Linearized theory of the fluctuation dynamics in two-dimensional topological lasers

We theoretically study the collective excitation modes of a topological laser device operating in a single-mode steady state with monochromatic emission. We consider a model device based on a two-dimensional photonic Harper-Hofstadter lattice including a broadband gain medium localized on the system edge. Different regimes are considered as a function of the value of the optical nonlinearity and of the gain relaxation time. The dispersion of the excitation modes is calculated via a full two-dimensional Bogoliubov approach and physically interpreted in terms of an effective one-dimensional theory. Depending on the system parameters, various possible physical processes leading to dynamical instabilities are identified and characterized. On this basis, strategies to enforce a stable single-mode topological laser operation are finally pointed out.

Figure 1

Harper-Hofstadter model. (a) Energy bands of the conservative Harper-Hofstadter Hamiltonian given by Eq. (1) with flux $\theta =1/4$ in a finite lattice of $n_y=399$ sites along $y$ with periodic boundary conditions along $x$. The dark-blue (dashed-green) lines indicate the dispersion of the edge modes localized on the $y=1$ ($y=n_y$) edge. The dark dot indicates the spatially most localized edge mode within the lower energy gap on the $y=1$ edge. (b) Spatial localization function $\mathrm{\Lambda }\left(k_x\right)$ (red line) and curvature of the dispersion (blue line) of the edge states. The left (right) part of the plot refers to the edge mode living on the $y=1$ edge in the lower (upper) energy gap. (c) Imaginary part of the Bogoliubov spectrum of the linearized dynamics around the vacuum solution for a pump strength right at the lasing threshold. The thick black lines show the prediction of the full 2D model, while the thin red ones show the prediction of the 1D effective theory for the excitations living on the $y=1$ edge.

Figure 2

Dispersion of the collective excitation modes on top of a class-A topological laser. The left (right) panels show the real (imaginary) part of the Bogoliubov dispersion for a system of size $n_y=23$ that is lasing on the maximally localized mode at $k_x^{\text{Las}}=-0.982$ for which gain is strongest. Black (red) dots indicate the results of the full 2D model (1D effective theory). G (A) in the right panel indicate the Goldstone (amplitude) branches. System parameters: $\gamma =0.02J$, adiabatic regime ${\gamma }_R/\gamma =+\infty , P/P_{\mathrm{th},1}=2, g=g_R=0$.

Figure 3

Scaling of the Goldstone branch at small k. The red, green, and blue lines show the Goldstone and the amplitude branches, calculated with the full 2D model for different hopping strengths and pumping parameters, as detailed in the legend. The thick black dashed line is a quadratic fit of $-\mathrm{Im}\left[{\omega }^{\mathrm{Bog}}\left(k\right)\right]$, on top of which all three dispersion relations collapse at small $k$. The thin black dotted lines correspond to the ${(k^2/2m_{*})}^2/\mathrm{\Gamma }$ scaling predicted by the 1D model. The magenta dash-dotted line indicates the relaxation rate at which Goldstone and amplitude branches stick for $P/P_{\mathrm{th},1}=2$. Other parameters: $k_x^{\text{Las}}=-0.982$, adiabatic regime ${\gamma }_R/\gamma =\infty , g=g_R=0$.

Figure 4

(a) Imaginary part of the elementary excitation spectrum with a slow reservoir ${\gamma }_R/\gamma =2.5$. Black (red) lines stand for results computed with the full 2D (1D effective) model. (b) Comparison of the small-$k$ scaling of the Goldstone branch for different values of the reservoir speed, calculated with the 2D theory. Parameters: $\gamma =0.2J, P/P_{\mathrm{th},1}=2, g=g_R=0, k_x^{\text{Las}}=-0.982, n_y=23$.

Figure 5

Imaginary part of the elementary excitation spectrum with small nonlinearities $g|{\psi }_1^0{|}^2,g_R{N}^0\ll J$ in the adiabatic regime, calculated with the 1D effective theory. Black (blue) corresponds to the unstable $g_{\text{eff}}/m_{*}<0$ (stable $g_{\text{eff}}/m_{*}>0$) regime. Parameters: $k_x^{\text{Las}}=-0.954, \gamma =0.2J, P/P_{\mathrm{th},1}=2, g/\beta =0.05J, g_R=0$. For these parameters, $g\mathrm{\Lambda }\left(k^{\text{Las}}\right){\left|{\psi }_x^{ss}\right|}^2=0.037J\ll J$.

Figure 6

(a) Imaginary part of the elementary excitation spectrum with slow reservoir relaxation rate ${\gamma }_R=0.5\gamma$ for sizable nonlinear interactions $g_R/R=-1.5$ between the lasing mode and the carriers and $g=0$. (b) Imaginary part of the elementary excitation spectrum in the adiabatic regime for a strong repulsive optical nonlinearity $g/\beta =3.5J$ and $g_R=0$ and a pump strength $P/P_{\mathrm{th},1}=1.25$. For these parameters, the lasing frequency is ${\omega }^{\text{Las}}-\epsilon \left(k_x^{\text{Las}}\right)\approx g\mathrm{\Lambda }\left(k^{\text{Las}}\right){\left|{\psi }^{ss}\right|}^2\approx 0.21J$. Lasing occurs at $k_x^{\mathrm{Las}}=-0.919$ in (a) and $k_x^{\mathrm{Las}}=-0.982$ in (b); the black lines are the result of numerical 2D calculations, while the red lines are the prediction of the effective 1D model. In both panels, $\gamma =0.2J$. $n_y=31$ for (a) and 23 for (b).