Spectrum with the quantum regression formula

In order to actually know which transitions are relevant and to what extent, we solve the complete master equation (2.71) for dissipative coupled modes under incoherent continuous pump. In this case, we will obtain the spectra $s_a(\omega)$ ^5.3 with the density matrix formalism, applying Eq. (2.108).

First, we obtain the steady state elements $\rho_{ij}^{(\mathrm{SS})}$ of the system. The labels and index the whole Hilbert space, namely, in our case of two oscillators, $i=\{mn\}$ and $j=\{\mu\nu\}$ . As a result, $M_{ij,kl}$ is a $n_\mathrm{max}^4\times n_\mathrm{max}^4$ matrix where $n_\mathrm{max}$ is the truncation of each oscillator's Hilbert space. In the computations, we have checked that the results were independent of this truncation once it is taken large enough. In Eq. (2.108), and are the creation and destruction operators of the transitions so in the case of cavity (normal) emission, where $A=\ud{a}$ and , we have $A_{mn;\mu\nu}=\sqrt{\mu+1} \delta_{n\nu}\delta_{m,\mu+1}$ and $B_{mn;\mu\nu}=\sqrt{\mu}\delta_{n\nu}\delta_{m,\mu-1}$ .

Again, the drawback of this method is the high computational cost to obtain and invert it for each point of the spectra. Therefore, we present here the spectra at low pump, after probing the first steps of the quantized energy levels, which presents only small deviations from the LM. We use pumping rates of , yielding average number of excitations of the order $\langle n_{a,b}\rangle \approx0.1$ with probability to have two excitons of the order of . For these figures, a truncation at the fourth manifold ( $n_\mathrm{max}=3$ ) is enough to ensure convergence of the results. The other parameters are fixed to the following values, motivated by experiments: provides the unit (experimental figures are of the order of tens of $\mu$ eV), $\gamma_a=0.1g$ , $\gamma_b=0.01g$ and .

**Figure 5.8:** Mean number of photons (dashed line) and excitons (solid line) as a function of $\Delta$ , for the case of cavity pumping (, ) in thin red ( $\times5$ ) and electronic pumping (, ) 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-strenght 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: $\omega_b=0$ , , $\gamma_a=0.1g$ , $\gamma_b=0.01g$ , .
$\includegraphics[width=0.55\linewidth]{chap5/brasilia/figure4.eps}$

**Figure 5.9:** Cavity emission spectrum $s_a(\omega)$ as a function of detuning $\Delta$ . The pumping is photonic (a) or excitonic (b). The case, which corresponds to the linear Rabi doublet, is shown in solid red and the bare exciton ( $\omega_b=0$ ) cavity ( $\omega_a=\Delta$ ) lines in dashed green. The probability of having more than two excitons is very low, mostly contributions from transitions $1\rightarrow 0$ (, vacuum-Rabi) and $2\rightarrow 1$ () appear. For high positive detuning, Coulomb interactions generate additional peaks close to $\omega_b+(p-1)U$ . Peaks originated from transitions $3\rightarrow 2$ () appear, although with a very small intensity, due to the nonzero probability to have three excitons in the system. All energies are in units of . Parameters: $\omega_b=0$ , , $\gamma_a=0.1g$ , $\gamma_b=0.01g$ , .
$\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure5a.eps}$ $\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure5b.eps}$

**Figure 5.10:** Spectra $s_a(\omega)$ for different detunings corresponding to vertical ``cuts'' in Fig 5.9. Both cases, of cavity pumping (, ) and electronic pumping (, ), 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, E3. Close to resonance, these result in satellites surrounding the linear vacuum Rabi doublet, that dominates because populations collapse at resonance. Parameters: $\omega_b=0$ , , $\gamma_a=0.1g$ , $\gamma_b=0.01g$ , .
$\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure6a.eps}$ $\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure6b.eps}$ $\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure6c.eps}$ $\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure6d.eps}$ $\includegraphics[width=0.45\linewidth]{chap5/brasilia/figure6e.eps}$

Mean numbers of excitons and photons are plotted in Fig. 5.8, for the two cases of cavity (only) and electronic (only) pumping. Close to resonance, $\Delta\approx0$ , both pumping yield approximately equal exciton and photon populations (note that the cavity pumping case has been magnified by a factor five). Detuning the modes results in a collapse (cavity pumping) or increase (electronic pumping) of the exciton population, as could be expected. Regardless of the kind of pumping, however, the cavity population is approximately constant. In the cavity pumping case, this is because the exciton gets decoupled and thus the cavity is pumped at a constant rate (one can actually see a small increase in its population). In the electronic pumping case, this is because although the coupling decreases, the exciton population increases in proportion so as to feed the cavity with a constant flux of photons. In both cases, an asymmetry is notable with detuning, because the interactions bring the cavity and the exciton modes closer or further from resonance, respectively, coupling them more efficiently for positive detuning and therefore allowing a larger production of excitons in that case. As a result, the nonlinear branches, , of the actual spectra, for positive and negative detunings, shown in Fig. 5.9, are not exactly as those shown in Fig. 5.7(b). The blueshifted peak is more clearly seen in the positive detuning case thanks to this exciton population asymmetry with detuning. However, an excellent qualitative agreement is obtained with the manifold method, if one superimposes the vacuum Rabi doublet to the lines arising from higher manifolds. Depending on the pumping scheme--cavity (a) or exciton (b)--only quantitative features are changed that consist mainly in different linewidths and intensities of the branches, that are otherwise well accounted for by the manifold method (see also Fig. 5.10). While comparing Figs. 5.9(a) and (b), it should be borne in mind how the total population changes with detuning, as shown in Fig. 5.8. For this reason, panel (b) has a more complex structure, but this is due to the higher manifolds that can be reached with the electronic pumping. It is in fact possible to identify the $3\rightarrow 2$ contribution by extracting the lines in Fig. 5.9 that do not appear in Fig. 5.7(b). These lines are clearly weaker due to the very low (but not vanishing) probability to have three excitons in the system. The transitions from even higher manifolds are too improbable to be seen in the spectra for the pumping considered here. The main differences between the two pumping schemes, if equal populations can be considered by adjusting the pumping, are therefore to be found in the linewidth and intensities of the lines. The positions of these lines embeds the most precious indications on the physical system.

In Fig. 5.10, spectra are displayed for particular detunings, in solid black for the case of electronic pumping, and thin red for cavity pumping. The electronic pumping, the most relevant case experimentally, yields the most interesting spectral shape. On top of the vacuum Rabi doublet, the interactions produce additional peaks that are clearly associated to the exciton at large detunings [panels (a) and (e)]. Three peaks, E1, E2 and E3 are obtained that correspond to one, two and three excitons coupling to the cavity mode, respectively. As these are brought in resonance with the cavity mode, the vacuum Rabi doublet dominates (essentially because the efficient coherent coupling collapses the exciton population) and satellite peaks are observed, that betray the quantum nature of the system, as the emission originates from transitions between quantized manifolds. In the absence of interactions (LM), the linear Rabi doublet is always observed independently of the total number of particles. Therefore, interactions are useful to evidence a quantum behavior linked to quantized energy transfers, in the spirit of such experiments as those used with atoms by Brune et al. (1996) in cavity to demonstrate quantization of the light field. Here, nonlinear features are observed directly in the optical spectrum, whereas in Brune et al. (1996), time-resolved measurements were used to probe anharmonic oscillations of the Rabi flops. This represents a notable experimental advantage, as measurements with cw incoherent pumping are typically easier to perform than time-resolved spectroscopy.