Spectral signatures of exceptional points and bifurcations in the fundamental active photonic dimer

The fundamental active photonic dimer consisting of two coupled quantum well lasers is investigated in the context of the rate-equation model. Spectral transition properties and exceptional points are shown to occur under general conditions, not restricted by parity-time symmetry as in coupled-mode models, suggesting a paradigm shift in the field of non-Hermitian photonics. The optical spectral signatures of system bifurcations and exceptional points are manifested in terms of self-termination effects and observable drastic variations of the spectral line shape that can be controlled in terms of optical frequency detuning and inhomogeneous pumping.

Figure 1

Real (left) and imaginary (right) parts of the normalized eigenvalues ${\lambda }_{1,2}/\mathrm{\Lambda }$ of the zero state for a zero ($\alpha =0$, top) and a nonzero ($\alpha =5$, bottom) linewidth enhancement factor. The existence of spectral transitions and exceptional points takes place along the line $\mathrm{\Delta }-\alpha \mathrm{\Delta }P=0$. A nonzero linewidth enhancement factor raises the restriction of zero detuning ($\mathrm{\Delta }=0$) for the existence of exceptional points.

Figure 2

Stability region (dark-blue shaded area) of the zero state in the ($P_1,P_2$) parameter space for zero $[\alpha =0$ (top)] and nonzero $[\alpha =5$ (bottom)] linewidth enhancement factor, under zero $[\mathrm{\Delta }=0$ (left)] and nonzero $[\mathrm{\Delta }=2$ (right)] detuning. The red dotted lines correspond to (a) $P_1=0.5$ and (d) $P_1=0.15$. Reversing of laser pump dependence and self-termination scenarios take place for increasing $P_2$ along the constant $P_1$ lines.

Figure 3

Eigenvalues trajectories corresponding to the cases of Fig. 2 (top, left) and Fig. 2 (bottom, right). The vertical dashed lines denote zero crossings of the real part and stability changes of the zero state. The rate-equation model predicts that self-termination scenarios can take place under configurations not restricted by exact or approximate fulfillment of $\mathcal{PT}$-symmetry conditions.

Figure 4

Eigenvalues (top, middle) and spectral line shape (bottom) of the phase-locked states for $\alpha =5$, $\mathrm{\Delta }=0$ and $log\mathrm{\Lambda }=-2$. The vertical yellow and green dotted lines correspond to zero crossings of the eigenvalue real part and exceptional points, respectively. Left: Phase-locked states under symmetric pumping $P_1=P_2$ ($\theta$ and $X_0$ are determined by $\rho$). The phase-locked states are stable between the vertical yellow dotted lines. Right: Phase-locked states under asymmetric pumping $P_1\ne P_2$ ($\theta$ is determined by $\rho$ and $X_0=0.01$). The phase-locked states are stable between the first and the second (left to right) vertical yellow dotted lines. The spectral line shape follows the imaginary part of the eigenvalues. Exceptional and bifurcation points are manifested in the spectral line shape through the emergence of side bands and intensity peaks, respectively.

Figure 5

Eigenvalues (top, middle) and spectral line shape (bottom) of the phase-locked states for $\alpha =5$, $\mathrm{\Delta }\ne 0$, and $P_1\ne P_2$. The vertical yellow and green dotted lines correspond to zero crossings of the eigenvalue real part and exceptional points, respectively. Left: Phase-locked states for $log\mathrm{\Lambda }=-1.9$, $X_0=\sqrt{0.5}$, and $\theta =\pi$. The phase-locked states are stable between the first and the second as well as between the third and the fourth (left to right) vertical yellow dotted lines. No exceptional points exist in this case. Right: Phase-locked states for $log\mathrm{\Lambda }=1$, $X_0=\sqrt{0.5}$, and $\theta =\pi /2$. The phase-locked states are stable on the left side of the vertical yellow dotted line. The spectral line shape follows the imaginary part of the eigenvalues. Exceptional and bifurcation points are manifested in the spectral line shape through the emergence of side bands and intensity peaks, respectively.