# Robustness of STIRAP Shortcuts under Ornstein-Uhlenbeck Noise in the Energy Levels
## 2 STIRAP Shortcuts in the Absence of Noise
Under the total Hamiltonian \(H^{\prime}=H^{\prime}_{0}+H^{\prime}_{cd}\), the system can track with perfect fidelity and for arbitrarily short times one of the adiabatic states, for example \(|\phi_{+}(t)\rangle\), and this is why this method is called the transitionless tracking algorithm. For an evolution along \(|\phi_{+}(t)\rangle\), the corresponding probability amplitudes are:  
\[\mathbf{b}^{\prime}(t)=\left( \begin{array}{c}b^{\prime}_{1}(t)\\ b^{\prime}_{2}(t)\end{array} \right)=e^{-i\frac{\hbar}{4}\int_{0}^{t}\Omega( \tau)d\tau}\left( \begin{array}{c}\cos\frac{\theta}{2}\\ \sin\frac{\theta}{2}\end{array} \right). \tag{10}\]  
The drawback of this technique is that the counter-diabatic Hamiltonian is proportional to \(\sigma_{y}\), while Hamiltonian \(H^{\prime}_{0}\) contains only terms proportional to \(\sigma_{x}\) and \(\sigma_{z}\). In the case of the original three-level system, the counter-diabatic term becomes:  
\[H_{cd}(t)=\hbar\left( \begin{array}{ccc}0&0&i\theta\\ 0&0&0\\ -i\theta&0&0\end{array} \right), \tag{11}\]  
i.e., introduces a direct coupling between states \(|1\rangle\) and \(|3\rangle\), which can be difficult or even impossible to implement. Nevertheless, there is a modification of this method that allows one to implement a shortcut of the desired adiabatic evolution using the original Hamiltonian (2) or (4) [40]. According to this technique, in the total Hamiltonian \(H^{\prime}=H^{\prime}_{0}+H^{\prime}_{cd}\), we apply the unitary transformation \(|\bar{\psi}\rangle=U^{\dagger}|\psi\rangle\), where:  
\[U(t)=e^{-i\frac{\theta(t)}{2}\sigma_{z}}=\left( \begin{array}{cc}e^{-i\phi(t )/2}&0\\ 0&e^{i\phi(t)/2}\end{array} \right), \tag{12}\]  
and obtain the modified Hamiltonian:  
\[\bar{H}^{\prime}=\frac{\hbar}{2}\left(\frac{\Omega_{\rm s}}{2}-\phi\right) \sigma_{z}+\frac{\hbar}{2}\left(\frac{\Omega_{p}}{2}\cos\phi+\dot{\theta}\sin \phi\right)\sigma_{x}+\frac{\hbar}{2}\left(\frac{\Omega_{p}}{2}\sin\phi-\dot{ \theta}\cos\phi\right)\sigma_{y}. \tag{13}\]  
If we set:  
\[\tan\phi=\frac{2\dot{\theta}}{\Omega_{p}}=\frac{2\dot{\theta}}{\Omega\sin \theta}, \tag{14}\]  
then the last term in Equation (13), proportional to \(\sigma_{y}\), vanishes, and the modified Hamiltonian becomes:  
\[\bar{H}^{\prime}=\frac{\hbar}{2}\frac{\Omega_{\rm s}}{2}\sigma_{z}+\frac{ \hbar}{2}\frac{\tilde{\Omega}_{p}}{2}\sigma_{x}, \tag{15}\]  
which has the same form as the two-level Hamiltonian \(H^{\prime}_{0}\) with controls:  
\[\tilde{\Omega}_{p}(t) = \Omega_{p}\cos\phi+2\dot{\theta}\sin\phi=\sqrt{\Omega_{p}^{2}+4 \dot{\theta}^{2}}=\sqrt{\Omega^{2}\sin^{2}\theta+4\dot{\theta}^{2}}, \tag{16a}\] \[\tilde{\Omega}_{\rm s}(t) = \Omega_{\rm s}-2\dot{\phi}=\frac{\Omega^{3}\sin^{2}\theta\cos \theta+4\dot{\Omega}\dot{\theta}\sin\theta+4\dot{\Omega}(2\dot{\theta}^{2}\cos \theta-\ddot{\theta}\sin\theta)}{\Omega^{2}\sin^{2}\theta+4\dot{\theta}^{2}}. \tag{16b}\]  
The modified probability amplitudes are \(\tilde{\mathbf{b}}=U^{\dagger}\mathbf{b}^{\prime}\), thus:  
\[\tilde{\mathbf{b}}(t)=\left( \begin{array}{c}\tilde{p}_{1}(t)\\ \tilde{b}_{2}(t)\end{array} \right)=\left( \begin{array}{c}b^{\prime}_{1}(t)e^ {i\phi(t)/2}\\ b^{\prime}_{2}(t)e^{-i\phi(t)/2}\end{array} \right). \tag{17}\]  
The modified Hamiltonian for the three-level system is:  
\[\bar{H}(t)=\frac{\hbar}{2}\left( \begin{array}{ccc}0&\tilde{\Omega}_{p}(t)&0 \\ \tilde{\Omega}_{p}(t)&0&\tilde{\Omega}_{\rm s}(t)\\ 0&\tilde{\Omega}_{\rm s}(t)&0\end{array} \right), \tag{18}\]the same as \(H_{0}\) in the resonant case \(\Delta=\delta=0\), but with modified controls \(\tilde{\Omega}_{s},\tilde{\Omega}_{p}\) instead of \(\Omega_{s},\Omega_{p}\).