Theory of quantum spin-Hall topological lasers

TL;DR

The paper addresses robust single-mode lasing in a time-reversal-symmetric quantum spin-Hall photonic lattice. It models a Taiji-resonator-based QSH array with S-shaped waveguides, saturable gain, and Kerr nonlinearity, using coupled-mode theory to capture edge-state dynamics and mode competition. The main finding is that an effective nonreciprocity emerges from parity-breaking and nonlinear gain, enabling lasing in a single topological edge mode that is resilient to backscattering, with Kerr nonlinearity further selecting the gap. This work advances robust topological lasers for integrated silicon photonics and points to future exploration of non-linear and nonreciprocal photonic platforms.

Abstract

We theoretically investigate a quantum spin-Hall topological laser formed by an array of dielectric ring resonators endowed of saturable gain. The system preserves time-reversal symmetry, the clockwise and counter-clockwise modes in each ring resonator acting as two pseudospin states that experience opposite synthetic magnetic fields. We consider ring resonators featuring an internal S-shaped waveguide asymmetrically coupling the two pseudospin states. In spite of the non-magnetic nature of the configuration, we show that an effective breaking of reciprocity is induced by the interplay of spatial parity breaking with saturable gain and a Kerr optical non-linearity. This enables robust single-mode topological lasing even in the presence of realistic levels of backscattering.

Figures (14)

