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UV frequency metrology on CO (a3Pi); isotope effects and sensitivity to a variation of the proton-to-electron mass ratio

UV frequency metrology has been performed on the a3Pi - X1Sigma+ (0,0) band of various isotopologues of CO using a frequency-quadrupled injection-seeded narrow-band pulsed Titanium:Sapphire laser referenced to a frequency comb laser. The band origin is determined with an accuracy of 5 MHz (delta \nu / \nu = 3 * 10^-9), while the energy differences between rotational levels in the a3Pi state are determined with an accuracy of 500 kHz. From these measurements, in combination with previously published radiofrequency and microwave data, a new set of molecular constants is obtained that describes the level structure of the a3Pi state of 12C16O and 13C16O with improved accuracy. Transitions in the different isotopologues are well reproduced by scaling the molecular constants of 12C16O via the common mass-scaling rules. Only the value of the band origin could not be scaled, indicative of a breakdown of the Born-Oppenheimer approximation. Our analysis confirms the extreme sensitivity of two-photon microwave transitions between nearly-degenerate rotational levels of different Omega-manifolds for probing a possible variation of the proton-to-electron mass ratio, \mu=m_p/m_e, on a laboratory time scale.

Recently, Bethlem and Ubachs [12] identified metastable CO as a probe for detecting a temporal variation of the proton-to-electron mass ratio, µ = m p /m e , on a laboratory time scale. Two-photon microwave transitions between nearly-degenerate rotational levels in different Π Ω spin-orbit manifolds were shown to be very sensitive to a possible variation of µ. As a measure of the inherent sensitivity of a transition to a drifting µ, the sensitivity coefficient, K µ , is defined via: Transitions between the J = 8, Ω = 0, the J = 6, Ω = 1 and the J = 4, Ω = 2 levels display sensitivities ranging from K µ = −300 to +200 [12]. For an overview on the topic of varying physical constants, we refer the reader to [13] and [14].
In this paper, we present high-precision UV measurements of the a 3 Π -X 1 Σ + (0,0) band in CO. In total 38 transitions in all six naturally occurring isotopes have been measured with MHz accuracy. All three Ωmanifolds have been probed, with J up to eight. A com-prehensive fit of the optical data combined with previously published rf [3] and mw [5,6] measurements was performed. The molecular constants found for 12 C 16 O are mass scaled and compared with the measured transitions in other isotopologues.

II. LEVEL STRUCTURE OF CO
The a 3 Π state is the first electronically excited state of CO, lying 6 eV above the X 1 Σ + ground state. CO in the a 3 Π state has two unpaired electrons, leading to a nonzero electronic spin, S, and orbital angular momentum L. For low rotational levels, the a 3 Π state is best described in a Hund's case (a) coupling scheme, with the good quantum numbers Λ and Σ, the projection of L and S on the molecular axis, respectively. The projections of the total angular momentum J on the molecular axis and on the space-fixed axis lead to the good quantum numbers Ω and M . The basis functions are |nΛ |v |SΣ |JΩM , representing the electronic orbital, vibrational, electronic spin and rotational components of the wave function, respectively. For higher rotational levels, the spin decouples from the electronic angular momentum and a Hund's case (b) coupling scheme becomes more appropriate. In Hund's case (b), the different Ωmanifolds are mixed.
The energies of the lower rotational levels of the a 3 Π and X 1 Σ + states are shown in lambda-doublet components can only be reached via P or R transitions.
The separation of the electronic motion and nuclear motion is not exact, leading to a splitting into lambdadoublet states of opposite parity, as indicated, not to scale, in Fig. 1. The lambda doubling in the Ω = 0 state is large and relatively independent of J. The lambda doubling in the Ω = 1 and Ω = 2 manifolds is much smaller. In Fig. 1 the total parity, i.e., the product of the symmetries of the rotational and electronic parts of the wavefunction, is indicated by the -and + signs. The electronic part of the wave function of the upper (lower) lambda-doublet levels has f (e) symmetry.
The spin-forbidden a 3 Π -X 1 Σ + system becomes weakly allowed due to spin-orbit mixing of singlet electronic character into a 3 Π, most significantly of the A 1 Π state lying 2 eV above the a 3 Π state. As the A 1 Π state consists of a single, Ω = 1, manifold, it only couples to the Ω = 1 levels in a 3 Π. Transitions to the Ω = 0, J > 0 and Ω = 2 manifolds become weakly allowed by mixing of the different Ω-manifolds. The Ω = 0, J = 0 level is not mixed with the other Ω-manifolds, hence, the transition to this level, i.e. the P 1 (1), does not obtain transition strength via coupling to the A 1 Π state. The a 3 Π (v = 0) state can only decay to the ground state, hence, the radiative lifetimes of the different rotational are inversely proportional to the transition strengths. The lifetimes are thus strongly dependent on J and Ω. For example the J = 2, Ω = 2 level has a lifetime of 140 ms, whereas the J = 1, Ω = 1 level has a lifetime of 2.6 ms [15].
Isotopes with an odd number of nucleons have a nonzero nuclear spin that leads to hyperfine structure. The two relevant odd-nucleon-number nuclei are 13 C and 17 O, with nuclear spin I = 1/2 and I = 5/2, respectively. Due to its zero electronic angular and spin momentum the hyperfine splitting in the ground-state of CO is small (≈50 kHz) [16]. The hyperfine splittings in the a 3 Π state vary between 30 and 500 MHz for the measured transitions.

