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The entrainment (or locking) phenomenon, by which an oscillator adapts its natural rhythm to an external periodic signal, is well-known in physics, chemistry, biology, etc.; however, controlling an stochastic nonlinear system with a small-amplitude signal is a challenging task, and systems that allow for low-cost experiments are scarce. Here we use a semiconductor laser with optical feedback, operated in the regime where it randomly emits abrupt spikes. We quantify the quality of the entrainment of the optical spikes to periodic, small-amplitude electric perturbations of the laser pump current. We use the success rate (SR) that counts the number of spikes that occur within a short time window after each perturbation, and the false positive rate (FPR) that counts the additional spikes that occur outside the window. The ROC curves (SR vs. FPR plots) uncover parameter regions where the electric perturbations fully control the laser spikes, entraining them, such that the laser emits, shortly after each perturbation, one and only one spike (i.e., SR=1 and FPR=0). We also characterize the locking-unlocking transitions when the perturbation amplitude and frequency vary.

Semiconductor lasers with time-delayed optical feedback display a wide range of dynamical regimes, which have found various practical applications. They also provide excellent testbeds for data analysis tools for characterizing complex signals. Recently, several of us have analyzed experimental intensity time-traces and quantitatively identified the onset of different dynamical regimes, as the laser's current increases. Specifically, we identified the onset of low-frequency fluctuations (LFFs), where the laser intensity displays abrupt dropouts, and the onset of coherence collapse (CC), where the intensity fluctuations are highly irregular. Here we map these regimes when both, the laser current and the feedback strength vary. We show that the shape of the distribution of intensity fluctuations (characterized by the standard deviation, the skewness, and the kurtosis) allows to distinguish between noise, LFFs, and CC, and to quantitatively determine (in spite of the gradual nature of the transitions) the boundaries of the three regimes. Ordinal analysis of the inter-dropout time intervals consistently identifies the three regimes occurring in the same parameter regions as the analysis of the intensity distribution. Simulations of the well-known time-delayed Lang-Kobayashi model are in good qualitative agreement with the observations.