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Abstract
A nonlinear non-Hermitian topological laser system based on the higher-order corner states of the 2-dimensional (2D) Su-Schrieffer-Heeger (SSH) model is investigated. The topological property of this nonlinear non-Hermitian system described by the quench dynamics is in accordance with that of a normal 2D SSH model. In the topological phase, all sites belonging to the topological corner states begin to emit stable laser light when a pulse is given to any one site of the lattice, while no laser light is emitted when the lattice is in the trivial phase. Furthermore, the next-nearest-neighbor (NNN) couplings are introduced into the strong-coupling unit cells of the 2D SSH model, which open a band gap in the continuous band structure. In the topological phase, similar to the case of 2D SSH model without NNN couplings, the corner sites can emit stable laser light due to the robustness of the higher-order corner states when the NNN couplings are regarded as the perturbation. However, amplitude of the stimulated site does not decay to zero in the trivial phase, because the existence of the NNN couplings in the strong-coupling unit cells make the lattice like one in the tetramer limit, and a weaker laser light is emitted by each corner.
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1. Introduction
Topological insulators (TIs) [1,2], including Chern insulators [3,4], time-reversal-invariant TIs [5,6] and topological crystalline insulators [7], have attracted much interest in recent years due to their robust topological edge states, forming a hot topic in condensed-matter physics. Then, the concept to higher-order TIs (HOTIs) has been generalized [8–18]. For example, a second-order TI is a $d$-dimensional ($d$D) insulator which has $(d-1)$D topological boundary states but also $(d-2)$D topological boundary states. These topological edge states have been investigated or observed in electric circuits [19–23], phonics [24–26] and photonics models [27–29].
Recently, laser systems based on topological edge states have been investigated through photonics models [30–47]. A typical photonics model is a coupled-ring array made of active resonators which has been widely used in topological photonics [48–50]. 2D TIs based on 2D lattices [32,33,38–47], including Chern insulators, valley topological insulators, and topological crystal insulators, provide a well scheme to realizing large-scale topological laser arrays with a higher power. These models generate laser light along the edge belonging to edge modes of a 2D lattice. Furthermore, topological laser based on the higher-order corner states of kagome lattice [46] as well as corner states of 2D lattice with non-Hermitian skin effect [47] has been investigated, where stable laser light is emitted by the sites belonging to the topological corner states.
In this paper, we consider a laser system based on the 2D Su-Schrieffer-Heeger (SSH) model [51], which is composed of coupled vertical and horizontal stacks of SSH chains [52]. Compared with the kagome lattice [17,46], the 2D SSH model has a simpler geometry but a more special band structure, where the topological phase transition happens with the band inversion but without a band gap opened on either side of the phase transition point. Due to the continuousness of the band structure near the zero energy, for a finite structure of 2D SSH model in the topological phase, the zero-energy eigenvalues are buried within continuous bulk bands, with the second-order topological corner states called the bound states in the continuum (BICs) [53]. Through the dynamics of the laser system in the 2D SSH model, stable laser light is emitted by each corner site of the lattice in the topological phase, while no laser light emitted in the trivial phase. Besides, the next-nearest-neighbor (NNN) couplings in a coupled-ring array made of active resonators may not be ignored sometimes. The NNN couplings are introduced into the strong-coupling unit cells of the 2D SSH model [54,55], and the dynamics of the topological laser system in this situation is discussed. When the NNN couplings exist in the strong-coupling unit cells, a band gap in the band structure of the 2D SSH model in the topological phase will be opened, and the topological zero-energy modes will be located in this band gap. This characteristic is also absent in the kagome lattice for the band gap is opened beside the phase transition point, and NNN couplings cannot be introduced into the unit cell with three sites in the kagome lattice. Similar to the result of the 2D SSH model without NNN couplings in the topological phase, the corner sites can emit stable laser light. While the results of the two cases are quite different in the trivial phase. The NNN couplings make the 2D SSH model more like a set of tetramers, and laser lights can also be emitted by the corner sites when any one site is stimulated.
The remainder of this paper is organized as follows. In Section 2, we describe the 2D SSH model and investigate the dynamics of a topological laser system with nonlinear non-Hermitian gain and loss on it. In Section 3, the NNN couplings are introduced into the strong-coupling unit cells of the 2D SSH model, and dynamics of the laser system on it is also discussed. Conclusions are presented in Section 4.
