” Phys. Rev. B 91(8), 085420 (2015).
[Crossref]

Wang, K.

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

Wang, Y.

A. J. X. Han, J. L. Wu, Y. Wang, Y. Y. Jiang, Y. Xia, and J. Song, “Multi-qubit phase gate on multiple resonators mediated by a superconducting bus,” Opt. Express 28(2), 1954–1969 (2020).
[Crossref]

Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
[Crossref]

X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017).
[Crossref]

Wang, Z.

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

F. Song, S. Yao, and Z. Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” Phys. Rev. Lett. 123(17), 170401 (2019).
[Crossref]

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

Weidemann, S.

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

Wenner, J.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

White, T. C.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

Williamson, I. A. D.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Wittek, S.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

Wojcik, C.

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

Wu, C.

Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
[Crossref]

Wu, J.

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

Wu, J. L.

Wu, T.

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

Wunner, G.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
[Crossref]

Xia, Y.

A. J. X. Han, J. L. Wu, Y. Wang, Y. Y. Jiang, Y. Xia, and J. Song, “Multi-qubit phase gate on multiple resonators mediated by a superconducting bus,” Opt. Express 28(2), 1954–1969 (2020).
[Crossref]

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

Xiao, L.

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

Xing, Y.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Xu, Z.

L. Li, Z. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

Xue, P.

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

Yan, L. L.

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
[Crossref]

Yang, C.

H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
[Crossref]

Yang, C. P.

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

Yao, S.

F. Song, S. Yao, and Z. Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” Phys. Rev. Lett. 123(17), 170401 (2019).
[Crossref]

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

Ye, J. Y.

J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
[Crossref]

Yi, W.

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

Yin, C.

C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
[Crossref]

Yoshida, T.

T. Yoshida, K. Kudo, and Y. Hatsugai, “Non-Hermitian fractional quantum hall states,” Sci. Rep. 9(1), 16895 (2019).
[Crossref]

You, J. B.

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

Yu, C. S.

Yu, D. M.

R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020).
[Crossref]

Yu, L.

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

Yu, X. L.

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

Yuce, C.

C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A 97(4), 042118 (2018).
[Crossref]

C. Yuce and Z. Oztas, “Pt symmetry protected non-Hermitian topological systems,” Sci. Rep. 8(1), 17416 (2018).
[Crossref]

Zhang, C.

C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019).
[Crossref]

Zhang, G.

C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017).
[Crossref]

Zhang, G. Q.

Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
[Crossref]

Zhang, L. L.

X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$PT symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020).
[Crossref]

Zhang, S.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Zhang, W. N.

Zhang, Y.

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

Zhao, X.

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

Zheng, Z. F.

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

Zhou, K.

C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019).
[Crossref]

Zhou, L. W.

H. L. Wang, L. W. Zhou, and J. B. Gong, “Interband coherence induced correction to adiabatic pumping in periodically driven systems,” Phys. Rev. B 91(8), 085420 (2015).
[Crossref]

Zhou, X.

X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017).
[Crossref]

Zhu, A. D.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Zhu, G.

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

Zhu, S. L.

H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
[Crossref]

F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
[Crossref]

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

Zhu, X. Y.

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

Zilberberg, O.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Zou, L. J.

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

J. Phys. B (1)

J. X. Liu, J. Y. Ye, L. L. Yan, S. L. Su, and M. Feng, “Distributed quantum information processing via single atom driving,” J. Phys. B 53(3), 035503 (2020).
[Crossref]

J. Phys.: Condens. Matter (1)

X. S. Li, Z. Z. Li, L. L. Zhang, and W. J. Gong, “$\mathcal {PT}$PT symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling,” J. Phys.: Condens. Matter 32(16), 165401 (2020).
[Crossref]

Nat. Commun. (1)

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
[Crossref]

Nat. Phys. (2)

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits,” Nat. Phys. 16(7), 747–750 (2020).
[Crossref]

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-Hermitian bulk-boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020).
[Crossref]

Nature (1)

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

New J. Phys. (3)

