Hyperfine resolved optical spectroscopy of the A$^2\Pi \leftarrow $X$^2\Sigma^+$ transition in MgF

We report on hyperfine-resolved laser spectroscopy of the A$^2\Pi \leftarrow $X$^2\Sigma^+$ transition of MgF, relevant for laser cooling. We recorded 25 rotational transitions with an absolute accuracy of better than 20 MHz, assigned 56 hyperfine lines and determined precise rotational, fine and hyperfine structure parameters for the A$^2\Pi$ state. The radiative lifetime of the A$^2\Pi$ state was determined to be 7.2(3) ns, in good agreement with \textit{ab initio} calculations. The transition isotope shift between bosonic isotopologues of the molecule is recorded and compared to predicted values within the Born-Oppenheimer approximation. We measured the Stark effect of selected rotational lines of the A$^2\Pi \leftarrow $X$^2\Sigma^+$ transition by applying electric fields of up to 10.6 kV cm$^{-1}$ and determined the permanent electric dipole moments of $^{24}$MgF in its ground X$^2\Sigma^+$ and first excited A$^2\Pi$ states to be $\mu_X=$2.88(20) D and $\mu_A=$3.20(22) D, respectively. Based on these measurements, we caution for potential losses from the optical cycling transition, due to electric field induced parity mixing in the excited state. In order to scatter $10^4$ photons, the electric field must be controlled to below 1 V cm$^{-1}$.


I. INTRODUCTION
Recently, magnesium monofluoride (MgF) has been identified as a promising candidate molecule for laser cooling and magneto-optical trapping experiments [1][2][3] . Compared to other group II monofluoride molecules that have been laser-cooled so far [4][5][6] , MgF is lighter and has a stronger optical cycling transition in the ultraviolet. These properties allow for exerting a large radiation force to rapidly slow and cool the molecules and produce a magneto-optical trap with a high capture velocity. The predicted low off-diagonal vibrational branching and the simple hyperfine structure of MgF reduces the complexity of the optical setup significantly.
Rotationally resolved optical spectra of MgF have been recorded in absorption [7][8][9][10][11] and emission 11 . The vibrationrotation emission in the electronic ground state has also been studied 12 . Precise hyperfine and rotational constants of MgF in its X 2 Σ + electronic ground state for vibrational states v = 0, 1, 2 and 3 were determined from its millimeter-wave spectrum 13,14 . Recently, Xu et al. 15 recorded optical absorption spectra of the A 2 Π ←X 2 Σ + transition, resolving a prior debate in the literature about the sign and value of the spinorbit coupling constant. However, this study suffered from a large systematic frequency offset of about −4.1 GHz and the Hamiltonian that was used to fit to the experimental data did not account for the presence of Λ-doubling and hyperfine structure (vide infra) in the excited state † . Recently, optical cycling experiments have been performed 17 , a first step towards laser cooling experiments. a) These authors contributed equally to this work † A further study from the same group 16 was submitted during the preparation of this article. We find a systematic frequency offset of about +2.3 GHz in the reported line centers.
MgF has also been studied theoretically using ab initio methods 1,2,18 . Pelegrini et al. 19 calculated various properties of MgF as part of a wider study of group II monofluorides. These predictions show good agreement with the available experimental data for CaF. For MgF, they predicted a radiative lifetime of 7.16 ns for the A 2 Π ,v ′ = 0 level, and a decay probability of 1.4% to the X 2 Σ + , v ′′ = 1 first vibrationally excited state. They also predicted the permanent electric dipole moments for the ground and excited states to be 2.67 D and 4.23 D respectively.
Here, we present hyperfine-resolved UV laser-induced fluorescence (LIF) spectra of MgF produced in a cryogenic buffer gas molecular beam. The large frequency calibration error present in the previous study by Xu et al. 15 is corrected by calibrating our wavemeter with known transition frequencies in Yb. The eigenvalues of an appropriate Hamiltonian are fitted to the measured hyperfine energy levels to derive precise spectroscopic constants for the A 2 Π state. We record and analyse transition isotope shifts between the two bosonic isotopologues and compare to predictions from mass scaling arguments. The spectral width of isolated lines is measured with high accuracy to determine the radiative lifetime of the A 2 Π state. The electric dipole moments of the ground and excited states are deduced from the Stark shifting of individual rotational lines in electric fields of up to 10.6 kV cm −1 . We then determine how opposite parity levels in the excited state mix in an external electric field. This effect can result in large losses from the optical cycling scheme, if stray electric fields are not well-controlled.