III. EXPERIMENTAL SETUP
The molecular beam setup used for frequency metrology on CO is schematically depicted in Fig. 2. A pulsed beam of CO is produced by expanding CO gas into vacuum, using a solenoid valve (General Valve series 9). A backing pressure of 2 bar was used for recording transi-tions at low J, while for recording transitions at higher J, the backing pressure was reduced to 0.5 bar. For recording transitions in 13 C 16 O, isotopically enriched CO (Linde Gas) was used. The enriched sample also contained slightly enhanced fractions of 13 C 18 O and 13 C 17 O, sufficient for obtaining signals. Spectra of 12 C 17 O and 12 C 18 O were measured using a natural CO sample.
After passing through a 1 mm skimmer, the molecular beam is crossed at right angles with laser radiation tunable near 206 nm. In the interaction region, a magnetic field of up to 200 Gauss can be applied by two coils in Helmholtz configuration. After being excited to the a 3 Π state, the molecules fly 60 cm downstream before being detected by an Electron Multiplier Tube (EMT). The resulting signal is recorded using a digital oscilloscope and the integrated signal is stored. The absolute detection efficiency of this method for metastable CO (6 eV internal energy) is estimated to be on the order of 10 −3 [17]. Note that the time of flight (750 µs) is short compared to the lifetime of the metastable state (>2.6 ms). An adjustable slit is mounted in front of the EMT to limit the divergence of the beam that reaches the detector, thereby limiting the Doppler width of the recorded transitions.
The spectroscopic measurements are performed with a narrow-band frequency-quadrupled titanium:sapphire (Ti:Sa) pulsed laser described in detail by Hannemann et al. [18,19]. A schematic drawing of the laser setup is shown in Fig. 3. A continuous wave (CW) Ti:Sa ring laser (Coherent 899) produces around 700 mW of laser power tunable near 824 nm. Its output is split into three parts, with all parts having approximately the same power. One part of the light is sent as a seed frequency to a pulsed Ti:Sa ring oscillator that is pumped at 10 Hz with 50 mJ of pulsed 532 nm light from an injection seeded Nd:YAG laser (Spectra Physics LAB-170). The oscillator is locked to the CW seed light using a Hänsch-Couillaud scheme. The pulsed IR light emanating from the oscillator is amplified in a bow-tie multi-pass Ti:Sa amplifier pumped with 300 mJ of pulsed 532 nm light from the same Nd:YAG laser that pumps the oscillator. After nine passes the laser power of the IR-beam is around 70 mJ in a 100 ns pulse. These pulses are then frequency doubled twice in two consecutive BBO crystals, resulting in pulses of 20 µJ at 206 nm.
In order to determine the absolute frequency of the CW Ti:Sa ring laser, the CW light is mixed with the light from an erbium-doped fiber frequency-comb laser (Menlo systems MComb at 250 MHz repetition frequency) that is locked to a global positioning system (GPS) disciplined Rb-clock standard. The optical interference beat signal is measured with a photodiode and an Agilent 53132A counter. The obtained beat frequency is then transferred via ethernet to the central computer at which the data is analyzed. Further details on the absolute frequency calibration can be found in Sec. V D.
By making a small portion of the pulsed light interfere with part of the CW light, possible small differences between the frequencies of the CW seed laser and the A ring laser is used to injection seed an oscillator cavity, which is pumped by a Nd:YAG laser at 10 Hz. The produced IR pulses are amplified in a multi-pass amplifier and subsequently quadrupled in two consecutive BBO crystals. The absolute frequency of the CW seed laser is determined using a fiber comb laser. OI: optical isolator; GP: glass plate; OC: output coupler; HR: high reflective mirror; PA: Piezo Actuator; HC-lock: Hänsch-Couillaud lock; SMF: single mode fiber; AOM: acousto-optical modulator.
central frequency of the pulsed output of the bow-tie amplifier are measured and corrected for [18]. In order to have a good fringe visibility, and to ascertain that the full wavefront of the pulsed output is mapped onto the CW reference beam, both beams are sent through a short single-mode fiber. The beat pattern is detected using a fast photodiode in combination with an oscilloscope and analyzed online.
The UV laser beam is split into two parts and sent through the molecular beam machine from opposite sides to limit the Doppler shift due to a possible imperfect perpendicular alignment of the laser beam. In order to ensure that the two beams are perfectly counterpropagating, the two laser beams are recombined after passing through the machine, forming a Sagnac interferome- ter. The paths through the molecular beam machine are aligned such that the two beams interfere destructively at the exit port (a dark fringe). The transition frequency is measured twice, using either the laser beam from the left or from the right hand side [20].

