Excitability and coherence resonance are studied in a semiconductor quantum dot laser under short optical self-feedback. For low pump levels, these are observed close to a homoclinic bifurcation, which is in correspondence with earlier observations in quantum well lasers. However, for high pump levels, we find excitability close to a boundary crisis of a chaotic attractor. We demonstrate that in contrast to the homoclinic bifurcation the crisis and thus the excitable regime is highly sensitive to the pump current. The excitability threshold increases with the pump current, which permits to adjust the sensitivity of the excitable unit to noise as well as to shift the optimal noise strength, at which maximum coherence is observed. The shift adds up to more than one order of magnitude, which strongly facilitates experimental realizations.
© 2014 Optical Society of America
Semiconductor quantum dot (QD) lasers [1, 2] are promising candidates for optical communication applications and high-speed data transmission, since they are singled out by a narrow linewidth  due to small phase-amplitude coupling [3–5], and by strongly suppressed relaxation oscillations. The latter can be attributed to the special carrier scattering dynamics of QD lasers [6, 7]. It results in an higher dynamical stability of these lasers with respect to perturbation, e.g., external optical injection  or optical feedback . This results in simpler bifurcation scenarios and therefore in a better observability of nonlinear effects in experiments, which will be crucial for the noise induced dynamics discussed in this paper.
Understanding noise-induced effects, e.g., due to spontaneous emission noise, is indispensable for a variety of semiconductor devices, to name just a few: the performance of QD optical amplifiers , the polarization dynamics of surface emitting lasers  or the synchronization properties of coupled laser systems [12, 13]. A special situation emerges for devices that can be operated as excitable systems, i.e., systems that rest in a stable steady state, but can be excited to emit a spike by a super-threshold perturbation (e.g. noise). Well-known examples for excitable systems are spiking neurons , cardiac dynamics , and nonlinear chemical reactions . Excitability in laser systems received considerable interest in the last years. It was observed experimentally [17–20] and studied theoretically [21–24] in lasers with optical injection. Furthermore, excitability was found in lasers with short optical feedback [25, 26] as well as in lasers with a long external cavity , and it was investigated theoretically in lasers with saturable absorbers [28, 29].
Recently, it has been demonstrated that an excitable optical unit may be used as an optical tongue wrench permitting to sense single perturbation events . Data transmission systems based on excitable optical units confer a high degree of robustness due to their inherent signal reshaping capabilities. Therefore, it has been suggested to use an excitable optical unit as optical switch for all-optical-signal processing where it only reacts on sufficiently high optical input signals  or for noise reduction in optical telecommunication applications : a noisy input pulse triggers a “clean” output pulse. Generation of nanosecond pulses by an excitable semiconductor laser in an integrated optoelectronic circuits was already experimentally demonstrated . However, in the conventional setup of a quantum well (QW) semiconductor laser under long external optical feedback bifurcation points lie very dense. This makes it hard to experimentally address the small regions of excitability, which occur only close to certain bifurcation points. Instead, the QD laser with short optical feedback studied in this paper is dynamically more stable, and showing a simpler bifurcation scenario, it is thus more promising for this kind of application.
The counter-intuitive effect that an increase of the noise can lead to an increase of correlation, i.e., to an increase of the regularity of the spikes observed in the excitable regime, is known as coherence resonance [34–36]. In contrast to stochastic resonance (see  for a review) the effect occurs without periodic forcing of the system. Coherence resonance is already an intensively studied effect and was shown theoretically in quantum well (QW) lasers with saturable absorber , in QD lasers under optical injection , in lasers subject to long optical feedback [32, 39], in laser systems with polarization instabilities [40, 41], in semiconductor super-lattices , as well as in non-excitable systems below a subcritical Hopf-bifurcation [43–45].
In this paper, excitability and coherence resonance close to a boundary crisis bifurcation found in a QD laser subject to short optical feedback are studied. It is known that coherence resonance can be controlled by delayed feedback, e.g. for neural systems in the framework of the FitzHugh-Nagumo model (type-II excitability) [46–48], for systems close to a saddle-node infinite period bifurcation (type-I excitability), and close to a subcritical Hopf-bifurcation , however so far it has not been investigated close to a boundary crises. Using a sophisticated microscopically motivated rate equation approach, we show that the interesting effect of tunable regularity of emitted spikes strongly depends on the operating pump current and is thus easily accessible in experiments.
