Optical Pumping of TeH+: Implications for the Search for Varying mp/me

Molecular overtone transitions provide optical frequency transitions sensitive to variation in the proton-to-electron mass ratio ($\mu\equiv m_p/m_e$). However, robust molecular state preparation presents a challenge critical for achieving high precision. Here, we characterize infrared and optical-frequency broadband laser cooling schemes for TeH$^+$, a species with multiple electronic transitions amenable to sustained laser control. Using rate equations to simulate laser cooling population dynamics, we estimate the fractional sensitivity to $\mu$ attainable using TeH$^+$. We find that laser cooling of TeH$^+$ can lead to significant improvements on current $\mu$ variation limits.


I. INTRODUCTION
The Standard Model has proven remarkably robust, but it fails to explain many known phenomena such as gravity, dark matter and dark energy. This quandary has motivated searches for physics beyond the Standard Model, including searches for space-time evolution of the dimensionless constants. Such an evolution could occur over cosmic time scales [1,2] and might be related to the problem of dark energy [3]. Alternatively, oscillatory or transient variations over shorter time scales would be expected to arise in certain proposed models for dark matter [4,5].
Molecular rotational and vibrational energies scale like E h (M/m e ) β where E h is the atomic unit of energy defined by the electronic energy scale, M is the reduced mass of the molecule, β = −1/2 for vibrations and β = −1 for rotations [6]. Neutrons and protons primarily derive their masses from the strong interaction such that m n ≈ m p ≈ 3Λ QCD , while electrons derive their mass from the weak scale via the Higgs field vacuum expectation value [6,7]. Consequently, rotational and vibrational transitions of molecules can act as a probe into the variation of µ ≡ m p /m e and therefore the ratio of the strong to weak energy scales. In many models, for example models assuming Grand Unification, µ varies by a factor of 30-40 more rapidly than the fine structure constant α. Based on these arguments, there is strong motivation for experimental searches for varying µ [8][9][10].
Atomic hyperfine transitions also have dependence on µ, but their sensitivity suffers compared with molecules because of the smaller energy interval [10]. However, despite orders of magnitude smaller absolute sensitivities to varying µ, the simpler atomic state preparation requirements have allowed atoms to set the current best laboratory constraints. Comparison of two different hyperfine transitions and an optical atomic clock has yielded a limit of ∼1×10 −16 /year [11,12]. The best experimental limit from a molecule, set by comparing a rovibrational transition in SF 6 to a Cs hyperfine transition, is 6 × 10 −14 /year [13].
In TeH + , a vibrational overtone transition has been identified as a potentially promising candidate for µ variation detection [14]. The systematic uncertainties for reasonable experimental conditions are projected at the 1 × 10 −18 level or below. Furthermore, TeH + is one of a small, but growing class of molecular ions identified as having so-called diagonal Franck-Condon factors (FCFs), offering the possibility of rapid state preparation through broadband rotational cooling [15][16][17][18][19][20].
We envision the µ variation experiment being performed on a single molecular ion, using quantum logic spectroscopy (QLS) [21]. Preparation of the initial spectroscopy state could be accomplished either by optical pumping [22][23][24] or projectively [25]. The speed at which one can initially prepare and reset the spectroscopy state has critical implications for the statistical uncertainty that can be obtained in a measurement. Here, we evaluate realistic optical pumping state preparation timescales for TeH + and draw conclusions about statistical uncertainties in the search for varying µ. We also discuss more generally the molecular ion qualities desirable for obtaining low statistical uncertainty and identify some molecular ion species, which can serve as benchmarks for µ variation searches.

