## 3 Rapid Spin Initialization with Optical Fields
### Constant Control  
When the control \(\Omega\) was constant, we could calculate analytically the final population (steady state) of the target state \(|2\rangle\). From the equilibrium of Equations (1)-(7), we find:  
\[\rho_{22}(\infty)=\frac{1+a^{2}+b^{2}}{4+2a^{2}+b^{2}}, \tag{15}\]  
where:  
\[a=\frac{F_{x}+F_{z}}{2}\frac{\gamma}{\Omega},\quad b=\frac{\omega_{21}+\omega _{43}}{\Omega}. \tag{16}\]  
The time needed to arrive at the steady state could be estimated following [16], where the out-of-resonance transition \(|2\rangle\rightarrow|3\rangle\) was momentarily neglected. Under this assumption, state \(|3\rangle\) was decoupled, while states \(|1\rangle,|2\rangle\), and \(|4\rangle\) formed a \(\Lambda\) system. Additionally, we could use equation \(\rho_{11}\ =\ 1-\rho_{22}-\rho_{44}\) to further eliminate \(\rho_{11}\), ending up with the reduced system:  
\[\dot{\rho}_{22} = \gamma_{42}\rho_{44}, \tag{17a}\] \[\dot{\rho}_{44} = -(\gamma_{41}+\gamma_{42})\rho_{44}+2\Omega\rho_{41}^{I},\] (17b) \[\dot{\rho}_{41}^{I} = -\frac{\gamma_{41}+\gamma_{42}}{2}\rho_{41}^{I}+\Omega(1-\rho_{22 }-2\rho_{44}). \tag{17c}\]  
Figure 2: The distance dependence of the Purcell factors for (**a**) \(x\)- and (**b**) \(z\)-orientations of the transition dipole moment. \(R\) is the distance between the QD and the MoS\({}_{2}\) monolayer, for three qualities of the semiconductor material: good (black dashed-dotted line), medium (blue dashed line), and bad (red solid line). The red stars highlight the values for the three examples of Figure 6.  
We could find an estimate of the necessary time to build \(\rho_{22}\) by eliminating adiabatically \(\rho_{44},\rho_{41}^{I}\), in which case, we obtained a rate equation for \(\rho_{22}\):  
\[\dot{\rho}_{22}=r(1-\rho_{22}), \tag{18}\]  
with:  
\[r=\frac{F_{z}}{2+a^{2}}\gamma \tag{19}\]  
and \(a\) as in Equation (16). We would like to reach high levels of \(\rho_{22}\) as fast as possible. The goal could be obtained by choosing system parameters in a way to make both the population \(\rho_{22}(\infty)\) and the rate \(r\) large, but unfortunately, these objectives were conflicting. Observe that \(\rho_{22}(\infty)\) is a decreasing function of both \(\Omega,F_{z}\), but \(r\) is an increasing function of \(\Omega\), while it depends non-monotonically on \(F_{z}\). The optimal parameters \(F_{x},F_{z},\Omega\) were eventually determined from a compromise between these conflicting objectives. Figure 3 displays contours of \(\rho_{22}(T)\) as a function of the distance \(R\) between the QD \(-\)MoS\({}_{2}\) monolayer and the Rabi frequency \(\Omega\), for the three semiconductor qualities and three final times \(T=1/\gamma,2/\gamma,3/\gamma\). Note that we used the Purcell factors \(F_{x},F_{z}\) displayed in Figure 2. The three rows of the figure correspond to the three qualities (good, medium, and bad from top to bottom), while each column corresponds to a specific duration \(T\). Observe from the second column of Figure 3, cases marked by red stars, that a high level \(\rho_{22}(T)>0.994\) could be reached in a short time \(T=2/\gamma=1.097\) ns and for moderate Rabi frequency \(\Omega=5.17\gamma\). The corresponding optimal Purcell factors are listed in \[a=\frac{F_{x}+F_{z}}{2}\frac{\gamma}{\Omega},\quad b=\frac{\omega_{21}+\omega _{43}}{\Omega}. \tag{16}\]0 In Figure 4, we display \(\rho_{22}(T)\) in the absence of the monolayer, versus the Rabi frequency \(\Omega\) and for the previously used durations. Note that in this case, \(\rho_{22}(T)\) was less than 0.9; thus, the layer substantially sped up spin initialization. Figure 5 shows contour plots similar to those in Figure 3, but for three different permittivities of the dielectric host, \(e_{1}=2,4,6\), when bad quality MoS\({}_{2}\) was used in the layer and the duration was set to \(T=2/\gamma=1.097\) ns. Observe that similar fidelity levels were obtained with the free-space vacuum case, although they slightly decreased with increasing permittivity.