Optical turbulence and transverse rogue waves in a cavity with triple-quantum-dot molecules

We show that optical turbulence extreme events can exist in the transverse dynamics of a cavity containing molecules of triple quantum dots under conditions close to tunneling-induced transparency. These nanostructures, when coupled via tunneling, form a four-level configuration with tunable energy-level separations. We show that such a system exhibits multistability and bistability of Turing structures in instability domains with different critical wave vectors. By numerical simulation of the mean-field equation that describes the transverse dynamics of the system, we show that the simultaneous presence of two transverse solutions with opposite nonlinearities gives rise to a series of turbulent structures with the capability of generating two-dimensional rogue waves.

Figure 1

(a) Cavity configuration for a typical triple-quantum-dot molecule with an injected pump and tunnelings. (b) Energy-level structure for dots in the molecule.

Figure 2

(a) Imaginary (solid line) and real (dashed line) parts of the complex susceptibility $\chi$ for $|F_i{|}^2=0.5, T_A=0.1, T_B=0.01, {\omega }_{12}=0.67$, and ${\omega }_{23}=0.3$. (b) Bistability and multistability in the input-output intensities for $\mathrm{\Sigma }=1, \delta =0.3$, and $\theta =-0.5$.

Figure 3

(a) Turing instability domains having two different critical wave vectors, indicated by $K_{c1}$ and $K_{c2}$, and (b) their map on the multistability curve for ${\omega }_{12}=0.8$. (c) Variation of the control parameter threshold versus the stationary intensity for the onset of patterned solutions and (d) the corresponding critical wave vectors. Other parameters are the same as in Fig. 2.

Figure 4

Average intensity for a sequence of transverse spatial solutions for the instability domains of (a) the small and (b) the large wave vectors, respectively. The rightward and leftward arrows depict the direction of change in the control parameter ${\omega }_{12}$, implying scans I and II. Note that the trajectories of solutions in scans I and II depart from each other at ${\omega }_{12}=0.36$ and ${\omega }_{12}=0.485$ for the first and second instability domains, respectively.

Figure 5

Linear and nonlinear refractive indices versus increasing values of the control parameter ${\omega }_{12}$. Hatched areas show the $n_2>0$ region where hexagons form; gray shaded areas, honeycombs of $n_2<0$.

Figure 6

Wave numbers associated with different solutions in different scans for the instability domains of (a) small and (b) large wave numbers. Gray areas depict the turbulent interval in either domain and the rightward (leftward) arrows show the direction of scan I (scan II). Note that for scan II in (b) no turbulent structure is excited.

Figure 7

Transverse structures (a), (d) slightly below, (b), (e) within, and (c), (f) slightly above the turbulent interval for the small (top row) and large (bottom row) wave-number instability domains.

Figure 8

Left: Widening of the intensity dip in the background pattern before the appearance of an intensity peak associated with the emerging pattern. Right: The same cell of the background pattern containing a peak of the new pattern, which shrinks after a short time when the emerging peak disappears. This breathing behavior continues until the control parameter is increased and more of these breathing cells engage in the competition.

Figure 9

Left: Maximum intensity (blue), transverse average intensity (orange) and standard deviation of the intensity (black) versus time, respectively, for ${\omega }_{12}=0.359$ and ${\omega }_{12}=0.363$, from top to bottom. Right: Corresponding PDF histograms for maximum intensity values. The solid red (dotted blue) line represents the best fit of the PDF data with a Gaussian (Weibull) distribution. Time is normalized to the photon lifetime in the cavity.

Figure 10

Intensity versus phase values of the peaks. Filled red circles, turbulent structure with no RWs at ${\omega }_{12}=0.359$; blue circles, turbulent structure with RWs at ${\omega }_{12}=0.363$; and green square, stable honeycomb pattern at ${\omega }_{12}=0.370$.

Figure 11

The effect of noise on the probability density functions: (a) ${\omega }_{12}=0.361$ and (b) ${\omega }_{12}=0.363$. The curves represent the best Weibull fits with (solid red curve) and without (dotted black curve) noise.