IV. EXPERIMENTAL RESULTS
In Table I the frequencies are listed for the measured transitions in the a 3 Π -X 1 Σ + (0,0) band. In view of the time-consuming measurement procedure, only a selection of transitions has been investigated. In total, 38 transitions have been recorded in all six stable isotopologues of CO. In 12 C 16 O, 13 C 16 O and 13 C 18 O, we have recorded low J transitions to each Ω-manifold and each parity level, to be able to give a full analysis of the level structure. Also, transitions to the J = 8, Ω = 0, J = 6, Ω = 1 and J = 4, Ω = 2, the near degenerate levels of interest to the search for µ variation, were measured. Furthermore, the R 2 (0) transition, the most intense line under our conditions, has been measured in the 12 C 17 O, 12 C 18 O and 13 C 17 O isotopologues. Note that the observed signal strengths for the different transitions varies over more than three orders in magnitude.
In Fig. 4 a typical recording of the R 2 (0) transition in 12 (7) transition and the R1(7) transition in 12 C 16 O. Both transitions originate from the same ground state level and connect to one of two nearly degenerate levels. Therefore, the combination difference of these two transition frequencies corresponds to the frequency of the transition between the two nearly-degenerate levels.
each pulse of the pulsed laser, the frequency of the CW laser is determined using the fiber comb laser and a possible shift between the pulsed laser and the CW laser is determined using the online CW-pulse offset detection. Subsequently, these data are combined with the metastable CO signal from the EMT. Typically, the recorded scans are not perfectly linear, resulting in an uneven distribution of the data points along the frequencyaxis. This has no influence on the peak determination. At the peak of the transition, the observed signal corresponds to typically a few thousand detected metastable CO molecules per laser pulse. Note that the fluctuations in the signal shown in Fig. 4 are due to pulse-to-pulse variations of the molecular beam and the UV power and not due to counting (Poisson) statistics. The solid lines in the figure show a Gaussian fit to the spectra. The transition frequency of the R 2 (0) transition is determined by taking the average of the measurements taken with the laser beam propagating from either side.
Transitions to the six near-degenerate levels in both 12 C 16 O and 13 C 16 O were measured. For 13 C 18 O transitions to four of the six near degenerate levels were obtained. In Fig. 5 a recording of the P 2 (7) and R 1 (7) transitions in 12 C 16 O is shown. Both transitions originate from the J = 7 ground state level, thus the combination difference is equal to the frequency of the J = 6, Ω = 1, + → J = 8, Ω = 0, + transition, which is measured to be 3256.6 MHz.
As the transition strengths of transitions in the spin forbidden a 3 Π -X 1 Σ + system originate from mixing of , the transition strengths of the different transitions are proportional to (the square of) the Ω = 1 character of the final rotational level and the Hönl-London factor. From this, we expect the P 2 (7) to be 3.2 times more intense than the R 1 (7). Experimentally, we find a ratio of 2.4 to 1. The deviation is explained by the fact that the J = 8, Ω = 0 has a longer lifetime than the J = 6, Ω = 1 (15.7 ms vs. 3.5 ms), and consequently, a smaller fraction of the metastable molecules decays back to the ground state before reaching the EMT. Taking the lifetime into account, we expect a ratio of 2.7 to 1, in reasonable agreement with the experiment.
The J = 0, Ω = 0 is the only J = 0 level, and is therefore not mixed with the Ω = 1 and Ω = 2 manifolds. Consequently, the P 1 (1) transition, connecting the X 1 Σ + J = 1 with the a 3 Π Ω = 0, J = 0, + parity level does not obtain any transition strength from coupling to the A 1 Π state. Nevertheless, we were able to observe this transition, shown in Fig. 6, albeit with low signal to noise. From our measurements, we estimate that the P 1 (1) transition is about 65 times weaker than Q 1 (1) and about 10 4 times weaker than Q 2 (1), resulting in a lifetime of 8(1) s [21]. The transition strength is ascribed to mixing of the a 3 Π state with a 1 Σ + state, most likely the X 1 Σ + ground state [22].
The rotational levels of isotopologues with one or two odd-numbered nuclei show hyperfine splitting. In Fig. 7 a recorded spectrum of the Q 2 (1) transition in 13 C 16 O is shown. The transitions to the hyperfine sublevels F = J − I = 1/2 and F = J + I = 3/2 of the excited state are clearly resolved.