The group II metal monofluorides are interesting candidates for laser cooling because of the single, unpaired, and metalcentered electron that is polarized away from the fluorine atom. The internuclear distance and potential energy curves of the ground and first excited states are very similar. This leads to a very diagonal Franck-Condon matrix which reduces the number of vibrational repump lasers required to scatter a large number of photons. Measurements of the fine and hyperfine structure and the electric dipole moments provide information about the spin density at the fluorine nucleus and the charge distribution. This provides information to better understand the bonding structure in these molecules 20 . We compare our results to the other group II monofluoride molecules, CaF, SrF and BaF, which have been studied in detail.

II. HAMILTONIAN
We use the following effective Hamiltonian, which operates in a given vibrational state with energy E 0 : It describes rotation (B, D), spin-orbit A, spin-rotation γ, Λdoubling (p, q) and magnetic hyperfine interactions (a, b F , c and d). The ground state of MgF is a X 2 Σ + state, for which Λ = 0 and therefore A = p = q = a = d = 0. Fig. 1 shows the relevant levels and transitions in absence of hyperfine structure. For a given value of J, the level with the lowest energy is labeled F 1 . We use ∆J F ′ F ′′ (N ′′ ) to label the transitions. The energies of the J-levels in the ground state can be calculated using, and in the A 2 Π state using the following formula, (3) Here, the minus (plus) sign applies for the F 1 (F 2 ) levels respectively. In the case of J ′ = 1/2, expression (3) reduces to When comparing transition frequencies of different isotopologues, an explicit form for the energy E 0 is required. Here we assume that with T e being the potential energy minimum of an electronic state, and the remaining terms describing the vibrational energy up to second order in the vibrational quantum number v.

III. EXPERIMENTAL SETUP
The spectrometer used for this study is similar to the one described previously 21,22 . MgF molecules are produced in a cryogenic helium buffer gas cell that is cooled to 2.7 K using a closed-cycle helium cryocooler. The cell's geometry is based on the design of Truppe et al. 23 ; it has a length of 40 mm with a bore diameter of 10 mm, and an aperture of 4 mm. A Mg rod is ablated by 20 mJ of pulsed Nd:YAG laser (Continuum Minilite II, 1064 nm) radiation focused to a waist diameter of 0.4 mm. The hot Mg atoms react with NF 3 gas (0.001 sccm flow rate, 100 K) to form MgF molecules. The molecules are cooled by collisions with the cryogenic He buffer gas which is flowing into the cell continuously at a rate of 1 sccm. This helium flow also extracts the pulse of molecules from the cell into a molecular beam with a rotational temperature of about 4 K and a mean forward velocity of typically 160 m s −1 . The forward velocity increases over several thousand ablation shots, but can be restored to its original value by cleaning the cell. The molecules are detected by laser-induced fluorescence (LIF) 44 cm downstream from the buffer gas cell aperture 21,22 . The transverse velocity spread of the molecular beam is reduced to about 1 m s −1 by a 2x2 mm 2 square aperture placed at the entrance to the LIF detector. Here, a continuous wave (CW) 359 nm laser beam from the second harmonic of a titanium sapphire laser intersects the molecular beam perpendicularly. We ensure that any Doppler FIG. 2. Laser induced fluorescence spectrum of the (6s6p) 1 P 1 ← (6s 2 ) 1 S 0 transition in Yb used to verify the accuracy of our wavemeter. The black dots represent the data recorded in this study and the blue curve shows a fit using Lorentzian lineshapes. The red sticks represent the line positions obtained by Kleinert et al. 24 with an absolute accuracy of better than 1 MHz. The inset shows the spectral line from the 174 Yb isotope in more detail.
shift arising from misalignment of the probe beam is below 10 MHz by measuring spectra with and without retroreflecting the laser. The LIF is imaged onto a photomultiplier tube (PMT, Hamamatsu R928) and the resulting photo-current is amplified to give a time-dependent fluorescence signal. We measure and stabilize the fundamental wavelength of the titanium sapphire laser using a wavemeter (HighFinesse WS8-10), which has an absolute accuracy of 20 MHz and a measurement resolution of 0.4 MHz. The wavemeter is calibrated using a temperature-stabilized HeNe laser (SIOS), whose absolute frequency is known to within 5 MHz. Additionally, we determine the (6s6p) 1 P 1 ← (6s 2 ) 1 S 0 transition frequency and isotope shifts of Yb by applying high-resolution laser spectroscopy to a pulsed buffer gas beam of Yb atoms. These frequencies are known with an absolute and relative uncertainty of better than 1 MHz 24 . Our experimental spectrum is presented in Fig. 2. The lines under the spectrum show the transition frequencies measured by Kleinert et al. 24 . In our spectrum, the line-centers are determined from a fit to multiple Lorentzian lineshapes and listed in Table I. We reproduce the absolute transition frequencies within 10 MHz and the relative frequencies (isotope shifts) to within 1 MHz over the range of several GHz. Since the deviation between our measured line centers and the published values is within the absolute accuracy of the wavemeter, we use the calibration from the HeNe laser without further correction. We found it was necessary to add a constant flow of dry nitrogen gas through the doubling cavity, in order to avoid absorption of the fundamental light by water vapor. This effect was particularly pronounced near the Q 1 (0) fundamental frequency of 834.3255 THz.