J. S. Pedernales, R. Di Candia, D. Ballester, and E. Solano, “Quantum simulations of relativistic quantum physics in circuit QED,” New J. Phys. 15(5), 055008 (2013).
[Crossref]

N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New J. Phys. 15(7), 073017 (2013).
[Crossref]

X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, “Optical isolation with nonlinear topological photonics,” New J. Phys. 19(9), 095002 (2017).
[Crossref]

Opt. Express (3)

Phys. Rev. A (11)

S. L. Su, F. Q. Guo, L. Tian, X. Y. Zhu, L. L. Yan, E. J. Liang, and M. Feng, “Nondestructive Rydberg parity meter and its applications,” Phys. Rev. A 101(1), 012347 (2020).
[Crossref]

F. Mei, J. B. You, W. Nie, R. Fazio, S. L. Zhu, and L. C. Kwek, “Simulation and detection of photonic chern insulators in a one-dimensional circuit-QED lattice,” Phys. Rev. A 92(4), 041805 (2015).
[Crossref]

C. Zhang, K. Zhou, W. Feng, and X. Q. Li, “Estimation of parameters in circuit QED by continuous quantum measurement,” Phys. Rev. A 99(2), 022114 (2019).
[Crossref]

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[Crossref]

F. Mei, G. Chen, L. Tian, S. L. Zhu, and S. Jia, “Robust quantum state transfer via topological edge states in superconducting qubit chains,” Phys. Rev. A 98(1), 012331 (2018).
[Crossref]

C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A 97(4), 042118 (2018).
[Crossref]

R. Li, D. M. Yu, S. L. Su, and J. Qian, “Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms,” Phys. Rev. A 101(4), 042328 (2020).
[Crossref]

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96(5), 052317 (2017).
[Crossref]

C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, “Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems,” Phys. Rev. A 97(5), 052115 (2018).
[Crossref]

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between $\mathcal {PT}$PT-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95(5), 053626 (2017).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous $\mathcal {PT}$PT-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96(4), 043810 (2017).
[Crossref]

Phys. Rev. B (11)

H. Jiang, L. J. Lang, C. Yang, S. L. Zhu, and S. Chen, “Interplay of non-Hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices,” Phys. Rev. B 100(5), 054301 (2019).
[Crossref]

P. M. Billangeon, J. S. Tsai, and Y. Nakamura, “Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits,” Phys. Rev. B 91(9), 094517 (2015).
[Crossref]

D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019).
[Crossref]

L. Jin and Z. Song, “Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry,” Phys. Rev. B 99(8), 081103 (2019).
[Crossref]

K. Padavić, S. S. Hegde, W. DeGottardi, and S. Vishveshwara, “Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems,” Phys. Rev. B 98(2), 024205 (2018).
[Crossref]

Y. Kuno, “Phase structure of the interacting Su-Schrieffer-Heeger model and the relationship with the Gross-Neveu model on lattice,” Phys. Rev. B 99(6), 064105 (2019).
[Crossref]

H. L. Wang, L. W. Zhou, and J. B. Gong, “Interband coherence induced correction to adiabatic pumping in periodically driven systems,” Phys. Rev. B 91(8), 085420 (2015).
[Crossref]

C. Li, S. Lin, G. Zhang, and Z. Song, “Topological nodal points in two coupled Su-Schrieffer-Heeger chains,” Phys. Rev. B 96(12), 125418 (2017).
[Crossref]

X. L. Yu, L. Jiang, Y. M. Quan, T. Wu, Y. Chen, L. J. Zou, and J. Wu, “Topological phase transitions, majorana modes, and quantum simulation of the Su-Schrieffer-Heeger model with nearest-neighbor interactions,” Phys. Rev. B 101(4), 045422 (2020).
[Crossref]

L. Li, Z. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

K. G. Fedorov, S. V. Shitov, H. Rotzinger, and A. V. Ustinov, “Nonreciprocal microwave transmission through a long josephson junction,” Phys. Rev. B 85(18), 184512 (2012).
[Crossref]