2. Topological laser in 2D SSH model
The lattice of the 2D SSH model consisting of four sites per unit cell with dimerized nearest-neighbor couplings is shown in Fig. 1, where $\kappa _A$ is the intracell hopping and $\kappa _B$ is the intercell hopping. The Hamiltonian in the real space is written as
Fig. 1. Schematics of a finite lattice of 2D SSH model. Each unit cell contains four sites, and $\kappa _A$ ($\kappa _B$) is the intracell (intercell) hopping.
It is known the 2D SSH model has chiral symmetry as well as $C_{4v}$ symmetry, and band structure in the lattice is in a topological phase for $-1<\lambda <0$ and in a trivial phase for $0<\lambda <1$. In the following, we take a square lattice with the size $L=6$. The energy eigenvalues $E_{n}$ ($n=1,2,\ldots,N$) of such a finite structure, in units of $\kappa$, are shown in Fig. 2(a). The second-order topological corner states corresponding to the zero-energy modes (red lines) emerge for $-1<\lambda <0$, and the density of a corner state is shown in Fig. 2(b), where the density of the state becomes arbitrarily small except for the four corners. These corner states are BICs due to these zero-energy modes existing in the continuous band structure. Figures 2(c) and 2(d) are two bulk states of the 2D SSH model in the trivial phase with $\lambda =0.5$, and the densities of them are distributed on both corner and bulk sites.
Fig. 2. (a) The energy spectrum of a finite lattice of 2D SSH model with the size $L=6$. (b) Density of a corner state corresponding to a zero-energy mode in the topological phase with $\lambda =-0.5$. (c),(d) Density of two bulk states in the trivial phase with $\lambda =0.5$.
The finite lattice of the 2D SSH model can be realized through a coupled-ring array made of active resonators. The Hamiltonian $H$ can be written as a $4L^{2}\times 4L^{2}$ matrix $\{H_{mn}\}_{4L^{2}\times 4L^{2}}$, then the dynamics of a laser system on the coupled-ring array is expressed by
The property of such a nonlinear non-Hermitian system can be expressed by the quench dynamics [46,56–59], where a pulse is given to one site, and its time evolution under Eq. (2) is explored. We numerically solve Eq. (2) under the initial condition $\psi _{1}(0)=1$, i.e. a pulse is given to the corner $A$ at $t=0$. A good signal to detect whether the system is topological or trivial is the saturated amplitude $|\psi _{1}|$ after enough time. With the loss or gain in each resonator, the system will reach a stationary state as the system evolves for enough time, and the time is chosen as $t=60$ here after our numerical analysis. The saturated amplitude $|\psi _{1}|$ for different losses $\gamma$ is plotted in Fig. 3. It is shown that the amplitude $|\psi _{1}|$ is a finite value when the 2D SSH model is in the topological phase, while it is almost zero in the trivial phase.
Fig. 3. The saturated amplitude $|\psi _{1}|$ at the corner $A$ after enough time. The parameters are set as $\kappa =1$, $\xi =2$ and $\eta =1$.
Time evolution of the amplitude $|\psi _{n}|$ of the 2D SSH model with the initial condition $\psi _{1}(0)=1$ is plotted in Fig. 4. Due to the existence of the loss or gain in each resonator, the amplitude $|\psi _{n}|$ of each site reaches a stationary solution. In the topological phase, the density of state is mainly localized at the four corner sites, and the four saturated amplitudes are almost identical. While in the trivial phase, the four saturated amplitudes are almost zero. In the trivial phase, the corner states do not exist, and the bulk states are distributed on both corner and bulk sites as shown in Figs. 2(c) and 2(d). The system evolves under the bulk state causing the energy flowing into the bulk site, and the loss in each resonator quickly dissipates the energy.
Fig. 4. Time evolution of the amplitude $|\psi _{n}|$ of the 2D SSH model. The parameters are set as $\kappa =1$, $\gamma =1/2$, $\xi =2$, $\eta =1$, (a1)-(a5) $\lambda =-0.5$ for the topological phase and (b1)-(b5) $\lambda =0.5$ for the trivial phase.
Figure 5 is density plot of time evolution of the amplitude $|\psi _{n}|$ of the side $AB$. For convenience, we re-mark the numbers of the sites along the side $AB$, which are marked from $1$ to $12$. The amplitudes $|\psi _{1}|$ and $|\psi _{12}|$ both reach to a finite value after enough time for the initial condition $\psi _{1}(0)=1$ and $\psi _{6}(0)=1$ in the topological phase in Figs. 5(a) and 5(c), while quickly decay to zero in the trivial phase in Figs. 5(b) and 5(d).
