SE analytical spectrum

**Figure 5.2:** SE spectra of emission of the AO (thick black line) from the state $\ket{5}$ as compared to the HO emission (a Lorentzian in dashed red). Large interactions (a) help resolving the five peaks of each transition (thin lines) in the de-excitation process, while small interactions (b) give rise to a broad asymmetric peak. The position of the individual peaks are marked with vertical guide lines at . $\gamma_b$ is the unit here and $\omega_b=0$ is the reference energy.
$\includegraphics[width=0.47\linewidth]{chap5/AO/SE.eps}$ $\includegraphics[width=0.45\linewidth]{chap5/AO/SE2.eps}$

The SE case truncates naturally the Hilbert space as the dynamics of decay only involves states with less number of excitations than the initial one ( $n_\mathrm{max}$ ). Therefore, the solution can be obtained analytically for the correlator of interest,

$\displaystyle C_1^\mathrm{SE}(t,\tau)=e^{-(\omega_b+\gamma_b/2)\tau}\sum_{n=1}^... ...{U}{U-i\gamma_b}(e^{-(iU+\gamma_b)\tau}-1)\right]^{n-1} C_n^\mathrm{SE}(t,t)\,,$

(5.7)

and its spectrum. The parameters that define the Lorentzian, the dispersive lineshapes and their weight for each peak $p=1,2,\hdots,n_\mathrm{max}$ in the expression for the spectra (2.105), are:

$\begin{subequations}\begin{align}&\omega_p=\omega_b+(p-1)U\,,\\ [.3cm] &\gamma_p... ...1}}{n (n-p-2)!}C_n^0\,. <tex2html_comment_mark>564 \end{align}\end{subequations}$

**Figure 5.3:** SS spectra of emission of the AO (thick black line) as compared to the HO emission (a Lorentzian in dashed red) for the parameters in inset (all in units of $\gamma_b$ ). In the first row (a)-(c) we fix the pumping to $P_b=0.3\gamma_b$ and decrease the interactions, loosing in resolution of the second and third manifold peaks (thin lines) and recovering the Lorentzian lineshape. The shifted positions of the peaks are marked with vertical guide lines in order to compare with the SE positions . In the second row (d)-(f), we fix the interactions to a large number, $U=5\gamma_b$ , and increase the pump in order to achieve (lasing regime). High manifold transitions melt into a broad shoulder for the AO while the HO Lorentzian, narrows with $\Gamma_b=\gamma_b-P_b$ . The shoulder is placed approximately at evidencing a transition from a ``quantum'' (with resolved individual transition) to a ``classical'' (a mean field broad peak) regime. (f) Contour plot of this effect as a function of pump (large emission in black). The peak positions, (vertical thin lines), the ever narrowing Lorentzian $\pm\Gamma_b/2$ (dashed red) and the value of (black thick line), are plotted for comparison. HO and AO populations diverge in the same way as pump approaches $\gamma_b$ but their spectra is qualitatively very different, the first one narrowing and the latter effectively broadening and blueshifting. $\gamma_b$ is the unit here and $\omega_b=0$ is the reference energy.
$\includegraphics[width=0.32\linewidth]{chap5/AO/Fig2a.eps}$ $\includegraphics[width=0.32\linewidth]{chap5/AO/Fig2b.eps}$ $\includegraphics[width=0.32\linewidth]{chap5/AO/Fig2c.eps}$ $\includegraphics[width=0.32\linewidth]{chap5/AO/Fig2d.eps}$ $\includegraphics[width=0.33\linewidth]{chap5/AO/Fig2e.eps}$ $\includegraphics[width=0.32\linewidth]{chap5/AO/Fig3.eps}$

The positions and broadenings are reproduced exactly by the manifold method of Sec. 2.5.2, meaning that we can associate each individual peak with the transitions from the manifold with to excitations. The peak is weighted in the total spectra by and , which are a sum of contributions from the dynamics of all the states above the ones of the transition, that is, those from to $n_\mathrm{max}$ . The only line that remains at $\omega_b$ is that corresponding to the decay from the first manifold, the linear transition with . The rest of lines are blue-shifted for each manifold , from $\omega_b$ (HO) to $\omega_b+(p-1)U$ and they are broadened from $\gamma_b$ to $\gamma_b+2(p-1)\gamma_b$ (the higher the manifold, the broader the peak as compared to the HO), as we can see in Fig. 5.2 (thin black lines). We consider $\omega_b=0$ as the reference energy from now on.

For large interactions, Fig. 5.2(a), the individual transitions can be resolved, while, if they are not very large ( $U<\gamma_b$ ), they stick together forming a broad asymmetric peak at the origin, as in Fig. 5.2(b). The asymmetry may result also in an additional effective blueshift, noticeable if we compare with the Lorentzian symmetric emission of the HO (in dashed red).