Phys. Rev. Lett. (9)

J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with purcell protection and tunable coupling,” Phys. Rev. Lett. 106(3), 030502 (2011).
[Crossref]

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

A. Y. Song, X. Q. Sun, A. Dutt, M. Minkov, C. Wojcik, H. Wang, I. A. D. Williamson, M. Orenstein, and S. Fan, “$\mathcal {PT}$PT-symmetric topological edge-gain effect,” Phys. Rev. Lett. 125(3), 033603 (2020).
[Crossref]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref]

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-Hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020).
[Crossref]

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref]

F. Song, S. Yao, and Z. Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” Phys. Rev. Lett. 123(17), 170401 (2019).
[Crossref]

Phys. Rev. X (1)

Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff, “Digital quantum simulation of spin models with circuit quantum electrodynamics,” Phys. Rev. X 5(2), 021027 (2015).
[Crossref]

Quantum Inf. Process. (2)

L. H. Ma, Y. H. Kang, Z. C. Shi, B. H. Huang, J. Song, and Y. Xia, “Shortcuts to adiabatic for implementing controlled phase gate with cooper-pair box qubits in circuit quantum electrodynamics system,” Quantum Inf. Process. 18(3), 65 (2019).
[Crossref]

M. D. Kim and J. Kim, “Scalable quantum computing model in the circuit-QED lattice with circulator function,” Quantum Inf. Process. 16(8), 192 (2017).
[Crossref]

Rev. Mod. Phys. (1)

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Sci. China Phys. Mech. Astron. (1)

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice,” Sci. China Phys. Mech. Astron. 61(8), 080313 (2018).
[Crossref]

Sci. Rep. (4)

Y. Wang, C. Guo, G. Q. Zhang, G. Wang, and C. Wu, “Ultrafast quantum computation in ultrastrongly coupled circuit QED systems,” Sci. Rep. 7(1), 44251 (2017).
[Crossref]

C. Yuce and Z. Oztas, “Pt symmetry protected non-Hermitian topological systems,” Sci. Rep. 8(1), 17416 (2018).
[Crossref]

T. Yoshida, K. Kudo, and Y. Hatsugai, “Non-Hermitian fractional quantum hall states,” Sci. Rep. 9(1), 16895 (2019).
[Crossref]

F. Munoz, F. Pinilla, J. Mella, and M. I. Molina, “Topological properties of a bipartite lattice of domain wall states,” Sci. Rep. 8(1), 17330 (2018).
[Crossref]

Science (3)

