Photodissociation spectra of single trapped CaOH⁺ molecular ions Open Access

Molecules possess various internal degrees of freedom that are both chemically and physically intriguing. Molecular ions in radio-frequency traps enable precision molecular spectroscopy on the single molecule scale and are, thus, a promising platform for applications of molecules in quantum technologies^1,2 and the exploration of fundamental physics.^3,4 For molecular ions, direct laser cooling via cycling transitions proves to be challenging and has not yet been realized experimentally.^5–7 While the translational motion of trapped molecular ions can be sympathetically cooled by laser cooling co-trapped atomic ions,^8–10 the absence of suitable cycling transitions poses a challenge in investigating the internal structure of molecular ions. Most spectroscopic experiments with trapped molecular ions rely on quantum logic methods utilizing co-trapped atomic qubits^11–14 or destructive detection methods based on, e.g., photodissociation channels.^15–20 

Various molecular ions can be generated from trapped atomic ions colliding with background gas molecules in a vacuum chamber. Photodissociation of molecular ions generated in this way leads to the recovery of atomic ions. This can be observed from the increase in the overall atomic ion fluorescence,^21,22 which provides a way to study the rovibrational structure of such molecular ions.

For trapped ion experiments based on ⁴⁰Ca⁺, which is one of the most well-studied and widely implemented ion species for quantum information processing,²³ molecular ions generated in chemical reactions with background gas are predominantly CaH⁺, CaO⁺, and CaOH⁺.²⁴ While extensive theoretical and experimental research has been conducted on the two diatomic species,^{12,14,20,25–27} there are few spectroscopic studies concerning the triatomic CaOH⁺ molecular ion.

As illustrated in Fig. 2, our experiment is based on a linear Paul trap³⁰ in an ultrahigh vacuum (UHV). A gate valve separates the chamber into two parts: the main experiment chamber and the molecular gas chamber. The main experiment chamber containing the ion trap is connected to a non-evaporable getter (NEG) pump and a NEG-ion combination pump to maintain a base pressure of ∼10⁻¹⁰ mbar. Ca⁺ ions in the ion trap originate from an integrated ⁴⁰Ca target mounted near the trap. By ablating the target with 532 nm laser pulses, neutral ⁴⁰Ca atoms are introduced to the ion trap, where they are photoionized in a two-step process using 375 and 422 nm lasers.³⁰ The generated Ca⁺ ions are then captured by the ion trap and Doppler-cooled by driving their 4²S_1/2 ↔ 4²P_1/2 cycling transition at 397 nm, as shown in the inset of Fig. 2. Additional lasers at 866 and 854 nm are applied to repump the ions that decay to the 3²D_3/2 and 3²D_5/2 states, respectively, back to the cyclic cooling manifold.

To the right of the main chamber, the molecular gas chamber is used to accumulate water vapor for the generation of CaOH⁺ molecular ions. When the gate valve is closed, this section is not pumped by the vacuum pumps attached to the main chamber, and the water vapor pressure inside increases over time due to outgassing from the chamber wall. To increase the partial pressure of water further, water vapor that is outgassing from an unbaked stainless steel hose is introduced to the molecular gas chamber via a leak valve.

To generate CaOH⁺ for the photodissociation measurement, water vapor is introduced to the main chamber by opening the gate valve. In our setup, the partial pressure of water vapor in the system cannot be directly measured. Instead, the increase in water vapor pressure in the main chamber is inferred from the change in total pressure estimated from the current of the ion pump. While waiting for the generation of CaOH⁺, the gate valve is opened every 5 min for ∼1 s, which results in the total pressure estimated by the ion pump increasing from $∼10−10$ to $∼10−9$ mbar.

As investigated by Okada et al. in 2003, the chemical reaction rate between H₂O and the trapped Ca⁺ ions increases when the ions are pumped into the 3²D_3/2 state by switching off the 866 nm repumping light during laser cooling.²¹ Therefore, the 866 nm laser is periodically switched off for 1 s every 2 s in the molecule generation phase. The generation rate of one CaOH⁺ molecular ion is typically 10–15 min for the ion crystals of 3–8 Ca⁺ ions investigated in this experiment.