To determine the electric dipole moment of MgF in the A 2 Π and X 2 Σ + states, we install transparent copper mesh electrodes below and above the molecular beam to apply electric fields to the molecules inside the LIF detector. The distance between the electrodes is measured to be 9.0(3) mm. I. The measured (6s6p) 1 P 1 ← (6s 2 ) 1 S 0 transition frequencies of Yb relative to the transition frequency of 174 Yb as determined by Kleinert et al. 24 . For the isotopes with a nuclear spin of I = 0, F ′ is given in brackets. The last column gives the absolute frequency differences of the isotope shifts between the two measurements with a standard deviation (SD) of 1.3 MHz.

Isotope
Isotope shift (MHz) 24  The voltage on the electrodes is supplied by a high-voltage power supply (Spellman SL1200) and measured with a calibrated high-voltage probe and multimeter with a combined relative accuracy of 10 −4 .

IV. ISOTOPE SHIFTS, SPECTROSCOPIC CONSTANTS, HYPERFINE STRUCTURE, AND Λ-DOUBLING
The vibrational, rotational, fine and hyperfine structure of the ground electronic state of MgF is well known. To improve the spectroscopic parameters for the rotational, fine and hyperfine structure of the A 2 Π state, we record low-J rotational lines of the A 2 Π ←X 2 Σ + transition. The isotope shifts between the 26 MgF and 24 MgF are discussed in section IV A. From section IV B onwards, we focus on the most abundant 24 MgF, summarizing the results of our measurements, and point out some important differences with the other group II monofluorides.

A. Isotope shifts
We use a Mg metal ablation target with a natural isotopic abundance of 79%, 10% and 11% for 24 Mg, 25 Mg, and 26 Mg, respectively. 24 Mg and 26 Mg are bosons with a nuclear spin I( 24 Mg) = 0 and I( 26 Mg) = 0 whereas 25 Mg is a fermion with a nuclear spin I( 25 Mg) = 5/2. Fluorine has one stable isotope with a nuclear spin of I( 19 F) = 1/2. Fig. 3 shows a typical spectrum when exciting the R 2 /Q 21 (1) line. In this example we scan over the three MgF isotopologues, and observe isotope shifts of −3.35 GHz ( 25 MgF) and −6.48 GHz ( 26 MgF), relative to 24 MgF. The inset of Fig. 3 shows the more complex hyperfine structure of 25 MgF. Each rotational line is split into two groups of hyperfine lines separated by about 1GHz, which are both further split by a few hundred MHz. The larger splitting arises from the Fermi interaction between the electron and 25 Mg nuclear spin in the ground electronic state 13 . We did not analyse the The shift in transition frequencies between isotopologues is characteristic to a molecular species. This molecular isotope shift can be used as an additional means of identification, and to reveal small deviations from the Born-Oppenheimer approximation. Since 24 MgF and 26 MgF exhibit the same hyperfine structure, the shift in the gravity center of a rotational line can be found straightforwardly by comparing the positions of equivalent hyperfine peaks. We define the transition isotope shift, δ ν i , as, with ν i ( j) being the absolute frequency of the optical transition i in isotopologue j. Within the Born-Oppenheimer approximation, the isotope shift comes about through changes to the relevant reduced masses in the molecular system. For the rovibrational constants, the relevant reduced mass is that computed from the two atomic masses, m Mg and m F , For the electronic contribution T e , it is that of the valence electron mass, m e , and the remaining molecular mass, We define ρ = m mol ( 24 MgF)/m mol ( 26 MgF), which has the value of about 0.983, and ρ el = m el ( 24 MgF)/m el ( 26 MgF), noting that 1−ρ el = 5.67×10 −7 . To predict the isotope shifts, we use equations (2), (3) and (5) to calculate the energy differences, applying the following relations, FIG. 4. Isotope shifts of the A 2 Π ←X 2 Σ + transition in MgF. The blue joined points are calculated as described in the text, and the red joined points are from our dataset. (a) The where the asterisks refer to the constants for 26 MgF. The constants ω e , ω e x e are taken from the work of Novikov and Gurvich 11 † . The remaining values are taken from tables II and III in the subsequent sections of this article. From the difference in vibrational constants we expect an isotope shift of −7.33 GHz, while the change in T e contributes +0.47 GHz. In Fig. 4 a and b, we plot the calculated δ ν i values for the four branches of the A 2 Π 1/2 ←X 2 Σ + and A 2 Π 3/2 ←X 2 Σ + transitions. We plot the predicted values up to N = 7, and compare with those available from our measurements. We observe a systematic difference between the measured and calculated values of 450 − 500 MHz; the mean of the differences is 470 MHz and their standard deviation is 30 MHz. Thus, while the rotational dependence of the isotope shift is well described by relations (9), the combined shift of the electronic and vibrational terms is not. An inaccuracy of 1.8 cm −1 in any one of the vibrational constants would be required to account for our observations, which seems unlikely. According to Hougen 25 , we should also include a term B L 2 ⊥ in the definition of E 0 , which accounts for the component of the electronic orbital angular momentum perpendicular to the internuclear axis. A non-zero value of L 2 ⊥ would increase the discrepancy, by up to 540 MHz. It is therefore likely that the † While Barber et al. 12 provided more accurate values for the ground state, it is only the difference in ground and excited state constants that matters.  13 . In the original article, the hyperfine structure parameter of the fluorine nucleus specific mass shift contribution to T e , the field shift of the Mg nucleus, or deviations from the Born-Oppenheimer approximation are responsible for the additional shift. These can only be derived by more sophisticated calculations, and our values provide an important benchmark in this regard.