V. MEASUREMENT UNCERTAINTIES
In this paragraph the main sources of uncertainty in the transition frequencies are discussed. The uncertainty budget is summarized in Table II.

A. Zeeman effect
The ground state of CO is a 1 Σ state, hence, its Zeeman shift is small. The a 3 Π state on the other hand has both electronic angular momentum and spin, and experiences a considerable Zeeman shift. In the Ω = 2 state the effect due to the electronic angular momentum and due to spin will add up, whereas in the Ω = 0 and Ω = 1 state these effects will partly cancel. Hence, we expect the largest Zeeman shift to occur in the Ω = 2 levels, in particular in the J = 2, Ω = 2 level.
In Fig. 8 the recorded spectra are shown for the R 1 (1), R 2 (1) and R 3 (1) transitions in 12 C 16 O in a magnetic field of 170 Gauss. In these measurements, the polarization of the 206 nm light is parallel to the applied magnetic field, hence, only ∆M J = 0 transitions are allowed. As all three measured transitions originate from the light that has been quadrupled in non-linear crystals. In the earth magnetic field, approximately 0.5 Gauss, the J = 2, Ω = 1, M J = −2 and M J = 2, will be shifted in opposite directions by less than 1 MHz. As the polarization is nearly perfect, the shift of the line center is estimated to be less than 1 kHz.

B. Stark effect
The Stark shift in the ground state of CO is negligibly small as the dipole moment is only 0.1 Debye (corresponding to 0.17 · 10 −2 cm −1 /(kV/cm)) and mixing occurs between rotational levels. The Stark shift in the a 3 Π state on the other hand is larger as the dipole moment is 1.37 Debye and mixing occurs between the lambda-doublet components [23]. We estimate that the electric field in the excitation region is below 1 V/cm. This corresponds to a Stark shift of 30 kHz for the J = 2, Ω = 2, M Ω = 4 and less for all other levels.

C. AC-Stark effect
A priori, the AC-Stark shift is difficult to estimate. We have measured the R 2 (0) transition in 12 C 16 O for different laser powers, reducing the laser power by over an order of magnitude, but found no significant dependence of the transition frequency on laser power. Thus, we estimate the AC-Stark shift to be less than 100 kHz.

D. Uncertainty in absolute frequency determination
The absolute frequency of the CW light is calibrated by mixing this light with the output of a frequency-comb laser and counting the resulting beat frequency f beat . The frequency f CW is then obtained by the relation with n the mode number of the frequency comb that is nearest to the frequency of the CW light, and f rep and f 0 the repetition and the carrier-envelope offset frequencies of the frequency comb, respectively. f rep is tunable over a small range around 250 MHz and f 0 is locked at 40 MHz. We infer that the sign of f beat is positive from the observation that f beat increases when the CW laser is scanned towards higher frequency. Likewise, the sign of f 0 is positive from the observation that f beat increases when f 0 is decreased. The beat note is averaged over a period of 100 ms. On this time scale, the accuracy of the Rb-clock standard is 10 −10 , equivalent to 150 kHz at the used frequencies. This uncertainty enters separately in each data point taken in a frequency scan. As each scan is approximately 1000 data points, this uncertainty averages out. The integer mode number n is determined by measuring the transition frequency of the R 2 (0) transition in 12 C 16 O using the frequency comb at different repetition frequencies and then determining at which transition frequency these three measurements coincide. This method gives an unambiguous transition frequency provided that the change in repetition frequency is much larger than the measurement uncertainty [24]. In our case this condition is well met as the uncertainty between two consecutive measurements is on the order of 500 kHz, while the repetition frequency of the comb is varied by 2 MHz. The absolute value obtained for the R 2 (0) transition of 12 C 16 O is used to calibrate a wavelength meter (Burleigh WA-1500, 30 MHz precision) on a daily basis. This wavelength meter is then used to determine the mode number for the measurements of the other transitions.