Fig. 5. Density plot of time evolution of the amplitude $|\psi _{n}|$ of the side $AB$. The sites on the side are marked from $1$ to $12$. (a),(b) The initial condition $\psi _{1}(0)=1$, and (c),(d) the initial condition is $\psi _{6}(0)=1$. The parameters are set as $\kappa =1$, $\gamma =1/2$, $\xi =2$, $\eta =1$, (a),(c) $\lambda =-0.5$ for the topological phase and (b),(d) $\lambda =0.5$ for the trivial phase.
Time evolution of several amplitudes $|\psi _{n}|$ is plotted in Fig. 6. In the topological phase, the amplitude $|\psi _{1}|$ at the corner site $A$ is rapidly saturated as shown in Fig. 6(a), and other amplitudes is saturated with a delay, which implies the propagation of a wave with a tiny amplitude through the square lattice although it is hard to see the propagation of the wave in Fig. 5(a). The propagation of this wave induces a stimulated emission at the other three corner sites. The delay time of amplitude $|\psi _{144}|$ is longer than that of amplitude $|\psi _{22}|$, which means that the wave propagates through a longer path from corner site $A$ to $B$ than from corner site $A$ to $D$. Delay time of the saturated amplitudes (a) $|\psi _{B}|$ of the corner site $B$ and (b) $|\psi _{D}|$ of the corner site $D$ with different values of the hopping amplitude $\kappa$ is plotted in Fig. 7, and it is shown that increasing the value of $\kappa$ can shorten the delay time. Besides, in the trivial phase, the amplitude $|\psi _{1}|$ quickly decays to zero in Fig. 6(b), and there is no excitation at the other three corner sites.
Fig. 6. Time evolution of several amplitudes $|\psi _{n}|$. The parameters are set as $\kappa =1$, $\gamma =1/2$, $\xi =2$, $\eta =1$, (a) $\lambda =-0.5$ for the topological phase and (b) $\lambda =0.5$ for the trivial phase.
Fig. 7. Delay time of the saturated amplitudes (a) $|\psi _{B}|$ and (b) $|\psi _{D}|$ with $\kappa =1/4, 1/2, 1,\ldots,2$. The parameters are set as $\kappa =1$, $\gamma =1/2$, $\xi =2$, $\eta =1$, and $\lambda =-0.5$
Moreover, we study the tetramer limit ($\kappa _{A}=1$ and $\kappa _{B}=0$) in the trivial phase. Then the system is decomposed into a set of tetramers with only the intracell hopping $\kappa _A$. Taking the initial condition $\psi _{1}(0)=1$, we obtain the equations
We numerically solve Eqs. (4)–(7) and plot the time evolution of the amplitudes $|\psi _{n}|$ ($n=1,2,3,4$) in Figs. 8(a) and 8(b). Amplitude $|\psi _{1}|$ approaches a saturated value, which is consistent with Fig. 3 when $\lambda$ is close to $1$. Analytically stationary solutions of Eqs. (4)–(7) are too complex, so we remove the losses on sites $2$, $3$ and $4$ to obtain a simple analytical solution. Removing the losses on sites $2$, $3$ and $4$, Eqs. (4)–(7) turn into
Then, the stationary solutions of Eqs. (8)–(11) have a simple form and are given by
The numerical solutions are shown in Figs. 8(c) and 8(d), amplitudes $|\psi _{1}|$ and $|\psi _{4}|$ approach a saturated value (12).
Fig. 8. Time evolution of the amplitudes $|\psi _{n}|$ of the tetramer in 2D SSH model (a),(b) with and (c),(d) without losses on sites $2$, $3$ and $4$. The parameters are set as $\kappa _{A}=1$, $\kappa _{B}=0$, $\gamma =1/2$, $\xi =2$ and $\eta =1$.
3. 2D SSH model with NNN couplings
The NNN couplings in a coupled-ring array made of active resonators may not always be ignored as a matter of fact. Photonics systems in such a 2D form can be realized by a 2D coupled waveguide array [22] or a coupled-ring array [32,33,48–50], and the sites are usually coupled by evanescent waves or link rings. In these cases, the coupling between two sites always depends on the distance between them. Hence, in the strong-coupling unit cells, the sites are close to each other, and the NNN couplings in a coupled-ring array may be enhanced. In this section, we consider the 2D SSH model with NNN couplings in the strong-coupling unit cells. As is shown in Fig. 9, the next-nearest-neighbor couplings $\chi$ exist in the unit cells with the hopping $\kappa _{B}$ for $\kappa _{A}<\kappa _{B}$ (topological), and in the unit cells with the hopping $\kappa _{A}$ for $\kappa _{A}>\kappa _{B}$ (trivial). In the following, we take $\chi =\kappa _{B}$ for $\kappa _{A}<\kappa _{B}$ and $\chi =\kappa _{A}$ for $\kappa _{A}>\kappa _{B}$. The size of the 2D SSH model NNN couplings is chosen to be $L=6$, which is the same as that of the finite lattice of 2D SSH model in Fig. 1. The sites numbers are also marked as in Fig. 1.
