Shifts due to distant neighboring resonances for laser measurements of $2{}^3{S}_1$-to-$2{}^3{P}_J$ transitions of helium

Quantum-mechanical interference between transitions from the metastable $2{}^3{S}_1{m}_J=0$ state to $2{}^3{P}_1{m}_J=\pm 1$ and to $2{}^3{P}_2{m}_J=\pm 1$ is shown to cause shifts in these resonances, despite the fact that the resonances are separated by more than 1000 natural widths. The $2{}^3{P}_1$-to-$2{}^3{P}_2$ fine-structure interval can be determined from the difference of these laser transitions, and a comparison between experiment and theory for this interval allows for precise tests of the quantum-electrodynamic (QED) theory used to calculate the interval. The shifts described here are large enough to be important for this test of QED and therefore to affect the continuing program of determining the fine-structure constant from comparison between accurate experimental measurements and theoretical calculations of the helium $2{}^{3}P$ energy intervals.

Figure 1

The $n=2$ triplet energy levels of helium. Population starts in $|1\rangle$ and interacts with a circularly polarized laser field. For ${\sigma }_{-}$, the states on the left ($|0\rangle$ to $|4\rangle$ in blue) can be populated. Quantum-mechanical interference between the $|1\rangle \to |2\rangle$ (solid arrow) and $|1\rangle \to |3\rangle$ (dashed arrow) transitions causes shifts in the measured resonances. Similar transitions with positive $m_J$ result from ${\sigma }_{+}$, as shown at the right in red. The cycling transition (dotted arrow) and the allowed radiative decay paths are also shown.

Figure 2

Timing for the atom traveling through a Gaussian laser beam with peak intensity $I_0$. The atom starts in $|1\rangle$ of Fig. 1 at a time $t_i$ before the atom enters the laser beam, and the final population of $|0\rangle$ is determined at a time $t_f$ after it has left the laser beam, and after the $2{}^{3}P$ atoms have had time to decay back to the $2{}^{3}S$ states.

Figure 3

Line shapes of the $|1\rangle \to |2\rangle$ (a) and $|1\rangle \to |3\rangle$ (b) resonances of Fig. 1 obtained from numerical integration of Eq. (3) for the laser beam of Fig. 2 with $T_L=4$ $\mu \mathrm{s}$ and $I_0=10$ $\mu \mathrm{W}/{\mathrm{cm}}^2$. The differences, (c) and (d), between these line shapes and those that would result if ${\gamma }_{23\to 1}=0$ are due to the interference effects calculated in this work.

Figure 4

Contour plot of the shift (in kHz) for the $2{}^3{P}_1$-to-$2{}^3{P}_2$ interval. The shift is in addition to a small ac shift of Eq. (4) and results from quantum-mechanical interference. The region below the dashed line has a FWHM of the resonance $<$1.8 MHz.

Figure 5

(a) Fine-structure interval inferred from signals at $\pm$$\Delta f$. The diamonds and uncertainties are from Ref. [15], and the solid line is the final result presented in that work. The squares are corrected for interference shifts calculated in this work. (b) A comparison of measurements and theory for the $2{}^3{P}_1$-to-$2{}^3{P}_2$ interval. The measurement of Ref. [15] is shown along with the corrected result from this work. The measurement of Ref. [17] may also be subject to interference corrections, but that calculation is beyond the scope of this work. The microwave measurement of Ref. [4] has only a small interference correction [2]. The QED theory [12] is also shown.