Past exam of the advanced quantum physics course of the University of Cambridge / 2019 / Section A / 3

Use the electronic weak-field Zeeman splitting model, neglecting hyperfine structure. The Landé g-factor is $g_g=2$ for $3S_{1/2}$ and $g_e=2/3$ for $3P_{1/2}$. The shift of an absorption frequency is thereforeHere the magnetic field is weak relative to spin-orbit coupling, so $J$ remains a useful quantum number, but the spectral line widths are small enough to resolve its magnetic substates.

For linear polarization parallel to $z$, the optical electric field has only the field-axis $\pi$ component. The electric-dipole selection rule is $\Delta m=0$. Its two allowed transitions are $m_g=m_e=\pm 1/2$, givingFor propagation perpendicular to $z$ with linear polarization also perpendicular to $z$, the optical electric field is an equal-amplitude combination of the two field-axis circular components. It drives ${\sigma }^{\pm }$ transitions with $\Delta m=\pm 1$. These connect $-1/2\to +1/2$ and $+1/2\to -1/2$, givingThus each orientation selects two lines, with twice the separation in the perpendicular-polarization case. Neither pair has an unshifted central component. This is polarization-selective Zeeman splitting of an alkali D1 line, an anomalous Zeeman effect rather than a normal orbital triplet.