II. MOLECULAR STRUCTURE
The four lowest lying electronic states ( Figure 1) of TeH + (X 1 0 + , X 2 1, a 2 , b0 + ) are well described by the Hund's case (c) basis. They are all predicted to have bond equilibrium distances within ∼0.1 pm of each other [26], implying nearly identical rotational constants and that each of the transitions will have highly diagonal FCFs. Consequently, each electronic transition will have well separated P, Q and R branches allowing a spectrally-shaped broadband laser to selectively cover transitions that remove rotational quanta [27]. Highly diagonal FCFs lead to suppressed vibrational excitation during the rotational cooling (Figure 2).
Since there does not exist experimental data for TeH + , we attempt to evaluate the accuracy of the TeH + multireference configuration interaction with single and double excitations and Davidson correction for higher excitations (MRCISD+Q/aV5Z) calculations [26] by comparing theoretical [28] and experimental [29][30][31] investigations of the isoelectronic species antimony hydride (SbH). Compared with the TeH + calculation, the MRCISD+Q calculation for SbH uses a smaller basis set (of quadruple zeta quality) and fewer configuration state functions and is expected to be less accurate. FCFs depend most strongly on the difference in equilibrium bond length between electronic states, and the equilibrium bond lengths for SbH were predicted to within 3 pm of the measured values. A comparison between the predictions of the MRCISD+Q/aV5Z level of theory for the CAs molecule [32] and experimental measurements [33] shows that calculated bond lengths are within 1 pm of experimental values; therefore, the calculations for TeH + should be more accurate than the   [34] and are predicted to be 15 µs for b0 + , 2.4 ms for a2 and 460 ms for X 2 1.
Transition moments between b0 + and a2 and between a2 and X 1 0 + will be insignificant as both are quadrupole transitions.

A. Magnetic Dipole Moments
The isoelectronic molecule SbH was observed to have significant magnetic dipole transition moments on X → b transitions [29]. Magnetic dipole transitions will connect states of the same parity, so these transitions are useful for state preparation of a single parity state, which would otherwise require an additional step to the cooling process. The TeH + magnetic dipole moments for the b0 + −X 2 1 (g s b0 + |S x |X 2 1 ) and X 2 1−X 1 0 + (g s X 2 1|S x |X 1 0 + ) transitions ( Figure 3) were computed using MOLPRO [35] and input into LEVEL 16 [34] to obtain the Einstein A coefficients. The magnetic dipole spontaneous emission rates for the b0 + − X 2 1 and X 2 1 − X 1 0 + transitions are 70-times slower and five-times faster than the corresponding E1 transitions, respectively. We therefore include these M1 transitions in our simulation of the cooling dynamics. to preserve their vibrational mode. This means that continuous pumping of rotational or vibrational energy removing transitions will efficiently populate the lowest energy rotational and vibrational states, i.e., efficient internal state cooling. In this paper, we consider variants of such a scheme on TeH + . When discussing rotational cooling, we assume a thermal population distribution at room temperature where ∼99% of the population is in J < 12 of the ground electronic and vibrational state.

III. INTERNAL STATE COOLING
The lifetime of the X 2 1 state is long compared to excited vibrational state lifetimes of X 1 0 + , so we do not consider rotational cooling via the X 1 0 + → X 2 1 transition. For any cooling scheme, however, X 2 1 will be important as it is the strongest decay channel of both b0 + and a2. The addition of this laser significantly reduces the complexity of the four-level system by (1) effectively reducing v = 0 of X 1 0 + and X 2 1 into a single state and (2) via the relatively strong M1 transition, providing different parity coupling than in E1 transitions ( Figure 4).
Because each rotational cooling scheme must involve the population in X 2 1, we propose coupling X 1 0 + and X 2 1 with a broadband laser on the Q branch. The requirements of the broadband source are simplified by the structure of X 1 0 + and X 2 1, which has the first 12 Q branch transitions within one wavenumber of each other. The X 1 0 + → X 2 1 transition is 9.6 µm [26], which allows for a single QCL to couple rotational states of the two ground vibrational states.
For cooling to proceed at the maximum rate set by upper state spontaneous emission, the X 2 1 − X 1 0 + transition, whose Einstein A coefficients are ¡ 2 s −1 , must be driven at well above saturation. For a2 or b0 + as the choice of the upper state, this requires coupling X 2 1 − X 1 0 + at ∼3 and five orders of magnitude above saturation, respectively. Given that saturation occurs with a spectral intensity of ∼130 µW/(mm 2 cm −1 ), a 1 cm −1 broad QCL with 50 mW of power focused onto the molecule is easily capable of meeting these requirements.
B. Rotational Cooling on X − b0 + at 600 nm The most rapid cooling scheme will involve the optical transitions between X and b0 + as b0 + has the shortest lifetime of the diagonal electronic states. We propose cooling by pumping from X 2 1 because the transition dipole moment of X 1 0 + and b0 + is expected to be an order of magnitude weaker than X 2 1 and b0 + .
The P branch of X 2 1 → b0 + has been predicted to span 612 nm-618 nm for J < 12, and the spectral intensity at saturation is estimated to be ∼500 mW mm −2 /cm −1 (10 W mm −2 /nm). Rapid progress on broadband commercial lasers in this spectral region suggests that a light source capable of saturating all the required transitions might soon be available.
We note that inclusion of P(1) in the coverage of the P branch with a broadband source will lead to sub-optimal cooling as decay from |b0 + , J = 0 can only increase rotational energy. Exclusion of P(1), however, will limit cooling by leaving J = 1 dark to the cooling laser. As seen in Figure 4, this can be avoided with the addition of a CW laser tuned to

