# Robustness of STIRAP Shortcuts under Ornstein-Uhlenbeck Noise in the Energy Levels
## 2 STIRAP Shortcuts in the Absence of Noise  
The derivation of STIRAP shortcuts follows the procedure described in [19; 40]. The time evolution of probability amplitudes in the three-level STIRAP system, \(\mathbf{c}(t)=(c_{1}(t),c_{2}(t),c_{3}(t))^{T}\) where superscript \(T\) denotes the matrix transpose, is governed by the following Schrodinger equation:  
\[i\hbar\frac{d}{dt}\mathbf{c}(t)=H_{0}(t)\mathbf{c}(t) \tag{1}\]  
with the STIRAP Hamiltonian:  
\[H_{0}(t)=\frac{\hbar}{2}\left( \begin{array}{ccc}0&\Omega_{p}(t)&0\\ \Omega_{p}(t)&\Delta&\Omega_{s}(t)\\ 0&\Omega_{s}(t)&\delta\end{array} \right), \tag{2}\]where \(\Omega_{p}(t),\Omega_{s}(t)\) are the Rabi frequencies for the pump and Stokes lasers, respectively, while \(\Delta,\delta\) are the one-photon and two-photon detunings, respectively. The desired shortcuts are derived for the case where both resonances hold,  
\[\Delta=\delta=0. \tag{3}\]  
Deviations from this ideal situation are later modeled through the considered noise mechanisms. For this resonant case, the three-level system can be reduced to the following effective two-level system:  
\[H_{0}^{\prime}=\frac{\hbar}{2}\left( \begin{array}{cc}\Omega_{s}/2&\Omega_{p} /2\\ \Omega_{p}/2&-\Omega_{s}/2\end{array} \right)=\frac{\hbar}{2}\frac{\Omega_{s}}{2 }\sigma_{z}+\frac{\hbar}{2}\frac{\Omega_{p}}{2}\sigma_{x}. \tag{4}\]  
The relation between the three-level probability amplitudes \(\mathbf{c}(t)\) and the two-level amplitudes \(\mathbf{b}(t)=(b_{1}(t),b_{2}(t))^{T}\) is:  
\[c_{1}(t) = |b_{1}(t)|^{2}-|b_{2}(t)|^{2}, \tag{5a}\] \[c_{2}(t) = 2i\text{Im}[b_{1}^{*}(t)b_{2}(t)],\] (5b) \[c_{3}(t) = -2\text{Re}[b_{1}^{*}(t)b_{2}(t)]. \tag{5c}\]  
If we parameterize the pump and stokes Rabi frequencies in terms of a (generally) time-dependent amplitude \(\Omega(t)\) and angle \(\theta(t)\),  
\[\Omega_{s}(t)=\Omega(t)\cos\theta(t),\quad\Omega_{p}(t)=\Omega(t)\sin\theta( t), \tag{6}\]  
then we find the following instantaneous eigenstates for the effective two-level system:  
\[|\phi_{+}(t)\rangle = \left( \begin{array}{c}\cos\frac{\theta}{2}\\ \sin\frac{\theta}{2}\end{array} \right), \tag{7a}\] \[|\phi_{-}(t)\rangle = \left( \begin{array}{c}\sin\frac{\theta}{2}\\ -\cos\frac{\theta}{2}\end{array} \right), \tag{7b}\]  
with corresponding eigenvalues:  
\[E_{\pm}=\pm\hbar\frac{\Omega(t)}{4}\,. \tag{8}\]  
Now, suppose that we vary the angle \(\theta\) from the initial value \(\theta(0)=0\) to the final value \(\theta(T)=\pi/2\) at the final time \(t=T\). If the change is slow enough (adiabatic), then the two-level system moves along the adiabatic state \(|\phi_{+}(t)\rangle\), from the initial state \(\mathbf{b}(0)=(1,0)^{T}\) to the final state \(\mathbf{b}(T)=(0,1)^{T}\). In this scenario, the three-level system moves from state \(|1\rangle\) to state \(|3\rangle\) along the dark state \(\cos\theta(t)|1\rangle-\sin\theta(t)|3\rangle\), as can be verified from Equation (5). If the change in the angle \(\theta\) is fast, the adiabatic evolution breaks down, and at the final time, there are in general unwanted residual populations, in state \(|2\rangle\) for the two-level system or in states \(|1\rangle\) and \(|2\rangle\) for the three-level system. In order to achieve the desired population transfer in arbitrarily short times, it is necessary to add in Hamiltonian \(H_{0}^{\prime}\) an extra counter-diabatic Hamiltonian \(H_{cd}^{\prime}\) to cancel the diabatic terms arising when \(H_{0}^{\prime}\) is transformed to the time-dependent adiabatic basis,  
\[H_{cd}^{\prime}(t) = i\hbar\sum_{n=\pm}\left[|\dot{\phi}_{n}(t)\rangle\langle\phi_{n} (t)|-\langle\phi_{n}(t)|\dot{\phi}_{n}(t)\rangle|\phi_{n}(t)\rangle\langle \phi_{n}(t)|\right] \tag{9}\] \[= i\hbar\sum_{n=\pm}|\phi_{n}(t)\rangle\langle\phi_{n}(t)|\] \[= \hbar\frac{\dot{\theta}}{2}\sigma_{y},\]where we note that the second term in the first line of Equation (9) is zero.