After verifying the generation of a single CaOH⁺, the measurement of the photodissociation spectrum of CaOH⁺ is performed by applying femtosecond pulse trains produced by an OPA. The spatial beam profile and position of the OPA light are measured by a beam profiler mounted on a rail. To assist in the alignment of the OPA light, an additional 397 nm Doppler cooling beam is applied co-propagating with the OPA light, which is aligned to the center of the ion trap by maximizing the 397 nm light scattered by the ion crystal. The OPA light is overlapped with this axial Doppler cooling beam at the position of the ions. We estimate the alignment precision of this procedure to be 10 μm, limited by the resolution of the beam profiler. The radius of the OPA light at the ion crystal was also estimated by the beam profiler to be 150(10) μm.

As shown in Eq. (3), a linear dependence of the dissociation rate on the average power of the OPA light is expected. This is verified by the measurement of the dissociation rate at a different I_avg at λ = 250 nm. Fitting the results to a power law function, as shown in Fig. 5(a), reveals an exponent of $α=0.9−0.3+0.8$⁠. Thus, to better estimate the mean dissociation time, the average power of the beam is adjusted between 6 and 60 μW in order for at least half of the dissociation trials to be successful within the maximum illumination time. The repetition rate of the OPA is fixed at 1 kHz. The photodissociation cross sections at wavelengths ranging from 245 to 275 nm are then measured and shown in Fig. 5(b).

In addition to the measurement of single-photon dissociation, we also investigated the photodissociation of CaOH⁺ at wavelengths around 500 nm. To achieve a dissociation rate that is comparable to the single-photon measurements, the repetition rate of the OPA is increased to 10 kHz, and the light intensity is increased by reducing the beam radius to 70(10) μm and increasing the average power to a few milliwatts. At λ = 500 nm, the dependence of the dissociation rate on the average intensity of the OPA light on the ions is investigated, as illustrated in Fig. 6(a). Fitting the data to a power law function gives an exponent of $α=1.9−0.8+1.2$⁠, which is consistent with two-photon processes.

To interpret experimental measurements and predict theoretical spectroscopic parameters of CaOH⁺, we use state-of-the-art quantum-chemical methods. We employ two wave-function approaches: the coupled cluster method restricted to single, double, and noniterative triple excitations, CCSD(T),³³ to describe the ground electronic state; and the internally contracted multireference configuration interaction method restricted to single and double excitations, MRCISD,³⁴ to describe excited electronic states. Electronic orbitals are constructed using the augmented correlation-consistent polarized weighted core-valence quintuple-ζ quality basis set (aug-cc-pwCV5Z-PP³⁵ for Ca, aug-cc-pwCV5Z³⁶ for O, and aug-cc-pV5Z³⁷ for H). We include the scalar relativistic effect in Ca using the small-core relativistic energy-consistent pseudopotential (ECP10MDF³⁸) to replace the inner-shell electrons. Electronic structure calculations are performed with the Molpro^39,40 package of ab initio programs. The uncertainties of present energy calculations should not be greater than 1%–2%.⁴¹ 

We optimize the ground-state geometry of linear CaOH⁺ and find the equilibrium internuclear distances $ReCaO=3.574$ Bohr and $ReOH=1.803$ Bohr. Such an equilibrium geometry gives the rotational constant B_e = 0.3655 cm⁻¹. Present theoretical harmonic frequencies ω_e for normal modes of CaOH⁺ are 3949.9 cm⁻¹ for OH stretching, 747.4 cm⁻¹ for CaO stretching, and 481.2 cm⁻¹ for doubly degenerate bending. The frequency for the OH mode is slightly larger as compared to our theoretical value of 3748 cm⁻¹ and the experimental value of 3737.8 cm⁻¹⁴² for free OH. In addition, using the finite-field approach, we predict the equilibrium permanent electric dipole moment of CaOH⁺ with respect to its center of mass to be 6.3 D. This relatively large dipole moment originates from the partially charge-transferred nature of the ground-state CaOH⁺, which can be formally described as Ca²⁺OH⁻.

Under room temperature (300 K), the molecular ion is likely to be in its vibrational ground state (97% for CaO stretching). Using the CCSD(T) method, we predict that the dissociation energy of CaOH⁺ in the ground rovibrational level (v = 0, j = 0) into Ca⁺ and OH (v = 0, j = 0) is 37 807 cm⁻¹, which corresponds to 264.5 nm. This value originates from 38 763 cm⁻¹ of the electronic contribution corrected by −956 cm⁻¹ of the zero point energy (ZPE) difference within the harmonic approximation, where the ZPE for CaOH⁺ and OH is 2830 and 1874 cm⁻¹, respectively.