Vibrational Repumping
Though branching from |b0 + , v = 0 into |X 2 1, v = 1 is slow, an additional CW laser and careful choice of the rotational cooling laser spectral cutoff can improve cooling time and fidelity. Because the vibrational constants of X 2 1 and b0 + are similar, the rotational The spectrum is such that the spectral cutoff can be placed between P(1) and P(2) for both vibrational states. Since the rotational cooling laser can connect states of the same parity via M1 transitions, any decays into |X 2 1, v = 1 will therefore be pumped into J = 1 where a CW laser can be used as a vibrational repump into v = 0 via the P (1) transition of |X 2 1, v = 1 → |b0 + , v = 0 (∼ 700 nm). Because decays from b0 + into X 1 0 + are more than an order of magnitude less frequent than into X 2 1, an extra laser coupling the X 1 0 + and X 2 1 v = 1 states is not necessary.
C. Rotational Cooling on X − a2 at 1300 nm Rotational cooling with IR frequencies can be done by optical pumping through a2. The relevant X 2 1 → a2 P branch transitions at room temperature are predicted to span ∼100 cm −1 from 1340 nm-1360 nm [26], within the telecom O-band. The spectral intensity for saturation of these transitions is < 25 mW mm −2 /cm −1 , meaning a 5 W broadband laser with a 1 mm 2 collimated beam area is sufficient for saturation.
Cooling via this transition will be limited by the 4-7 ms branching decay times of A cartoon of the transitions involved in the X − a2 cooling scheme(s) can be seen in Figure 5. In a more careful analysis of the cooling time scale, we note that the X 2 1 → a2 transition has no P branch transitions for J < 3. In a cooling scheme relying on a QCL coupling the X states via the Q branch and a broadband laser covering the P branch of X 2 1 → a2, the lack of P branch transitions for J < 3 implies rotations will cease being cooled once the population has been pumped into J = 0, 1, 2. If the broadband laser includes the Q branch of X 2 1 → a2, then at the cost of a reduced cooling rate, the broadband laser will pump J = 2 such that the population will transfer into J = 0, 1. Over much longer time scales (seconds) determined by the |X 2 1, J = 1 → |X 1 0 + , J = 0 branching time, the X coupling laser will pump the remaining population into |X 1 0 + , J = 0 . The fidelity of this final step will be limited by the much slower rate of blackbody redistribution.

CW Assist
With assistance from the b0 + state, it is possible to avoid the rate-limiting steps that were not included in our rough estimate of the cooling time scale. In the scheme relying on pumping the P and Q branch of X 2 1 → a2, an additional CW laser tuned to