B. Spectroscopic constants of the A 2 Π state
To determine the spin-orbit, rotational, spin-rotation, Λdoubling, and hyperfine constants of the A 2 Π state, we record 25 hyperfine-resolved rotational lines of the A 2 Π 1/2 ←X 2 Σ + and A 2 Π 3/2 ←X 2 Σ + transitions. The lines are slightly broadened by a small, uncompensated ambient magnetic field of 0.8 G in the LIF detector, and the effect of optical pumping between hyperfine and rotational states (discussed in section V); they are fitted using a sum of Lorentzian lineshapes. Using a Voigt profile did not change the fit residuals significantly. In this way, we determine the centers of 56 hyperfine lines for N ′′ ≤ 4, the observed frequencies of which are listed in Table VI of the Appendix. These line centers were used in a least-squares fit to determine spectroscopic parameters for the A 2 Π state, and we list the best fitted values in Table III. In our analysis, we fixed the ground state parameters to those determined by Anderson et al. 13 , and reproduce these in Table II for reference. The A 2 Π state is well approximated by a Hund's case (a) coupling scheme for the angular momenta. Relevant details regarding the Hamiltonian are provided in Appendix A. The Λ-splitting is determined by the linear combination p + 2q, and the hyperfine splittings are determined by the parameters a, b F + 2c/3 and d 26 † . To independently measure p and q, or b F and c, requires exciting to higher J levels where the Hund's case (a) approximation breaks down. These parameter pairs are otherwise strongly correlated when fitted separately and so we state the linear combinations in Table III. The same reasoning applies for the parameters A and γ. † Note that b F is related to b and c in equation 6.5 of Frosch and Foley 26 by III. Experimentally determined spectroscopic constants of the A 2 Π state of MgF and their standard deviation (SD). The Λdoubling parameters p and q, and the spin-orbit (A) and spin rotation (γ) constants are strongly correlated. We state their linear combinations p + 2q and A + γ that are well constrained by the fit.

Parameter
Value show hyperfine-resolved spectra of the P 1 /Q 12 (1) and Q 1 (0) lines of the A 2 Π 1/2 ←X 2 Σ + transition respectively. These lines originate from different rotational levels in the X 2 Σ + ground state and reach opposite parity levels in the same J ′ = 1/2 level of the excited state. The structure of the P 1 /Q 12 (1) line is dominated by the ground state fine and hyperfine interactions, and the excited state hyperfine interaction is not resolved. However, in the negative parity Λ-doublet, the excited state splitting is 179 MHz, and we resolve both the ground and excited state hyperfine structure in our spectra of the Q 1 (0) line (Fig. 5 b). This is caused by a dependence of the magnetic hyperfine interaction on the sign of Λ, and therefore a difference in the linear combinations of Λ states 26 . The magnetic hyperfine constant d(F) encapsulates this effect, and its influence on the hyperfine splittings is illustrated in the level scheme of Fig. 5 c.
The hyperfine parameters a, b F , c and d are related to properties of the electronic wavefunction in the molecule. A firstorder approximation was initially described by Frosch and Foley 26 and then subsequently simplified and corrected by Dousmanis 28 ‡ . The interaction between the electron and nuclear magnetic moment can be split into two parts: one part is sensitive to the electron density at the nucleus, contained in b F and the other is sensitive to the orbital and spin wavefunction away from the nucleus, which determines a, c and d. The coordinates of the electron relative to the interacting nucleus are expressed in the form (r 1 , χ), where r 1 is the distance from the nucleus and χ is the opening angle subtended with respect to the internuclear axis. According to Dousmanis § , ‡ In particular, we note the correction to the value of d, acknowledged by Frosch and Foley. § Here we use SI units, and our equations relate to the CGS units of Dousmanis by a factor µ 0 /(4π).
