Shifts due to quantum-mechanical interference from distant neighboring resonances for saturated fluorescence spectroscopy of the $2{}^{3}S$ to $2{}^{3}P$ intervals of helium

Quantum-mechanical interference with distant neighboring resonances is found to cause shifts for precision saturated fluorescence spectroscopy of the atomic helium $2^{3}S$-to-$2^{3}P$ transitions. The shifts are significant (larger than the experimental uncertainties for measurements of the intervals) despite the fact that the neighboring resonances are separated from the measured resonances by 1400 and 20$$000 natural widths. The shifts depend strongly on experimental parameters such as the angular position of the fluorescence detector, the intensity and size of laser beams, and the properties of the atomic beam. These shifts must be considered for the ongoing program of determining the fine-structure constant from the helium $2^{3}P$ fine structure.

Figure 1

The $n=2$ triplet energy levels of helium. The metastable states are labeled $|g_{\mu }\rangle$ and the $2^3{P}_J$$(m_J=m)$ states are labeled $|e_{Jm}\rangle$. The transitions being considered are shown as dashed and dotted red arrows, and all radiative decay paths are shown, with ${\gamma }_{Jm}^{\left(\mu \right)}$ being the decay rate from $|e_{Jm}\rangle$ to $|g_{\mu }\rangle$. Interference between decays that have the same $\Delta m_J$ (the same color in the figure) and end in the same $|g_{\mu }\rangle$ state leads to the shifts discussed in this work.

Figure 2

The fluorescence signals $F_0$ as a function of detuning $\Delta f$ from the $2{}^3{S}_1$-to-$2{}^3{P}_1$ resonance. Individual Doppler groups $\Delta f_D$, (a)–(c), give resonances at $\Delta f=\pm \Delta f_D$. For small $\Delta f_D$, $F_0$ depends on the relative phase, as shown for four values of $\Delta \varphi$ in (a). At larger $\Delta f_D$, the dependence is reduced substantially, as shown for the same values of $\Delta \varphi$ for $\Delta f_D=0.4$ MHz in (b). The solid curves represent averages over $\Delta \varphi$. The average of $F_0$ over all Doppler groups leads to the saturated fluorescence dip, as shown in (d) for $\Delta f_D^{\mathrm{HWHM}}=45$ MHz (thin red line) and 80 MHz (thick line). An expanded view of the saturated fluorescence dip within the dashed area in (d) is shown in (e), along with the amplitude (A) and full width at half maximum (FWHM) of the dip.

Figure 3

The amplitude and width of the $2{}^3{S}_1$-to-$2{}^3{P}_1$ saturated fluorescence dip [Fig. 2] as a function of intensity $I_0$ for three choices of interaction time $T_L$, as labeled in (a). The width of the dips is significantly broader than the natural width [thin dashed line in (b)] for intensities that lead to dips of substantial amplitude.

Figure 4

The ac Stark shifts (a) and interference shifts (b) and (c) of the $2{}^3{S}_1$-to-$2{}^3{P}_1$ saturated fluorescence dips. The graphs show integrated shifts due to Doppler groups ranging from $-|\Delta f_D|$ to $|\Delta f_D|$ and show that the shifts change sign for larger Doppler groups. Here a Gaussian distribution of Doppler groups with $\Delta f_D^{\mathrm{HWHM}}=80$ MHz is assumed. The shifts are shown for three choices of intensity ($I_0$) and interaction time ($T_L$), as labeled in (a). The total shift (from all Doppler groups) is the value at the right of each curve.

Figure 5

Contour graphs for ac Stark shifts and interference shifts vs interaction times $T_L$ and laser intensities $I_0$. The shifts shown in (a), (c), and (e) are for the $2{}^3{S}_1$-to-$2{}^3{P}_0$ transition, and those in (b), (d), and (f) are for $2{}^3{S}_1$-to-$2{}^3{P}_1$. Plots (a) and (b) are ac Stark shifts. Plots (c) and (d) are interference shifts (which are to be added to the ac Stark shift) for $\Delta m_J=0$ decays [$F_0$ in Eq. (8)], and (e) and (f) are interference shifts for $F_{\pm 1}$. A Doppler profile with $\Delta f_D^{\mathrm{HWHM}}=80$ MHz is used. The shift values given along the black contour lines are in kHz. Also shown on the plots (white dashed lines) are the widths of the saturated fluorescence dips in MHz.