The paper is structured as follows: At first, in Sec. 2 a dimensionless version of the dynamic equations is introduced and the structure of the basic continuous wave (cw) solutions is discussed. Next, in Sec. 3 the bifurcation structure of the deterministic system is analyzed and its dynamics in the bistable regimes close to the loci of the bifurcation points, which render the system excitable, is studied in detail. Then, in Sec. 4 coherence resonance of the system subject to Gaussian white spontaneous emission noise is discussed in dependence of the pump current, before concluding in Sec. 4. Eventually, in Appendix A the dimensionless version of the model equations is derived.
2. Quantum dot laser model
The microscopically based rate equation model for the QD laser under optical feedback was previously discussed in [9, 50, 51]. Here a dimensionless form of the dynamical equations is used, which is derived in Appendix A. A sketch of the edge-emitting single-mode laser device is shown in Fig. 1(a). The light in the cavity is modeled by a semiclassical Lang-Kobayashi-type  equation for the slowly varying complex amplitude ℰ of the electric field. Taking into account only one roundtrip of the light in the external cavity, the field amplitude ℰ (t − τ) delayed by the external cavity roundtrip time τ is coupled back into the laser with feedback strength k and rotated by the external cavity phase C.
The energy-band diagram of the dot-in-a-well structure under consideration is sketched in Fig. 1(b). The carriers are first injected in the InGaAs quantum well (QW), which acts as a carrier reservoir, with the dimensionless pump rate J. Within the QDs formed by pyramidal structures of InGaAs, localized, discrete electron and hole ground states are considered that lead to a wavelength of the optical transition of λopt = 1.3μm. The occupation probabilities of electrons and holes in these states are denoted by ρe and ρh, respectively.
Coulomb scattering (nonlocal Auger scattering) is the dominating scattering process for high carrier densities in the lasing regime . Therefore, electron-phonon scattering is neglected for the carrier exchange between QW and QDs, but it is taken into account for the intraband transitions within the carrier reservoir. In the model the carrier exchange between QW and QDs is mediated by non-constant microscopically calculated Coulomb in- ( ) and out-scattering ( ) rates [54–56], which are nonlinear functions of the dimensionless carrier densities of electrons (We) and holes (Wh) in the carrier reservoir, and therefore depend on J. Note, that we use only dimensionless quantities for the rates (details of the non-dimensionalization can be found in [7, 51]). The scattering rates also strongly depend on the energy spacings between the QW band edges and the discrete QD levels, which are given by ΔEe = 210meV and ΔEh = 50meV for electrons and holes, respectively. The latter strongly depend on the size of the QDs and also on their material composition. In comparison to conventional QW lasers the carrier lifetimes τe/h (in their dimensionless form with W = 0.7ns−1 being the Einstein coefficient of spontaneous emission) constitute additional timescales, which are responsible for the strong suppression of the relaxation oscillations (ROs) of QD lasers  mentioned in the introduction. The order of magnitude of τe and τh can be tuned by the pump current J, which permits to tune the turn-on damping of the laser.
In the subsystem of the carriers, different dynamics is taken into account for ρe and ρh as well as for We and Wh. Therefore, the system of coupled delay differential equations reads4, 5]. The α-factor is defined as the variation of the real refractive index, which is proportional to the real part of the complex susceptibility with the carrier density divided by the variation of the gain, which is proportional to the imaginary part of the complex susceptibility, with the carrier density. However, each charge carrier transition in the band structure under consideration contributes differently to the complex susceptibility. While the resonant transitions of the QD carriers mainly affect the gain, the main contribution to the change of the refractive index is given by the off resonant carriers of the surrounding carrier reservoir (QW) (see  for details). For shallow QDs the timescale of the intensity pulsations, which are, as the ROs, on the timescale of nano-seconds, is large compared to the timescale of the carrier exchange between QW and QDs (given by τe and τh). The carrier exchange is then fast enough to synchronize QD and QW carriers, and the approximation of a constant α-factor holds. But, for very deep dots the carrier exchange between QD and QW can become so slow that it takes place on a similar timescale than the intensity pulsations. As a result, QDs and QW carriers can become desynchronized in feedback regimes, in which intensity pulsations are observed. In this regimes, the approximation of a constant α-factor then fails. Nevertheless, for the band structure discussed in this paper, the approximation of a constant α-factor yields reliable results.