Here, µ(F) = 5.25µ N is the magnetic moment of the fluorine nucleus 29 , µ N is the nuclear magneton, and µ B is the Bohr magneton. The angled brackets denote expectation values of the A 2 Π electronic wavefunction, and ψ 2 (0) represents its probability density at the fluorine nucleus. From the value of a we find a typical radiusr = r −3 1 −1/3 = 0.88 Å, roughly half the internuclear equilibrium separation of 1.75 Å. Approximating sin 2 χ/r 3 1 ≈ sin 2 χ /r 3 leads to a typical valuẽ χ = 65.3°. A wavefunction uniformly distributed over χ has χ = 54.5°. These observations suggest that, much like in the ground electronic state, there is appreciable electron density between the two nuclei. This is in stark contrast with the other group II monofluorides CaF, SrF and BaF, where there is no resolvable interaction with the fluorine spin 20,30-33 , and upper limits of a few MHz have been inferred for d(F). It is only for the fermionic isotopologues of these molecules, where the metal nucleus has non-zero spin, that structure has been observed, and d( 87 Sr), d( 135 Ba), and d( 137 Ba) could be determined 34,35 . Finally, we note that the hyperfine interaction causes mixing between the X 2 Σ + (N = 3, F = 2) and X 2 Σ + (N = 1, F = 2) states, and between the A 2 Π(J = 1/2, F = 1) and A 2 Π(J = 3/2, F = 1) states. This mixing results in losses from the P 11 /Q 12 (1) optical cycling transition. For 24 MgF we calculate a branching ratio of 1.6 × 10 −6 for 24 MgF to the N = 3 states. For 25 MgF these losses are estimated to be an order of magnitude larger. A more detailed analysis of this effect is given in Appendix A.

D. Λ-doubling
In their discussion of the Λ-doubling in MgF, Walker and Richards derived values of p and q for MgF by extrapolation of the Λ-splitting observed at J ′ + 1/2 > 16 18 . Their analysis gives p + 2q = −50.3MHz, which disagrees with our result in both the sign of the interaction and the magnitude. This discrepancy is likely due to the omission of the spin-rotation interaction in their analysis of the 2 Σ + − 2 Σ + bands of MgF, and due to insufficient resolution in the A 2 Π ← X 2 Σ + absorption spectra. We here provide an improved measurement and update their discussion of the origin of Λ-doubling in the group II monofluorides. In Table IV, we compare measured values of p + 2q with the values obtained according to Van Vleck's pure precession approximation 36,37 . The approximation is valid when the Λ-doubling is dominated by the interaction with a single 2 Σ state, whose σ molecular orbital is In these studies, only p is reported, and q is assumed to be zero. mainly derived from an atomic p orbital 38 . Under this assumption, Here, l = 1 is the orbital angular momentum of the unpaired electron, and E Π − E Σ is the energy difference between the interacting states. Equations (11) apply for the interaction with a Σ + state, with the sign reversing when the interaction is with a Σ − state. In the table we give p vv + 2q vv values for pure precession with the nearby B 2 Σ + states which are higher in energy, and this appears to work well for the heavier monofluorides CaF, SrF and BaF. The trend is primarily due to the decreasing spin-orbit interaction moving up the group.
In the case of MgF, the interaction changes sign and is an order of magnitude smaller than expected from the interaction with the B 2 Σ + state. Therefore, Λ-doubling in MgF is more complex than for the heavier group II monofluorides and may comprise interactions with many Σ-states. Ab initio calculations by Kang et al. 1 imply that the B 2 Σ + state of MgF is of mixed character, which may explain the marked difference in Λ-doubling compared to the heavier group II monofluorides.
V. RADIATIVE LIFETIME OF THE A 2 Π ,v ′ = 0 LEVEL So far, the radiative lifetime of the A 2 Π state is known only theoretically 19 . We determine the lifetime experimentally by measuring the Q 1 (0) spectral line shape (Fig. 5 b) at low laser intensity. This line is convenient because the hyperfine structure in both ground and excited states is fully resolved, and the ground N = 0 rotational level contains the largest population of slow molecules for which the residual Doppler broadening is smallest. For these measurements, we reduced the expected Doppler broadening to 1 MHz by replacing the 2x2 mm 2 square molecular beam aperture in the detector with a 1 mm slit. We measure spectra at several probe laser intensities, and extract the full width at half-maximum (FWHM) for each line with a Lorentzian fitting function. (solid lines). Linear fits to the data show that each line broadens differently with increasing laser intensity. This broadening occurs at laser intensities well below the predicted twolevel saturation intensity, I s = πhcΓ/3λ 3 = 62 mW cm −2 , and is the result of optical pumping between rotational and hyperfine states of the molecule, which we discuss in the following paragraph.
