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Abstract
The non-Hermitian skin effect (NHSE) on the non-Hermitian Haldane model with gain and loss on the honeycomb lattice with the outline of a triangle is discussed. The NHSE only occurs on the edge of the lattice, transforming the edge modes into the higher-order corner modes. The NHSE can also occur on a lattice with only loss, which can be treated as a lattice with gain and loss as well as a global loss added to it. When the saturated gain is added to the three corner sites of the dissipative lattice, a single-mode laser system is obtained. When any one site is stimulated initially, the system will reach a saturated state depending on the distribution of the corner modes, and the stable laser light is emitted by sites at the corners.
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1. Introduction
Topological insulators (TIs) [1,2], as fascinating topics in condensed-matter physics, have attracted much attention in recent years. The novel topological states, including edge states and higher-order corner states of topological insulators, have been theoretically investigated and experimentally found in many photonic systems [3–12], resulting in the vigorous development of topological photonics [13].
Recent years, nonlinear topological photonics becomes an emerging field, and one of the hot topics is the topological insulator laser [14–19]. The topological states have special distributions, which are localized on the edges or at the corners, with the energy level in the band gap of the topological structures. For this reason, it is easy to distinguish these edge or corner modes from the bulk modes. Then, a single-mode laser system can be obtained by adding saturated gain to the edge or corner sites and adding loss to the bulk sites, with the topological modes being the gain modes and the bulk modes the loss modes.
On another hand, the higher-order corner states have been widely investigated recently [20–23]. Take the second-order TI as an 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. Generally, a $d$D $n$th-order TI supports $(d-n)$D boundary states. One of the methods to achieve the corner states is the non-Hermitian skin effect (NHSE) [24–36], where the asymmetric couplings between the sites on the lattices are usually considered. However, the asymmetric couplings are not easy to realize in some systems. Another way to obtain NHSE is adding gain and loss to the sites on the topological lattices [37,38]. Such gain-loss-induced NHSE can be experimentally realized through photons in optical coupled-cavity arrays or cold atoms in optical lattices.
In this work, we investigate the nonlinear dynamics of a laser system on a non-Hermitian Haldane model [39,40] with the corner modes induced by gain and loss. For a 2D structure, compared with the topological edge states, the higher-order corner states are easier to pick out, which are usually the zero-energy modes, while the energy levels of edge modes are always the continuous bands in the band gap of the structure. For the topological laser with the topological edge modes, the imaginary parts of all the eigenvalues of edge modes is increased with saturated gain added to the edge sites. So, to let the lasing remains in a single mode, the gain (pumping) should be high above the threshold in a finite range when it is increased [14]. However, for the topological laser with the higher-order corner modes, when the saturated gain is added to the corner sites, only the imaginary parts of zero-energy corner modes are significantly affected by the gain [17,18]. In this case, the laser remains single-mode even the gain is much higher than the threshold. Besides, the corner modes not only can be induced by gain and loss, but also can be realized on the lattice bipartite with two kinds of sublattice sites, where the two kinds of sites have different losses. This case can be treated as the lattice not only has gain and loss on the bipartite sites but also a global loss, and the global loss does not effect the eigenstates of the system but adds a extra decay factor to the eigenvalues. In this way, the dissipative lattice is easier to realized experimentally. The nonlinear dynamics of the laser system is explored. It is found, before the laser system getting saturated, the transmission of the energy on the lattice has phenomenon of the one-way propagation when a corner site is stimulated initially, corresponding to the quantum Hall effect of the Haldane model [39,40]. When the system gets saturated, the distribution of the saturated state has a similar form of the higher-order corner modes. Moreover, when any one site is stimulated initially, the system will reach a same saturated state, resulting in a single-mode laser with the stable laser light emitted by sites at the corners.