The linear gain coefficient is denoted by g. The value of g takes into account that due to the inhomogeneous broadening of the gain medium only a subgroup of all QDs matches the mode energies for lasing. Further, the rescaled feedback strength is denoted by k ≡ Kτph/τin, where K is the dimensionless feedback strength ranging from zero to one (see Appendix A), τin is the roundtrip time of the light in the internal cavity, and the dimensionless roundtrip time of the light in the external cavity is given by τ. The process of spontaneous emission is modeled by a complex Gaussian white noise term ξ (t′), i.e.,58]. The small parameter γ multiplying the right hand sides of Eqs. (1b)–(1e) expresses the timescale separation between the fast field equation and the slow subsystem of the carriers. The terms model the contributions of the scattering rates, where the Pauli blocking is described by the terms 1 − ρe/h. In- and out-scattering rates are related by a detailed balance relations [55, 59]. Fit functions for can be found in Ref. . Further, rw describes the ratio of the Einstein coefficients of induced and spontaneous emission, which was denoted by w in Ref. , and c is the band-band recombination coefficient in the QW. Parameter values used in the simulations are given in Tab. 1.
It is crucial to note for the subsequent analysis that the carrier equations (1b)–(1e) are not independent but contain carrier conservation, which can be seen by verifying that ρ′e + W′e = ρ′h + W′h holds. Thus, ρe − ρh + We −Wh is a constant, which is zero in the intrinsic case . Due to the carrier conservation, one dynamical variable can be eliminated by expressing one carrier type in terms of the others, i.e.,
2.1. External cavity modes–stationary solutions
In this section, the basic solutions of the dynamical equations (1) without noise (β = 0) are discussed. These external cavity modes (ECMs) organize the phase space of the system and provide a “backbone” for more complex, e.g., chaotic, dynamics observed in these systems , and therefore it is crucial to understand their bifurcation structure. They are cw solutions with constant photon number and carrier densities , , and a phase ϕ ≡ δωst of the electric field amplitude that varies linearly in timeequations (1) with feedback are denoted by the superscript s, and δωs ≡ τph(ω − ωth) is the deviation of the frequency ω of the ECM from the threshold frequency of the solitary laser ωth. Inserting the ECM-ansatz (3) into Eqs. (1), we find the following expressions for the non-zero intensity solutions ( ) Eq. (4a) into Eq. (4b), we obtain a transcendental equation for δωs in terms of α, τ, and C 51]. Taking advantage of the carrier conservation (Eq. (2)), we can reformulate Eq. (4a) to express and in terms of and Eqs. (4c) and (4d) into the sum of Eqs (4e) and (4f) Eqs. (4e) and (4f) self-consistently, which has to be done numerically, because are nonlinear functions of We and Wh.
