The dressed state Hamiltonian is given by

$$ \hat{H}{\lambda} = \begin{bmatrix} \lambda+ && i\hbar\frac{\dot{\theta}}{2}\\ -i\hbar\frac{\dot{\theta}}{2} && \lambda_-\end{bmatrix} $$

An off-diagonal term $\hbar\dot{\theta}/2 \hat{\sigma}_y$ drives the diabatic process, where

$$ \dot{\theta} = \frac{-\dot{\Delta}\Omega+\Delta\dot{\Omega}}{\Delta^2+\Omega^2} $$

The approach of the RAP is to make $\dot{\Delta}=0$ at every point where it makes $\dot{\theta}=0$. It requires a huge amount of time $T_{\rm adia}$, where it satisfy $\dot{\Delta}=(\Delta_f-\Delta_i)/T_{\rm adia}=0$.

The shortcut-to-adiabaticity (STA) method is developed to reduce the time budget while maintaining the adiabatic character. The STA aims to make $\dot{\theta}=0$, by properly tuning time-dependent variables $(\Omega,\Delta)$.

• The first time, the shortcut-to-adiabatic passage is called SHAPE (X. Chen, 2010).

D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to Adiabaticity: Concepts, Methods, and Applications, Rev. Mod. Phys. 91, 045001 (2019).

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Shortcut to Adiabatic Passage in Two- and Three-Level Atoms, Phys. Rev. Lett. 105, 123003 (2010).

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A. Lukin et al., Quantum Quench Dynamics as a Shortcut to Adiabaticity, arXiv:2405.21019.