# arc.alkali_atom_functions.AlkaliAtom.getDipoleMatrixElementHFS#

AlkaliAtom.getDipoleMatrixElementHFS(n1, l1, j1, f1, mf1, n2, l2, j2, f2, mf2, q, s=0.5)[source]#

Dipole matrix element for hyperfine structure resolved transitions $$\langle n_1 l_1 j_1 f_1 m_{f_1} |e\mathbf{r}|\ n_2 l_2 j_2 f_2 m_{f_2}\rangle$$ in units of $$a_0 e$$

For hyperfine resolved transitions, the dipole matrix element is $$\langle n_1,\ell_1,j_1,f_1,m_{f1} | \ \mathbf{\hat{r}}\cdot \mathbf{\varepsilon}_q \ | n_2,\ell_2,j_2,f_2,m_{f2} \rangle = (-1)^{f_1-m_{f1}} \ \left( \ \begin{matrix} \ f_1 & 1 & f_2 \\ \ -m_{f1} & q & m_{f2} \ \end{matrix}\right) \ \langle n_1 \ell_1 j_1 f_1|| r || n_2 \ell_2 j_2 f_2 \rangle,$$ where $$\langle n_1 \ell_1 j_1 f_1 ||r|| n_2 \ell_2 j_2 f_2 \rangle \ = (-1)^{j_1+I+F_2+1}\sqrt{(2f_1+1)(2f_2+1)} ~ \ \left\{ \begin{matrix}\ F_1 & 1 & F_2 \\ \ j_2 & I & j_1 \ \end{matrix}\right\}~ \ \langle n_1 \ell_1 j_1||r || n_2 \ell_2 j_2 \rangle.$$

Parameters
• l1 (n1.) – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 1

• j1 – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 1

• f1 – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 1

• mf1 – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 1

• l2 (n2.) – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 2

• j2 – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 2

• f2 – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 2

• mf2 – principal, orbital, total orbital, fine basis (total atomic) angular momentum, and projection of total angular momentum for state 2

• q (int) – specifies transition that the driving field couples to, +1, 0 or -1 corresponding to driving $$\sigma^+$$, $$\pi$$ and $$\sigma^-$$ transitions respectively.

• s (float) – optional, total spin angular momentum of state. By default 0.5 for Alkali atoms.

Returns

dipole matrix element( $$a_0 e$$)

Return type

float