Optimally symmetrical cases:

Optimally symmetrical cases: $g<G\leq \sqrt {2}g$

As a first example, we study the situation where the parameters are equal only in a crossed way: $P_{E1}/g=\gamma_{E2}/g$ and $P_{E2}/g=\gamma_{E1}/g$ . The system has a total input that is equal to the total output, $P_\mathrm{TOT}=\gamma_\mathrm{TOT}$ , and also equal effective decoherence rates, $\Gamma_1=\Gamma_2$ , and, therefore, equal Purcell rates, . This is a very special situation where the symmetry is not total but exists between the effective rates and there is a total compensation of the flows with the exterior. It leads to $1<D^s<\sqrt{2}$ and a positive renormalization of the coupling strenght, from to . This is a very unexpected effect to be completely induced by decoherence, more precisely, by the optimal interplay between dissipation and incoherent pump. Interestingly, the present configuration of parameters, that can make the coupling more effective, is not accessible in the LM where $\Gamma_+$ vanishes and the system does not have SS. In the LM, nothing seems to indicate that the coupling gets renormalized at resonance, decoherence has the only effect of diminishing the splitting of the dressed states ( $g\rightarrow\sqrt{g^2-\Gamma_-^2}$ ).

In Fig. 4.7 we have plotted the phase space of SC as a function of $P_{E1}/g=\gamma_{E2}/g$ and $P_{E2}/g=\gamma_{E1}/g$ with the usual color code. This configuration is in FSC only when all parameters are equal, , (blue line) and total symmetry is recovered. Otherwise, the possibility of reaching all other coupling regimes opens as the coupling is effectively improved . The SSC and MC regions are linked to the absence of total symmetry. In the inset we can see that, as a consequence of this special configuration, the system is richer in spectral shapes than the ones previously studied. The lineshape can be a doublet (area in white), a distorted doublet (light grey), a distorted singlet (dark grey) and a singlet (black), as listed in Table 4.1, although it never reaches a fully formed triplet or quadruplet.

**Figure 4.7:** Phase space of FSC/SSC/MC/WC as function of $P_{E1}/g=\gamma_{E2}/g$ and $P_{E2}/g=\gamma_{E1}/g$ . The color code is that of Fig. 4.4. In inset, the possible lineshapes of $S_1(\omega)$ : a doublet (in white), a distorted doublet (light grey), a distorted singlet (dark grey) and a singlet (black), as in Table 4.1.
$\includegraphics[width=0.7\linewidth]{chap4/examples/PS1.ps}$

Table 4.1: Correspondence between the lineshape of the spectrum and the value of functional

. A lineshape can be defined by the product $L_1\times L_2$ :

(

) is the number of times that $S_1(\omega)$ changes slope (concavity), that is, the number of real solutions to the equation $dS_1/d\omega=0$ ( $d^2S_1/d\omega^2=0$ ).

Lineshape			$L_1\times L_2$
singlet	1	2	2	1
distorted singlet	1	6	6	1.3
distorted doublet	3	8	24	1.6
doublet	3	4	12	2
triplet	5	6	30	3
quadruplet	7	8	56	4

The -axis in Fig. 4.7, with $P_{E1}=\gamma_{E2}=\gamma$ and $P_{E2}=\gamma_{E1}=0$ , is interesting enough with all the possible regions and lineshapes, to be analyzed in more detail. This is the limit of maximum renormalization of the coupling, $G=\sqrt{2}g$ ( $D^s=\sqrt{2}$ ), where the populations and mean values read

$\displaystyle n_2=\frac{1}{2+x^2}\,,\quad n_1=1-n_2\,,\quad n_B=\frac{1/2}{2+x^2}\neq n_1 n_2\,,\quad n_{12}=-i\frac{x/\sqrt{2}}{2+x^2}\,,$

(4.39)

with $x=P_{E1}/G=\gamma_{E2}/G$ . The two QDs are sharing one excitation only. The Rabi also simplifies to

, because $\Gamma_-=0$ , and $z_{1,2}=\sqrt{G^2-(g\pm\frac{\gamma}{2})^2}$ . In Fig. 4.8, we can see all these magnitudes in the different regimes.

