Optical cycling, radiative deflection and laser cooling of barium monohydride (BaH)

We present the first experimental demonstration of radiation pressure force deflection and direct laser cooling for barium monohydride (BaH) molecules resulting from multiple photon scattering. Despite a small recoil velocity (2.7 mm/s) and a long excited state lifetime (137 ns), we use 1060 nm laser light exciting the $X\rightarrow A$ electronic transition of BaH to deflect a cryogenic buffer-gas beam and reduce its transverse velocity spread. Multiple experimental methods are employed to characterize the optical cycling dynamics and benchmark theoretical estimates based on rate equation models as well as solutions of the Lindblad master equation for the complete multilevel system. Broader implications for laser cooling and magneto-optical trapping of heavy-metal-containing molecules with narrow transition linewidths are presented. Our results pave the way for producing a new class of ultracold molecules -- alkaline earth monohydrides -- via direct laser cooling and trapping, opening the door to realizing a new method for delivering ultracold hydrogen atoms (Lane 2015 \textit{Phys. Rev. A} 92, 022511).


I. INTRODUCTION
Various approaches have been developed for controlling atomic motion [1][2][3]. Radiation pressure force and associated laser cooling and trapping methods in particular have revolutionized atomic physics [4,5] and have since become invaluable tools for modern quantum science and engineering [6]. While primarily relying on rigorous angular momentum selection rules present only in a handful of atoms, laser-cooled atomic samples have provided an extremely fruitful experimental platform for probing important condensed matter models [7], studying exotic phases of matter [8], and developing new quantum sensors and ultra-precise clocks [9]. Already for diatomic molecules, the increased complexity of internal structure combined with various couplings between electronic, vibrational, and rotational dynamics preclude the existence of a pure two-level substructure required for repeated photon scattering [10].
In order to extend laser cooling to molecules, the use of quasi-closed cycling rovibrational transitions together with extra vibrational repumping lasers has been proposed [11,12].
Spurred by such conceptual advances as well as by the latest developments in continuouswave laser technology, major inroads in direct laser cooling of diatomic and even polyatomic molecules have been made in the last decade [13], culminating with the demonstration of three-dimensional magneto-optical trapping at sub-millikelvin temperatures for several molecular species [14][15][16][17]. Full control of molecular degrees of freedom at ultracold temperatures provides unique opportunities for realizing new applications, including quantum simulations of strongly interacting many-body systems [18], low-energy precision searches for physics beyond the Standard Model [19,20], experimental tests of fundamental chemical processes [21], and implementation of quantum computation protocols [22,23]. However, before these and many other possible applications [24] can be explored to the fullest extent, robust methods for generation of ultracold molecular samples with diverse properties and constituents must be realized and carefully characterized. A critical step towards this goal for each new molecular species is the demonstration of sustained optical cycling without considerable loss to unaddressed dark states [25], and control over molecular motion using laser light [26].
Here, we demonstrate such optical cycling and radiation pressure force milestones for diatomic barium monohydride (BaH), the first metal monohydride molecule to be directly laser cooled. Experimental techniques for achieving ultracold heavy-metal monohydrides have potential to open applications in probing fundamental symmetry-violating interactions [27,28] as well developing a novel method for indirect production of ultracold atomic hydrogen via precision photodissociation [29]. Our results are provided in order of increasing implementation complexity, presenting a roadmap for future experiments: i) detailed analysis of optical cycling (∼ 10 − 80 photons) using both depletion and deflection of a cryogenic beam, ii) laser compression of a molecular beam along one transverse dimension ( 100 photons), and iii) sustained optical cycling in a longitudinal slowing configuration ( 1, 000 photons). Our observations clearly show spatial manipulation of the BaH molecular beam, and demonstrate understanding of the optical cycling process out to several thousand photons. In addition to achieving the first optical manipulation of a heavy-metal monohydride molecule, we provide theoretical modeling of the optical cycling and magneto-optical trapping process for BaH. Taken together, our experimental and theoretical results provide important insights into the challenges of laser cooling and trapping for heavy molecules with reasonably narrow transition linewidths (Γ nat 2π × 10 6 s −1 ), a molecular class frequently encountered in fundamental physics applications [27,30,31].

II. EXPERIMENTAL APPARATUS
Compared to previously laser cooled molecules, which have excited state lifetimes comparable to alkali metals (τ sp ≈ 20 − 30 ns), BaH has a much longer excited state lifetime for any of the possible optical cycling transitions (τ sp > 100 ns). A small photon recoil velocity (v recoil 3 mm/s) together with a low possible scattering rate (R scat 1 × 10 6 s -1 ) require a use of a slow cryogenic buffer-gas beam [32] in order to provide enough moleculelaser interaction time to observe radiation pressure effects. Therefore, each of the following experiments was performed using a cryogenic molecular beam of BaH, whose construction, optimization and operation have been described in our previous publication [33]. Because of the small capture velocity for the radiation pressure force v rad ∼ Γ sp /k ≈ 1 m/s for BaH, we installed a 5-mm circular collimator 48 cm away from the cryogenic cell aperture to match the transverse velocity spread of the molecular beam to v rad .
In order to allow for a sufficient atom-molecule interaction time to observe transverse laser cooling of the BaH cryogenic beam, we used 15-cm long vacuum viewport windows anti-reflection coated for a reflectivity of r window < 0.5% per surface and out-of-vacuum mirrors with dielectric high-reflectivity coatings with r mirror > 99.8%. Therefore, with 10 round-trip passes the light intensity drops to about 80% and with 20 passes to 60%, which presents a challenge for maintaining a uniform scattering rate across the entire region, requiring a higher incident laser power. For the radiative deflection experiments, we used hollow rooftop mirrors (HRM) out of vacuum since the photon momentum kicks have to come from a single direction. Gold mirror coatings provided r HRM ≈ 97% which limited the number of passes with sufficiently high laser intensity to achieve efficient photon scattering.

