Collective Rabi dynamics of electromagnetically coupled quantum-dot ensembles

Rabi oscillations typify the inherent nonlinearity of optical excitations in quantum dots. Using an integral kernel formulation to solve the three-dimensional Maxwell-Bloch equations in ensembles of up to ${10}^4$ quantum dots, we observe features in Rabi oscillations due to the interplay of nonlinearity, nonequilibrium excitation, and electromagnetic coupling between the dots. This approach allows us to observe the dynamics of each dot in the ensemble without resorting to spatial averages. Our simulations predict synchronized multiplets of dots that exchange energy, dots that dynamically couple to screen the effect of incident external radiation, localization of the polarization due to randomness and interactions, as well as wavelength-scale regions of enhanced and suppressed polarization.

Figure 1

Nonorthogonal and $C^0$-continuous temporal basis function $T(t/\mathrm{\Delta }t)$ constructed from intervals of third-order Lagrange polynomials.

Figure 2

Population dynamics of adjacent quantum dots (top, shown with the trajectory of a quantum dot driven exclusively by the external laser in black) and a 1024-particle neighborhood [dots (1) and (2) maintain their separation and orientation between simulations]. The interaction between particles gives a greatly diminished response to the external pulse through a dynamical detuning of the two-dot system. The majority of the neighboring particles in the 1024-dot system follow trajectories nearly identical to that prescribed by the laser (omitted for clarity); many-dot effects, however, produce significant oscillatory modes in the evolution of selected quantum dots (shown in gray). Note that ${\rho }_{00}^{\left(1\right)}$ and ${\rho }_{00}^{\left(2\right)}$ appear to have two coherent modes in the presence of multiple dots: a high-frequency oscillation between the pair as well as a low-frequency oscillation of the group about a decaying envelope.

Figure 3

Spatial distribution of $|{\tilde{\rho }}_{01}|$ for 1024 quantum dots as an indicator of polarization. Recorded at $t=2\mathrm{ps}$ relative to the peak of a 1-ps-wide $\pi$ pulse, the color and size of each sphere indicates the location of each quantum dot and its polarization. Following a $\pi$ pulse, a single quantum dot would have no remnant polarization; here, due to the near-field interactions between particles, clusters of quantum dots remain in highly polarized states depending on their separation.

Figure 4

Inverse participation ratio (IPR) for the system shown in Fig. 3. The dashed line indicates the sample time for Fig. 3.

Figure 5

Coloration of $|{\tilde{\rho }}_{01}|$ at $t_1=-0.05, t_2=0, t_3=0.05$, and $t_4=0.10\mathrm{ps}$ relative to the peak of a 1-ps-wide pulse. 10 000 quantum dots randomly distributed throughout a $0.2\text{(radius)}\times 4\mu \mathrm{m}$ cylinder oriented along ${\mathbf{k}}_L$ demonstrate the near-field effects of Fig. 2 as distinct, outlying bright (dark) quantum dots. Additionally, the size of the system allows for wavelength-scale phenomena that appear here as five standing regions of enhanced polarization. (Note that we model quantum dots as point objects; the size of the spheres here has no physical interpretation.)

Figure 6

Coloration of $|{\tilde{\rho }}_{01}|$ for 10 000 quantum dots arranged in a finite planar geometry (in units of $\lambda$). The slab displays a prominent polarization pattern $1.25\mathrm{ps}$ after the peak of a $1\mathrm{ps}$ pulse.

Figure 7

The $z$ distribution of polarization $|{\tilde{\rho }}_{01}|$ for the geometry in Fig. 5. In each simulation, 10 000 quantum dots had random Gaussian noise (with width parameters of 0.0, 0.2, 0.4 and $0.6\mathrm{meV}$) added to a resonant $\hslash {\omega }_0=1500\mathrm{meV}$. The induced long-range patterns in the polarization remain for mild detunings $\lesssim 0.5\mathrm{meV}$ but have completely washed out at $0.6\mathrm{meV}$. Additionally, each simulation displayed the characteristic near-field coupling effects of Fig. 2 (not visible here).