qspec.simulate. Interaction.rabi  (  m = None  )[source]

The Rabi-frequency matrices $\Omega_m$ generated by laser m with direction $\hat{k}$ and polarization $\vec{\varepsilon}$ $$\begin{aligned} (\Omega_m)_{ij} &= (\Omega_m)_{ji}^\ast = \frac{E_m}{\hbar}\sum\limits_{(Xk)_{ij}}\sum\limits_\lambda (-1)^{k_{ij} + \lambda}\ d_{ij,k_{ij}}^\lambda \left[(-\mathrm{i})^{k_{ij} - X_{ij}}\,\vec{\varepsilon}_m \cdot\vec{Y}_{k_{ij}\lambda}^{(X_{ij})}(\hat{k}_m)\right]\\[2ex] d_{ij,k_{ij}}^\lambda &= d_{ji,k_{ij}}^\lambda = a_{ij,k_{ij}}^\lambda \sqrt{A_{ji}^{Xk_{fj}}\,8\pi^2\varepsilon_0\hbar\left(\!\frac{c}{\omega_{ij}}\!\right)^{\!3}} \quad\text{for all }\omega_i < \omega_j\\[2ex] a_{ij,k_{ij}}^\lambda &= (-1)^{F_i + I + k_{ij} + J_j}\sqrt{(2F_i + 1)(2J_j + 1)} \langle F_im_ik_{ij}\lambda|F_jm_j\rangle\begin{Bmatrix}J_j & J_i & k_{ij} \\F_i & F_j & I\end{Bmatrix}\\[2ex] X_{\!fi} &= \begin{cases}+1, & \text{if electric } (\mathrm{E}k_{fi}) \\ \ \,0, & \text{if magnetic } (\mathrm{M}k_{fi})\end{cases}\\[2ex] E_m &= \sqrt{\frac{2I_m}{\varepsilon_0c}}, \end{aligned}$$ where $A_{if}^{Xk_{fi}}$ is the Einstein coefficient for the electric (magnetic) decay $|i\rangle\rightarrow|f\rangle$ and the rank-$k_{fi}$ multipole order, $\vec{Y}_{k_{fi}\lambda}^{(X_{fi})}(\hat{k})$ is the vector spherical harmonic (see p. 215, Eq. (35) in [1]), $\hat{k}_m$ is the direction, $\vec{\varepsilon}_m$ the polarization, and $I_m$ the optical intensity of laser m.

Parameters:

mint

The laser index m. If None, an array of the Rabi frequencies of all lasers is returned.

Returns:

Omegandarray

The Rabi-frequency matrix $\Omega_m$ generated by laser m or an array for all lasers with shape (nl, atom.size, atom.size), where nl is the number of lasers of the Interaction ($2\pi\,\mathrm{MHz}$).