Figure 7 presents the ground and excited electronic states of the CaOH⁺ in the linear geometry as a function of the distance between Ca and the center of mass of OH. OH is described within the rigid-rotor approximation with an experimental internuclear distance of r_OH = 1.8332 Bohr.⁴³ The one-photon dissociation of CaOH⁺ is driven by the allowed electric dipole transition from the ground ¹Σ⁺ state to the excited ¹Π state, which is slightly repulsive around the ground-state equilibrium geometry and dissociates into Ca⁺ and OH. The corresponding vertical transition moment obtained with the MRCISD method is 0.8 D. The photodissociation cross sections are given by the convolution of the transition moment function and the overlap of the ground rovibrational and excited continuum nuclear wave functions. The exact values of the overlap and resulting cross sections are sensitive to the precise location of the classical turning point of the excited ¹Π state with respect to the equilibrium geometry of the ground ¹Σ state. While their alignment is accidental and beneficial for efficient resonant photodissociation, it is also challenging to predict accurately. Nevertheless, the employed ab initio method robustly predicts the vertical alignment to be within the width of the ground rovibrational wave function. The refined full-dimensional calculations based on the present methods should allow for predicting photodissociation cross sections in future dedicated studies.

In summary, we measured the photodissociation spectrum of single trapped CaOH⁺ molecular ions sympathetically cooled in Ca⁺ ion crystals at λ = 245–275 nm and λ = 500–540 nm. The photodissociation processes at wavelengths in these two ranges are verified to originate predominantly from a single-photon and a two-photon dissociation channel, respectively. While the one-photon dissociation cross section started to drop off around λ = 265 nm, a similar decrease was observed in the two-photon dissociation spectrum around λ = 525 nm. This implies that both processes involve the excitation of the molecular ion to its unbound first electronic excited state, which leads to its dissociation. Compared to the previous work²⁹ on single-photon dissociation of CaOH⁺ by Okada et al. in 2006, we report a more precise measurement of the dissociation cross sections at different wavelengths, which is consistent with the previous estimation of σ_eff ≈ 10⁻¹⁸ cm² at λ = 255 nm. The measured single-photon dissociation spectrum is also consistent with the photodissociation of CaOH⁺ at 272 nm observed by Hansen et al.²⁴ 

As for the two-photon dissociation of CaOH⁺ around λ = 500 nm, the dissociation process is found to be significantly less efficient compared to the single-photon dissociation process, as expected since higher peak intensity is typically required for higher-order processes. Yet femtosecond lasers at such wavelengths have proven to be alternatives to continuous wave lasers with λ < 265 nm for photodissociation of the molecular ion. The question arises as to whether nanosecond lasers that are used for ablation loading in trapped atomic ion experiments at λ = 515 nm³⁰ are able to dissociate CaOH⁺. Achieving a mean photodissociation time of 1 min for a single CaOH⁺ ion is estimated to require an average power of 0.2–0.5 W, considering a laser with f_rep = 2 kHz, τ_p = 1 ns, and the light focused onto the ions with a beam waist of w = 60 μm.

We employed advanced ab initio molecular electronic structure methods to theoretically characterize the ground-state properties of CaOH⁺ and to investigate the spectrum of excited electronic states involved in the measured photodissociation. We predicted the photodissociation threshold for CaOH⁺ → Ca⁺ + OH at 265 nm in very good agreement with the one-photon experimental observations. Efficient photodissociation is possible because of the coincidence of the slightly repulsive part of the excited ¹Π electronic state close to its inner classical turning point, which is accessible by a decent electric dipole transition moment.