**Figure 4.8:** (A) Positions of the peaks ( $\omega_1$ in green, $\omega_2$ in orange) and populations ( in blue, in purple, in brown, $\vert n_{12}\vert$ in dashed yellow) as function of $P_{E1}/g=\gamma_{E2}/g$ , for $P_{E2}=\gamma_{E1}=0$ . The vertical lines mark the transitions from SC (at 0) to SSC, to MC, to WC, also clear from the evolution of the ``bubble'' in positions. The function (in black) tracks the lineshape of the spectra as coded in Table 4.1. The most interesting lineshapes, that can only appear in SSC and MC, are the distorted doublet (h) and singlet (g), for parameters marked in plot (A). The total spectra (in black) is decomposed in inner (green) and outer (orange) peaks coming from the transitions in Fig. 4.1(c).
$\includegraphics[width=0.7\linewidth]{chap4/examples/Case1.eps}$ $\includegraphics[width=0.45\linewidth]{chap4/examples/Case1g.eps}$ $\includegraphics[width=0.45\linewidth]{chap4/examples/Case1h.eps}$

In the limit $x\ll 1$ , there is FSC with all the levels equally populated ( , ) and $n_{12}=-ix/\sqrt{2}$ . Soon the SSC opens a bubble in the eigenenergies with the splitting of inner and outer peaks. The transition into MC, with the collapse of the inner peaks, takes place at $\gamma=2(\sqrt{2}-1)g$ , and into WC, closing the bubble, at $\gamma=2(\sqrt{2}+1)g$ . The maximum of $z_2=\sqrt{2}g$ (in orange) takes place at $\gamma=2g$ , when the coherence $\vert n_{12}\vert=1/4$ is maximum. This is a special point where the splitting of the dressed mode is the largest possible, $2\sqrt{2}g$ , even though the final lineshape is a singlet. Finally, when the coupling becomes very weak, $x\gg 1$ , the first dot saturates and $n_2=n_B=n_{12}=0$ .

**Figure 4.9:** Spectra for positive varying detuning (anticrossings) in SSC (first row, (a)-(b)) and in MC (second row, (c)-(d)), with the parameters in Fig. 4.8(g) and (h). The first column corresponds to the spectra as it is observed and the second to the profile of the decomposition in the transition lines, revealing the underlying PL structure (quadruplets in the first case and triplets in the second). The lines in (d) are not well behaved because the dispersive part of the SC peaks is playing an important role. The first dot (that is pumped) emits at $\omega_{E1}=0$ and dominates over the second dot (that decays) with emission at $\omega_{E2}=-\Delta$ .
$\includegraphics[width=0.45\linewidth]{chap4/anticrossing/Ganticrossing.eps}$ $\includegraphics[width=0.45\linewidth]{chap4/anticrossing/Ganticrossing2.eps}$ $\includegraphics[width=0.45\linewidth]{chap4/anticrossing/Hanticrossing.eps}$ $\includegraphics[width=0.45\linewidth]{chap4/anticrossing/Hanticrossing2.eps}$

The spectra acquire interesting lineshapes: a doublet in SSC that gets distorted due to the underlying quadruplet structure, in Fig. 4.8(g), and then a singlet distorted due to the underlying triplet structure, in (h), as the two inner peaks stick together. Before reaching WC, the spectra has become a plain singlet. The way to distinguish mathematically the different possible shapes is explained in Table 4.1. The anticrossing that this lineshapes form when detuning between the modes is varied from zero to $\Delta_\mathrm{max}$ , is also peculiar. In Fig. 4.9(a) and (d) we can see that the distorted doublet and singlet (resp.) keep their features up to $\Delta_\mathrm{max}=g$ and $\Delta_\mathrm{max}=2g$ (resp.). Note that the singlet maxima oscillates around the origin: first, to the left, then, right and finally center, when the detuning is larger than in the plot. In all cases, the emission at $\omega_{E1}=0$ is dominant over $\omega_{E2}=-\Delta$ because the first dot is being pumped and the second dissipates the excitation. The underlying structure is plotted next to them, in (b) and (d), showing the profile of the four and three peaks that form the spectra.