In general, the spectrum of an open transition will broaden when the number of photon scattering events is sufficient to optically pump the molecule to a state not addressed by the laser. This broadening can occur at an arbitrarily low laser intensity I, provided that the interaction time t i is large enough. In the absence of hyperfine structure, molecules are pumped to N ′′ = 2 on the Q 1 (0) line after an average of three scattering events. With the inclusion of hyperfine structure, molecules are pumped to both N ′′ = 2 and also between hyperfine levels of N ′′ = 0, further reducing the number of scattering events. For our experiments where the typical laser interaction time is t i = 10 µs, this effect becomes significant even when I ∼ 10 −3 I s . To verify our understanding, we simulate the interaction with the laser using rate equations, the measured hyperfine splittings, and the branching ratios for each hyperfine decay channel. We assume all three polarization components are excited with equal probability; this is a reasonable approximation given the magnetic field in the detector mixes the ground states by spin precession during the interaction time with the laser. We fit the simulated spectra with the same Lorentzian model, and find that the model predicts the relative broadening rates within the experimental uncertainties. A complete quantitative treatment of this effect requires detailed information about the laser profile and the collection optics, which is beyond the requirements of this paper. We refer the interested reader to the work of Wall et al. 43 as an example.
To estimate an upper bound for the true Lorentzian linewidth Γ/(2π), where Γ = 1/τ 0 and τ 0 is the radiative lifetime of the A 2 Π ,v ′ = 0 level, we use the spectrum taken at a peak intensity of 80 µW cm −2 , shown in Fig. 5 b. Here the FWHM of the individual hyperfine lines are consistent within the uncertainties, indicating that the effect of optical pumping is small. We fit the data to a sum of three Voigt profiles, finding a Lorentzian FWHM of Γ/(2π) = 22.0(5)MHz and a Gaussian FWHM of 3.8(1.8)MHz. The ground state splitting can be estimated from our experimental spectra, and compared to the precise measurements of Anderson et al. to estimate an uncertainty on the linearity of the laser scan. We find agreement within ±1 MHz, consistent with the Yb measurements presented in Section III. The Gaussian contribution to the lineshape arises from the Doppler effect, residual Zeeman shifts and laser frequency instability. To estimate its systematic uncertainty, we measured the (3s 2 7s) 2 S 1/2 ← (3s 2 3p) 2 P 1/2 narrow transition in a buffer gas beam of atomic aluminum. The atoms are produced in the same beam machine, with a forward velocity similar to the MgF molecules. The probe-light at 225.8 nm is generated from the fourth harmonic of the same titanium sapphire laser, which increases the laser frequency noise by at least a factor two. The sensitivity to Doppler shifts arising from the shorter probe wavelength is increased by a factor 1.6 relative to the Q 1 (0) line in MgF. The Al transition has a natural linewidth of 2.5 MHz 44 , and using this value we arrive at an upper bound of the Gaussian FWHM contribution of 9.2(2) MHz at the Al detection wavelength. We therefore vary the Gaussian contribution in the final fit between 1 and 4.6 MHz, and find that this changes the fitted value of Γ/(2π) by at most 1 MHz. From this, we estimate a lower bound for the radiative lifetime of the A 2 Π ,v ′ = 0 state to be τ 0 = 1/Γ = (7.23 ± 0.16 stat ± 0.33 sys ) ns, in agreement with the theoretical prediction by Pelegrini et al. 19 of 7.16 ns. The uncertainty is dominated by the systematic uncertainty in the Gaussian contribution to the spectral lineshape.

VI. ELECTRIC DIPOLE MOMENT MEASUREMENTS
The application of an external electric field E in the detector introduces an additional term, to the Hamiltonian given in Equation (1). Here, µ i is the vector dipole moment operator in electronic state i. From the Stark splitting and shifting of the LIF spectra we can determine the magnitude of the dipole moments |µ A | and |µ X |. The Stark Hamiltonian (12) couples nearby molecular states of opposite parity having the same total angular momentum projection M F onto an axis parallel to E. To calculate the energies under an applied field, we diagonalize the Hamiltonian matrix constructed using the relevant Hund's case basis functions, including all levels with J ≤ 9/2. In the electronic ground state the Stark Hamiltonian mainly mixes states separated by one unit of the quantum number N ′′ . For the electric fields applied in this study of up to 10.6 kV/cm, the Stark shift is quadratic. For the excited states, the dominant interaction is between the closely spaced Λ-doublet levels. The interaction is strong compared to this splitting and results in a lin-ear Stark shift even for low electric field strengths. The Stark effect overcomes the hyperfine interaction at modest electric field strengths of about 0.1 kV/cm, after which the levels separate by their value of the angular momentum projection M J ′ . Fig. 6 a and b show spectra of the R 21 (0) and Q 1 (0) lines, respectively, under field-free conditions and with an electric field of 4.44 kV cm −1 applied in the detector. The total span of each spectrum is determined by the Stark effect of the excited state, and the small shift in the gravity center is due to the ground state Stark shift. With the laser polarization oriented perpendicular to the electric field, we excite and observe all four M J ′ components of the R 21 (0) lines in Fig. 6 a. We measure spectra at various applied fields and fit the modeled spectra to extract best fit values for the dipole moments, obtaining µ A = 3.20 ± 0.01 stat ± 0.22 sys D and µ X = 2.88 ± 0.03 stat ± 0.20 sys D. The systematic uncertainty is dominated by the determination of the electric field between the mesh electrodes. We assume a measurement uncertainty of 0.3 mm in the mesh separation of 9 mm, and with finite element modelling we estimate a reduction of 2% in the electric field strength relative to infinite plate electrodes of the same separation. The combination of these effects far exceeds the statistical uncertainty from the fitting procedure. Simulated spectra using the best fit values are shown inverted in Fig. 6 a and b, demonstrating good quantitative agreement with the measurements. The FWHM of the lines measured at low and high field are consistent within 1 MHz, from which we deduce that spatial inhomogeneity of the electric field across the probe beam is below 0.1%.