The remainder of this paper is organized as follows. In Section 2, we describe the NHSE on the non-Hermitian Haldane model with gain and loss, where the higher-order corner modes are generated. In Section 3, the single-mode laser system based on the higher-order corner modes of the non-Hermitian Haldane model is discussed, and the nonlinear dynamics of a laser system on the non-Hermitian Haldane model is explored. Conclusions are presented in Section 4.
2. Non-Hermitian Haldane model with gain and loss
The Hamiltonian of the Haldane model in the momentum space can be written as [39,40]
where
Fig. 1. Schematic of the non-Hermitian Haldane model on the honeycomb lattice with the outline of a triangle of the size $L=4$. Loss or gain are added to the bipartite sites as $\mathrm {i}\gamma _{1}$ and $\mathrm {i}\gamma _{2}$, and the solid and dashed lines represent the nearest-neighbor hoping $t_{1}$ and the next-nearest-neighbor hoping $t_{2}\mathrm {e}^{\mathrm {i}\phi }$ respectively.
One of the methods to achieve the higher-order corner states is the NHSE, which is usually realized by nonreciprocal hoppings. There exists another way to realize the non-Hermitian skin effects with only gain and loss on sites $a$ and site $b$, and this method is easier to implement in experiments than the former with asymmetric coupling strengths.
The Hamiltonian (1) can be written in the form of a $2\times 2$ matrix,
Next, we consider the two sites having different gain or loss $\mathrm {i}\gamma _{1}$ and $\mathrm {i}\gamma _{2}$, and then, the non-Hermitian Hamiltonian is
If we let $\gamma _{1}=\gamma$ and $\gamma _{2}=-\gamma$, the Hamiltonian (3) is the one with the parity-time ($PT$) symmetry [38].
In Fig. 2, we plot the band structure of a nanoribbon of the non-Hermitian Haldane model with 40-site width. Sites $a$ have the gain $\mathrm {i}\gamma _{1}=\mathrm {i}\gamma$ and sites $b$ have the loss $\mathrm {i}\gamma _{2}=-\mathrm {i}\gamma$. Figures 2(a) and 2(b) are the real part and the imaginary part of the energy band. In the band gap in Fig. 2(a), there exist two topological edge modes corresponding to gain sites $a$ and loss sites $b$ respectively, with the positive and negative parts of the two edge mode shown in Fig. 2(b). Figures 2(c) is the combination of the real part and the imaginary part of the energy band in Figs. 2(a) and2(b).
Fig. 2. The band structure of a nanoribbon of the non-Hermitian Haldane model with 40-site width. (a) The real part and (b) the imaginary part of the energy band. (c) The combination of the real part and the imaginary part of the energy band. The parameters are $M=0$, $t_{1}=1$, $t_{2}=0.2$, $\phi =\pi /2$, $\gamma _{1}=\gamma$, $\gamma _{2}=-\gamma$ and $\gamma =0.5$.
In Fig. 3, considering a non-Hermitian Haldane model on the honeycomb lattice with the outline of a triangle in Fig. 1, we show the NHSE acting on the topological edge modes with the increase of $\gamma$. The size $L$ of the triangle is defined as the number of sites $b$ (with $\mathrm {i}\gamma _{2}$) along an edge. Figure 3(a) is the distribution of a topological edge state of the normal Haldane model with no gain and loss on a triangle with the size $L=12$. In Figs. 3(b)–3(d), with the increase of the parameter $\gamma$, the edge state is localized into the corners opposite to the direction of the next-nearest-neighbor hopping $t_{2}\mathrm {e}^{\mathrm {i}\phi }$ between the loss sites $b$ along the edge of the triangle. The next-nearest-neighbor hopping $t_{2}\mathrm {e}^{\mathrm {i}\phi }$ in the bulk sites $a$ and $b$ are balanced, but on the edge only the next-nearest-neighbor hopping between the loss sites exist, so the NHSE acting only on the topological edge states but not on the bulk states.
Fig. 3. The evolution of the NHSE acting on the topological edge modes with the increase of $\gamma$. Other parameters are the same as in Fig. 2.
