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Matrix Elements

Goal

Compute the multipole matrix elements with compute_matrix_element.

Multipole matrix elements determine the coupling between transitions. The Rabi frequencies is defined as

\[ \Omega = \begin{cases} \frac{eE_0}{\hbar} \langle 1|\vec{r}_e \cdot \hat{\epsilon}|0 \rangle \, \text{for electric transitions} \\ \frac{B_0}{\hbar} \langle 1|\vec{\mu} \cdot \hat{b}|0 \rangle \, \text{for magnetic transitions} \end{cases} \]

In this section we'll describe how these \(\langle 1|\vec{r}_e \cdot \hat{\epsilon}|0 \rangle, \langle 1|\vec{\mu} \cdot \hat{b}|0 \rangle\) elements are computed. Currently, TrICal supports matrix element computation for:

Let's consider a transition from level 0 to 1 in an ion with nuclear spin \(I\) and associated quantum numbers:

symbol definition
\(M_i\) magnetization (a.k.a. \(m_F\))
\(J_i\) spin-orbital
\(F_i\) total angular momentum
\(q = M_1 - M_0\) change in magnetization for the transiton

For dipole transitions, \(q\) can be \(0, \pm 1\), each of which corresponds to a required polarization \(\hat{q}\):

\(q\) \(\hat{q}\)
\(\sigma_{-}\) -1 \(\frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ i\\ 0 \end{pmatrix}\)
\(\pi\) 0 \(\frac{1}{\sqrt{2}} \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}\)
\(\sigma_{+}\) 1 \(\frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ -i\\ 0 \end{pmatrix}\)

As a result, the matrix element will depend on the overlap between the laser's polarization (electric field direction) \(\hat{\epsilon}\) and the required \(\hat{q}\).

Important

\[ \langle 1|\vec{r}_e \cdot \hat{\epsilon}|0 \rangle = \frac{1}{\omega_0 e}\sqrt{\frac{3\pi\epsilon_0\hbar c^3 A_{10}}{\omega_0 }} \sqrt{(2F_1 + 1)(2F_0 + 1)} \begin{Bmatrix} J_0 & J_1 & 1\\ F_1 & F_0 & I \end{Bmatrix} \nonumber \\ \sqrt{2J_1+1} \hat{q}\cdot\hat{\epsilon}\begin{pmatrix} F_1 & 1 & F_0\\ M_1 & -q & -M_0 \end{pmatrix} \]

where \(\{\}\) refers to the Wigner-6j symbol and \(()\) refers to the Wigner-3j symbol.

For dipole transitions, \(q\) can be \(0, \pm 1\), each of which corresponds to a required polarization \(\hat{q}\):

\(q\) \(\hat{q}\)
\(\sigma_{-}\) -1 \(\frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ i\\ 0 \end{pmatrix}\)
\(\pi\) 0 \(\frac{1}{\sqrt{2}} \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}\)
\(\sigma_{+}\) 1 \(\frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ -i\\ 0 \end{pmatrix}\)

As a result, the matrix element will depend on the overlap between the laser's magnetic field direction \(\hat{b} = \hat{k} \times \hat{\epsilon}\) and the required \(\hat{q}\).

Important

\[ \langle 1|\vec{\mu} \cdot \hat{b}|0 \rangle = \sqrt{\frac{3 \hbar \epsilon_0 c^5 A_{10}}{16 \pi^3 \omega_0^3}} \sqrt{(2F_1 + 1)(2F_0 + 1)} \begin{Bmatrix} J_0 & J_1 & 1\\ F_1 & F_0 & I \end{Bmatrix} \nonumber \\ \sqrt{2J_1+1} \hat{q}\cdot\hat{b}\begin{pmatrix} F_1 & 1 & F_0\\ M_1 & -q & -M_0 \end{pmatrix} \]

where \(\{\}\) refers to the Wigner-6j symbol and \(()\) refers to the Wigner-3j symbol.

For quadrupole transitions, the laser's unit wavevector \(\hat{k}\) becomes relevant for coupling (in addition to its polarization). In particular, the coupling strength is now proportional to the overlap between \(\hat{k}\) and \(\hat{Q}\hat{\epsilon}\).

\(\hat{Q}\) is a matrix that depends on the value for \(q\), which can now be one of \(0, \pm 1, \pm 2\):

\(q\) \(\hat{Q}\)
-2 \(\frac{1}{\sqrt{6}}\begin{pmatrix} 1 & i & 0\\ i & -1 & 0\\ 0 & 0 & 0\end{pmatrix}\)
-1 \(\frac{1}{\sqrt{6}}\begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & i\\ 1 & i & 0\end{pmatrix}\)
0 \(\frac{1}{3}\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 2\end{pmatrix}\)
1 \(\frac{1}{\sqrt{6}}\begin{pmatrix} 0 & 0 & -1\\ 0 & 0 & i\\ -1 & i & 0\end{pmatrix}\)
2 \(\frac{1}{\sqrt{6}}\begin{pmatrix} 1 & -i & 0\\ -i & -1 & 0\\ 0 & 0 & 0\end{pmatrix}\)

Important

\[ \langle 1|\vec{r}_e \cdot \hat{\epsilon}|0 \rangle = \frac{1}{\omega_0 e}\sqrt{\frac{15\pi\epsilon_0\hbar c^3 A_{10}}{\omega_0}} \sqrt{(2F_1 + 1)(2F_0 + 1)} \begin{Bmatrix} J_0 & J_1 & 2\\ F_1 & F_0 & I \end{Bmatrix} \nonumber \\ \sqrt{2J_1+1} \hat{k}\cdot\hat{Q}\hat{\epsilon}\begin{pmatrix} F_1 & 2 & F_0\\ M_1 & -q & -M_0 \end{pmatrix} \]