E. Doppler effect
In a molecular beam experiment, the first-order Doppler effect is reduced by aligning the laser-beam perpendicular to the molecular beam. Two residual effects remain: (i) The finite transverse temperature of the molecular beam leads to a broadening of the transition, (ii) A possible imperfect perpendicular alignment of the laser beam leads to a shift of the center frequency.
To limit the Doppler width, we have placed a variable slit in front of the EMT, which limits the divergence of the beam hitting the detector. In Fig. 9 recordings of the Q 2 (1) transition of 12 C 16 O using a slit width of 10 mm and 3 mm are shown. When the slit width is reduced below 3 mm the width of the transition remains the same while the signal decreases further. The minimum full width at half maximum (FWHM) observed is 23 MHz, which is attributed to the line width of the UV-radiation. In our measurements, we have used a slit width of 6 mm, as a compromise between signal intensity and line width.
In order to eliminate the Doppler shift in the measurement, the UV laser beam is aligned in the geometry of a Sagnac interferometer. The angle between the two counterpropagating laser beams may be estimated to be smaller than λ/d, with λ being the wavelength of the light in nm and d being the diameter of the laser beam in mm [20]. In our case this results in a maximum Doppler shift of 200 kHz. We have verified this by comparing measurements of the R 2 (0) transition of 12 C 16 O in a pure beam of CO (velocity of 800 m/s) and a beam of 5% CO seeded in He (longitudinal velocity of 1500 m/s). The second-order Doppler shift is sub-kHz. The recoil shift is ≈ 25 kHz.

F. Uncertainty in peak determination
The number of metastable molecules detected (at the peak of) the R 2 (0) transition of 12 C 16 O, the strongest transition observed, is on the order of 10 4 per laser pulse, while at the P 1 (1) transition of 12 C 16 O, the weakest transition observed, it is only a few molecules per laser pulse. The ability to determine the line center from the measurement is limited by pulse-to-pulse variations of the molecular beam and the UV power. Typically, the un-certainty in the peak determination is 300 kHz (corresponding to 1% of the line-width).