3. Bifurcation structure and excitable dynamics
To understand the noise induced excitations, we first have to characterize the bifurcation structure of the deterministic system, and we then have to discuss which phase space configurations lead to excitability. In principle, this structure was already reported elsewhere, [50, 51, 60]. Here, we focus on the dependence of the bifurcations on the pump current, which is crucial for the discussion of the excitable dynamics. Figures 2(a) and 2(c) depict the bifurcation diagrams of the local maxima of Nph versus feedback strength K for low J = 2Jth (red dots) and for higher J = 3Jth (black dots) and J = 4Jth (gray dots), respectively. Note that for the subsequent discussion we use the feedback strength K, which is more intuitive because it ranges from zero to one. The bifurcation diagrams have been obtained by increasing K stepwise using in each step the last τ-interval of the time series of the previous run as initial condition. Figure 2(b) depicts the frequency deviation δωs of the ECMs. Solid and dashed lines indicate stable and unstable solutions, respectively. For low K, only one ECM (blue line) exists, which initially is stable. For J = 2Jth this ECM is destabilized in a supercritical Hopf bifurcation at KH(J = 2Jth) = 0.085 (red dot in Fig. 2(b)), which results in a stable solution with a periodically modulated Nph (see leftmost inset in Fig. 2(a)). Increasing K further, this periodic orbit undergoes a cascade of period doubling bifurcations. After a large period-2 window, the system becomes chaotic at K = 0.21 (see middle inset in Fig. 2(a) for a time series of Nph). At the end of the region with complex dynamics, the chaotic attractor collapses onto a limit cycle, and periodic pulse packages are observed in the time series of Nph, which will be discussed in detail below (see rightmost inset in Fig. 2(a)). At Ksn = 0.2290 a new pair of ECMs is created at in a saddle-node bifurcation (limit point) indicated by an open black circle in Fig. 2(b). The position of the limit point is determined by Eq. (6) only, and is thus independent of J. In Fig. 2(b), the stable 2nd ECM is depicted by a green and the unstable anti-mode by a black dashed line. The photon number of the stable parts of the first and the 2nd ECMs and of the unstable anti-mode are plotted in Figs. 2(a) and 2(c), by thick blue, thick green, and black dashed lines, respectively. For J = 2Jth, bi-stability between the periodic orbit and the 2nd ECM is observed upon its creation at Ksn, until eventually at Khom = 0.22920, the periodic orbit is annihilated in a homoclinic bifurcation (brown vertical arrow) with the saddle (anti-mode) of the 2nd ECM-pair. For K > Khom, the laser emits in stable cw operation on the 2nd ECM.
With varying current, the bifurcation scenario changes. In Ref. , it was shown that the RO damping increases linearly with the pump current. This is the reason why for higher J = 3Jth and J = 4Jth, the Hopf-bifurcation points KH shift toward higher K-values  (red dots and blue arrows in Fig. 2(b). Further, for pump currents larger than J > 2.8Jth the end of the bifurcation cascade is not marked by a homoclinic bifurcation, but by a boundary crisis  of the chaotic attractor that collides at Kcris with the saddle (anti-mode) of the 2nd ECM pair. Bi-stability is now observed in the interval [Ksn, Kcris]. The feedback strengths Khom and Kcris, at which homoclinic bifurcation and boundary crisis occur, have been found by up- and down-sweeping K with a very small stepsize of ΔK = 1 · 10−5. For up-sweeping K, the system remains on the periodic orbit (chaotic attractor), up to Khom (Kcris), while for down-sweeping K the laser emits on the 2nd ECM down to Ksn. Therefore, Khom and Kcris are determined by the upper limit of the bi-stability region. In contrast to the homoclinic bifurcation that is independent of the pump current for J ∈ [Jth, 2.8Jth], the feedback strength Kcris, at which the boundary crisis occurs, increases with the pump current (see dark and light arrow in Fig. 2(c)).
The laser is excitable for K-values little larger than Khom for J < 2.8Jth and analogously for K-values little above Kcris for J > 2.8Jth. In both cases, the short unstable manifold of the anti-mode acts as perturbation threshold. For J ≤ 2.8Jth, the response of the system to a super-threshold perturbation is a large excursion of the trajectory in phase space close to the “ghost” of the limit-cycle that is destroyed in the homoclinic bifurcation. For J > 2.8Jth, the excursion in phase space is guided by the ruin of the chaotic attractor that collapses at Kcris. In Figs. 2(a) and 2(c) this situation is elucidated, the threshold is given by the difference of the photon numbers of the 2nd ECM (thick green line) and of the anti-mode (black dashed line). The threshold is very low for K = Khom and increases with K. This implies that for J > 2.8Jth, when the system re-stabilizes in a boundary crisis, the threshold can be tuned by varying the pump current and with it the critical feedback strength Kcris = Kcris(J).
