Oscillations of the photon echo intensity in a pulsed magnetic field: Zeeman splitting in LiLuF$_4$:Er$^{3+}$

A method of high-resolution time-resolved optical spectroscopy using oscillations of the photon echo intensity in the presence of a perturbation, which splits the optical frequencies of the transitions of two or more ion subgroups, has been proposed and demonstrated. This method has been applied to systems in which the Zeeman effect is manifested. The transition frequencies of ions are switched by a pulsed magnetic field. Oscillations of the photon echo intensity were observed in LiLuF$_4$:Er$^{3+}$ and LiYF$_4$:Er$^{3+}$. The first minimum corresponding to the accumulated phase of the electric dipole moment $\pi/2$ is reached in the pulsed magnetic field with an amplitude of $\sim2$ G at a duration of $30$ ns. The Zeeman splitting in this field is $\approx10$ MHz, which is much less than the laser spectral width ($0.15$ Å${}\approx 9$ GHz). The $g$ factor of the $^4$F$_{9/2}$(I) excited state of the Er$^{3+}$ ion in the LiLuF$_4$ matrix has been determined in zero magnetic field. The comparison with the $g$-factor value found from the measurement of the absorption spectrum in a magnetic field of $8$ kG has been performed.