G. Frequency chirp in the pulsed laser system
The frequency of the pulsed IR light differs slightly from the frequency of the CW seed laser mainly due to two effects: (i) Mode pulling in the oscillator and (ii) Frequency chirp and shift in the amplifier. The pump laser induces a change of the refractive index of the Ti:Sa crystal and therefore a variation of the optical path length of the cavity on a time scale much shorter than the response time of the electronic system used for locking the cavity. Therefore, at the instant the pulse is produced the cavity resonances are shifted from the frequency they are locked to. The size of the frequency shift due to mode pulling depends on the pump power. The optical properties of the Ti:Sa crystal in the bow-tie amplifier change depending on the population inversion, which decreases during the multi-step amplification of the IR pulse. The combined effect of both phenomena results in a shift on the order of a few MHz. We measure and compensate for these effects by making a part of the pulsed light interfere with part of the CW light. The CW-pulse offset can be determined within 200 kHz. However, as a result of wavefront distortions of the pulsed beam, it depends very critically on the alignment. We have measured the chirp at different positions in the wavefront by moving a small pinhole through the pulsed beam, and observed deviations of a few MHz. These deviations are amplified in the harmonic generation stages. As it is unclear which part of the pulsed laser beam gives rise to the observed metastable CO signal, we cannot compensate for this effect. Consequently, we find rather large day-to-day deviations in the measurements; measured transition frequencies taken on the same day, with a specific alignment and setting of the Ti:Sa oscillator and bow-tie amplifier, agree within a 500 kHz. Measured transition frequencies taken on different days, on the other hand, may deviate by a few MHz. In order to compensate for this systematic effect, we chose the R 2 (0) transition in 12 C 16 O as an anchor for each measurement session. With this anchoring, measurements of any given transition (other than the R 2 (0) transition) taken on different days agree within 500 kHz, showing that the consistency of the positions of the rotational levels of a 3 Π is sub-MHz. The relatively large systematic uncertainty due to mode pulling and chirp will only enter in the value of the band origin. The root-mean-square deviation of 14 measurements of the 12 C 16 O R 2 (0) transition is about 2 MHz. We have set the uncertainty of the R 2 (0) transition, and hence the systematic uncertainty on all transitions, conservatively at 5 MHz. Note that Salumbides et al. [25] have circumvented this systematic offset of the absolute calibration by putting a small pinhole in the pulsed beam, therewith selecting a portion of the laser beam with smaller wavefront distortions. We could not apply this method here, The effective Hamiltonian for a 3 Π state has been derived by several authors [10,26]. We have used the effective Hamiltonian from Field et al. [10] with the additions and corrections discussed by Carballo et al. [5]. The matrix elements for this effective Hamiltonian are listed in Table III. As discussed by Carballo et al. [5], this Hamiltonian is equivalent to the effective Hamiltonian derived by Brown and Merer [26], but the molecular constants used in these Hamiltonians have a slightly different physical meaning, which will have consequences for the mass scaling discussed in Sec. VI B. In Table IV,  A least-squares fitting routine was written in Mathematica to obtain the molecular constants of the effective Hamiltonian. We have verified that our fitting routine exactly reproduces the results of Carballo et al. [5] and that it is consistent with PGopher [27]. For 12 C 16 O, we have fitted our optical data simultaneously with lambdadoubling transition frequencies in the rf domain measured by Wicke et al. [3] and the rotational transition frequencies in the mw domain measured by Carballo et al. [5] and Wada and Kanamori [6]. The fitted set consists of 9 rf transitions, 28 mw transitions and 14 optical transitions. The different data sets were given a weight of one over the square of the measurement uncertainties, taken as 50 kHz for the mw and rf data and 1 MHz for the optical data. The molecular constants for the X 1 Σ + state of 12 C 16 O are taken from Winnewisser et al. [28]. As discussed by Carballo et al. [5], γ and A J can not be determined simultaneously from data of a single isotopologue, therefore γ was fixed to zero. The first column of Table V lists the different constants obtained from our  fit for 12 C 16 O. The deviations between the observed and fitted transition frequencies are listed in Table I. All measured transitions could be fitted to approximately their respective uncertainties. However, the total root mean square (rms) of the residuals of the rf and mw data is significantly increased when the optical data is included; Carballo reported rms residuals of 27 kHz, while we find rms residuals of 52 kHz. The rms of the residuals of the fitted 12 C 16 O optical transitions is equal to 3.3 MHz.
The molecular constants found from the fit to the combined data agree well with the constants found from a fit to the rf and mw data alone, but with largely decreased uncertainties. The uncertainty of A, C and C δ , which are poorly constrained by the rotational and lambdadoubling transitions alone, are reduced by more than a factor of 10. Somewhat unexpectedly, the uncertainty of several other constants, including B, are also substantially decreased by the fit to the combined data. This can be understood from the fact that the mw data does not constrain B, but rather a combination of A and B. This also explains why the uncertainty of B as obtained from a fit to the rotational and lambda-doubling transitions is ∼300 kHz, whereas the rotational transitions have a quoted uncertainty of 5-8 kHz and are fitted with an rms uncertainty of 27 kHz [5]. Our optical data directly probe A, and the more precise value of A results in turn in a more precise value of B. Adding the optical data results in an uncertainty in B of 30 kHz, much closer to the value one would expect from the precision of the recorded mw transitions. Altogether, the new set of constants is more balanced and adequately describes the a 3 Π state.
As seen from Table I, the residuals of the optical transitions probing the Ω = 1 manifold are larger than those to the Ω = 0 and Ω = 2 manifolds, which is surprising as these transitions are the strongest transitions in the spectra and are measured with a higher signal-to-noise ratio than the other transitions. We have investigated whether this might be explained by perturbations arising from the a 3 Σ + or D 1 ∆ state. When perturbations with the a 3 Σ + state were included, using the perturbation parameters from Carballo et al. [5], the residuals decreased marginally. A slight improvement was obtained by including a perturbation with the D 1 ∆ state. However, as the perturbation parameters of the coupling between the a 3 Π and D 1 ∆ states are unknown, it is unclear if this improvement is genuine.
A similar analysis has been performed for 13 C 16 O. We have included the hyperfine interaction in the effective Hamiltonian following Brown et al. [29]. Only terms that are diagonal in J were included, since contributions from off-diagonal terms are estimated to be smaller than 100 kHz [11]. We have fitted our optical data simultaneously with lambda-doublet transition frequencies measured by Gammon et al. [11] and the rotational transition frequencies measured by Saykally et al. [4]. The fitted set consisted of 19 rf transitions, 4 mw transitions and 24 optical transitions. The molecular constants for the X 1 Σ + are taken from Klapper et al. [16]. The molecular constants resulting from the fit are listed in Table V. The difference between the observed transition frequencies and the frequencies from the fit are listed in Table I. The rms of the residuals of the fitted 13 C 16 O optical transitions is equal to 0.9 MHz.
For 13 C 18 O, as for the other isotopologues, no previous measurements on the a 3 Π were found in the literature, except for four mw-transitions in 12 C 18 O [4]. Hence, no fit has been attempted.