D. Vibrational Cooling
Depending on the choice of excited spectroscopy state in a µ variation measurement, vibrational cooling may be beneficial. Specifically, we envision our spectroscopy states to be of the form |X 1 0 + , v ′′ = 0, J ′′ = 0 and |X 1 0 + , v ′ , J ′ = 1 . As the excited state spontaneously decays, the rotational state population will slowly diffuse as the molecule vibrationally relaxes.
For v ′ = 1, decay can only leave the population in the vibrational ground state, and so, vibrational cooling is not necessary to minimize state re-preparation time.
For v ′ > 1 we propose active vibrational cooling by driving ∆v = −1 transitions of the form |X 2 1, v → |b0 + , v − 1 (see Figure 6). Similar to the rotational cooling schemes, |X 1 0 + , v → |X 2 1, v must also be coupled, and this is accomplished via the Q branch. However, because there is no Q branch transition for J = 0, we must include R(0) of each vibrational level. As the X 1 0 + → X 2 1 transitions only span ∼100 cm −1 for v = 1 to v = 7 and the rotational spacing is large, a b0 + v-1 In the 1300 nm cooling schemes, we observe a significant reduction in the cooling rate compared to the 600 nm schemes. The simplest and slowest 1300 nm scheme (solid red line The cooling time for the vibrational cooling stage is determined by minimizing the following expression: where τ is the interrogation time (assumed to be equal to the excited state lifetime), T VC is the amount of time the vibrational cooling lasers are on and ρ v=0 (t) is the fraction of the population in v = 0 at time t. Vibrational cooling was simulated assuming broadband coverage approximately at the saturation intensity of the relevant |X 2 1, v → |b0 + , v − 1 transitions. The cooling times for the first eight excited vibrational states can be seen in Table II. In every case, the vibrational cooling lasers pumped > 99% of the population into the ground vibrational state. It is noteworthy that vibrational cooling will not contribute significantly to the overall duty cycle as T VC ≪ τ for any choice of vibrational state.
Assuming the rotational cooling stage is applied for time T RC , the average time for a successful experimental cycle is estimated to be: where T p is the total time necessary for state readout and hyperfine state preparation, the term ρ J=0 (t) is the fraction of the population in |X 1 0 + , v = 0, J = 0 at time t after the start of the rotational cooling stage and the factor of two arises from needing to measure two points to estimate the offset from the line center. The optimal rotational cooling time will thus be the time that minimizes T c .

VI. µ VARIATION MEASUREMENT
In a Ramsey measurement on a single ion, the Allan deviation is given by: where C is the fringe visibility, T R is the Ramsey time, T c is the cycle time and T is the total measurement time [37,38]. Optimal cycling occurs for T c = 2T R and T R set to about the upper state lifetime τ , for which C ≈ 0.6 [37]. Laser cooling of the internal molecular state opens up the possibility for efficient state preparation, which can allow for repeated interrogation of the same molecular sample and low dead time. To evaluate the benefit of  The vibrational interval from v = 0 to v ′ = n at frequency Ω will vary in response to changing µ as described by: Before statistics are considered, the absolute sensitivity coefficient S = ∂Ω/∂(lnµ) provides the most important figure of merit for the transition, since it expresses the shift in the measured frequency [39,40]. It is also convenient to define a relative sensitivity coefficient K µ [10] given by: We must also account for detrimental statistical effects of the finite upper state lifetime.
Fluctuations in the frequency measurements are described by an Allan deviation σ y (T ) for some overall measurement time T . The vibrational frequency measurements yield values for µ itself (albeit with a large theoretical uncertainty), and the square root of the two-sample variance in µ is: Statistical uncertainty in µ variation can be related to σ tions for µ variation measurements using polar molecule overtone transitions are discussed in [14].

A. Single-Ion TeH + Measurement
In our simulated results for statistical sensitivity of a ∆µ measurement using a single TeH + ion, the spectroscopy interval is probed using Ramsey's method, and we take T R = τ and C = 0.6 [37]. Results for various state preparation schemes are shown in Figure 8.
The results suggest that spectroscopy on a single TeH + ion can be used for a significantly improved search for varying µ.
We find that the attainable precision is most sensitive for the larger overtone transitions.
The ultimate decision for which vibrational interval to choose for spectroscopy will depend on how much vibrational cooling laser power is available. In the extreme case where no vibrational cooling is used, v ′ = 1 is the optimal choice. At the other extreme, with enough vibrational cooling laser power to saturate all the transitions, the best simulated statistical sensitivity to ∆µ after one day of averaging is described by σ (µ) y = 3.6 × 10 −17 . For this transition, the 600 nm cooling scheme significantly outperforms the 1300 nm cooling scheme.