Besides, it should be noticed, if sites $a$ and $b$ are all with loss, where $\gamma _{1}<0$ and $\gamma _{2}<0$ (assuming $\gamma _{2}<\gamma _{1}<0$ here), the Hamiltonian (3) can be written as
3. Topological laser with higher-order corner states
The non-Hermitian Haldane model above can be realized through a coupled-ring array made of microring resonators [4,5,14,15]. For a laser system on the lattice, the dynamics of the system is expressed by a set of coupling equations [14]
First, we analyze the linear model with the parameter $\eta \rightarrow \infty$ in Eq. (5). The size of the triangle considered here is $L=12$, and the loss are set as $\gamma _{1}=0$ and $\gamma _{2}=-2\gamma =-1.8$. Figure 4(a) is the real part of the energy spectrum of the linear system with $\xi =0$, where the three zero-energy modes corresponding to the corner states are marked with the red points. The evolution of the real and imaginary parts of the spectrum of the linear system with the increase of parameter $\xi$ is plotted in Fig. 4(b). For $\xi <1$, all the eigenstates of the system are loss modes, and only the imaginary part of the corner modes has a obvious increase with the pump strength rising. When $\xi >1$, the imaginary part of the corner modes becomes positive, while the imaginary part of other modes remains negative. This means the corner modes are gain modes and other modes are loss modes, and $\xi =1$ is the threshold gain of the laser system. It can be found this system maintains single-mode lasing even the gain $\xi$ is much lager than the threshold gain.
Fig. 4. (a) The real part of the energy spectrum of the linear system with $\eta \rightarrow \infty$ and $\xi =0$ in Eq. (5). (b) The energy spectrum of the linear system with the increase of $\xi$. The points with a same color in (b) represent the energy spectrum corresponding to the parameter $\xi$ with the same color. The parameters are $M=0$, $t_{1}=1$, $t_{2}=0.2$, $\phi =\pi /2$, $\gamma _{1}=0$, $\gamma _{2}=-1.8$, and $\gamma =0.9$.
Next, we explore the nonlinear dynamics of the laser system in Fig. 5 through Eq. (5). Figure 5(a) shows the saturated amplitude $|\psi _{A_{1}}|$ of site $A_{1}$ show in Fig. 1 with the increase of the pump strength $\xi$ after the laser system evolving for enough time, with the initial condition $\psi _{n}(0)=\delta _{n,n_{A_{1}}}$. As is shown in Fig. 5(a), the saturated amplitude $|\psi _{A_{1}}|$ is zero when $\xi <1$ and nonzero when $\xi >1$, which agrees with the result of the threshold gain in Fig. 4(b). For the pump strength $\xi >1$, the amplitude $|\psi _{A_{1}}|$ reaches a finite value with the saturated gain and becomes larger with the increase of the pump strength. In Fig. 5(b), we plot the time evolution of several sites at the corner of the triangle, which are marked as $A_{i}$, $B_{i}$ and $C_{i}$ ($i=1,2,3$) in Fig. 1. As is shown in the figure, the initially stimulated site $A_{1}$, as well as the two nearest sites $A_{2}$ and $A_{3}$, is rapidly saturated, but with different saturated amplitudes of the three sites. After a delay, the amplitudes of sites $B_{i}$ at another corner reach the saturation, while the saturation of the amplitudes of sites $C_{i}$ at the third corner takes a longer delay.
Fig. 5. (a) The saturated amplitude $|\psi _{A_{1}}|$ of site $A_{1}$ with the increase of pump strength $\xi$. (b)Time evolution of the amplitudes of sites $A_{i}$, $B_{i}$ and $C_{i}$ at the corner of the triangle. The parameters are $M=0$, $t_{1}=1$, $t_{2}=0.2$, $\phi =\pi /2$, $\gamma _{1}=0$, $\gamma _{2}=-1.8$, $\gamma =0.9$, $\xi =2$ and $\eta =1$.
