拉比频率
贡献者： addis
1双态系统中
式 12
\begin{equation}
A_{mn} = \left\langle \phi_m(t) \right\rvert H'(t) \left\lvert \phi_n(t) \right\rangle = \left\langle \psi_m \right\rvert H'(t) \left\lvert \psi_n \right\rangle \mathrm{e} ^{ \mathrm{i} (E_m-E_n)t/\hbar}~,
\end{equation}
\begin{equation}
\boldsymbol{\mathbf{A}} \boldsymbol{\mathbf{c}} = \mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d}{t}} \boldsymbol{\mathbf{c}} ~.
\end{equation}
若微扰哈密顿矩阵的对角元消失（例如束缚态有 $ \left\langle \psi_n \middle| \boldsymbol{\mathbf{r}} \middle| \psi_n \right\rangle = 0$），有 $\omega_0 = E_2 - E_1$
\begin{equation}
\left\{\begin{aligned}
\dot c_1 &= \frac{1}{ \mathrm{i} \hbar} H_{12}' \mathrm{e} ^{- \mathrm{i} \omega_0 t} c_2\\
\dot c_2 &= \frac{1}{ \mathrm{i} \hbar} H_{21}' \mathrm{e} ^{ \mathrm{i} \omega_0 t} c_1
\end{aligned}\right. ~.\end{equation}
拉比频率（Rabi Frequency）
若矩阵 $ \boldsymbol{\mathbf{H}} '$ 是一个常数矩阵乘以一个 $ \cos\left(\omega t\right) $，当 $\omega \to \omega_0$ 时有
拉比频率为
\begin{equation}
\omega_r = \frac{1}{2} \sqrt{(\omega - \omega_0)^2 + ( \left\lvert V_0 \right\rvert /\hbar)^2}~.
\end{equation}
解
\begin{equation}
c_2(t) = \frac{2A_{21} ^\dagger }{ \mathrm{i} \hbar \omega_r} \exp\left[- \mathrm{i} (\omega - \omega_0) t / 2\right] \sin\left(\omega_r t/2\right) ~,
\end{equation}
\begin{equation}
\left\lvert c_1(t) \right\rvert ^2 + \left\lvert c_2(t) \right\rvert ^2 = 1~,
\end{equation}
\begin{equation}
P_{21}(t) = \left\lvert c_2(t) \right\rvert ^2 = \frac{4 \left\lvert A_{12} \right\rvert ^2}{\hbar^2 \omega_r^2} \sin^2(\omega_r t/2)~.
\end{equation}
1. ^ 参考文献：Griffiths [1]、Bransden [2] 和 Sakurai [3]。
[1] ^ David Griffiths, Introduction to Quantum Mechanics, 4ed
[2] ^ Bransden, Physics of Atoms and Molecules, 2ed
[3] ^ J.J. Sakurai. Modern Quantum Mechanics Revised Edition