# Robustness of STIRAP Shortcuts under Ornstein-Uhlenbeck Noise in the Energy Levels
## 2 STIRAP Shortcuts in the Absence of Noise
The relation between the three-level and two-level probability amplitudes for evolution under the modified Hamiltonians \(\tilde{H},\tilde{H}^{\prime}\) is:  
\[\tilde{c}_{1}(t) = |\tilde{b}_{1}(t)|^{2}-|\tilde{b}_{2}(t)|^{2}=\cos\theta, \tag{19a}\] \[\tilde{c}_{2}(t) = 2i\text{Im}[\tilde{b}_{1}^{*}(t)\tilde{b}_{2}(t)]=i\sin\theta \sin\phi,\] (19b) \[\tilde{c}_{3}(t) = -2\text{Re}[\tilde{b}_{1}^{*}(t)\tilde{b}_{2}(t))]=-\sin\theta \cos\phi. \tag{19c}\]  
In order to achieve the population transfer from level \(|1\rangle\) to level \(|3\rangle\) using the modified Hamiltonian \(\tilde{H}\), it is necessary to impose some boundary conditions on angle \(\theta\) and its derivatives, since we note from Equation (14) that angle \(\phi\) is a function of \(\theta,\dot{\theta}\). In order to start from \(|\tilde{c}_{1}(0)|^{2}=1\), we can choose \(\theta(0)=0\). In order to end at \(|\tilde{c}_{3}(T)|^{2}=1\), we can pick \(\theta(T)=\pi/2\) and \(\phi(T)=0\). We also require that \(\tilde{\Omega}_{p}(0)=0,\tilde{\Omega}_{s}(T)=0\), so the initial and the desired target states are stationary states of the Schrodinger equation \(i\hbar\dot{\mathbf{c}}=\dot{H}\mathbf{\dot{c}}\). From Equation (16a), we see that the first requirement leads to the condition \(\dot{\theta}(0)=0\), while from Equation (16b), the second requirement leads to \(\ddot{\theta}(T)=0\). We additionally impose \(\dot{\theta}(T)=0\) for smoothness. Summarizing, the boundary conditions are:  
\[\theta(0)=0,\quad\theta(T)=\pi/2,\quad\phi(T)=0,\quad\dot{\theta}(0)=\dot{ \theta}(T)=\ddot{\theta}(T)=0. \tag{20}\]  
We emphasize that the modified fields \(\tilde{\Omega}_{s},\tilde{\Omega}_{p}\) are the actual controls that should be applied in the laboratory setting in order to implement the desired shortcut to adiabaticity. We now use the above theory to derive two different shortcuts to adiabaticity, which we will compare in the presence of energy level noise in the next section. In the first case, we consider a polynomial time dependence for angle \(\theta\). It can be easily verified that the lowest order polynomial to satisfy the boundary conditions (20) is:  
\[\theta(t)=3\pi s^{2}-4\pi s^{3}+\frac{3\pi}{2}s^{4},\quad s=\frac{t}{T}, \tag{21}\]  
where note from Equation (14) that for \(\dot{\theta}(T)=0\), it is also \(\phi(T)=0\), as long as \(\Omega_{p}(T)\
eq 0\). For \(\Omega(t)\), we can use a constant value \(\Omega(t)=\Omega_{0}\) or a ramp \(\Omega(t)=\Omega_{0}t/T\), and for both cases, \(\Omega_{p}(T)=\Omega_{0}\). In Figure 1, we plot the controls \(\tilde{\Omega}_{p}(t),\tilde{\Omega}_{s}(t)\) and the corresponding time evolution of populations \(p_{i}(t)=|\tilde{c}_{i}(t)|^{2},i=1,2,3\), for \(\Omega_{0}=15T^{-1}\) and arbitrary duration \(T\). The second shortcut that we consider is obtained using Gaussian pulses for the original pump and Stokes fields:  
\[\Omega_{p}(t) = \Omega_{0}e^{-\left(\frac{t-T/2-\tau}{\sigma}\right)^{2}}, \tag{22a}\] \[\Omega_{s}(t) = \Omega_{0}e^{-\left(\frac{t-T/2+\tau}{\sigma}\right)^{2}}, \tag{22b}\]  
thus:  
\[\tan\theta = e^{\frac{4\pi}{\sigma^{2}}(t-T/2)},\] (23a) \[\dot{\theta} = \frac{4\tau}{\sigma^{2}}\frac{\tan\theta}{1+\tan^In Figure 3a, we plot the final population \(p_{3}(T)\) versus parameter \(\Omega_{0}\) (in units of \(T^{-1}\)), in the absence of noise and for the various shortcuts considered so far: those derived from polynomial \(\theta(t)\) with constant (blue solid line) and ramp (orange dashed line) \(\Omega(t)\) and the shortcut derived from Gaussian pulses (green dotted line). We also plot the efficiency of the simple STIRAP method with the Gaussian pulses of Equation (22) (solid purple line), which obviously lies well below that of the shortcuts, indicating that STIRAP requires substantially longer times or larger amplitudes in order to achieve comparable efficiency levels.