The reported measurement is implemented on single CaOH⁺ molecular ions in small trapped ion crystals, which enables the relatively precise evaluation of the photodissociation cross section due to the repeatable positioning of the molecule. The scale of the system is also crucial for investigating a two-photon dissociation process that requires tightly focused light to achieve sufficient intensity. By utilizing a femtosecond laser with a broad spectral range, the photodissociation channel of CaOH⁺ is studied without dedicated rotational state preparation. The reported photodissociation spectrum can be a basis for studying the rovibrational structure of CaOH⁺ via dissociation-based spectroscopy methods. Designed to investigate the interaction between femtosecond laser pulses and a linear ion crystal with a single CaOH⁺, this experiment represents a precursor for performing quantum logic spectroscopy with polyatomic molecular ions.⁴⁴ 

See the supplementary material for the numerical data for the calculation of one-dimensional potential energy surfaces for the ground and excited electronic states of CaOH+ presented in Fig. 7.

This research was funded by Grant No. ERC-2020-STG 948893, ESQ Discovery project SDEF: State-dependent force spectroscopy for trapped ions, FWF 1000 Ideas Project No. TAI-798, and the Foundation for Polish Science within the First Team program. We acknowledge Poland’s high-performance computing infrastructure, PLGrid (HPC Center: ACK Cyfronet AGH), for providing computer facilities and support within computational Project No. PLG/2023/016115. The authors would like to acknowledge AQT for the support with the ion trap and vacuum pumps, in particular Daniel Nigg and Georg Jacob; Markus Teller for assistance with vacuum work; electronics workshop technicians Kilian Prokop and Wolfgang Kuen for lab infrastructure support; Milan Oncak for discussions on quantum chemistry calculations; Christian Marciniak for general discussions; Marco Valentini for Ca⁺ lasers implementation; and the Quantum Optics and Spectroscopy Group and associated ion trapping groups at the Universität Innsbruck for general assistance.

The authors have no conflicts to disclose.

Z.W., S.W., B.F., and P.S. designed the experiment. Z.W., S.W., and B.F. carried out the dissociation experiment and analyzed the data. M.I.M., E.M., and B.F. performed the characterization of the OPA system. P.G. and M.T. performed electronic structure calculations. Z.W., S.W., B.F., and M.T. contributed to the manuscript. Z.W., S.W., V.P., M.I.M., E.M., G.M., R.N., B.F., and P.S. contributed to the experimental setup. All authors reviewed the manuscript. B.F. and P.S. supervised the project.

Zhenlin Wu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (supporting); Resources (equal); Software (supporting); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Stefan Walser: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (supporting); Methodology (supporting); Resources (equal); Software (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Verena Podlesnic: Resources (equal). Mariano Isaza-Monsalve: Data curation (equal); Resources (equal); Writing – review & editing (equal). Elyas Mattivi: Data curation (equal); Resources (equal); Visualization (equal); Writing – review & editing (equal). Guanqun Mu: Resources (equal). René Nardi: Resources (equal); Writing – review & editing (equal). Piotr Gniewek: Data curation (equal); Formal analysis (equal). Michał Tomza: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Brandon J. Furey: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (supporting); Methodology (supporting); Resources (equal); Software (supporting); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Philipp Schindler: Conceptualization (equal); Funding acquisition (equal); Methodology (lead); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available at http://doi.org/10.5281/zenodo.11109790.

The photodissociation process considered in this paper is treated as a consequence of a photon absorption event that excites the molecular ion to its unbound first electronic excited state. We thus define the effective photodissociation cross section σ_eff = ησ in terms of the absorption cross section σ and the quantum yield η. We assume all of the photon absorption events lead to dissociation of the molecular ion, i.e., η = 1, and the molecular rovibrational structure is not considered for simplicity.

To verify the memorylessness of the photodissociation process, the illumination for photodissociation is divided into several time intervals with various fixed durations. From this data, we can estimate the probability of dissociation for a fixed illumination duration as well as the cumulative time until dissociation. If the photodissociation process has no memory effect, the probability of the molecule being dissociated after being illuminated for a fixed time T will be described by the exponential distribution CDF, as in Eq. (2).

We can, therefore, also estimate the mean dissociation time using the estimated dissociation probability for fixed times. These probabilities are fitted to the exponential distribution CDF as indicated in Fig. 8, e.g., λ = 270 nm and λ = 500 nm. We estimate the photodissociation cross sections via this analysis for all data and compare them to the cumulative dissociation time, as shown in Fig. 9. We find that the cross sections obtained from both methods agree with the uncertainties, which indicates that the single- and two-photon dissociation processes are memoryless.

