Broadening and intensity redistribution in the $\text{Na}\left(3p\right)$ hyperfine excitation spectra due to optical pumping in the weak excitation limit

Detailed analysis of spectral line broadening and variations in relative intensities of hyperfine spectral components due to optical pumping is presented. Hyperfine levels of sodium $3p_{1/2}$ and $3p_{3/2}$ levels are selectively excited in a supersonic beam at various laser intensities under the conditions when optical pumping time is shorter than transit time of atoms through the laser beam. The excitation spectra exhibit significant line broadening at laser intensities well below the saturation intensity, and redistribution of intensities of hyperfine spectral components is observed, which in some cases is contradicting with intuitive expectations. Theoretical analysis of the dynamics of optical pumping shows that spectral line broadening sensitively depends on the branching coefficient of the laser-driven transition. Analytical expressions for branching ratio dependent critical Rabi frequency and critical laser intensity are derived, which give the threshold for onset of noticeable line broadening by optical pumping. The critical laser intensity has its smallest value for transitions with branching coefficient equal to 0.5, and it can be much smaller than the saturation intensity. Transitions with larger and smaller branching coefficients are relatively less affected. The theoretical excitation spectra were calculated numerically by solving density matrix equations of motion using the split propagation technique, and they well-reproduce the observed effects of line broadening and peak intensity variations. The calculations also show that presence of dark (i.e., not laser coupled) Zeeman sublevels in the lower state results in effective branching coefficients which vary with laser intensity and differ from those implied by the sum rules, and this can lead to peculiar changes in peak ratios of hyperfine components of the spectra.

Figure 1

Collimation of the sodium beam by skimmers and apertures. The laser beam crosses the atomic beam at right angles 23 cm downstream from the nozzle.

Figure 2

Hyperfine energy levels of the $3s$ and $3p$ states of Na.

Figure 3

Excitation spectra of the $3s_{1/2}$, $F^{″}=2\to 3p_{1/2}$, $F^{\prime }=1,2$ transitions in Na. Residual Doppler width due to finite collimation angle is $\Delta {\nu }_D=11.2\text{MHz}$ (at $b=2\text{mm}$). The expected peak ratio is 1:1. Saturation intensities of the lhs and rhs components are 7.5 and $12.5\text{mW}/{\text{cm}}^2$, respectively.

Figure 4

Excitation spectra of the $3s_{1/2}$, $F^{″}=2\to 3p_{3/2}$, $F^{\prime }=1,2,3$ transitions in Na. Residual Doppler width due to finite collimation angle is $\Delta {\nu }_D=15.9\text{MHz}$ (at $b=3\text{mm}$). The expected peak ratio is 1:5:14. Saturation intensities of the three components are 37.4, 12.5, and $6.2\text{mW}/{\text{cm}}^2$, respectively.

Figure 5

Excitation spectra of the $3s_{1/2}$, $F^{″}=1\to 3p_{1/2}$, $F^{\prime }=1,2$ transitions in Na. Residual Doppler width due to finite collimation angle is $\Delta {\nu }_D=11.2\text{MHz}$ (at $b=2\text{mm}$). The expected peak ratio is 1:5. Saturation intensities of the lhs and rhs components are 37.4 and $12.5\text{mW}/{\text{cm}}^2$, respectively.

Figure 6

Line strengths ${\tilde{S}}_i^{\left(j\right)}$ (square frames) and branching ratios ${\Pi }_i$ (circular frames) for (a) $3s_{1/2}\to 3p_{1/2}$ and (b) $3s_{1/2}\to 3p_{3/2}$ hyperfine transitions.

Figure 7

Signal $J$ [Eq. (17)] as a function of the reduced detuning $\delta /{\Gamma }_e$ for different values of the pumping parameter $P_{pump}$ (shown as labels of the curves).

Figure 8

The reduced width $\Delta {\nu }_{pum}/\Delta {\nu }_{nat}$ given by Eq. (19) as a function of the pumping parameter $P_{pump}$.

Figure 9

(a) The FWHM widths of the lhs $\left(F^{\prime }=1\right)$ and rhs $\left(F^{\prime }=2\right)$ peaks of Fig. 3 as a function of laser intensity $I_{las}$. Solid squares, experiment, $F^{\prime }=1$; open triangles, experiment, $F^{\prime }=2$; solid curves, theory; and dashed curves, pure power broadening neglecting the broadening due to optical pumping. Arrows indicate saturation intensities of both transitions. (b) Calculated peak ratio $\mathfrak{R}$ of the rhs $\left(F^{\prime }=2\right)$ and the lhs $\left(F^{\prime }=1\right)$ peaks of Fig. 3 as a function of laser intensity $I_{las}$. The calculation was performed with and without taking the $\Delta m_F=\pm 1$ cascades into account.

Figure 10

(Color online) Zeeman sublevels involved in (a) the $3s_{1/2}$, $F^{″}=2\to 3p_{1/2}$, $F^{\prime }=1$ transition, and (b) the $3s_{1/2}$, $F^{″}=2\to 3p_{1/2}$, $F^{\prime }=2$ transition. Spontaneous emission leads to the population loss to the $F^{″}=1$ level of the ground state, and to the dark $m_{F^{″}}=\pm 2$ levels in (a) and to $m_{F^{″}}=0$ in (b). The effective branching coefficient in case (a) is therefore ${\Pi }_{eff}\left(2,1\right)<5/6$ and in case (b) it is ${\Pi }_{eff}\left(2,2\right)<1/2$.