# Robustness of STIRAP Shortcuts under Ornstein-Uhlenbeck Noise in the Energy Levels
## 3 Performance of the Shortcuts in the Presence of Ornstein-Uhlenbeck Noise in the Level Energies
In Figure 7, we display similar results for the two shortcuts derived from Gaussian pulses, with the first column corresponding to the usual shortcut and the second column to the modified one obtained by truncating the negative part of the Stokes pulse, while the three rows correspond to different noise correlation times. Observe that the behavior of these shortcuts was more robust than in the previous case. As before, the performance was degraded with increasing correlation time, but here, the reduction was milder. Note that the modified shortcut behaved better for smaller values of \(\Omega_{0}\), while for larger values, the two shortcuts had the same efficiency. The reason was that for larger \(\Omega_{0}\), the truncated negative part of the Stokes pulse became smaller, and the two shortcuts actually coincided. Note that, since the two shortcut types used in Figures 6 and 7 had different time dependence, the same value of parameter \(\Omega_{0}\) may correspond to different areas of the pump and Stokes pulses. In order to make a more fair comparison between them, we used two different \(\Omega_{0}\) resulting in similar pulse areas for the two shortcuts. Specifically, for the shortcut with constant \(\Omega(t)=\Omega_{0}\), we used the value \(\Omega_{0}=15T^{-1}\), while for the standard shortcut derived from Gaussian pulses, we used \(\Omega_{0}=10\pi T^{-1}\). The corresponding pulses are displayed in Figures 1a and 2a, respectively. The areas of the pump and Stokes pulses were 11.34, 10.52 in the first case and 10.36, 12.36 in the second case, i.e., they had similar values. In Figure 8, we plot the time evolution of populations \(p_{i}(t),i=1,2,3\), in the interval \(0\leq t\leq T\). The first column shows results for the pulses of Figure 1a, while the second column for the pulses of Figure 2a. As before, the three rows correspond to different values of correlation time,\(\tau_{c}/T=0.008,0.08,0.8\), while we set \(\sigma_{n}=5T^{-1}\). Observe that for all cases, the shortcut derived from Gaussian pulses (second column) was more robust to noise than the shortcut derived from polynomial \(\theta(t)\) with constant \(\Omega(t)=\Omega_{0}\) (first column). This difference became more pronounced for larger correlation times. A possible explanation of this behavior might be the observation that for the shortcut with constant \(\Omega(t)=\Omega_{0}\) (first column), the major changes in the populations took place during longer time intervals than the corresponding changes for the shortcut derived from Gaussian pulses (second column). Consequently, the evolution of populations in the first case was probably affected more by the low frequency noise corresponding to increasing correlation time \(\tau_{c}\). Figure 6: Final populations \(p_{i}(T),i=1,2,3\), as functions of parameter \(\Omega_{0}\), for the STIRAP shortcuts derived from polynomial \(\theta(t)\) with (**a,c,e**) constant \(\Omega(t)=\Omega_{0}\) and (**b,d,f**) ramp \(\Omega(t)=\Omega_{0}t/T\). The three rows correspond to three different values of the noise correlation time, \(\tau_{c}/T=0.008,0.08\), respectively, while the standard deviation is set to \(\sigma_{n}=5T^{-1}\). For the cases corresponding to \(\tau_{c}/T=0.8\) (last row), where the averages still fluctuate, we also display the 98% confidence intervals. Figure 7: Final populations \(p_{i}(T),i=1,2,3\), as functions of parameter \(\Omega_{0}\), for the STIRAP shortcuts derived from Gaussian pulses. In (**a**,**c**,**e**) standard shortcut has been used and in (**b**,**d**,**f**) the modified shortcut has been used. The three rows correspond to three different values of the noise correlation time, \(\tau_{c}/T=0.008,0.08,0.8\), respectively, while the standard deviation is set to \(\sigma_{n}=5T^{-1}\). This observation can be further exploited to provide a rough explanation about the somehow strange dependence of efficiency versus \(\Omega_{0}\), for the shortcut with constant \(\Omega(t)=\Omega_{0}\) and noise correlation \(\tau_{c}=0.8T\), displayed in Figure 6e.