During the photodissociation measurement, the cooling laser at 397 nm and the repumpers at 866 and 854 nm are constantly applied for Doppler cooling of the ion crystal. The upper limit of the power applied for each laser is 0.6 mW (397 nm), 0.4 mW (866 nm), and 0.3 mW (854 nm). The beam radii of the lasers at the ion position are larger than 50 μm. To provide axial confinement of the ion crystal, the voltage on the endcap electrodes of the ion trap is set to 205 V. This leads to an axial COM motional frequency of ω₀ = 2π × 526.8(2) kHz for all the experiments in this paper, based on the measurement discussed in  Appendix D.

The femtosecond OPA system used in this work is an ORPHEUS-HP OPA seeded and pumped by a Carbide CB5 amplifier from Light Conversion. As knowledge of the pulse duration is required to obtain the two-photon dissociation cross section, we characterized it using a combination of measurements of the spectra and autocorrelation measurements. While the pulse duration at λ = 500–540 nm could not be directly measured with our current setup, the pulses produced by the OPA are guaranteed to be transform-limited by the manufacturer. To verify this, the pulse duration and the spectrum of the OPA at λ = 590 nm were measured, and the time-bandwidth product was inferred, as shown in Fig. 10.

To measure the pulse duration, we used an autocorrelator setup that splits the light from the OPA into two paths, where one path has an adjustable time delay. Overlapping both branches on a phase-matched BBO crystal induces type I second-harmonic generation. Assuming a Gaussian temporal profile, the FWHM of the SHG signal was measured to be $τ̃ac=206(4)$ fs, corresponding to a pulse duration of $τp=τ̃ac/2=146(3)$ fs.

The spectra of the OPA light were measured with an Ocean Insight Ocean HR spectrometer. Fitting the spectrum at λ = 590 nm to a Gaussian distribution, the FWHM spectral width was determined to be Δν = 2.973(6) THz. This gives a time-bandwidth product of TBP = τ_p Δν = 0.433(8), which is in agreement with the time-bandwidth product of a transform-limited Gaussian pulse, i.e., TBP_Gaussian = 0.441.

Thus, we assume the pulses are transform-limited at λ = 500–540 nm in the two-photon dissociation experiments. The measured center wavelengths, their corresponding spectral bandwidths, and the inferred pulse durations are reported in Table I.

In addition to the statistical uncertainties in the measured parameters of OPA pulse duration, the following sources of uncertainty are considered: The uncertainties of the mean dissociation times are given by Bayesian analysis and are significantly larger than the uncertainties of the measured dissociation times. The uncertainty of the average intensity of the OPA light at the ion crystals derives from the uncertainty in average power (0.1 μW in single-photon measurements, 0.1 mW in two-photon measurements), the uncertainty of beam positioning (10 μm), and the uncertainty of the beam radius. The beam radius is estimated by the beam profiler with a precision of 10 μm at a position corresponding to the real ion crystal position in the beam path. The measured optical path length can have a deviation from the path length to the ion crystal of <5 mm, which results in a possible systematic error in the measured beam radius at the ion crystal of <10 μm. The total uncertainty of the beam radius is thus considered to be 10 μm.

The uncertainties presented in this paper are based on a 68% confidence level.

The photodissociation experiments in this paper are carried out with an automated script according to the process tree illustrated in Fig. 11.

The photodissociation experiment requires real-time evaluation of the ion crystal configuration such that the number of fluorescing (bright) Ca⁺ and non-fluorescing (dark) molecular ions can be monitored. Knowledge of the ion crystal configuration allows a script to run automatic measurements and take actions such as (re)loading the trap or emptying it if more than one dark ion is created. The ion crystal configuration is identified using solely image data of the ions taken by an imaging system with 10× magnification using an Andor iXon^EM+ 885 camera with a pixel size of 8 μm. For this process, a region of interest (ROI) in the image centered around the ions is converted into two normalized data sets: the column sum, where the ROI is summed along the columns of the image array; and the row sum, where the summation is done along the rows (see Fig. 12). All analysis to identify the ion crystal configuration is based on these two data sets.

The identification of the ion crystal configuration is performed in real-time and runs continuously throughout the measurements. An image of the ions is taken with an exposure time of 0.2 s, from which the ion crystal configuration is identified. The identification process takes $∼0.2$ s, then the next image is taken and analyzed. Three algorithms are used to determine the ion crystal configuration: (i) check for the presence of bright ions, (ii) count of bright ions, and (iii) determination of the ion crystal configuration via a template fitting method.