Next, the dynamics in phase space is discussed to gain a better understanding of the difference between the excitable behavior close to the homoclinic bifurcation and close to a boundary crisis. Figures 3(a) and 3(b) depict time series and phase space projections onto the (Nph, We)-plane for K = 0.2290, i.e., just below Khom, where the periodic orbit still exists. A subthreshold perturbation of the system from the stable 2nd ECM, i.e, the lasing fixed point, (green line) decays rapidly back to this steady state, while a super-threshold perturbation (blue line) yields strictly periodic pulse package, i.e., a motion along the periodic orbit. The green and the blue triangles in the closeup of Fig. 3(a) denote the starting point of the trajectories and the photon number of the anti-mode is plotted as a black line. In Ref.  we showed that the inter-pulse interval time TISI scales logarithmically with the distance from the bifurcation point, i.e., TISI ∼ ln|K − Khom|, as it is expected close to a homoclinic bifurcation . In the phase space projection in Fig. 3(b), it can be seen that after a power dropout at the end of each pulse package (nearly vertical part of the trajectory), the trajectory at first performs pronounced damped oscillations spiraling around the point in phase space, where the 2nd pair of ECMs has been created at the nearby saddle-node bifurcation (the green dot in Fig. 3(b) indicates the stable 2nd ECM). Afterwards, it is re-injected into the high gain region during the power dropout.
The lower panel of Fig. 3 depicts the excitability of the laser close to the boundary crisis for K = 0.23, which is a little below Kcris(J = 3). A super-threshold perturbation (blue line in Fig. 3(d)) yields rather regular pulse packages, although they are not strictly periodic as the ones observed close to the homoclinic bifurcation. Furthermore, the inter-spike interval time does not obey a specific scaling law as the pulse packages described before. From the phase space projection in Fig. 3(b), we see that the trajectory has essentially the same shape observed close to the homoclinic bifurcation, but does not close up, which yields a certain width of the chaotic attractor in phase space. Note that these regular pulse packages are similar to those observed by Heil et al. in a QW laser with short optical feedback [64, 65]. Comparing the distance of of the 2nd ECM (green line) and of the anti-mode (black line) in Figs. 3(a) and 3(c), we see that the excitation threshold is much larger close to the crisis than close to the homoclinic bifurcation. Thus, close to the crisis larger perturbations (higher noise levels) are needed to excite the system and cause a phase space excursion.
4. Coherence resonance
In this section, we analyze the phenomenon of coherence resonance close to the end of the first bifurcation cascade discussed in the previous section. As a measure for the regularity of the pulse packages, the correlation time tcor is used. For a stationary stochastic process y, it was introduced by Stratonovich  as67], we calculate Ψy from the ensemble averaged power spectral density. Here, we take the photon number as stochastic process, i.e., y = Nph. Another measure for the regularity of the pulse packages is the normalized standard deviation of the inter-spike interval time TISI  35]. In our laser system, the noise is applied only to the optical equations. Therefore, measuring TISI not directly from the timeseries of Nph but from the carrier inversion yields more robust results, because the latter are only indirectly effected by the noise. The dropout of Nph before the first intensity spike of each pulse package (cf. Fig. 3(a)) corresponds to a spike of the inversion ρinv defined in Eq. (5), which is followed by a damped oscillation towards its steady state value . To determine TISI, a threshold value is chosen, which is crossed during the first dropout of each pulse package, but not during the subsequent damped oscillations. To find the exact timing position of the first spike of the inversion of the n-th pulse package, we define a probability density by Figs. 3(c) and 3(d)). Therefore, for the deterministic system it is always possible to find an appropriate threshold value , that is only passed by the first spike of each pulse package.
