Optical spectra of a quantum dot in a microcavity in the nonlinear regime

The optical emission spectrum of a quantum dot in strong coupling with the single mode of a microcavity is obtained in the nonlinear regime. We study how exciton-exciton interactions alter the emission spectrum of the system, bringing the linear Rabi doublet into a multiplet structure that is strongly dependent on the cavity-exciton energy detuning. We emphasize how nonlinearity can be used to evidence the genuine quantum nature of the coupling by producing satellites peaks of the Rabi doublet that originate from the quantized energy levels of the interactions.

Figure 1

(Color online) Energy levels of the eigenstates of Eq. (1) up to the second manifold (two excitations) all at zero detuning $\left(\Delta =0\right)$, left panel for weak (or no-) coupling $\left(V_R=0\right)$, and central and right panel in strong coupling with right panel, also including interactions $U$ varying on the $x$ axis. The transitions between levels account for the spectral features. Red lines correspond to the vacuum-field Rabi doublet that turns into a single green line in weak-coupling regime. Blue lines superimpose to the Rabi doublet when higher manifolds are probed. Without interactions $U=0$ these transitions do not appear in the spectra. Transitions from $n=2$ to $n=1$ in the presence of interactions are plotted in Fig. 3 as a function of the detuning and $U$. New qualitative features appear, thanks to the interactions. Black dashed lines are new transitions previously forbidden although they remain weak.

Figure 2

Excitonic component at resonance of the eigenvectors corresponding to each energy level of the manifold $n=2$ (see Fig. 1) as a function of the interaction strength $U$. Varying the detuning also changes the character of the lines. In the inset, the exact energies of the eigenstates are plotted as a function of $U$ that are sketched in the right-upper part of Fig. 1.

Figure 3

(Color online) Cavity emission spectra from second to first manifold transitions: (a) Resonant case as a function of interactions $U$ ($y$ axis). (b) Case of fixed interactions $\left(U=2\right)$ as a function of detuning ($\Delta ={ϵ}_1-{ϵ}_2$ in $x$ axis). The noninteracting case $U=0$, where only Rabi doublet arises, is also shown (red superimposed lines) as well as the bare cavity and excitonic lines (dashed green lines). Lines are labeled in blue corresponding to the transitions of Fig. 1. Parameters in both plots are ${ϵ}_2=0$, ${\Gamma }_1=0.1$, and ${\Gamma }_2=0.01$, all are in units of $V_R$.

Figure 4

(Color online) Mean number of photons (dashed line) and excitons (solid line) as a function of the detuning between cavity and excitonic modes $\left(\Delta ={ϵ}_1-{ϵ}_2\right)$ for the case of cavity pumping ($P_1=P$ and $P_2=0$) in thin red $\left(\times 5\right)$ and electronic pumping ($P_1=0$, $P_2=P$) in thick black. Regardless of the detuning, an approximately constant and equal population is obtained for the cavity intensity due to the balance between the effective coupling strength and the exciton population. An asymmetry is observed with detuning due to the interactions that bring the exciton closer or further to resonance with the cavity mode. Parameters are ${ϵ}_2=0$, $U=2$, ${\Gamma }_1=0.1$, ${\Gamma }_2=0.01$, and $P=0.01$.

Figure 5

(Color online) Cavity emission spectrum as a function of detuning $\Delta$. The $U=0$ case for photons, which corresponds to the Rabi splitting, is also shown in red solid lines as well as the uncoupled ($V_{\text{R}}=0$ and $U=0$) cavity and excitonic lines in green dashed lines. As the mean number of excitons is very low (smaller than 1), the probability of having more than two excitons is very low; only the second to first and first to zero manifold transitions appear. For high positive detuning, Coulomb interactions generate an additional peak close to ${ϵ}_2+U\left(n_2-1\right)$. Note that peaks originated from transitions between manifolds with $n>2$ appear with a very small intensity due to the nonzero probability to have three excitons in the system. All energies are in units of $V_R$. Parameters are ${ϵ}_2=0$, $U=2$, ${\Gamma }_1=0.1$, ${\Gamma }_2=0.01$, and $P=0.01$.

Figure 6

(Color online) Spectra for different detunings corresponding to vertical “cuts” in Fig. 5. Both cases of cavity pumping ($P_1=P$ and $P_2=0$) and electronic pumping ($P_1=0$ and $P_2=P$) are represented in thin red and thick black, respectively. At very large detunings [(a) and (e)], multiple excitons occupancy is observed through the peaks E1, E2, and E3. Close to the resonance, these result in satellites surrounding the linear Rabi doublet that dominates because populations collapse at resonance. Parameters are ${ϵ}_2=0$, $U=2$, ${\Gamma }_1=0.1$, ${\Gamma }_2=0.01$, and $P=0.01$.

Figure 7

(Color online) Cavity spectral emission as a function of detuning $\Delta$ obtained via Green’s function technique. Parameters are similar to those in Fig. 5.