# 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  
We consider the system evolution under the total Hamiltonian:  
\[H(t)=\tilde{H}(t)+H_{n}(t), \tag{24}\]  
where \(\tilde{H}(t)\) is given in Equation (18) and the noise term is:  
\[H_{n}(t)=\frac{\hbar}{2}\left( \begin{array}{ccc}n_{1}(t)&0&0\\ 0&n_{2}(t)&0\\ 0&0&n_{3}(t)\\ \end{array} \right), \tag{25}\]  
with \(n_{i}(t),i=1,2,3\), being independent Ornstein-Uhlenbeck noise processes. They are defined through the relations [33]:  
\[\hat{n}_{i}=-\frac{1}{\tau_{c}}n_{i}+\frac{1}{\tau_{c}}w_{i}, \tag{26}\]  
where \(w_{i}(t),i=1,2,3\), are independent Gaussian white noises with zero mean and correlations:  
\[\langle w_{i}(t)w_{j}(t+\tau)\rangle=2\sigma_{n}^{2}\tau_{c}\delta_{ij}\delta( \tau). \tag{27}\]  
The resulting Ornstein-Uhlenbeck processes are also zero mean Gaussian with correlations:  
\[R_{n_{i}n_{j}}(\tau)=\langle n_{i}(t)n_{j}(t+\tau)\rangle=\sigma_{n}^{2} \delta_{ij}e^{-|\tau|/\tau_{c}} \tag{28}\]  
Figure 5: **(a)** Pump and Stokes pulses (in units \(T^{-1}\)) for the shortcut derived from Gaussian pulses with \(\Omega_{0}=0.5T^{-1}\). **(b)** Corresponding time evolution of the level populations. **(c,d)** Controls (in units \(T^{-1}\)) and populations for the modified shortcut where the final negative part of the Stokes pulse is truncated. and steady state probability distributions:  
\[P_{i}(n)=\frac{1}{\sqrt{2\pi\sigma_{n}^{2}}}e^{-\frac{\pi^{2}}{2\sigma_{n}^{2}}}. \tag{29}\]  
Obviously, parameter \(\tau_{c}\) is the noise correlation time. For each process, the power spectral density is:  
\[S_{n_{i}n_{i}}(\omega)=\int_{-\infty}^{\infty}R_{n_{i}n_{i}}(\tau)e^{-i\omega \tau}d\tau=\frac{2\sigma_{n}^{2}\tau_{c}}{1+(\omega\tau_{c})^{2}} \tag{30}\]  
and the expectation value of the total power is:  
\[\langle n_{i}^{2}(t)\rangle=R_{n_{i}n_{i}}(0)=\frac{1}{2\pi}\int_{-\infty}^{ \infty}S_{n_{i}n_{i}}(\omega)d\omega=\sigma_{n}^{2}. \tag{31}\]  
In the following simulations, the Ornstein-Uhlenbeck processes were implemented using the algorithm described in [33]. We used a fixed standard deviation \(\sigma_{n}\), corresponding to fixed noise power, and various correlation times \(\tau_{c}\). From Equation (30), corresponding to a low-pass filter with cut-off frequency \(\omega_{c}=1/\tau_{c}\), it is obvious that for larger correlation times, the fixed noise power was distributed to lower frequencies \(\omega\). Thus, for larger \(\tau_{c}\), the noises in the energy levels were low frequency signals, which brought the system out of resonance for larger time intervals, leading to a degraded performance. Note that the noise model considered here, with independent noises of equal strength in each energy level, applies directly to the physical system of molecules in a liquid; see [32]. In atomic systems, usually the second level is more noisy than the others. Obviously, the first model with independent noises present in all levels provided a harder test for the efficiency of STIRAP shortcuts, and for this reason, we used it here. In Figure 6, we plot the final populations \(p_{i}(T)=|\bar{x_{1}}(T)|^{2},i=1,2,3\), versus parameter \(\Omega_{0}\) (in units \(T^{-1}\)) for the shortcuts derived from polynomial \(\theta(t)\), with constant \(\Omega(t)=\Omega_{0}\) (first column) and ramp \(\Omega(t)=\Omega_{0}t/T\) (second column), in the presence of the aforementioned noise processes in the energy levels. We fixed \(\sigma_{n}=5T^{-1}\) and used three different values of the correlation time, \(\tau_{c}/T=0.008,0.08,0.8\), corresponding to each of the three rows from top to bottom. The diagrams were produced by averaging over the results of 200 simulations. Observe that for both shortcuts, the efficiency \(p_{3}(T)\) had a similar robust behavior to this kind of noise, which was degraded as the noise correlation time increased.