The first method checks for the presence or absence of bright ions in the trap. Here, a moving average is applied to the row sum of the image data before applying a discrete Fourier transform (DFT). If the trap is empty, the moving average is mainly flat and the amplitude of the longest spatial wavelength in the DFT is small; if there are bright ions in the trap, the moving average exhibits a broad peak and, thus, the amplitude of the longest spatial wavelength in the DFT is larger. A threshold is then set to differentiate between an empty or a non-empty trap.

The number of bright ions is counted from the column sum via a peak finder algorithm, which can determine the number and positions of bright ions without requiring a fluorescence-based threshold. This algorithm extracts and sorts local maxima of column sum over subsets of length comparable but smaller than the ion separation. From the largest amplitude difference and their amplitudes, the algorithm classifies peaks into two distinct groups: large-amplitude peaks originating from bright ions and small-amplitude peaks due to noise. Knowledge of the positions of the bright ion peaks can be utilized to identify the presence of dark ions , e.g., determining the trap center position, as it requires an ion crystal of solely bright ions.

To determine the molecular mass, a radio frequency (RF) “tickling” signal with an amplitude of 30 mV modulates the DC voltage on the endcap electrodes, which defines the axial confinement. The signal excites the COM motion of the ions, which peaks when the signal frequency is in resonance with the COM motional frequency. Due to Doppler broadening, the ions fluoresce less when their motional state is excited.⁴⁶ The COM motional frequency can then be determined by scanning the RF tickling frequency to minimize the fluorescence.

In crystals with mixed ion species, the COM motional frequency depends not only on the different ion masses but also on the specific ion crystal configuration.⁴⁷ For instance, the two ion configurations (Ca⁺, Ca⁺, CaOH⁺) and (Ca⁺, CaOH⁺, Ca⁺) have a different COM motional frequency, with a deviation of 1.7 kHz in our system. When applying an RF signal that is close to the COM motional frequency of the current ion configuration, the ions reconfigure their respective positions due to the induced motion. After the reconfiguration, the RF signal is no longer resonant with the COM motion, which makes it difficult to perform a precise measurement of the resonance frequency. Thus, we perform measurements with crystals consisting of exactly two trapped ions, as there exists only a single COM motional frequency due to symmetry. However, the dissociation measurements are performed with larger ion crystals to speed up the CaOH⁺ generation process. To prevent ion loss during the tickling scan, a Doppler cooling cycle of about ∼1 s is applied to the ions, with the RF signal being turned off after each tickling measurement in the scan. For every single Ca⁺ and molecular ion pair, we performed and summed up to five measurement runs.

In order to determine the COM motional frequency from the images taken with the EMCCD camera, the measured photon counts at each pixel are summed in a small ROI around the ions. This ROI is 4σ wide along the y-axis of the images, where σ is the waist of a Gaussian fitted to the row sum dataset, as shown in Fig. 12. Along the x-axis, this small ROI has a length that is 5 times that of a two ion crystal with an endcap electrode voltage of 205 V, i.e., 8.5 μm. The total photon counts integrated over the ROI as a function of the RF tickling frequency are normalized, and the COM motional frequency is determined by fitting a Gaussian function to the data, as shown in Fig. 13.

The fitted COM motional frequencies from 65 Ca⁺ and molecular ion pairs and the corresponding reference COM motional frequencies of Ca⁺ gave masses for the molecular ions ranging from 56.7 to 57.2 amu. This confirms that all 65 molecular ions measured were CaOH⁺.

In order to estimate the statistical uncertainty of the measured COM motional frequency of trapped atomic Ca⁺ ions, we evaluated 34 data sets taken over the course of hours on two different days. For a voltage of 205 V applied to the endcap electrodes, a COM motional frequency of 526.8 kHz with a standard deviation of 200 Hz was determined. Since the errors in the fitted COM motional frequency of all 65 processed datasets of Ca⁺ and molecular ion pairs and all reference datasets are all smaller than 200 Hz, this value is utilized to obtain an upper limit for the uncertainty of COM motional frequencies. This results in an estimated uncertainty in the derived molecular ion masses of 0.1 amu.