Metrology of high-$n$ Rydberg states of molecular hydrogen with $\mathrm{\Delta }\nu /\nu =2\times {10}^{-10}$ accuracy

Precision spectroscopy of molecular Rydberg states of high principal quantum number ($n\ge 50$) followed by Rydberg series extrapolation currently represents the most accurate method of determining the ionization energies of neutral molecules. Until recently, such determinations relied on the use of pulsed laser radiation and supersonic molecular beams, which limited the spectral linewidths and induced ac Stark shifts, thus causing undesirable statistical and systematic uncertainties. We report on the combination of single-mode continuous-wave laser radiation, calibrated with a frequency comb, with slow supersonic beams expanding from low-temperature reservoirs, to record transitions to high-$n$ molecular Rydberg states with unprecedented accuracy. As illustration, we present the result of a measurement, with a relative frequency accuracy of $\mathrm{\Delta }\nu /\nu =2\times {10}^{-10}$ and an absolute accuracy of 64 kHz, of the transition from the $GK$ ${}^1\mathrm{\Sigma }_g^{+}(v=0,N=2)$ state of ${\mathrm{H}}_2$ to the $n=50$ $f$ Rydberg state belonging to a series converging on the $X^{+}{}^2\mathrm{\Sigma }_g^{+}(v^{+}=0,N^{+}=0)$ ground state of ${{\mathrm{H}}_2}^{+}$. The accuracy of the result $[{\nu }_{\text{NIR}}=380132832.236{\left(0.035\right)}_{\text{stat}}{\left(0.054\right)}_{\text{sys}}\text{MHz}]$ approaches the accuracy that can be achieved in similar measurements on ultracold alkali-metal-atom samples, although the dominant sources of uncertainties are different. The implications of our result for precision measurements of the adiabatic ionization energy of ${\mathrm{H}}_2$ and of the energy-level structure of ${{\mathrm{H}}_2}^{+}$ are discussed.

Figure 1

Excitation scheme to study the transitions from the $GK$(0,2) state to $nf\left[X^{+}(0,0)\right](S=0)$ Rydberg states (see text for details).

Figure 2

Schematic overview of the experimental setup (see text for details).

Figure 3

Measurement chamber and detection sequence. (a) Electrode stack with photoexcitation region, surrounded by a mumetal magnetic shield. The molecular beam (gray line) crosses the laser beams (colored arrows) at right angles. (b) Rod electrodes used for the compensation of stray electric fields in the plane perpendicular to the stack axis. (c) Electric-field pulse sequence used for the pulsed-field ionization of Rydberg states (see text for details).

Figure 4

Spectrum of the $50f{0}_3\leftarrow GK(0,2)$ transition with the analysis using a Lorentzian line-shape model. Upper panel: fit of the line model (blue lines) to the raw data (black points). Lower panel: weighted residuals (black points) and relative weights (blue lines) used in the fitting procedure. The fitted line-shape parameters are given in the inset.

Figure 5

Measured frequencies ${\nu }_{\text{NIR}}$ for the $50f{0}_3\leftarrow GK(0,2)$ transition. The lower (upper) panel contains the measured frequencies ${\nu }_{\text{NIR,a}}$ (${\nu }_{\text{NIR,b}}$). The mean values of pairs of measurements are displayed in the middle panel, where the vertical bars indicate one standard deviation. The standard deviation of the entire measurement set is indicated by the dashed magenta lines and the standard deviation of the mean of the different measurements by the cyan area. The different symbols indicate measurements taken on different days after full realignment of the laser and molecular beams.