Excitable interplay between lasing quantum dot states

The optically injected semiconductor laser system has proven to be an excellent source of experimental nonlinear dynamics, particularly regarding the generation of excitable pulses. Typically for low-injection strengths, these pulses are the result of a small above-threshold perturbation of a stable steady state, the underlying physics is well described by the Adler phase equation, and each laser intensity pulse is accompanied by a $2\pi$ phase rotation. In this article, we show how, with a dual-state quantum dot laser, a variation of type I excitability is possible that cannot be described by the Adler model. The laser is operated so that emission is from the excited state only. The ground state can be activated and phase locked to the master laser via optical injection while the excited state is completely suppressed. Close to the phase-locking boundary, a region of ground-state emission dropouts correlated to excited-state pulses can be observed. We show that the phase of the ground state undergoes bounded rotations due to interactions with the excited state. We analyze the system both experimentally and numerically and find excellent agreement. Particular attention is devoted to the bifurcation conditions needed for an excitable pulse as well as its time evolution.

Figure 1

Setup for unidirectional injection experiment where the slave laser (SL) is a QD laser and the master laser (ML) is a tunable laser source (TLS). PC is the polarization controller, OSC is the oscilloscope, and OSA is the optical spectrum analyzer. The red lines represent light at approximately 1300 nm close to the GS emission, the blue is ES only, and the purple is both GS and ES.

Figure 2

Time trace of excitability. (a) shows the 300-ps dropout, (b) shows the 80-ps ES pulse, and (c) is the phase. Detuning is $-8$ GHz.

Figure 3

Real and imaginary parts of the electric field showing the phase evolution. The steady state is at approximately (0.6,0). When noise triggers a pulse it results in a bounded trajectory that does not go around the origin. The spiral close to the origin suggests the presence of a saddle focus.

Figure 4

A three-dimensional figure showing the phase and the GS and ES intensities. When the ES turns on, it pulls the phase away from the origin, stopping a $2\pi$ rotation. The saddle focus has two oscillations as in Fig. 3 but it is not visible from this reference angle.

Figure 5

Numerical bifurcation diagram. (a) shows the GS (red) and ES (blue) intensities vs $\varepsilon$. (b) shows the phase of the GS vs $\varepsilon$. Continuous (dashed) lines correspond to stable (unstable) branches. LP and H denote limit points and Hopf bifurcation points, respectively. BP denotes an unstable bifurcation point. $\text{LP}1$ denotes a saddle-node homoclinic (SNH) bifurcation and arises at $\varepsilon =4.4373$. The fixed parameters are $\eta =0.01$, $\mathrm{\Delta }=0$, $\alpha =3$, $\beta =2.4$, $g_0^g=g_0^e=0.55$, $J=56$, $B_{e,h}=B_{e,h}^w=100$, $C_e^w=0$, $C_h^w={10}^2$, $C_e=B_{e}exp(-2)$, $C_h=B_h$. $I_g$ is normalized by its value ($I_g=223.83$) at $\varepsilon =8$.

Figure 6

Numerically obtained noise ($D=0.25$) induced excitable events. (a) shows the GS (blue) and ES (red) intensities. (b) shows the corresponding phase evolution. The parameters are the same as in Fig. 5, where the vertical line indicates the point of operation.

Figure 7

Simulated 3D plots. (a) has a low-noise ($D=0.25$) level and consequently displays many rotations in the spiral. (b) has a more realistic noise level ($D=1$) and is in close agreement with the experimental results in Fig. 4.

Figure 8

Zoom of Fig. 5, where LP1 denotes the limit point which is a SNH bifurcation. The blue dot labeled SF represents the saddle focus. The green dotted line marks the operating injection strength where the noise induced events in Figs. 6 and 7 are found. The dashed lines are unstable branches and the solid lines are stable. The same operating parameters described in Fig. 6 are used here.