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. Schematic diagram of 1D nonreciprocal coupled SSH chain with alternating coupling strength.
Fig. 2.
Fig. 2. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\theta$ for 1D nonreciprocal coupled SSH chain, where we set $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the nonreciprocal coupling strengths (a) $\Delta =0.1$, (b) $\Delta =0.5$, (c) $\Delta =0.8$, (d) $\Delta =1$, (e) $\Delta =1.5$, (f) $\Delta =2$, (g) $\Delta =2.5$, and (h) $\Delta =3$, respectively.
Fig. 3.
Fig. 3. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\Delta$ for 1D nonreciprocal coupled SSH chain. Here, we set $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the periodic parameters (a) $\theta =0$, (b) $\theta =\pi /2$, and (c) $\theta =\pi$, respectively.
Fig. 4.
Fig. 4. The degree of complex energy levels of the system, where $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the periodic parameters (a) $\theta =0$, (b) $\theta =\pi /2$, and (c) $\theta =\pi$, respectively. The degrees of complex energy levels with blue line and pink line are equal 0.98 and 1.
Fig. 5.
Fig. 5. (a) The state distributions $|\psi _{(50)}|^{2}$, $|\psi _{(51)}|^{2}$ (two red zero-energy edge states in Fig. 2) and $|\psi _{(49)}|^{2}$, $|\psi _{(52)}|^{2}$ (two red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically nontrivial regime $\theta =0.35\pi$. (b) The state distributions $|\psi _{(50)}|^{2}$, $|\psi _{(51)}|^{2}$ (two red bound states in Fig. 2) and $|\psi _{(49)}|^{2}$, $|\psi _{(52)}|^{2}$ (two new red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically trivial regime $\theta =0.65\pi$. The other parameters are selected as $L_\textrm {I}=50$, $L_\textrm {II}=50$, and $\delta =0.5$, respectively.
Fig. 6.
Fig. 6. The state distributions of localized photonic states for the system, where $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$. (a) $\theta =0$ and $\Delta =0.1$, (b) $\theta =0$ and $\Delta =3$, (c) $\theta =\pi$ and $\Delta =0.8$, (d) $\theta =\pi$ and $\Delta =3$. The red line, blue line, blue dotted line, green line, and green dotted line represent the state distributions of the zero-energy edge state, interface states with pure imaginary energy eigenvalues, new interface states with pure imaginary energy eigenvalues, bound states with complex energy eigenvalues, and bound states with pure imaginary energy eigenvalues, respectively.
Fig. 7.
Fig. 7. The dynamic process of the photonic states in 1D nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, and $\Delta =0.5$, respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively.
Fig. 8.
Fig. 8. The dynamic process of the photonic states in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, and $\Delta =3$, respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively.
Fig. 9.
Fig. 9. The dynamic process of the photonic states in Hermitian SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, (a) $\Delta =0.5$, and (b) $\Delta =3$. The excited positions are both the 1st and 20th lattice sites for Figs. (a)-(b).
Fig. 10.
Fig. 10. The dynamic process of the photonic states with on-site defect potentials in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $t=1$, $\theta =0$, and $\Delta =0.5$, respectively.The excited positions are the 1st, 2nd, 3rd, and 4th lattice sites, corresponding to Figs. (a)-(d), respectively.
Fig. 11.
Fig. 11. Schematic diagram of 1D non-Hermitian circuit QED lattice system consisting of an array of transmission line resonators denoted by $a_{j}$ and $b_{j}$, each of them coupled to a two-level superconducting flux qubits $Q_{j}^{a}$ and $Q_{j}^{b}$. The nonreciprocal coupling between chain $\textrm {I}$ and chain $\textrm {II}$ can be implemented by using a long Josephson junction operates in the flux-flow regime.

Equations (5)

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H = j = 1 L 2 ( t 1 a j a j + 1 + t 2 a j + 1 a j + 2 + b j b j + 1 + t 2 b j + 1 b j + 2 + H . c . ) + Δ a L b 1 Δ b 1 a L ,
H = H I + H II + H I-II ,
H I = j = 1 L ( ω 1 2 σ z , j a + ω a a j a j + g a , j σ a , j + a j + g a , j σ a , j a j ) + j = 1 L 2 ( t 1 a j a j + 1 + t 2 a j + 1 a j + 2 + H . c . ) , H II = j = 1 L ( ω 2 2 σ z , j b + ω b b j b j + g b , j σ b , j + b j + g b , j σ b , j b j ) + j = 1 L 2 ( t 1 b j b j + 1 + t 2 b j + 1 b j + 2 + H . c . ) , H I-II = Δ a L b 1 Δ b 1 a L ,
H = j = 1 L 2 ( t 1 a j a j + 1 + t 2 a j + 1 a j + 2 + t 1 b j b j + 1 + t 2 b j + 1 b j + 2 + H . c . ) + j = 1 L ( Δ a g a , j 2 Δ 1 ) a j a j + j = 1 L ( Δ b g b , j 2 Δ 2 ) b j b j + Δ a L b 1 Δ b 1 a L ,
H = j = 1 L 2 [ ( 1 δ cos θ ) a j a j + 1 + ( 1 + δ cos θ ) a j + 1 a j + 2 + ( 1 δ cos θ ) b j b j + 1 + ( 1 + δ cos θ ) b j + 1 b j + 2 + H . c . ] + j = 1 L ξ a j a j + j = 1 L ξ b j b j + Δ a L b 1 Δ b 1 a L .