Quantum interference effects in saturated absorption spectroscopy of $n=2$ triplet-helium fine structure

Over the past decade, precision measurements of the helium $2^{3}P$ fine-structure intervals have been performed using a variety of experimental techniques. We have shown in previous work [Phys. Rev. A 89, 043403 (2014), and references therein] that these measurements can be significantly shifted by quantum interference with neighboring transitions, despite the fact that the neighboring transitions are more than a thousand natural widths away from the transition being measured. The shifts depend on the experimental technique used, since different techniques allow for different quantum-mechanical interference paths. Here we consider measurements using the saturated absorption technique, and find that quantum interference effects cause substantial shifts for the $2^3{P}_1–2^3{P}_2$ interval. We find that four independent measurements for this interval are now in agreement when interference shifts are taken into account.

Figure 1

The $n=2$ triplet energy levels of helium, illustrating the measurement of the $2^3{S}_1,m_J=1–2^3{P}_1,m_J=1$ interval. The population is assumed to start in the $2^3{S}_1,m_J=1$ metastable state ($|1\rangle$), and is excited (solid arrow) by a linearly polarized 1083-nm laser tuned to the $2^3{P}_1,m_J=1$ state ($|2\rangle$). There is a small amplitude of also driving (dotted arrow) to the far-off-resonance $2^3{P}_2,m_J=1$ state ($|3\rangle$). The atoms can then decay back to $|1\rangle$, or to the $2^3{S}_1,m_J=0$ state ($|0\rangle$). The $|0\rangle$ state can be considered to be a dark state due to the electric-dipole-forbidden $2^3{S}_1,m_J=0–2^3{P}_1,m_J=0$ transition, and thus only the four numbered states play a significant role in the measurement.

Figure 2

Phase- and Doppler-averaged saturated-absorption lineshape calculated for the $2^3{S}_1–2^3{P}_1$ resonance, for a laser intensity of $I=0.1 \mathrm{mW}/{\mathrm{cm}}^2$ and an interaction time of $T=1 \mu \mathrm{s}$. The points defining the FWHM of the lineshape (in this case 2.8 MHz) are indicated.

Figure 3

Widths of the calculated saturated-absorption line shapes (solid lines) for the $2^3{S}_1–2^3{P}_1$ resonance, over a range of interaction times $T$, and for various laser intensities $I$. The data points are experimental widths [22], which are used to calibrate the pressure scale (bottom axis) with the time scale (top axis). The natural width of the resonance is shown as a dashed line.

Figure 4

Shifts in the $2^3{S}_1–2^3{P}_1$ resonance as a function of included Doppler groups. The shifts are calculated using uniformly distributed Doppler groups ranging from $-|\mathrm{\Delta }{f}_D|$ to $|\mathrm{\Delta }{f}_D|$. Results are shown for a laser intensity of 0.1 $\mathrm{mW}/{\mathrm{cm}}^2$, with four choices of atom-laser interaction times $T$. The shift has converged when all Doppler groups with $|\mathrm{\Delta }{f}_D|<20$ MHz are included, and thus the total shift from all Doppler groups is the value at the right of each curve.

Figure 5

Total shifts (solid thin lines) in the $2^3{P}_1–2^3{P}_2$ interval over a range of interaction times $T$ (which result from helium cell pressures $P$) and laser intensities $I$. The thin dashed lines give the effect when only the ac shift term is included (and the $\frac{{\mathrm{\Omega }}_3{\gamma }_{23\to 1}}{2{\omega }_{23}}$ term in Eq. (2) is artificially set to zero) The thick lines show least-squares fits of the shifts over the pressure range of 18 to 38 mTorr, extrapolated to zero pressure. The circle indicates the intensity and pressure at which the most significant data in [17] were taken.

Figure 6

Measurements and theory for the $2^3{P}_1–2^3{P}_2$ fine-structure interval in helium. The points labeled a [10], b [16], c [23], and d [17] are measurements, with each using a different experimental technique. The point labeled e is the calculation of Pachucki and Yerokhin [7], adjusted for the CODATA 2010 [24] value of $\alpha$. The solid error bars are the values without interference effects. The correction for interference effects have now been estimated for all of these measurements (in [20], [16], [18], and in this work, respectively), and the dashed error bars give the corrected measurements. The numbers shown on the $x$ axis are frequencies relative to a $2291170$ kHz offset.