Alignment-to-orientation conversion in a magnetic field at nonlinear excitation of the $D_2$ line of rubidium: Experiment and theory

We studied alignment-to-orientation conversion caused by excited-state level crossings in a nonzero magnetic field of both atomic rubidium isotopes. Experimental measurements were performed on the transitions of the $D_2$ line of rubidium. These measured signals were described by a theoretical model that takes into account all neighboring hyperfine transitions, the mixing of magnetic sublevels in an external magnetic field, the coherence properties of the exciting laser radiation, and the Doppler effect. In the experiments, laser-induced fluorescence components were observed at linearly polarized excitation and their difference was taken afterwards. By observing the two oppositely circularly polarized components, we were able to see structures not visible in the difference graphs, which give deeper insight into the processes responsible for these signals. We studied how these signals are dependent on intensity and how they are affected when the exciting laser is tuned to different hyperfine transitions. The comparison between experiment and theory was carried out fulfilling the nonlinear absorption conditions. The theoretical curves described the experimental measurements satisfactorily, reproducing even small features in the shapes of the curves.

Figure 1

Absorption from the ground-state hyperfine magnetic sublevel $m_{F_g}$ and creation of $\mathrm{\Delta }{m}_F=1$ and $\mathrm{\Delta }{m}_F=2$ coherences in the excited state when the magnetic field $B=0$.

Figure 2

Frequency shifts of the magnetic sublevels $m_F$ of the excited-state fine-structure level $5^2{P}_{3/2}$ as a function of magnetic field for ${}^{85}\mathrm{Rb}$. Zero-frequency shift corresponds to the excited-state fine-structure level $5^2{P}_{3/2}$. The numbers above the lines correspond to the values of $m_F$. Level crossings are marked by squares for $\mathrm{\Delta }{m}_F=1$ and circles for $\mathrm{\Delta }{m}_F=2$.

Figure 3

Frequency shifts of the magnetic sublevels $m_F$ of the excited-state fine-structure level $5^2{P}_{3/2}$ as a function of magnetic field for ${}^{87}\mathrm{Rb}$. Zero-frequency shift corresponds to the excited-state fine-structure level $5^2{P}_{3/2}$. The numbers above the lines correspond to the values of $m_F$. Level crossings are marked by squares for $\mathrm{\Delta }{m}_F=1$ and circles for $\mathrm{\Delta }{m}_F=2$.

Figure 4

Excitation and observation geometry.

Figure 5

Fine and hyperfine energy-level splittings for the $D_2$ transitions of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$.

Figure 6

Top view of the experimental setup. Although in the top view it appears that the beam is parallel to the $y$ axis, in fact it enters the coils at an angle of 45° with respect to the $y$ axis in the $yz$ plane.

Figure 7

Signal dependence on the excited-state hyperfine level $F_e$ to which the laser is tuned when excited from the ground-state hyperfine level $F_g=2$ of ${}^{85}\mathrm{Rb}$. The left-most plot shows the difference between the two oppositely circularly polarized components $\left(I_{{\sigma }^{+}}-I_{{\sigma }^{-}}\right)$ for the $F_g=2⟶F_e=1$ transition, the center plot shows the difference $\left(I_{{\sigma }^{+}}-I_{{\sigma }^{-}}\right)$ for the $F_g=2⟶F_e=2$ transition, and the third plot (right-most) corresponds to the $F_g=2⟶F_e=3$ transition. The smooth, red curve is theory and the black filled circles connected by a black line are the experimental data, with only every tenth point shown.

Figure 8

(a),(b) Relative intensities of the two oppositely circularly polarized fluorescence components, (c) their difference, and (d) the circularity value for the $F_g=2⟶F_e=2$ transition of ${}^{85}\mathrm{Rb}$ (80 scans averaged). Arrows denote the positions of peaks and maximum (or minimum) values of broader structures. The smooth, red curve is theory and the black filled circles connected by a black line are the experimental data, with only every tenth point shown.

Figure 9

Energy shifts of the magnetic sublevels $m_F$ as a function of magnetic field for ${}^{85}\mathrm{Rb}$ in the magnetic field range $0<B<10\text{G}$. The $m_F$ values are written next to the curves. Squares denote $\mathrm{\Delta }{m}_F=1$ crossings and circles denote $\mathrm{\Delta }{m}_F=2$ crossings.

Figure 10

Signal dependence on intensity for excitation of the $F_g=2⟶F_e=2$ transition of the $D_2$ line of ${}^{85}\mathrm{Rb}$. The plots are organized in columns: relative intensities of the two oppositely circularly polarized fluorescence components are shown in the left and center columns and their difference is shown in the right-most column. The smooth, red curve is theory and the black filled circles connected by a black line are the experimental data, with only every tenth point shown.

Figure 11

Signal dependence on intensity for excitation of the $F_g=1⟶F_e=1$ transition of the $D_2$ line of ${}^{87}\mathrm{Rb}$. The plots are organized in columns: relative intensities of the two oppositely circularly polarized fluorescence components are shown in the left and center columns and their difference in the right-most column. Arrows denote the positions of peaks and maximum values of broader structures. The smooth, red curve is theory and the black filled circles connected by a black line are the experimental data, with only every tenth point shown.

Figure 12

Energy shifts of the magnetic sublevels $m_F$ as a function of magnetic field for ${}^{87}\mathrm{Rb}$ in the magnetic field range $0<B<40\text{G}$. The $m_F$ values are written next to the curves. The square denotes a $\mathrm{\Delta }{m}_F=1$ crossing.

Figure 13

Dependence of the squared Rabi frequency ${\mathrm{\Omega }}_R^2$ on the intensity $I$ together with a linear fit for the transition $F_g=2⟶F_e=2$ in ${}^{85}\mathrm{Rb}$ and the transition $F_g=1⟶F_e=1$ in ${}^{87}\mathrm{Rb}$.