To study coherence resonance, the QD laser is operated on the stable 2nd ECM just behind the bifurcation cascade, where the deterministic system is not bistable anymore. For instance, this implies that the deterministic system would respond to a super-threshold excitation by an excursion in the phase space along the “ghosts” of the attractors destroyed in the homoclinic bifurcation (J ≤ 2.8Jth) and the boundary crisis (2.8Jth < J), respectively, and would then return to the stable 2nd ECM. Meaning that in contrast to the timeseries shown in Figs. 3(a) and 3(c) each super-threshold excitation is followed only by one pulse package. Subject to noise, the system can be excited if the perturbation introduced by the noise is large enough to overcome the excitability threshold. Figure 4(d) depicts tcor (red triangles, right y-axis) and RT (blue dots, left y-axis) as functions of the noise strength β for K = 0.22921 and J = 2Jth, i.e., for a K-value closely above the homoclinic bifurcation at Khom = 0.22920. Furthermore, tcor is shown for K = 0.2314 and J = 3Jth (black stars, right y-axis) as well as for K = 0.24515 and J = 4Jth (gray hexagons, right y-axis), i.e., for K-values closely above the crisis of the chaotic attractor at Kcris(J = 3Jth) = 0.23324 and Kcris(J = 4Jth) = 0.24514, respectively (cf. Fig. 2). A clear maximum of tcor can be observed in all three cases indicating coherence resonance. Figures 4(a)–4(c) visualize the respective dynamics for values of β below (β = 1 · 10−10, Fig. 4(a)), at (βopt = 5 · 10−9, Fig. 4(b)), and above (β = 6.5 · 10−8, Fig. 4(c)) the noise strength βopt, at which the maximum of tcor is observed for J = 2Jth. Analogously, Figs. 4(e)–4(g) depict time series below (β = 0.02, Fig. 4(e)), at (βopt = 0.038, Fig. 4(f)), and above (β = 0.08, Fig. 4(g)) the noise strengths βopt = 0.23325 of the coherence maximum for J = 3Jth. The β -values, at which the time series are taken, are indicated by gray dashed vertical lines in Fig. 4(d).
Let us first discuss the coherence resonance close to the homoclinic bifurcation. Generally, the time between two excitations TISI can be decomposed into the time needed to activate the system ta and the refractory time tr, which the system needs to settle back to the rest state. In our system, the rest state is the stable 2nd ECM, and the refractory time is given by the time the system needs to spiral back to the 2nd ECM after one excitation. This means that tr is fixed by the internal dynamics of the system, while ta depends on the noise strength β. For low values of β, the activation time ta is long compared to tr (see Fig. 4(a)). Increasing β, it becomes easier for the system to overcome the excitation threshold and the pulse packages arise more regularly (see Fig. 4(b)). This is indicated by an increase of tcor and a decrease of RT. Increasing β further, pulse packages are excited more often, but the regularity of their appearance decreases and they are additionally deformed by the noise (see Fig. 4(c)). This leads to a decrease of tcor and an increase of RT. The maximum of tcor does not coincide exactly with the minimum of RT. This is expected, because tcor accounts for coherence in periodicity of the pulse packages as well as coherence in amplitude fluctuations, while RT only measures the periodicity of the pulse packages.
Higher pump currents of J = 3Jth and J = 4Jth lead to higher excitability thresholds (see Fig. 2(c)). Thus, a maximum of the correlation is therefore expected at a higher level of the noise. This is the reason why the maximum of tcor shifts to higher values of the noise strength β with increasing J (see black stars and gray hexagons in Fig. 4(d) for J = 3Jth and J = 4Jth, respectively). By comparing the time traces taken at the maxima of tcor for J = 2Jth and J = 3Jth, which are depicted in Figs. 4(b) and 4(f), respectively, two effects are prominent. On the one hand, the higher noise level in Fig. 4(f) becomes obvious, and, on the other hand, we see that the peak heights of the pulse package are varying more strongly in Fig. 4(f) than in Fig. 4(b), i.e., the amplitude jitter of the pulse packages is larger. However, the measure RT fails at higher values of the noise strength, because there is an ambiguity in distinguishing the peak position from positions of extreme noise events. The trajectory is just distorted so much by the noise that the first dropout in ρinv crossing is not necessarily the beginning of a pulse package. Therefore, RT has not been depicted for J = 3Jth and J = 4Jth. That for J = 3Jth and J = 4Jth the dynamics beyond the coherence maximum is dominated by the noise can be seen in Fig. 4(g) depicting for J = 3Jth a time trace right to the maximum of tcor.
