Spectrum in the Steady State

When the pump is taken into account and the spectra computed in the SS, the situation changes, as we can see in Fig. 5.3, and there are several regimes appearing. Plot (a) corresponds the most to the ``quantum'' regime, with low pump ( $P_b<\gamma_b$ ), so that only the first manifolds are probed, and large interactions ( $U>\gamma_b,P_b$ ), so that the individual transitions are distinguishable. In this regime, where the individual peaks can be well resolved, each peak narrows with pump, behaving like a multimode laser. The larger the interactions, the more separated and narrower the peaks are and the less interferences between them exist.

When the pump is of the order of the decay [Fig. 5.3(a)-(d)], the individual peaks from the different manifold transitions (centered at the vertical guide lines) are further shifted from the SE positions due to the pump/decay interplay. If the interactions are large, still the peaks can be resolved, but when they are small, the peaks start to overlap and interfere (they grow a dissipative contribution) resulting in a broad shoulder on the right side of the central linear peak . They end up forming a distorted asymmetrical Lorentzian when $U<\gamma_b$ . Therefore, decreasing interactions at low pump induces a transition from the ``quantum'' to ``classical'' regime in the sense that, even with $n_b\ll 1$ , without interactions, the system turns into an HO.

The second possibility to induce the quantum to classical transitions, is to increase pump, as it is done in the Fig. 5.3(d)-(f). For the HO (dashed red lines), the effective linewidth can only decrease with pump as $\Gamma_b=\gamma_b-P_b$ (this is a bosonic signature). Its population, the same as the AO in this model, reaches when $P_b=0.5\gamma_b$ . Although for a thermal mixture the vacuum is always the most probable state, at this point one can say that there is an ``inversion'' of the population: the probability to have excitons in the SS (given by $1-P_b/\gamma_b$ ) overcomes the probability of vacuum (given by $P_b/\gamma_b$ ), as in the 2LS. The HO starts ``lasing'' with a noticeable narrowing of the linewidth if pump is further increased. For the AO, $P_b=0.5\gamma_b$ is also the point at which the second to first manifold transition energy, with , is the same as the mean transition energy, [vertical line in Fig. 5.3(d)]. This means that manifolds of excitations () start to behave as an ensemble of emitters with inhomogeneous broadenings. Two lines can be resolved, the broad envelope of an increasing number of peaks from transitions , and the first manifold transition , corresponding to the linear transition. The broad emission peak is placed approximately at the mean manifold energy , while the linear peak is at 0. Fig. 5.3(e) shows an example of high pump, where this transition has taken place and the linear peak has been almost ``swallowed'' by the mean field of manifold emissions. The result is a blueshifted peak at with a broadening of the order of as well. Increasing pump or interactions in the AO has a ``saturation'' effect, as pump does in the two-level system (2LS). There, as we know, $P_\sigma=0.5\gamma_\sigma$ is also the point of population inversion and where the broadening increases as $\Gamma_\sigma=\gamma_\sigma+P_\sigma=n_\sigma P_\sigma$ .

The transition from quantum to classical regimes is better appreciated in the contour plot of Fig. 5.3(f). The well distinguished peak at $\omega=5\gamma_b$ , melts into the mean field peak at around $P_b\approx 0.5\gamma_b$ . The mean field peak blueshifts then following (the black bending line) and becomes soon more important than the linear peak .

**Figure 5.4:** Comparison between the truncation of the QRF in the Hilbert space of correlators (a) and in the Hilbert space of states (b), equal for both HO and AO. In (a), respectively (b), we plot the average values $\langle\ud{b}^nb^n\rangle$ and the SS populations $\rho_{nn}$ , as a function of , for the whole range of physical parameters $P_b/\gamma_b$ . Pump increases in the sense of the arrow as $P_b/\gamma_b=0.001,0.1,0.2,\hdots,0.9$ .
$\includegraphics[width=0.47\linewidth]{chap5/AO/Fig4a.eps}$ $\includegraphics[width=0.47\linewidth]{chap5/AO/Fig4b.eps}$

It is interesting to note that for $P_b> 0.5\gamma_b$ (classical regime) the quantum regression formula expressed in terms of correlators, becomes numerically unstable and infinite precision is needed to obtain a solution for an adequate truncation. The best way to compute the spectra here is applying the QRF in its density matrix form, Eq. (2.108). The fact that this method is blind to the peak structure (that follows from a quantized structure), is already telling us that the best description of the system is not anymore in terms of manifolds of correlators but rather some mean field approximation. The truncation that Eq. (2.105) involves is of a different nature than that of Eq. (2.108), for the same $n_\mathrm{max}$ . The first one takes place in the Hilbert space of correlators and therefore requires that all averages $\langle\ud{b}^nb^n\rangle =n!n_b^n$ with $n>n_\mathrm{max}$ are dispensable. The second truncation, takes place in the Hilbert space of states and therefore requires that all the populations $\rho_{nn}=(1+n_b)^{-(1+n)}n_b^n$ with $n>n_\mathrm{max}$ are negligible. In order to understand the implications of this difference, we plot in Fig. 5.4 the average values $\langle\ud{b}^nb^n\rangle$ , in (a), and SS populations $\rho_{nn}$ , in (b), as a function of , for the whole range of physical parameters $P_b/\gamma_b\in (0,1)$ .

We can see that the truncation imposed on the density matrix is always safe, as long as the $n_\mathrm{max}$ is taken high enough ( $n_\mathrm{max}\approx 10$ for $P_b<0.8\gamma_b$ , for instance), because thermal populations decrease always with . The problem of using Eq. (2.108) to compute the spectra, however, is the high computational cost, as we already pointed out. On the other hand, the truncation in correlators looks only good at vanishing pump (the thick curve) as it is the only one that does not grow up again at some . In fact, the criteria to find $n_\mathrm{max}$ , so that one can safely neglect the correlators with $n>n_\mathrm{max}$ , is a bit more complicated than looking at this graph. The best criteria is simply that the solution does not change when increasing the truncation. However, Fig. 5.4 is representative of the increasing precision that one must use to solve the system as the pump is increased. The sooner the mean values $\langle\ud{b}^nb^n\rangle$ start to increase with (it does always increase at some for $P_b\neq 0$ ), the more difficult for the final result is to converge into the finite solution. We will find again this kind of increasing numerical complexity in the JCM as we cross to its classical regime.