# Robustness of STIRAP Shortcuts under Ornstein-Uhlenbeck Noise in the Energy Levels
## 2 STIRAP Shortcuts in the Absence of Noise
Observe that both the shortcuts with constant \(\Omega(t)=\Omega_{0}\) and  
Figure 1: (**a**) Pump and Stokes pulses (in units \(T^{-1}\)) for the shortcut to adiabaticity where the mixing angle is a polynomial function of time and \(\Omega(t)=\Omega_{0}\) is a constant. (**b**) Corresponding time evolution of the level populations. (**c**,**d**) Controls (in units \(T^{-1}\)) and populations for the case where \(\Omega(t)=\Omega_{0}t/T\) is a ramp. In both cases, the value \(\Omega_{0}=15T^{-1}\) is used. Figure 2: (**a**) Pump and Stokes pulses (in units \(T^{-1}\)) for the shortcut to adiabaticity derived from Gaussian pulses. (**b**) Corresponding evolution of populations. Gaussian parameters: \(\Omega_{0}=10\pi T^{-1}\), \(\tau=T/8\), and \(\sigma=T/6\). ramp \(\Omega(t)=\Omega_{0}t/T\) achieve perfect population transfer to level \(|3\rangle\) even for very small values of \(\Omega_{0}\). This is not surprising since the corresponding controls \(\tilde{\Omega}_{p},\tilde{\Omega}_{s}\) contain the terms \(4\dot{\theta}^{2},2\dot{\phi}\), respectively, which do not vanish as \(\Omega_{0}\to 0\). This is clearly demonstrated in Figure 4, where we display the controls and the evolution of populations for the shortcut with constant \(\Omega(t)=\Omega_{0}=0.1T^{-1}\). On the other hand, the shortcut derived from Gaussian pulses has a decreasing efficiency with decreasing \(\Omega_{0}\). For this shortcut, the variables \(\theta,\dot{\theta},\ddot{\theta}\) are independent of \(\Omega_{0}\) and depend only on parameters \(\tau,\sigma\); see Equation (23), which have been selected such that the corresponding boundary conditions in Equation (20) are satisfied with high precision. What actually fails for decreasing \(\Omega_{0}\) is the boundary condition \(\phi(T)=0\), as displayed in Figure 3b. In Figure 5a,b, we plot the controls and corresponding populations obtained for this shortcut when the small value \(\Omega_{0}=0.5T^{-1}\) is used. Observe that the performance (value of \(p_{3}\)) degrades when the Stokes pulse \(\tilde{\Omega}_{s}(t)\) acquires negative values. If we truncate the negative final part of the Stokes pulse, a better transfer can be achieved, as demonstrated in Figure 5c,d. The efficiency of this modified shortcut versus \(\Omega_{0}\) is also displayed in Figure 3a (red dashed-dotted line). In the following section, we evaluate the performance of the presented STIRAP shortcuts in the presence of Ornstein-Uhlenbeck noise in the energy levels of the system. \[\tilde{c}_{2}(t) = 2i\text{Im}[\tilde{b}_{1}^{*}(t)\tilde{b}_{2}(t)]=i\sin\theta \sin\phi,\]0 (a) Controls (in units \(T^{-1}\)) and (b) time evolution of populations for the shortcut to adiabaticity derived from polynomial \(\theta(t)\) with constant \(\Omega(t)=\Omega_{0}=0.1T^{-1}\). \[\tilde{c}_{2}(t) = 2i\text{Im}[\tilde{b}_{1}^{*}(t)\tilde{b}_{2}(t)]=i\sin\theta \sin\phi,\]1 (a) The efficiency (final population \(p_{3}(T)\) of level \(|3\rangle\)) of the various shortcuts to diabaticity versus parameter \(\Omega_{0}\), in the absence of noise. The shortcuts derived from polynomial \(\theta(t)\) with constant (blue solid line) and ramp (orange dashed line) \(\Omega(t)\) have a unit efficiency independent of \(\Omega_{0}\), while the efficiency of the shortcut derived from Gaussian pulses (green dotted line) and its modification (red dashed-dotted line) increases gradually to unity with increasing \(\Omega_{0}\). We also plot for reference the efficiency of the simple STIRAP method with Gaussian pulses (22) (purple solid line). (b) Final angle \(\phi(T)\) versus \(\Omega_{0}\) for the shortcut derived from Gaussian pulses. Obviously, the boundary condition \(\phi(T)=0\) is severely violated for smaller \(\Omega_{0}\).