Figure 2

(a) The driving protocol as in Eq. (28a)–(28b) for \(\Omega _{0}=2\) and \(T_{0}=40\), the black and gray solid lines represent \(\Omega _{P}(t)\) and \(\Omega _{S}(t)\) respectively. (b) The gray lines from top down (and from darker to lighter) show the occupation on the dark state \(\vert d\rangle \), \(\rho _{dd}\), as a function of time t for \(\gamma =0,2,4\) respectively. (c) The gray lines from down to top (and from darker to lighter) show \(\rho _{dd}\) as a function of time t for \(\Omega _{0}=2,4,6\), respectively. (d) The gray lines from down to top (and from darker to lighter) show \(\rho _{dd}\) as a function of time t for \(T_{0}=40{,}120{,}200\) respectively. In (b), (c), (d) the solid and dashed lines represent the exact numerical solutions from Eq. (14) and the analytic solutions from Eq. (29), respectively. The darkest lines in panels (c), (d) exactly correspond to the particular driving protocol in (a). The other parameters used are \(\gamma =2\), \(\Omega _{0}=2\), \(T_{0}=40\) if not particularly specified