Spectrum with detuning

The last relevant magnitude we consider that can be measured experimentally, is the power spectrum $s_c(\omega)$ ^6.3 of both the cavity () and the excitonic direct emission ( $c=\sigma$ ) in the steady state. The computation of the spectra in this case involves correlators of the kind $\langle\ud{c}(t^\mathrm{ss})[\ud{a}^{m}a^n\ud{\sigma}_1^{\mu_1} \sigma_1^{\nu_1}\ud{\sigma}_2^{\mu_2}\sigma_2^{\nu_2}](t^\mathrm{ss}+\tau)\rangle$ at all orders in the photonic indexes , . The fermionic ones $\mu_i$ , $\nu_i$ can only take values 0, . They in turn require as initial conditions at $\tau=0$ , mean values in the steady state ( $t^\mathrm{ss}$ ) of the form $\langle \ud{c}\ud{a}^ma^n\ud{\sigma}_1^{\mu_1}\sigma_1^{\nu_1}\ud{\sigma}_2^{\mu_2}\sigma_2^{\nu_2}\rangle^\mathrm{ss}$ that can be found applying again the quantum regression theorem, in a more efficient way than using the density matrix obtained in the previous Section. A truncation in the photonic indexes is needed again to close the equations. The number of correlators needed to obtain $\langle\ud{c}(t)c(t+\tau)\rangle$ goes like $16n_\mathrm{max}$ where $n_\mathrm{max}$ is the maximum value that , are allowed to take. Truncation must be good enough so that the result is independent of it. As in the previous cases, the spectra is of the form of Eq. (2.105), with a sum of $16n_\mathrm{max}$ peaks, each composed of a Lorentzian and a dispersive part.

**Figure 6.24:** Set of (a) cavity and (b) excitonic spectra for the same parameters as in Fig. 6.22 varying the cavity energy also in the same rage of energies. The contour plot for the exciton emission is plotted in (c) in order to appreciate the anticrossings between cavity and QD modes. The cases corresponding to the three resonances are highlighted in thick red (from top to bottom): $\Delta=0$ (1PR), $\Delta=20g$ (2PR) and $\Delta=40g$ (1PR). (We recall that $\omega_a-\omega_E=-\Delta/2$ )
$\includegraphics[width=0.6\linewidth]{chap6/2P/Spectra/Cavity-P_1.eps}$ $\includegraphics[width=0.6\linewidth]{chap6/2P/Spectra/Exciton-P_1.eps}$ $\includegraphics[width=0.5\linewidth]{chap6/2P/Spectra/Exciton-c-P_1.eps}$

In Fig. 6.24 we put together a set of cavity (a) and excitonic (b) spectra for various detunings between the cavity and the exciton mode, varying in the same range as in Fig. (6.22). The cavity spectra is clearly dominated by the emission at the cavity energy, that moves in diagonal from left bottom to right-top corners of the plots. At each of the three resonances the cavity emission is enhanced (thick red lines), and notably more at the 2PR (middle one at $\Delta=\chi=20g$ ). The lasing behavior of the cavity emission is very different to the direct excitonic one [Fig. 6.24(b)], where one can learn more on the level structure. The QD frequencies pin some of the emission at $\omega=\omega_E$ (lowest transition) but mostly at $\omega=\omega_E-\chi$ (highest one), due to the little contribution of the ground state to the 1P dynamics [see Fig. 6.22(b)]. There are some line anticrossings and interference patterns around the resonant points giving away the SC physics that we discussed in previous Chapters, only now displaying more complicated structures. This is better appreciated in the contour plot of Fig. 6.24(c).