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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 [2–4] and topological states [5–9]. 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 [16–19], non-Hermitian skin effect [20–22], and fractional topological number [23–25]. 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 [27–30] 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 [32–35].
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 [36–43], 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 [48–50]. 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 [47,49,51–53]. 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
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.
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.
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.
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.
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.
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.
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.
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).
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.
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 ($\hbar =1$)
withFig. 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.
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
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.
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