Spectrum with the manifold method

In order to obtain an intuitive picture on the optical spectrum of our system, as we have proceed with the other models, we first study the allowed transitions between the possible energy levels with the manifold method at vanishing pump. The bare energies of Fig. 5.5 acquire an imaginary part ( $\omega_{a,b}\rightarrow\omega_{a,b}-i\gamma_{a,b}/2$ ) and we diagonalize again, applying the formula $i[E_k-(E_{k-1})^*]=i\omega_p+\gamma_p/2$ to obtain positions and broadenings associated to each transition. We consider the amplitude of probability, $I^{k\rightarrow k-1}_a$ , for the processes of annihilation of a photon, that is, the transitions corresponding to the normal mode emission, as in Eq. (2.97). We know from previous results that, if the coupling and interactions are strong enough, the contribution of each transition to the optical spectrum $s_a(\omega)$ can be approximated by an independent Lorentzian weighted by the probability of its initial state times the amplitude of probability of the transition. The results are plotted for transitions between manifolds 2 and 1 only, as a function of the nonlinearity strength in Fig. 5.7(a) and as a function of the detuning in Fig. 5.7(b). In the latter case, the exciton bare energy is kept constant, equal to zero, while the cavity mode is brought in or out of resonance with the exciton. As I explained in Chapter 1, detuning is varied experimentally through a variety of techniques, like changing temperature or applying a magnetic field (shifting the energy of the dot with negligible perturbation on the cavity) or growing a thin film (shifting the cavity mode energy without affecting the dot). In Fig. 5.7(b) we see the intensity, width and location of spectral individual lines for smooth changes in the detuning.

**Figure 5.6:** Excitonic component at resonance of the eigenvectors of , corresponding to the three energy levels of the manifold (see Fig. 5.5) as a function of the interaction strenght . Varying the detuning also changes the character of the lines. In inset are plotted the exact corresponding eigenenergies as a function of , that are sketched in the right-upper part of Fig. 5.5.
$\includegraphics[width=0.55\linewidth]{chap5/brasilia/figure2.eps}$

**Figure 5.7:** Cavity emission spectra of transitions between manifolds $2\rightarrow 1$ : (a) Resonant case as a function of interactions . (b) Case of fixed interactions () as a function of $\Delta$ . The noninteracting case , where only linear Rabi doublet arises, is also shown (red superimposed lines) as well as the bare cavity and excitonic lines (dashed green lines). Lines are labelled in blue corresponding to the transitions of Fig. 5.5. Parameters in both plots are: $\omega_b=0$ , $\gamma_a=0.1g$ , $\gamma_b=0.01g$ , all in units of .
$\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure3a.eps}$ $\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure3b.eps}$

Peaks appearing in Fig. 5.7(a) correspond to the transitions plotted in Fig. 5.5. They are labelled in blue: lower lines are the transitions ``A'' and upper lines the transitions ``B''. Comparing with the linear Rabi doublet, which is superimposed in red, we observe the aforementioned blueshift of both groups of lines, as it happened in the AO of previous Section. It is more important for the exciton-like mode (especially line B-2 at resonance when $U\gg g$ ) while the photon-like mode has a better resolved fine-structure splitting. At various detunings [Fig. 5.7(b)], complicated structures are found with crossing or anticrossing of the lines, as shown on the figure. Lines with the same bare-excitation (photon or exciton) character cross, whereas lines of a different character exhibit anticrossing. At large detunings, the bare photon and exciton modes (in green) are recovered but with an additional blueshifted bare exciton line.

Satellite peaks arise at very low and high energies from transitions that are forbidden in the LM. They enter the dynamics through nonlinear channels opened by the interactions. The dashed arrows in the right panel of Fig. 5.5 represent these two transitions, with two excitons as the initial state that release one excitation and leave one photon as the final state. They appear dimly in Fig. 5.7.

Here, as in the previous models, the manifold method allows an understanding of the composition of the spectra, as illustrated in Fig. 5.7 where the spectral lines have been labelled according to their corresponding transitions in Fig. 5.5. From this overall picture, the excitonic fraction is clearly associated to the blueshift.