B. Multi-Ion Spectroscopy
Besides searching for µ variation with a single-ion QLS measurement, an alternative approach using laser coolable polar molecules is to perform multi-ion spectroscopy. In principle, QLS can be extended to N molecular ions with only log(N) overhead in readout time and logic ions [41]. A simpler fluorescence readout scheme is normally not possible for molecular ions, since they usually lack cycling transitions. However, for molecules that can be rapidly laser cooled, there exist quasi-cycling transitions capable of scattering enough light for fluorescence detection. Additionally, negative differential (static) polarizabilities are ubiquitous in polar molecules for transitions starting from the ground rotational state [14]. A negative differential polarizability allows for choosing of a magic RF trap-drive frequency such that the Stark shift and micro-motion second order Doppler shifts cancel one another [14,42].
In TeH + , there do exist quasi-cycling transitions amenable to state detection via fluorescence. For example, the population in |X 1 0+, J = 0 can be left dark, while |X 1 0+, J = 1 can be driven in a quasi-cycling scheme by using one laser driving E1 and M1 coupling between |X 1 0+, J = 1, − ↔ |X 2 1, J = 1, ± and a second laser to couple |X 2 1, J = 1, + ↔ |b0 + , J = 0, + . The simulated results, using the same QCL discussed previously for the first laser and a CW laser at saturation for the second are plotted in Figure 9. On average, there will be approximately 400 spontaneously emitted photons at a rate of ∼5 photons per ms before an off-diagonal ∆v > 0 decay occurs. In a large ensemble, the result would be a rapid decrease in the scattering rate after ∼80 ms.

C. Homonuclear Molecule Benchmarks
It is interesting to note that the logic of choosing the optimal overtone transition in TeH + also sets a bound on the statistical sensitivity attainable for any molecule. The strongest known chemical bond is that of CO, with D = 90,000 cm −1 [43]. Molecular ion dissociation energies can approach this range; N + 2 and O + 2 have D = 54,000 cm −1 and D = 74,000 cm −1 , respectively. For a Morse potential, the upper bound on the sensitivity S is given by D/4, where D is the dissociation energy [39]. Although calculations are not generally available to describe broadening of overtone linewidths from coupling to other electronic states, the measured linewidths are expected to be limited by laser coherence. Statistical sensitivity of these species, using probe times set by currently available laser coherence, is shown in Table ??. Stark shifts for nonpolar species are favorably small, and other systematic uncertainties can be low, as well [44][45][46].
Homonuclear molecules can be loaded into the trap in the desired quantum state [47,48], and one can imagine an experimental cycle approaching zero dead time using a quantum logic protocol. Simple projective measurements within the two-level manifold can be used to reset to the lower spectroscopy state at the beginning of each cycle [25]. Trapped N + 2 prepared in its ground rotational state lifetime has been demonstrated to have lifetimes as long as 15 minutes, limited by the collisions with background gas [49]. After a collision changes the rotational state, a new molecule could be loaded. Alternatively, one could use a quantum logic state preparation approach that sequentially transfers the population from all possible populated states [50,51]. In the latter approach, the problem of recovery of the precollision parity must also be addressed, possibly by two-photon excitation of a short-lifetime electronic transition and then cleanup of resulting vibrational excitation. Since either state recovery approach might be time consuming, it could be preferable to operate at cryogenic temperatures to reduce the rate of collision with background gases.
Comparing the ideal zero dead time performance of TeH + and the homonuclear bench-   Table ??, we find that the best TeH + statistical uncertainty is nearly two orders of magnitude larger. However, since simpler optical pumping state preparation is available for TeH + , its experimental statistical uncertainty should be less sensitive to the vacuum environment. Furthermore, the quasi-cycling transitions of TeH + or other polar species offers the possibility of fluorescence readout in multi-ion spectroscopy.

VII. CONCLUSIONS
We have identified vibrational overtone transitions in TeH + as candidates for a spectroscopic search for varying µ, taking advantage of the optical pumping protocols for state preparation. Rate equation simulations show that TeH + can be optically pumped from room temperature to the rotational ground state in ∼100 ms using telecom wavelengths or ∼10 ms using optical wavelengths. In an overtone spectroscopy experiment, we find that realistically achievable experimental cycle times yield a statistical uncertainty as low as 4 × 10 −17 for a day of averaging. This demonstrates the possibility for significant improvement on the best laboratory limit of ∼1 ×10 −16 /year [11,12] and the current limit set by a molecule at 6 × 10 −14 /year [13].
We primarily limited our investigation to the performance of single ion spectroscopy using quantum logic, but simulations also support the potential for fluorescence state read-out of TeH + . Large Coulomb crystals of polar molecules, with state detection performed by fluorescence, could have favorably small systematic uncertainties because negative differential polarizabilities can allow for cancellation of Stark and second order Doppler shifts [42]. Our analysis suggests that the possibility of searching for µ variation using multi-ion spectroscopy on laser-coolable polar species warrants further investigation.