In Fig. 5(a) the dependence of the feedback strengths Kcris, at which the boundary crisis occurs, is depicted as a function of the pump current. It reveals that Kcris increases linearly with the pump current J. As mentioned in Section 3, it was shown in previous works [7, 69] that the RO damping increases linearly with J. Further, the feedback strengths KH of the first Hopf bifurcation marking the beginning of the first bifurcation cascade also reveals a linear dependence on J as discussed in [9, 51]. The linear dependence of Kcris on J shown in Fig 5(a) now suggest that the linear increase of Kcris with J is also due to the pump dependence of the RO damping. From Fig. 2(c), it can be presumed that this linear dependence of Kcris on J results in a square-root like increase of the projection of the excitability threshold onto the photon number with J. This can be seen as follows: is given by the difference of the photon number of the 2nd ECM (thick green lines in Fig. 2(c)) and its anti-mode (dashed black lines in Fig. 2(c)). The former increases square-root like with J, while the latter decreases with J in the same way, which causes the square-root like increase of depicted in Fig. 5(b). Since the increase of Kcris on J is relatively small in the current range plotted in Fig. 5(b), the increase of is nearly linear. Note that the threshold has always been determined at the same (very small) distance to Kcris, more precisely at K = Kcris + 1 · 10−5. Eventually, the dependence of the noise strength at the coherence maximum βopt on J is depicted in Fig. 5(c). The optimal noise level βopt increases with J as expected from the increase of the threshold. To our knowledge, this is the first time coherence resonance has been observed close to a boundary crisis. Further, in contrast to coherence resonance close to a homoclinic bifurcation studied in [25, 32], the pump current dependence of the coherence maximum observed close to a crisis should facilitates the experimental accessibility of the excitable regime.
In the remainder of this section, some practicalities of finding excitability and coherence resonance in laser systems are discussed. Experimentally it has been shown that spontaneous emissions noise is sufficient to excite a semiconductor laser under optical injection operating in a stable locked cw state close to the boundary of the locking tongue [17–19]. Further, excitability and coherence resonance has been experimentally observed in semiconductor lasers under optical feedback by adding broadband Gaussian white noise to the pump current [39,43]. Moreover, excitability close to a homoclinic bifurcation  and close to a crises  has been verified experimentally in an integrated multi-section semiconductor QW integrated feedback laser by perturbing the laser with short external optical pulses. Both methods, adding noise to the pump current and external optical pulses cause well tunable perturbations of the trajectory in the phase space. In our simulations, for simplicity we use the spontaneous emission noise in the field equation, i.e., the coefficient β, to excite the system. Since this is also a perturbation of the trajectory in the phase space, we expect that our results can be verified experimentally by the two methods mentioned above.
Further, our simulations have been performed for a fixed amplitude phase coupling α, a fixed band structure, and a fixed feedback phase C, but they are robust under changes of these parameters as discussed in the following. From the transcendental Eq. (6), it can be seen that the number of ECMs and thus the number of bifurcation cascades increases with τ and α (See  for a detailed discussion.). We have focused on the simplest scenario, of a short external cavity and a small α-factor, where only one instability region is observed. However, excitable regimes and crises can also be found for larger values of α and τ for which several bifurcation cascades occur [50,70]. The band structure mainly impacts on the damping of the ROs, which in terms influences the dynamical stability of the laser. For experimental realizations the enhanced dynamical stability of QD lasers is a big advantage, because the structure of the bifurcation cascade is simpler [50, 71], and they are thus less sensitive to perturbations unavoidable in experiments, e.g., small temperature fluctuations. Therefore, we expect that with a QD lasers it should be easier to detect the excitable regime and stay close to a homoclinic bifurcation or a boundary crises. Eventually, our results are robust under changes of the feedback phase C, that mainly shifts the range of K values at which the bifurcations occur. As mentioned above excitability has been demonstrated in multi-section integrated feedback laser, which reveals that the feedback parameters, especially the phase C and the feedback strength K, can be controlled well enough to stay close to the homoclinic bifurcation or the crises of the chaotic attractor. Indeed it has been shown that the bifurcation cascade can be scanned by careful tuning of external cavity phase C and pump current [72, 73].