To examine the behavior of the energy levels in MgF in electric fields in more detail, we show the simulated Stark shifts for low-J levels at high, moderate and small electric field strengths in Fig. 6 c. Between 10 and 100 kV/cm (left panel), different rotational levels interact significantly; this leads to avoided crossings between the excited states, indicated by circles in the Fig.6. At intermediate fields up to 10 kV/cm (center panel), the Stark effect of the ground state becomes comparable to the excited state. At electric fields below 0.25 kV/cm (right panel), mixing in the ground state is negligible, whereas in the excited state the nearby Λ-states mix significantly.
The X 2 Σ + ground state dipole moment has been predicted theoretically 19,[45][46][47][48][49] . The values obtained by different methods range from 2.67 D 19 to 3.126 D 48 . We note that our measured value of µ X is in good agreement with the theoretical value obtained by Fowler and Sadlej 46 , µ X = 2.8611D, using the complete active space self-consistent field (CASSCF) method. Pelegrini et al. also calculated µ A = 4.23 D, which is in poor agreement with our experimental results.

VII. ELECTRIC FIELD INDUCED ROTATIONAL BRANCHING
The laser cooling scheme for MgF relies on the parity (P) and angular momentum (J) selection rules of electric dipole transitions. In zero electric field, optical cycling is possible on the P 1 /Q 12 (1) transition, which excites molecules from the N ′′ = 1 ground states to the J ′ = 1/2, P ′ = + excited states. However, a small electric field E results in an exchange of population, ε 2 , between excited states of opposite parity. The relative transition strength from N ′′ = 1 becomes 1 − ε 2 when exciting the states of mostly positive character, and these states decay to N ′′ = 0, 2 with probability ε 2 . In addition, excitation to the states with mostly negative parity character becomes weakly allowed, with a relative transition strength ε 2 . These states decay to N ′′ = 0, 2 with near unit probability. This second loss channel cannot be neglected, because efficient optical cycling requires frequency sidebands to address the ground state hyperfine splitting, causing the weak transitions to be driven near resonance. As a result, we estimate the loss probability from the cooling cycle due to uncon-trolled electric fields, to lowest order in ε, as P loss ≈ 2ε 2 .
In Fig. 7, we plot P loss versus the electric field strength for the two possible values of F ′ , using our measured spectroscopic constants. We fit the data below 5 V/cm to P loss = α F ′ |E| 2 , finding that α 0 = 4 × 10 −5 cm 2 /V 2 and α 1 = 2 × 10 −4 cm 2 /V 2 . For the P 1 /Q 12 (1) cooling transition, electric fields of 18 V cm −1 and 9 V cm −1 lead to a mixing of the parity eigenstates of about 1.4 %, for F ′ = 0 and 1, respectively. At this level, losses from the optical cycle due to parity mixing match losses to the v ′′ = 1 vibrational manifold in the X 2 Σ + state predicted by Ref. 19 . We note that such electric field induced losses have already been observed experimentally in SrF 50 and AlF 22 .