• Figure 1: (a) Sketch of a $2\times 2$ example of a QSH array. For each site resonator, the field amplitude inside each counterpropagating mode is labeled $a_{\pm}$. The resonators are evanescently coupled with transmission and coupling amplitudes $t_{\rm w,s}$ and $ik_{\rm w,s}$, respectively, where the subindices w and s stand for the couplings between site and link resonators, and those between site resonators and S-shaped waveguides, respectively. $\Delta x$ is the vertical displacement introduced to generate a synthetic magnetic field for photons with opposite signs for the two $\pm$ pseudospin states. As these correspond to light propagating in the clockwise or counter-clockwise direction around each site resonator, they can be probed by means of coherent signals $F_{\pm}$ injected from either side of a bus waveguide coupled to the top left site resonator. (b) Sketch of the ribbon configuration used to compute the frequency bands of a QSH insulator with flux per plaquette given by $\alpha=1/4$. Each circle represents a site resonator (link resonators are omitted for clarity). The blue circles indicate one magnetic unit cell (with size $4\times 1$ for $\alpha=1/4$). The boundary conditions are periodic in the $x$ direction and open in the $y$ direction with $N_y=60$ site resonators. (c) Frequency bands of the QSH insulator shown in panel (b). The vertical axis represents the frequency $\omega$ in units of the tunneling rate $J$ with respect to the resonance frequency $\omega_0$ of the site resonators. The horizontal axis corresponds to the $k_x$ wavevector, which is a good quantum number. The edge modes are plotted only for the upper edge of the lattice.
• Figure 2: Coherently driven passive QSH insulator formed by an array of ring resonators without S-shaped elements. Left panels show the temporal Fourier transform $|$FT$(a_{\pm})|$ of the fields on the top-left resonator as a function of the frequency $\omega_{\rm FT}$. Central (right) panels show the intensity $| a^{(n_x,n_y)}_{\pm}|^2$ in the $+$ ($-$) pseudospin at each site resonator (normalized by the maximum intensity in the lattice and in logarithmic scale). The small residual intensity visible in the non-pumped pseudospin originates solely from numerical noise. In panels (a-e) no backscattering is present and (a) $\omega=\omega_0-1.75J$, $F_+=1$, and $F_-=0$; (b) $\omega=\omega_0$, $F_+=1$, and $F_-=0$; (c) $\omega=\omega_0+1.75J$, $F_+=1$, and $F_-=0$; (d) $\omega=\omega_0-1.75J$, $F_+=0$, and $F_-=1$; (e) $\omega=\omega_0$, $F_+=0$, and $F_-=1$. Panel (f) has $\omega=\omega_0+1.75J$, $F_+=0$ and $F_-=1$, and includes a Hermitian backscattering $|\beta^{\rm (BS)}_{\pm,\mp}|=0.04J$ with a random phase at each site resonator; furthermore a few sites are removed around the left edge.
• Figure 3: Passive QSH insulator formed by an array of TJRs in the presence of Hermitian backscattering. The array is coherently driven through the bus waveguide by means of a signal coupling to the CCW- edge mode. We plot the ratio $|a^{(1,1)}_+|^2/|a^{(1,1)}_-|^2$ between the intensities in the $+$ and $-$ pseudospin of the top-left $(1,1)$ resonator as a function of the S-waveguide coupling loss rate $\gamma_{\rm s} = ck^2_{\rm s} / L_\circ n_{\rm L}$ for three values of the backscattering modulus $|\beta^{\rm (BS)}_{\pm,\mp}|$ (in units of the tunneling rate $J$). Squares, circles, and triangles are calculated as the average over 10 realizations of the system. The backscattering phase for each site resonator is chosen randomly in every realization. Error bars represent the standard deviation. Lines are added as a guide to the eye.
• Figure 4: (a) Total intensity $I^{(\pm)}_{\rm T}$ in the $\pm$ pseudospin state as a function of the pump rate $P$. Data are calculated for a trivial array of ring resonators (blue squares), a QSH array of ring resonators (green circles), and a QSH lattice with TJRs asymmetrically coupling the $+$ into the $-$ pseudospin (red triangles) with S-waveguide coupling loss rate $\gamma_{\rm s}=0.2J$. Solid (empty) markers correspond to the $-$ ($+$) pseudospin. No backscattering is considered. Each data set corresponds to a single realization starting from random initial conditions for the field amplitudes. The dashed black line is the expected dependence of the total intensity $I^{(-)}_{\rm T}/n_{\rm s}=N_{\rm P}(P/\gamma_{\rm T}-1)$ for $N_{\rm P}$ independent pumped resonators. (b) Laser operation in a state with equal intensities in the two pseudospins. The total intensity in each pseudospin $I^{(\pm)}_{\rm T}$ is plotted as a function of the pump rate $P$. In this case, a Hermitian backscattering of magnitude $|\beta^{\rm (BS)}_{\pm,\mp}|=0.05J$ and random phases at each site resonator is considered.
• Figure 5: Frequency-dependent spectrum $|$FT$(a_{\pm})|$ of the field amplitudes $a_{\pm}$ in the top-left (1,1) resonator of an active QSH lattice, as a function of the Fourier transform frequency $\omega_{\rm FT}$. The pump rate is fixed to $P=4\gamma_{\rm T}$. No backscattering is considered (i.e., $\beta^{\rm (BS)}_{\rm \pm,\mp}=0$). Data is calculated using 40 realizations, each starting from random initial conditions. Dark (light) blue solid lines correspond to the $+$ ($-$) pseudospin. The shaded regions represent the bulk bands of the QSH array in the linear regime. (a-c) QSH laser of ring resonators with different values of the Kerr non-linearity: (a) $n_{\rm NL}=-10^{-5}n^{-1}_{\rm s}$, (b) $n_{\rm NL}=0$, and (c) $n_{\rm NL}= 10^{-5}n^{-1}_{\rm s}$. (d-f) QSH laser of TJRs with $\gamma_{\rm s}=0.2J$ and different values of the Kerr non-linearity: (d) $n_{\rm NL}=-10^{-5}n^{-1}_{\rm s}$, (e) $n_{\rm NL}=0$, and (f) $n_{\rm NL}= 10^{-5}n^{-1}_{\rm s}$. In all panels, dashed-dotted lines represent the shift of the topological bandgaps due to the Kerr non-linearity.