Fig. 9. 2D SSH model with NNN couplings $\chi$ in the strong-coupling unit cells. (a) The strong-coupling unit cells are cells with hopping $\kappa _{B}$, and (b) the strong-coupling unit cells are cells with hopping $\kappa _{A}$.
Figure 10(a) is the energy spectrum of a finite lattice of 2D SSH model with NNN couplings in cells with coupling $\kappa _{A}$, and Fig. 10(b) is the energy spectrum of the model with NNN couplings in cells with coupling $\kappa _{B}$. The corner states corresponding to the zero-energy modes also emerge for $-1<\lambda <0$ and disappear for $0<\lambda <1$. Figures 10(c) and 10(d) give the energy eigenvalues of 2D SSH model without NNN couplings and with NNN couplings in the topological phase with $\lambda =-0.5$. It is known the corner states of 2D SSH model in the topological phase are BICs, and it can be found the energy spectrum in Fig. 10(c) is almost continuous and inset with the topological zero-energy modes (red points). It is not easy to distinguish the topological zero-energy eigenvalues and the zero-energy eigenvalues of the bulk states. Differently, the topological zero-energy modes are located in a band gap in Fig. 10(d). Moreover, introducing NNN couplings is also a method used to open a band gap in the continuous band structure of 2D SSH model. The density of a corner states corresponding to one of the zero-energy eigenvalues in Fig. 10(d) are shown in Fig. 11(a), and Fig. 11(b) is a bulk state of the trivial 2D SSH model with NNN couplings.
Fig. 10. Energy spectrum of a finite lattice of 2D SSH model (a) with NNN couplings $\chi$ in cells with coupling $\kappa _{A}$ and (b) with NNN couplings $\chi$ in cells with coupling $\kappa _{B}$. Energy eigenvalues of 2D SSH model of size $L=6$ (c) without NNN couplings and (d) with NNN couplings in the topological phase. Red points are the zero-energy eigenvalues of the topological corner states. The parameters in (c),(d) are set as $\kappa =1$ and $\lambda =-0.5$.
Fig. 11. (a) The density of a corner state corresponding to one of the zero-energy eigenvalues in Fig. 10(d). (b) The density of a bulk state of the 2D SSH model with NNN couplings in the trivial phase ($\lambda =0.5$).
With the initial condition $\psi _{1}(0)=1$, time evolution of the amplitude $|\psi _{n}|$ of the 2D SSH model with NNN couplings is plotted in Fig. 12. Similar to the results of the topological phase in Fig. 4, the amplitude $|\psi _{n}|$ of each site also reaches a stationary solution due to the robustness of the higher-order corner states. While in the trivial phase, the four saturated amplitudes of the corner sites do not decay to zero, and also reach a nonzero stationary solution, which is different from that in Fig. 4 for trivial phase. The results of trivial phase are caused by the corner-like state shown in Fig. 11(b). As is shown in Fig. 11(b), although this eigenstate is a bulk state, its density is mainly distribute on the sites of the cells at four corners. And the system in the trivial phase will evolve into a stationary state like this corner-like state when the first site is stimulated.
Fig. 12. Time evolution of the amplitude $|\psi _{n}|$ of the 2D SSH model with NNN couplings. The parameters are set as $\kappa =1$, $\gamma =1/2$, $\xi =2$, $\eta =1$, (a1)-(a5) $\lambda =-0.5$ for the topological phase and (b1)-(b5) $\lambda =0.5$ for the trivial phase.
The density plot of time evolution of the amplitude $|\psi _{n}|$ of the side $AB$ of the 2D SSH model with NNN couplings is shown in Fig. 13. The amplitudes $|\psi _{1}|$ and $|\psi _{12}|$ also reach to a finite value after enough time in the topological phase in Figs. 13(a) and 13(c), like the results in Figs. 5(a) and 5(c). However, this phenomenon also occurs in the trivial phase when NNN couplings are introduced into the model, which is different from the results of model without NNN couplings.