In Fig. 6, we plot the evolution of the state distribution of the laser system on the triangular lattice. Figures 6(a1)–6(a5) are the evolution of the system with a corner site stimulated initially. The energy of site $A_{1}$ is transmitted through the lattice in the form of a wave with a tiny amplitude to the sites at the other two corners. The amplitudes of the bulk sites are almost zero during the evolution. The direction of the transmission of the wave is $A_{i}\rightarrow B_{i}\rightarrow C_{i}$, corresponding to the different delay time of the saturation in Fig. 5(b), which is caused by the quantum Hall effect of the Haldane model with the one-way propagation along the edge. Figures 6(b1)–6(b5) are the evolution of the system with a edge site stimulated initially. It can be seen that the energy on the stimulated site is dissipated at the beginning, and then the amplitudes of sites $B_{i}$ and $C_{i}$ reach saturated values. After a longer delay, the amplitudes of sites $B_{i}$ also become saturated. It seems that the phenomenon of the one-way propagation vanishes when the edge site is stimulated initially. The evolution of the system with the site at the center of the triangle stimulated initially is shown in Figs. 6(c1)–6(c5). There is no gain on the bulk sites, so the energy is dissipated in the bulk of the lattice and propagates in the form of a wave with a tiny amplitude to the three corners. Due to the rotational symmetry of the triangle, the amplitudes of sites at the corners reach saturation at the same time. In fact, due to the NHSE on the edge modes of the Haldane model, the edge modes are transformed in to the corner modes. Hence, when we stimulate a edge site at the beginning in Fig. 6(b1), the energy does not propagation along the edge due to the absences of the edge modes but propagates in the bulk like the form of wave in Figs. 6(c1)–6(c5). The time evolution of the amplitudes of corner sites corresponding to the initial state in Fig. 6(b1) is plotted in Fig. 7(a), and it can be found the amplitudes of sites $C_{i}$ reach saturation before sites $B_{i}$ although the initially stimulated site is closer to the sites $B_{i}$. So, the one-way propagation also exists but does not play the leading role in Figs. 6(b1)–6(b5). More over, the time evolution of the amplitudes of corner sites corresponding to the initial state in Fig. 6(c1) is shown in Fig. 7(b). So, when any one site is stimulated, the sites at the corners begin to emit stable laser light, and the distribution of the saturated state depends on the corner mode in Fig. 3(d).
Fig. 6. Time evolution of the state distribution of the laser system on the lattice with (a1)-(a5) a corner site, (b1)-(b5) an edge site and (c1)-(c5) a bulk site stimulated initially. The parameters are the same as in Fig. 5.
Fig. 7. Time evolution of the amplitudes of sites $A_{i}$, $B_{i}$ and $C_{i}$ at the corner of the triangle with the initial condition corresponding to (a) Fig. 6(b1) and (b) Fig. 6(c1). The parameters are the same as in Fig. 5.
4. Conclusions
In summary, based on the corner modes of a non-Hermitian Haldane model with gain and loss, we have explored the nonlinear dynamics of a laser system on the honeycomb lattice with the outline of a triangle. With gain and loss on the bipartite sites of the Haldane model, the NHSE occurs on the edge of the lattice, and the edge modes are transformed into the corner modes. These corner modes can also be obtained by adding a global loss to the lattice with gain and loss, resulting in a lattice with only loss. By adding the saturated gain to the three corner sites of the dissipative non-Hermitian Haldane model, a single-mode laser system is obtained, with the corner modes the gain modes and the other modes the loss modes. When a corner site is stimulated initially, stable laser light is emitted by sites at the corners with the saturated state depending on the distribution of the corner modes. Moreover, the system will reach the same saturated state when any one site is stimulated initially, although the phenomenon of the one-way propagation is not obvious compared with the case of corner sites being stimulated initially.
Funding
National Natural Science Foundation of China (12174288, 12274326); National Key Research and Development Program of China (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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