VIII. CONCLUSION
We recorded hyperfine-resolved CW LIF spectra of 25 low-J lines of the A 2 Π ←X 2 Σ + transition in MgF, and analyzed the 24 MgF isotopologue in detail. By fitting the eigenvalues of the effective Hamiltonian to the measured line positions, we  (2)  determined the spectroscopic parameters of the A 2 Π state that are relevant for laser cooling experiments: rotational, fine-and hyperfine structure constants. We calibrated our wavemeter using the precisely known Yb (6s6p) 1 P 1 ← (6s 2 ) 1 S 0 transition frequencies 24 , and correct a −4.1 GHz systematic error in the line frequencies presented by Xu et al. 15 . Transition isotope shifts between the 24 MgF and 26 MgF isotopologues were recorded, and we observe an unexplained 470 MHz transition frequency shift which may indicate deviations from the Born-Oppenheimer approximation. We studied the broadening of the hyperfine lines due to optical pumping and recorded highresolution spectra of the Q 1 (0) line to determine the radiative lifetime of the A 2 Π , v ′ = 0 level to be τ 0 = 7.23(36)ns. By studying the fluorescence spectra under an applied electric field, we experimentally determined the dipole moments of the X 2 Σ + and A 2 Π states. Our value for the ground state, µ X = 2.88 ± 0.03 stat ± 0.20 sys D is in good agreement with the value predicted by ab initio calculations [45][46][47][48][49] . Using our value of µ A = 3.20±0.01 stat ±0.22 sys D, we predict the electric field strength at which parity-mixing in the excited states limits optical cycling on the P 1 /Q 12 (1) line. We find that 9 V cm −1 is sufficient for unwanted rotational branching to match the expected vibrational branching. To scatter more than 10 4 photons, stray electric fields have to be controlled to below the 1 V cm −1 level. Coincidentally, the hyperfine structure in the J ′ = 1/2, P ′ = +1 level is less than 1 MHz, which simplifies the laser cooling scheme significantly. On the contrary, the hyperfine splitting in the J ′ = 1/2, P ′ = −1, level is large, which increases the separation between opposite parity hyperfine levels. This reduces the sensitivity of the optical cycling scheme to stray electric fields substantially.
There are a number of notable differences between MgF and the heavier group II monofluorides CaF, SrF and BaF. First, the sign of the Λ-splitting is inverted in MgF, and its magnitude is about 100 times smaller. Second, MgF has the largest dipole moment in the A 2 Π state, whereas it has the smallest dipole moment in its ground state. Third, the interaction of the electronic angular momentum with the fluorine nuclear spin leads to a resolvable hyperfine structure in the excited state of MgF, and from our measured hyperfine constants we can infer that the electronic wavefunction has significant probability density between the nuclei. This supports the conclusions of Anderson et al. 13 regarding the greater covalency of the chemical bonding in MgF.
Finally, we note that our measurements form a stringent set of benchmarks for precise quantum chemical calculations on MgF. The hyperfine constants and dipole moment measurements are strong benchmarks for molecular orbital calculations of the X 2 Σ + and A 2 Π states, while the transition isotope shifts presented constrain vibrational constants and deviations from the Born-Oppenheimer approximation. be within subspaces with the same P and F due to parity and angular momentum selection rules. The pure spin-orbit and rotational Hamiltonian, H Ω i ,Ω k , has diagonal elements, (A2) Each F = 0, P subspace contains a single state and these do not mix. There are three states in a |Ω, J, P = ±, F = 1 subspace, and we order these states as |Ω, J = |1/2, 1/2 , |1/2, 3/2 , |3/2, 3/2 . The combined Hamiltonian H(F = 1, P = ±), which includes the hyperfine and Λ-doubling interactions, is given by, Here,b = b F + 2c/3, ≈ b = b F − c/3,p = p + 2q, and h.c. refers to the Hermitian conjugate. By applying perturbation theory to the Hamiltonian (A3), mixing of the |Ω = 1/2, J = 1/2 and |Ω = 1/2, J = 3/2 states due to the hyperfine interaction can be calculated. To second order, the population mixing ζ is, A4) The resulting loss from the cycling transition due to hyperfine mixing in the excited state, η A is, η A = S(P 12 ) S(P 12 ) + S(R 12 ) + S(Q 1 ) ζ (A5) Here, S(L) is the Hönl-London factor for decay from Ω = 1/2, J = 3/2 by transition L, and we note that only the P 12 path results in loss from the N = 1 ground states. Using our spectroscopic parameters, the loss probability from F = 1 is η A = 1.2 × 10 −6 . Loss can also occur via mixing with the |Ω = 3/2, J = 3/2 states, but this is suppressed by about a factor (A/B) 2 and we neglect this term. A similar calculation for the ground state can be used to calculate the hyperfine mixing between N ′′ = 1 and N ′′ = 3, and results in a total loss η X ≈ 0.4 × 10 −6 from the cooling cycle. Therefore, the total loss probability from the |Ω = 1/2, J = 1/2, P = +1, F = 1 excited states from hyperfine mixing is η = 1.6 × 10 −6 . To complete our discussion, we consider now the subspaces with F ≥ 2, each of which contains four states. For F = 2, P = ±, the combined Hamiltonian matrix is where the ordering of states is |Ω, J = |1/2, 3/2 , |1/2, 5/2 , |3/2, 3/2 , |3/2, 5/2 . The eigenenergies of equation (A3) and (A6) are, to first order, the diagonals of the matrices, and can be used to accurately determine a,b, d,p. The pa-