Fig. 13. Density plot of time evolution of the amplitude $|\psi _{n}|$ of the side $AB$ of the 2D SSH model with NNN couplings. The sites on the side are marked from $1$ to $12$. (a),(b) The initial condition $\psi _{1}(0)=1$, and (c),(d) the initial condition is $\psi _{6}(0)=1$. The parameters are set as $\kappa =1$, $\gamma =1/2$, $\xi =2$, $\eta =1$, (a),(c) $\lambda =-0.5$ for the topological phase and (b),(d) $\lambda =0.5$ for the trivial phase.
Time evolution of several amplitudes $|\psi _{n}|$ is plotted in Fig. 14. The results of the topological phase with NNN couplings in Fig. 14(a) are almost the same as the results without NNN couplings in Fig. 6(a), which implies the robustness of the higher-order corner states by regarding the NNN couplings as the perturbation. However, different from the trivial case in Fig. 6(b), the amplitudes $|\psi _{n}|$ do not decay to zero. Moreover, the amplitude $|\psi _{2}|$ of site near the corner site $A$ is increased by the NNN couplings. A possible reason is the NNN couplings enhance the interaction of the sites in a same unit cell, making the model is more like a 2D SSH model in the tetramer limit.
Fig. 14. Time evolution of several amplitudes $|\psi _{n}|$ of the 2D SSH model with NNN couplings. The parameters are set as $\kappa =1$, $\gamma =1/2$, $\xi =2$, $\eta =1$, (a) $\lambda =-0.5$ for the topological phase and (b) $\lambda =0.5$ for the trivial phase.
Next, we compare the result of Fig. 14(b) with that of the tetramer with NNN couplings. In the tetramer limit, the equations of the tetramer are
The numerically solutions of Eqs. (13)–(16) with the initial condition $\psi _{1}(0)=1$ are plotted in Fig. 15(a). Amplitude $|\psi _{1}|$ approaches a saturated value, as well as the amplitude $|\psi _{2}|$. Due to the symmetry of the tetramer with NNN couplings, we have $|\psi _{2}|=|\psi _{3}|=|\psi _{4}|$. Comparing the solutions of amplitude $|\psi _{1}|$ and $|\psi _{2}|$ in Fig. 15(a) with that in Fig. 14(b) for the trivial phase, we find the results of the two cases are very close, indicating the NNN couplings make the tetramerization play a leading role in the dynamics of 2D SSH model in the trivial phase when taking $\chi =\kappa _{A}$. Equations of the tetramer by removing the losses on sites $2$, $3$ and $4$ are written as
Then, the stationary solutions of Eqs. (17)–(20) are given by
Fig. 15. Time evolution the amplitudes $|\psi _{n}|$ of the tetramer with NNN couplings. Tetramer (a) with and (b) without losses on sites $2$, $3$ and $4$. The parameters are set as $\kappa _{A}=1$, $\kappa _{B}=0$, $\gamma =1/2$, $\xi =2$ and $\eta =1$.
4. Conclusions
In conclusion, we have investigated a topological laser system in coupled-ring array made of active resonators corresponding to the 2D SSH model. Through the quench dynamics, we find the nonlinear non-Hermitian 2D SSH model is in the topological phase for $-1<\lambda <0$ and in the trivial phase for $0<\lambda <1$, as same as the normal 2D SSH model. When a pulse is given to any one site of the lattice in the topological phase, the amplitudes of all sites belonging to the topological corner states will reach a finite value after a delay time, meaning that these sites begin to emit stable laser light. The time delay can be controlled by tuning the hopping amplitude $\kappa$. In the trivial phase, the amplitude of the stimulated site decays rapidly, and no laser light is emitted by other sites. However, the stationary solutions of the amplitudes of the four sites in the tetramer limit are nonzero.
Furthermore, for a 2D coupled waveguide array [22] or a coupled-ring array [32,33,48–50], the coupling between two sites always depends on the distance between them. So, in the strong-coupling unit cells, where sites are close to each other, the NNN couplings in a coupled-ring array may not be ignored. We introduce NNN couplings into the strong-coupling unit cells of 2D SSH model and discuss the dynamics of the topological laser system based on it. In the topological phase, it is found that the corner sites begin to emit stable laser light when any one site is stimulated, which is similar to the results of the 2D SSH model without NNN couplings. While in the trivial phase, the corner sites can also emit stable laser light although weaker than that in the topological phase. The reason is that tetramerization plays a leading role in the dynamics of the laser system due to the existence of the NNN couplings in the strong-coupling unit cells.
Funding
National Key Research and Development Program of China (2021YFA1400602); National Natural Science Foundation of China (12174288, 12204352, 12274326).
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grants No. 12174288, No. 12204352 and No. 12274326) and the National Key R & D Program of China (Grant No. 2021YFA1400602).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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