B. Mass scaling
An important motivation for this work was to validate the mass scaling of the energy levels of the a 3 Π state, and to confirm the sensitivity to a possible variation of the proton-to-electron mass ratio for a selected number of level spittings. In the literature, the reduced mass of the molecule is frequently denoted by the symbol µ.
In this paper, we will use µ to denote the proton-toelectron mass ratio, and will denote the reduced mass of the molecule by µ red . As we will see in Sec. VI C, µ red is linearly proportional to µ which is defined for the various isotopologues x C y O as The molecular constants as determined from the fits are effective molecular constants for the v = 0 level of the a 3 Π state. In general, an effective molecular constant X e,v can be expressed as Note that for the constants A and B the second term of Eq. (4) has a minus sign by convention [7]. Thus, the mass dependence of every constant consists of the mass dependence of X e and a correction due to the vibrational dependence of X e,v . The second column of Table V lists the dependence of the molecular constants, X e , on the reduced mass of the molecule, µ red . The effective molecular constant, X e,v , of an isotopologue with a reduced mass µ red then becomes where X e , α X,1 and α X,2 are the constants for the isotopologue with reduced mass µ red and Note that we use µ in Eq. (6) rather than µ red , see below.
As we have only measured transitions in the v = 0 band, the vibrational dependencies of the molecular constants cannot be extracted from our data. Hence, we have used the ratios between X e , α X,1 and α X,2 determined by Havenith et al. [7] to scale our constants. The molecular constants for 13 C 16 O and 13 C 18 O, found by scaling the constants of 12 C 16 O via the outlined procedure, are listed in the third and fourth column of Table V, respectively. For 13 C 16 O, the value of the band origin and the hyperfine constants were determined by fitting the data while the other constants were fixed at the scaled values. For 13 C 18 O, the hyperfine constants were taken to be identical to those of 13 C 16 O. The data for the X 1 Σ + state was calculated from Puzzarini et al. [30] and only the value of the band origin was fitted.  between the observed transition frequencies and the frequencies calculated using the scaled molecular constants are listed in Table I. As is seen the correspondence is satisfactory. The rms of the residuals of the 13 C 16 O and 13 C 18 O data with the frequencies found by scaling the molecular constants is equal to 5.1 MHz and 8.3 MHz, respectively. Until now the value of the band origin, E Π , was treated as a free parameter without considering its proper mass scaling. The value of the band origin consists of: (i) A pure electronic part, that scales as (µ red ) 0 except for a small correction due to the finite mass of the nuclei, known as the normal mass shift, or Bohr-shift, which is proportional to the reduced mass of the nuclei-electron system. (ii) A vibronic part that can be expanded in a power series of (v + 1/2) and contains the difference in zero-point energies in the a 3 Π and X 1 Σ + states. (iii) A rotational part, equal to B − D, that was absorbed in the value of the band origin in our definition of the effective Hamiltonian. (iv) The specific mass shift, dependent on the electron correlation function. (v) Nuclear-size effects, dependent on the probability density function of the electrons at the nucleus.
In the upper panel of Fig. 10, the derived values of the band origin of the a 3 Π state of the six stable isotopologues of CO are plotted as a function of the reduced mass. The solid line shows how the value of the band origin scales when effects (i-iii), which are expected to be dominant, are included. The used formulas for the normal mass shift, the vibrational and the rotational parts are: with where ∆E Π is the shift in the value of the band origin as a function of the reduced mass, µ red , with respect to a given isotopologue with reduced mass µ red , band origin E Π0 and rotational constants B 0 and D 0 . m el is the mass of the electron. The shift was calculated with respect to 12 C 16 O. For the X 1 Σ + state the vibrational constants were obtained from fitting to the data from Coxon et al. [31], while for the a 3 Π state the constants from Havenith et al. [7] were used. In the lower panel, the difference between the experimental and calculated values of the band origins for the different isotopologues are plotted. The experimental and scaled values of the band origins deviate by a few GHz, which is more than expected given the precision of the data used in this analysis. Most surprising is the difference between the values of the band origins of 13 C 16 O and 12 C 18 O, which have nearly equal reduced masses (7.18 vs. 7.21 amu, respectively). This suggests a breakdown of the Born-Oppenheimer approximation.
We have re-analyzed experimental data pertaining to all six stable isotopologues of CO for the E 1 Π (v = 1) state [32] and the C 1 Σ + (v = 1) state [33] and found a similar deviation as found for the a 3 Π, both in size and in direction. This suggests that the ground state is probably the source of the discrepancy. Calculations using the LEVEL program [34] were performed to estimate the effects of the breakdown of the Born-Oppenheimer approximation in the ground-state of CO following the approach by Coxon and Hajigeorgiou [31]. The calculated energy difference between the 12 C 18 O and 13 C 16 O isotopologues was approximately 50 times smaller than observed in our measurements and is thus insufficient to explain the observed effect [35].
The specific mass shift was not included in our analysis. It is however proportional to µ red , and can thus not explain the observed large difference between 12 C 18 O and 13 C 16 O. Nuclear size effects could in principle cause a similar isotope shift as observed, but the difference in nuclear charge radius between 17 O and 18 O is approximately 10 times larger than the difference between 16 O and 17 O, whereas the observed difference between the values of the band origins in the isotopologues with these oxygen isotopes is similar in size [36]. It is therefore unlikely that the observed isotopic effect is due to a nuclearsize effect.
In Fig. 11, the energies of the two-photon mw transitions between the near degenerate levels of the different Ω-manifolds are shown as a function of µ red . The solid lines show the values obtained from scaling the molecular constants of 12 C 16 O using the mass-scaling relations discussed in Sec.VI B. The crosses indicate the reduced masses of the six stable isotopologues, whereas the boxes show the values directly obtained from the measurements. The strong energy dependence on the reduced mass is indicative of a high sensitivity to the proton-to-electron mass ratio. The measured and calculated values for the two-photon microwave transitions between the nearly-degenerate levels are listed in Table VI. Our measurements reduce the uncertainties of the twophoton transitions to ≈1 MHz, a factor 20 better than before.
The mass of the proton is much larger than the masses of the constituent quarks, consequently, the mass of the proton is related to the strength of the forces between the quarks; Λ QCD , the scale of quantum chromodynamics [37]. As the same argument holds for the neutron-toelectron mass ratio, a possible variation of the protonto-electron mass ratio is expected to be accompanied by a similar variation of the neutron-to-electron mass ratio [38]. With this assumption, it follows that The sensitivity of a transition to a possible variation of the proton-to-electron mass ratio µ can now be calculated using the mass-scaling relations discussed before. The sensitivities for the two-photon microwave transitions are listed in the last column of Table VI. These coefficients have been calculated with an accuracy of 0.3-0.05%.
It is instructive to compare the sensitivity to what one would expect in a pure Hund's case (a). For the transition at 1628.3 MHz we expect a sensitivity of K µ = A/2ν = 1242751.3/2 × 1628.3 ≈ 382 which is 12% larger than calculated using the model that includes the coupling between the different Ω-manifolds [39].

