Theory of chiral edge state lasing in a two-dimensional topological system

We theoretically study topological laser operation in a bosonic Harper-Hofstadter model featuring a saturable optical gain. Crucial consequences of the chirality of the lasing edge modes are highlighted, such as a sharp dependence of the lasing threshold on the geometrical shape of the amplifying region and the possibility of ultraslow relaxation times and of convectively unstable regimes. The different unstable regimes are characterized in terms of spatiotemporal structures sustained by noise and a strong amplification of a propagating probe beam is anticipated to occur in between the convective and the absolute (lasing) thresholds. The robustness of topological laser operation against static disorder is assessed.

Figure 1

Energy bands of the conservative Harper-Hofstadter Hamiltonian (1) with flux $\vartheta =1/4$ in a lattice of $N_x=399$ sites along $x$ and periodic boundary conditions along $y$. Blue vs red color scale quantifies localization on the left or right edges. The horizontal black and orange lines indicate the WEG and PEG lasing frequencies shown in Fig. 3.

Figure 2

Topological lasing in a $25\times 25$ HH lattice with a flux $\vartheta =1/4$ per plaquette. (Left) Total intensity $I_T={\sum }_{m,n}{\left|a_{m,n}\right|}^2$ normalized to the number of amplifying sites vs gain strength for different configurations: single resonator (SR), whole system gain (WSG), and whole edge gain (WEG). (Right) Snapshot of the typical intensity distribution at an arbitrarily chosen time $t=1000{\gamma }^{-1}$ in a WSG configuration. If not differently specified, we have taken $\beta =1$.

Figure 3

Topological lasing in a $25\times 25$ HH lattice with $\vartheta =1/4$ in a one-site-thick WEG configuration. (a) Snapshot of the steady-state intensity distribution. The green rectangle indicates the amplifying sites. (b) Cuts of the intensity distribution along the $x=1$ line at times (from top to bottom) $\gamma t=43.85,51.65,59.45,67.20,75.00$. (c) Normalized realization-averaged power spectral density (PSD). The dashed lines indicate the center of mass of the distribution. For comparison, the orange arrows indicate the lasing frequency for a $1\times 15$ PEG. The gray shading indicates the density of states of the bands in Fig. 1.

Figure 4

[(a)–(c)] Normalized spatially integrated power spectral density (PSD) in the WEG configuration without disorder (a) and with a disorder strength equal to $\sigma \left(U\right)/J=0.4$ (b) and to $\sigma \left(U\right)/J=1.2$ (c). The PSD for a single random realization of lasing is displayed in green, while the average over multiple realizations [5000 in (a) and 2500 in (b) and (c)] is shown in blue. The shaded areas indicate the density of states of the bands in the absence of disorder and the central band region has been cut out for visualization convenience. [(d)–(f)] Snapshots of the typical intensity distribution at an arbitrarily chosen late time $t=500{\gamma }^{-1}$ for the same values of disorder as in (a)–(c). (g) Realization of the disorder used (upon rescaling) for all other simulations in the figure. Colors indicate the frequency shift of the different sites for a disorder strength $\sigma \left(U\right)/J=0.1$. (h) normalized emitted intensity as a function of gain strength for a single resonator (blue dashed line), for the nondisordered case (solid red line) and for a few disordered cases with $\sigma \left(U\right)/J=0.4$ (solid yellow line) and $\sigma \left(U\right)/J=0.4$ (violet line).

Figure 5

(a) Steady-state intensity distribution for a $1\times 15$ PEG in a large $11\times 51$ lattice. The green rectangle indicates the amplifying sites. (b) Lasing threshold for $1\times N$ PEG with different $J/\gamma =20,10$ (solid lines). Lasing threshold for a one-site-thick WEG case (dashed line). [(c)–(e)] Cuts of the intensity distribution along the $x=1$ line for different PEG geometries (see legends). The shaded area indicates the amplifying sites. The different curves in (c) and (d) refer to the steady state for different gain strengths $P/P_0=4$ (solid blue) and $P/P_0=2$ (dotted, light blue; to facilitate reading, these curves have been rescaled to have the same maximum as the solid blue ones); the different curves in (e) refer to different times separated by $0.05{\gamma }^{-1}$.

Figure 6

Spatiotemporal intensity patterns on the left border of a $11\times 51$ lattice with a $1\times 21$ amplified strip located at the center of the left side and indicated in all panels by the dashed white lines. In the first row, there is a noisy seed only at $t=0$, while on the third one there is noise at all times. In the second row, there is no noise, but there is a Gaussian pulse at frequency $\omega /J=1.9$ localized on the central site and centered in time at $t=15$, with standard deviation ${\sigma }_t=0.3$. All the times are measured in units of ${\gamma }^{-1}$. For homogeneity, in the AI case, we restricted to realization of lasing into the clockwise propagating topological mode localized on the left edge. In this figure, $\beta =0.01$ was taken.

Figure 7

[(a)–(c)] Normalized spatially integrated power spectral densities for different values of the disorder strength $\sigma \left(U\right)/J=0$ (a), 0.2 (b), and 0.5 (c); the shaded areas indicate the density of states of the bands in the absence of disorder. [(d)–(f)] Snapshots of the intensity distribution at a late time $t=500{\gamma }^{-1}$ for the same configurations of (a)–(c). (g) Frequency shift of the different sites for a disorder strength $\sigma \left(U\right)/J=0.1$; upon a suitable rescaling, this realization of the disorder is used in all panels. Same geometry with a $1\times 15$ amplifying strip in a $11\times 51$ lattice as in Fig. 5.

Figure 8

(a) Incident-frequency-dependent transmission spectrum for a $1\times 7$ PEG and different gain strengths (from bottom to top) $P/P_0=2.04,2.05,2.06,2.07,2.08,2.085,2.09$ approaching the lasing threshold. (b) Peak transmittivity as a function of gain strength for incident amplitude $E_0/\sqrt{J}={10}^{-7}$ (upwards triangles) or ${10}^{-8}$ (downwards triangles). Red lines indicate the result of the linearized calculation based on the input-output formalism of [36, 50].