<aside> 💡 Rapid adiabatic passage (RAP)

$$ \left|\lambda_-\right>: \left|g\right> \rightarrow \left|e\right> \\ \Delta: -\infty \rightarrow +\infty $$

Landau-Zener probability

$$ P_{\rm LZ}=|c_g(t\rightarrow \infty)|^2 = \exp\left(-2\pi \frac{|\Omega|^2}{ 4\dot{\Delta}}\right) $$

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In the dressed state picture, the Hamiltonian under the eigenbasis is

$$ \hat{H}{\lambda} =\underbrace{\hat{U}\hat{H}{\rm RWA}\hat{U}^{\dagger}}{\rm diagonal}\underbrace{-i\hbar \hat{U}(\partial_t \hat{U}^{\dagger})}{\rm off-diagonal}= \begin{bmatrix} \lambda_+ && i\hbar\frac{\dot{\theta}}{2}\\ -i\hbar\frac{\dot{\theta}}{2} && \lambda_-\end{bmatrix} $$

If the $\dot{\theta}\rightarrow 0$, only diagonal parts survive. It means, under the condition $\dot{\theta}\rightarrow 0$ the population of the each eigenstate is trapped. This is called adiabatic theorem; the n-th eigenstate of the initial Hamiltonian evolve to the n-th eigenstate of the final Hamiltonian under the adiabatic condition where $\dot{\theta}\rightarrow 0$. More precisely, adiabatic condition is written as

$$ \Delta_{\rm gap} \gg \left|i\hbar\frac{\dot{\theta}}{2}\right| $$

where $\Delta_{\rm gap} = \left|\lambda_+-\lambda_-\right| = \hbar\sqrt{\Delta^2+\Omega^2}$.

The ground state of the Hamiltonian $\hat{H}_{\lambda}$ is

$$ \left|\lambda_-\right> = -\sin\frac{\theta}{2}\left|g\right>+\cos\frac{\theta}{2}\left|e\right> $$

If the mixing angle adiabatically ($\dot{\theta}\rightarrow 0$) change as $\theta: \pi \rightarrow 0$, the ground state is also evolve $\left|\lambda_-\right>: \left|g\right> \rightarrow \left|e\right>$. The time of adiabatic evolution ($T_{\rm adia}$, $\Delta\theta/T_{\rm adia} \ll 1$) is shorter than the lifetime of the quantum system ($T_{\rm life}$) $T_{\rm adia} \ll T_{\rm life}$, the significant probability transfer from $\left|g\right>$ to $\left|e\right>$. This kind of quantum operation is called rapid adiabatic passage (RAP).

• Rapid: more rapid than the lifetime

The condition of the mixing angle $\theta: \pi \rightarrow 0$ could be re-written as $\tan\theta=\Omega/\Delta: -0 \rightarrow +0$.

Under the finite and positive Rabi frequency $\Omega > 0$, the condition of detuning is $\Delta: -\infty \rightarrow +\infty$.

By adiabatic driving, the quantum state $\left|g\right>$ is perfectly evolve to the $\left|e\right>$. If the diabatic term proportional to the $\dot{\theta}$ is significantly large so that hard to ignore, the leakage is dominant more than population trapping comes from adiabatic. So adiabatic theorem is not valid, therefore, the population of $\left|g\right>$ is not transfer to the $\left|e\right>$ but still remain at the $\left|g\right>$. It could be understood that there is a crossing from $\left|\lambda_-\right>$ to the $\left|\lambda_+\right>$. The leakage comes from diabatic transition is hard to analytically calculate in general because it has variety behavior depending on the form of $\Omega(t)$ and $\Delta(t)$. There are few analytically solved model: Landau-Zener, Allen-Eberly, etc.

• Adiabatic theorem is working not only the two-level system, but also all of the quantum system. In the many-body system, finding the ground state is not easy in the general case. For example, finding the ground state of the quantum Ising Hamiltonian is NP-hard problem. If the initial Hamiltonian is easy to prepare in the experimental setup and adiabatically controllable, the ground state of the Hamiltonian can be achieved by RAP. It means, RAP of the many-body system can be expected to solve NP-hard problem efficiently. This is the starting point of the adiabatic quantum computing (AQC).

(Majorana-) Landau-Zener (-Stuckelberg) ((M)LZ(S)) transition

Landau-Zener (LZ) path is one of the characteristic model of the RAP. It is simple and exactly solvable two-level system dynamics. It consist of the linear detuning sweep with constant Rabi frequency.