We found that a QD laser subject to optical feedback can be operated in an excitable regime, where the regularity of the emitted spikes is sensitive to the noise strength as well as to the pump current. More precisely, we have shown that coherence resonance exists close to a boundary crisis of a chaotic attractor. In contrast to coherence resonance close to a homoclinic bifurcation, which was theoretically predicted previously, this type of coherence resonance has the advantage to be highly sensitive to variations of the pump current. This permits to shift the excitability threshold and, consequently, the maximum of the coherence found in the emitted spikes. Further, our findings are robust over a large range of pump currents facilitating the experimental accessibility of the excitable regime. Since the operating pump current is easily accessible in an experiment, it opens up the possibility to experimentally observe coherence resonance in semiconductor QD lasers. Further, we connect the pump current induced shift of the boundary crisis to the damping of the turn-on relaxations, as it also increases linearly with the pump strength.
A. Derivation of the dimensionless model
In this section, the dimensionless version of the dynamical equations (1) used in the main text is derived from the physical model. The optical subsystem of the QD laser model with feedback is described by a Lang-Kobayashy type delay differential equation for the normalized slowly varying complex amplitude ℰ (t) of the electric field , where νth is the optical frequency at lasing threshold, and c.c denotes the complex conjugate. Since different dynamics is taken into account for electrons and holes the carrier subsystem consists of four coupled differential equations for the occupation probabilities ρe and ρh of electrons and holes in the discrete QD ground states, and the carrier densities for electrons, we, and holes, wh, in the surrounding QW acting as a carrier reservoir65, 74, 75].
The spontaneous emission is modeled by bimolecular recombination , where β is the spontaneous emission factor measuring the probability that a spontaneously emitted photon is emitted into the lasing mode. The Einstein coefficient for spontaneous emission is denoted by W. It can be determined by calculating the coherent interaction of a two-level system, i.e., a single QD, with all resonator modes in the framework of the second quantization . Note that the coefficients W̄ and W differ by three orders of magnitude (See Refs. [8,60] for details of their derivation.). In a semiclassical approach, the process of spontaneous emission is modeled by a complex Gaussian white noise term ξ = ξ (t), i.e.,
In the model, the carrier exchange between QW and QDs is mediated by non-constant microscopically calculated Coulomb in- ( ) and out-scattering ( ) rates [54, 56], which are nonlinear functions of the carrier densities of electrons (we) and holes (wh) in the carrier reservoir, and therefore depend on the pump current density j. The latter is normalized by the elementary charge e0 in the equations for the reservoir densities (Eqs. (11d) and (11e)). The inverse of the sum of the scattering rates yields the carrier lifetimes in the QD levels. Note that in thermodynamic equilibrium, there is a detailed balance between in- and out-scattering rates, which allows one to relate the coefficients of in- and out-scattering even away from the thermodynamic equilibrium . The detailed balance relation for in- and out-scattering rates for the quasi-equilibrium then reads 60] for fit functions for the in-scattering rates ). Analogously to the spontaneous emission in the field equations, the spontaneous emission in the QW is incorporated by the bimolecular term BSwewh, where BS is the band-band recombination coefficient (see Eqs. (11d) and (11e)). All physical parameters used are summarized in Table 2.
In the main text, a dimensionless form of the dynamical equations is used, which emphasizes the different timescales involved. As it is usually done for rate equation models of semiconductor lasers, time is rescaled with respect to the photon lifetime τph . Introducing the dimensionless time t′ ≡ t/τph as well as the dimensionless reservoir populations We ≡ we/(2NQD) and Wh ≡ wh/(2NQD), the set of dimensionless dynamical equations (1) discussed in the main text can be derived. Where the dimensionless linear gain coefficient g, the rescaled feedback strength k, the dimensionless delay time τ, the dimensionless coefficient of the spontaneous emission rsp, the ratio of photon and carrier lifetimes γ (Here the lifetime of the carrier subsystem is represented by W−1.), the ratio of the Einstein-factors of induced and spontaneous emission rw, the dimensionless pump rate J, the coefficient of spontaneous and non-radiative losses in the carrier reservoir c, and the dimensionless scattering rates have been introduced asTable 2, are listed in Table 1. Note, that the small parameter γ multiplying the right hand sides of Eqs. (1b)–(1e) expresses the timescale separation between the fast field equation and the slow subsystem of the carriers, i.e., the QD laser is a slow-fast system if the scattering rates are not to large.
We thank R. Aust and A. Zakharova for fruitful discussions. This work was supported by DFG in the framework of SFB 910 and SFB 787.
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