VII. CONCLUSION
UV frequency metrology has been performed on the a 3 Π -X 1 Σ + (0,0) band of various isotopologues of CO using a frequency-quadrupled injection-seeded narrowband pulsed Ti:Sa laser referenced to a frequency comb laser.
We have fitted our optical data for 12 C 16 O together with the lambda-doubling transitions of Wicke et al. [3] and the rotational transitions of Carballo et al. [5] and Wada and Kanamori [6]. Adding the optical data resulted in a large decrease of the uncertainties of A, C and C δ , and a smaller decrease in uncertainty in the other constants.  Table V. The sensitivities to a possible variation of the proton-to-electron mass ratio are listed in the last column.
From our measurements we obtain the value of the band origin with an uncertainty of 5 MHz, a 30-fold improvement compared to the value obtained from absorption measurements by Field et al. [10]. We have also measured the value of the band origin in different isotopes and found an unexpected behavior of the isotope shifts, probably due to a breakdown of the Born-Oppenheimer approximation.
Our main motivation for this study was to obtain more accurate values for the 2-photon transitions between near degenerate rotational levels in different Ω-manifolds and validate the large sensitivity coefficients predicted for these transitions. The calculated values of the transitions agree to within a few MHz with the measured values, giving confidence in the calculated values of K µ .