Resonator neuron and triggering multipulse excitability in laser with injected signal

Semiconductor lasers with coherent forcing are expected to behave similarly to simple neuron models in response to external perturbations, as long as the physics describing them can be approximated by that of an overdamped pendulum with fluid torque. Beyond the validity range of this approximation, more complex features can be expected. We perform experiments and numerical simulations which show that the system can display resonator and integrator features depending on parameters and that multiple pulses can be emitted in response to larger perturbations.

Figure 1

Schematic of the experimental setup. The injection setup is highlighted in green, and the electrical perturbation setup is highlighted in blue.

Figure 2

Simplified depiction of the shape of the periodic electrical perturbation sent to the EOM. The delay times between each couple of pulses increases gradually from a minimum of 0.08 ns to a maximum of 1.05 ns. Each couple is separated by its neighbors by a 20 ns delay. The same perturbation is repeated periodically after completion, where each period takes $2.5\mu \mathrm{s}$ in total.

Figure 3

Experimentally measured time traces of the response of the system when the two perturbations are separated by 0.10 ns (top) and 0.50 ns (bottom). The emitted power (dc level) is about $500\mu \mathrm{W}$ and the injected power is $4.6\mu \mathrm{W}$. Insets: shape of the perturbations. Pumping current: 1.023 mA. Forty realizations are superimposed and show that on the bottom trace, no excitable pulse was observed. One example realization is shown as the black trace.

Figure 4

Experimental response time histogram and efficiency curve of the perturbation for around 4000 events for each delay ($S=1.023A$). The histogram is normalized so that each vertical slice for a single delay sums up to the corresponding efficiency value.

Figure 5

Integrator behavior of the Adler equation (top) and resonator behavior in the presence of inertia (bottom). Top: Numerical response time histogram and efficiency curve of Eq. (1), with $\omega =0.01$ and $\beta =0.08$, constructed from $10000$ events for each delay. Inset: Shape of the perturbation for a delay of 1.3 (at approximately 0.5 efficiency). Each of the two Gaussians has an amplitude of 2.8 Hz and standard deviation of 0.35. Bottom: Numerical response time histogram and efficiency curve of Eq. (2), with $\omega =0.01, I=10$, and $\beta =0.08$, constructed from $20000$ events for each delay. The perturbation is made of two Gaussians with amplitude of 0.96 Hz and standard deviation of 5.65. Inset: Shape of the perturbation for a delay of 20.

Figure 6

Bifurcation diagram of principal codimension 1 bifurcations, with $\alpha =4, \sigma =50$, and $\mu =15$. The diagram shows a Fold-Hopf bifurcation, where a saddle-node and a Hopf bifurcation collide on a single point. The parameters for the simulations are chosen in the region in between the Hopf and the saddle node bifurcation. The pink (right) and cyan (left) points correspond to the parameter set in which the resonator behavior is analyzed (pink being closest to the experimental observation) while the orange square shows the parameter set we chose for the integrator regime.

Figure 7

Critical manifold and fixed points. The green (downward) continuous part of the critical manifold is stable; the red dashed part (upward) is unstable. The red (bottom) dot is the saddle point and the green (top right) dot is stable. The blue dot (left) is the unstable focus, which cannot be seen in the upper panel as it lies much higher in the $D$ direction of phase space. Starting very close to the red saddle point, the system can relax back to the green stable fixed point following either of the two trajectories depending on the exact initial conditions (purple or blue lines, simulations without noise starting close to the red saddle point, ${\mathrm{\Delta }}^{\prime }=4.2$).

Figure 8

Left: Integrator example (${\mathrm{\Delta }}^{\prime }=2.45, |E_I|=0.3$). First column: time series showing the perturbations, the emitted intensity, the relative phase, and the population inversion. Second column: trajectory in the Argand plane. Third column: phase-space projections on the $\text{Re}\left(E\right),D$ plane. Black triangles indicate the occurrence of the second perturbation. Blue (left), red (bottom), and green (right) dots: unstable focus, saddle, and stable fixed point. Top row: two perturbations separated by 0.2 ns trigger a response (no noise). Middle row: two perturbations separated by 0.6 ns do not (no noise). Bottom row: efficiency of repeated perturbations with varying delay between them ($\beta =0.01$). Right: Resonator example (${\mathrm{\Delta }}^{\prime }=4.2$). Two perturbations separated by 0.35 ns do not trigger a pulse (top, no noise), but perturbations separated by 0.42 ns do (middle row, no noise). Bottom row: the efficiency shows several maxima depending on the time separation between perturbations ($\beta =0.01$).

Figure 9

Resonator example and efficiency (${\mathrm{\Delta }}^{\prime }=4.8, \beta =0.01$).

Figure 10

3D plot of the critical manifold, saddle-node pair, and the separatrix in phase space $\text{Re}\left(E\right)-\text{Im}\left(E\right)-D$. Bottom figures are a top-down view of top figures. Left figures: separatrix manifold in the integrator regime (${\mathrm{\Delta }}^{\prime }=2.45, |E_I|=0.3$). Right figures: separatrix manifold in the resonator regime (${\mathrm{\Delta }}^{\prime }=4.2, |E_I|=0.8$). The color coding of the surface is proportional to $D$ to improve readability in the $\left[\text{Re}\left(E\right),\text{Im}\left(E\right)\right]$ plane.

Figure 11

Multipulse response obtained when applying strong perturbations outside of the integrator regime. Larger perturbations cause a larger number of spikes, but there is a clear stochastic component in the phenomenon. Beyond ${140}^{\circ }$ all perturbations elicit a response but already at ${120}^{\circ }$ the double-spike response is the most probable one.

Figure 12

Multipulse response consisting of $6\pi$ phase rotation in response to perturbation (${\mathrm{\Delta }}^{\prime }=4.2$, delay of 0.27 ns, $\beta =0.01$).