Improved measurement of the hyperfine structure of the laser cooling level $4f^{12}(^3H_6)5d_{5/2}6s^2$ $(J=9/2)$ in ${}^{169}_{\,\,69}{{\mathrm {Tm}}}$

Abstract

We report on an improved measurement of the hyperfine constant of the $4f^{12}(^3 H_6)5d_{5/2}6s^2$ $(J=9/2)$ excited state of ${}^{169}_{\,\,69}{{\mathrm {Tm}}}$ which is involved in the second-stage laser cooling of Tm. To measure the absolute value of the hyperfine splitting interval, we used Doppler-free frequency modulation saturated absorption spectroscopy of Tm atoms in a vapor cell. The sign of the hyperfine constant was determined independently by spectroscopy of laser-cooled Tm atoms. The hyperfine constant of the level was determined to be $A_J=-422.112(32)\,\hbox {MHz}$ from the energy difference between the two hyperfine sublevels, $-2110.56(16)\,\hbox {MHz}$. In relation to the saturated absorption measurement, we quantitatively treat contributions of various mechanisms to the line broadening and shift. We consider power broadening in the case when Zeeman sublevels of atomic levels are taken into account. We also discuss the line broadening due to frequency modulation and relative intensities of transitions in saturated absorption experiments.

Appendix: An effect of optical pumping on the intensity relations of transitions between hyperfine sublevels

In linear spectroscopy, the intensities of transitions between hyperfine components of two levels are given by their line strengths S

$$\begin{aligned} S\propto (2F_d+1)(2F_u+1) \begin{Bmatrix} J_u&F_u&I\\ F_d&J_d&1 \\ \end{Bmatrix}^2, \end{aligned}$$

(16)

where u and d denote the upper and lower levels of the transition. Using this relation, the quantum numbers belonging to different hyperfine sublevels may be readily identified in experiment. However, in saturated absorption experiments, the relation (16) may yield qualitatively wrong results.

Our experiment provides an example where the observed most intense line is not the one expected from Eq. 16. We calculate the transition $F^{{\prime }}=5\rightarrow F=4$ to have the largest line strength

$$\begin{aligned} \left( \frac{I_{F^{{\prime }}=5\rightarrow F=4}}{I_{F^{{\prime }}=4\rightarrow F=3}}\right) _{{{\mathrm {theor}}}}=\frac{44}{35}>1. \end{aligned}$$

(17)

In contrast, from the experiment we find (see Fig. 4)

$$\begin{aligned} \left( \frac{I_{F^{{\prime }}=5\rightarrow F=4}}{I_{F^{{\prime }}=4\rightarrow F=3}}\right) _{{{\mathrm {exp}}}}\approx 0.9<1. \end{aligned}$$

(18)

The observed intensities appear to be the consequence of optical pumping (for more discussion of the effect see Refs. [21, 26]). When the non-cyclic $F=3\rightarrow F^{{\prime }}=4$ resonance is exited, some fraction of atoms spontaneously decay from the upper $F^{{\prime }}=4$ to the lower $F=4$ sublevel. In this way, the number of atoms interacting with light effectively decreases, thus giving additional contribution to the Lamb dip and increasing the resonance amplitude. For the cyclic $F=4\rightarrow F^{{\prime }}=5$ resonance, the pumping is absent and its amplitude is of purely saturational origin.

An estimation of the enhancement factor f for the $F^{{\prime }}=4\rightarrow F=3$ transition intensity is given by the ratio of atoms pumped into $F=4$ state during the time of flight:

$$\begin{aligned} f=1+\varGamma _{F^{{\prime }}=4\rightarrow F=4}\times \tau _{{{\mathrm {flight}}}}. \end{aligned}$$

(19)

This estimation gives a factor of 2.5 that reasonably agrees with the experimental value of 1.5.