Effects of an Oscillating Electric Field on and Dipole Moment Measurement of a Single Molecular Ion

We characterize and model the Stark effect due to the radio-frequency (rf) electric field experienced by a molecular ion in an rf Paul trap, a leading systematic in the uncertainty of the field-free rotational transition. The ion is deliberately displaced to sample different known rf electric fields and measure the resultant shifts in transition frequencies. With this method, we determine the permanent electric dipole moment of ${\mathrm{CaH}}^{+}$, and find close agreement with theory. The characterization is performed by using a frequency comb which probes rotational transitions in the molecular ion. With improved coherence of the comb laser, a fractional statistical uncertainty for a transition line center of as low as $4.6\times {10}^{-13}$ was achieved.

Figure 1

Population in the initial $|J=2,m=-5/2,-⟩$ rotational state vs the Raman difference frequency offset to the center of the line model (dashed red line) of the $|2,-5/2,-⟩\to |4,-7/2,-⟩$ rotational transition at approximately 2 THz. Error bars indicate a 68% confidence interval. The number of probes ranges for each data point from 10 to 65 and was given by the probabilistic preparation of the initial state of the molecule. Dashed line shows the least squares fit of a Rabi line shape with free parameters of center frequency offset, remaining population, contrast, and Rabi rate, with pulse duration fixed. The fit yields a 0.9 Hz statistical uncertainty (95% confidence). The 128 ms probe time is a substantial fraction of the observed lifetime of the initial state, resulting in reduced contrast of the line.

Figure 2

An ideal linear rf trap forms an electric quadrupole field with a null along the trap axis. Ions positioned off of the null experience a rf electric field. Because of the electrode geometry of the trap used in our experiments, the rf-electric field along the ion-ion axis has a finite minimal amplitude. By adjusting the voltages applied to the dc electrodes and the more distant dc bias electrode, the ions can be intentionally shifted from this minimum to sample different rf electric field magnitudes and directions. Three 729 nm laser beams are used to measure micromotion along approximately mutually orthogonal directions, ${\widehat{k}}_1$ parallel to the magnetic quantization axis $\overrightarrow{B}$, ${\widehat{k}}_2$, and ${\widehat{k}}_3$.

Figure 3

Red markers show the measured shifts in frequencies of the $|2,-3/2,-⟩$ to $|0,-1/2,-⟩$ transition as a function of the squared rf electric field amplitude at the molecular ion location. Spectroscopic probe times varied from 2.5 to 20.5 ms for different data points. The blue marker shows the rf electric field-free transition frequency (95% confidence interval $[-82,80]\mathrm{Hz}$) obtained through extrapolation to zero field with uncertainties determined by parametric bootstrapping. The dashed green line is a linear fit to the data. Its slope can be used to determine the permanent dipole moment of the ion.