III. MOLECULAR STRUCTURE AND OPTICAL CYCLING SCHEME
There are a number of important differences between the relevant structure of BaH and that of other previously laser cooled diatomic and triatomic molecules. Except for YO [34], all of the diatomic and triatomic molecules laser cooled thus far consisted of an alkalineearth-metal like atom (Ca, Sr or Yb) ionically and monovalently bonded to an electronegative fluorine (F) [35][36][37] or hydroxyl (OH) [38][39][40] ligand. Such ligands with a moderate electronwithdrawing capability lead to a strong localization of a single unpaired electron on the metal atom with electronic excitations resembling those of an ionized alkaline earth metal atom M + [41]. However, because the partial charge of the metal increases with the electron affinity of the ligand [42], upon electronic excitation the M-L bond length change is largest for monohydrides and decreases for MF and MOH compounds [41]. More covalent nature of the metal-ligand bond for hydrides has been further confirmed by the estimated values of the quantum defect across a range of ligands attached to the same alkaline earth metal [41]. While previous theoretical studies [43,44] have indicated that a large number of strongly ionically bonded compounds (e.g. MF, MOH and MOR where R is a functional group) are well suited for laser cooling, other molecules with different geometries (MCH 3 ) and constituents (MSH) have more covalent nature of M-L bonds [45], with BaH potentially serving as an important stepping stone for understanding optical cycling and laser cooling in covalently-bonded systems [46].
One of the unique aspects of the BaH level structure is that it supports optical cycling on three different electronic transitions within a technically convenient near-infrared wavelength regime but with distinct laser cooling characteristics: i) X 2 Σ + ↔ A 2 Π 1/2 at 1061 nm (τ sp ≈ T D ≈ 30 µK), and iii) X 2 Σ + ↔ H 2 3/2 at 1110 nm (τ sp ≈ 10 µs, T D ≈ 0.4 µK) electronic transitions [47]. Due to the highly diagonal nature of the Frank-Condon factor (FCF) matrix (F v v ) for the X 2 Σ + ← A 2 Π 1/2 electronic decay 1 (F 00 > 0.987 [47]), we choose to use this transition for optical cycling, and repump molecules from excited vibrational levels v > 0 back into the cycle with an off-diagonal transition through the B 2 Σ + excited electronic state ( Fig. 1(a)). By separating the cycling and repumping transitions (i.e. they are not directly coupled with laser light via a common vibronic manifold), we increase the maximum achievable scattering rate by a factor of 1.75.
Following previous experiments [13,48], we drive all optical transitions from the first excited rotational state N = 1 of the electronic ground state X to the ground rotational level N = 0 in each excited electronic state (A or B) in order to ensure rotational closure (i.e. N = 1 ↔ N = 0). Parity as well as angular momentum selection rules for the electric dipole allowed transitions ensure that molecules decay back to the N = 1 rotational manifold of states [11], forming a quasi-closed cycling transition required for repeated photon scattering.
Optical cycling on a ∆N = −1 transition has the additional requirement that dark states need to be destabilized [48], which we achieved by the addition of a static magnetic field.
Because vibrational, rotational, and spin-rotational molecular constants in 2 Σ + electronic states scale as ω vib ∝ µ −1/2 red , B rot ∝ µ −1 red and γ SR ∝ µ −1 red with reduced molecular mass µ red [49], correspondingly large energy spacings in BaH (compared to MF or MOH molecules) result in additional challenges for optical cycling and laser cooling. Because both J-sublevels of the spin-rotation splittings in the N = 1 states (1.5γ SR,v =0 = 8.64 GHz [50] and 1.5γ SR,v =1 = 8.41 GHz) cannot be easily addressed with the same light source, we use two separate external cavity diode lasers (ECDLs) to excite J = 1/2 and J = 3/2 states ( Fig. 1(b) and 1(c)), with 1.5γ SR,v offsets. The two pairs of resulting ECDLs are co-aligned with matching linear polarization to seed two tapered amplifiers producing ∼ 100 mW of 1060 nm light and ∼ 30 mW of 1009 nm light in the beam cooling region. For the (0, 0) X 2 Σ + → A 2 Π 1/2 transition 2 , the hyperfine sidebands are generated using a 40 MHz AOM, which leads to one additional off-resonant sideband for the J = 1/2 hyperfine states ( Fig. 1(b)). For the 1 Following the established convention in the field of molecular spectroscopy we mark quantum numbers with double (single) primes to refer to the electronic ground (excited) state, correspondingly. 2 Following the convention in the spectroscopic literature for diatomic molecules [51], we use the (v , v ) X → A notation to indicate the change in the vibrational quantum number from v in the X electronic state to v in the A state.
(0, 1) X 2 Σ + → B 2 Σ + vibrational repumper, the hyperfine structure is addressed using a 20 MHz EOM driven with a modulation depth set to optimize the power in the ± 1st order.
Each laser frequency is stabilized via referencing to a HighFinesse WS7 wavemeter, which provides a short term (∼ 1 s) instability of ∼ 1 MHz and a longer term (∼ 5 h) instability of 5 MHz, consistent with performance achieved in other experiments [52]. Frequency stability of the reference wavemeter was additionally verified by monitoring the wavelength of a frequency-comb stabilized ECDL over the course of an hour and by daily calibration of the wavemeter with a frequency-stabilized HeNe laser. We use the X 2 Σ + → E 2 Π 1/2 excitation at 683 nm to collect the time-of-flight (ToF) data and spatial distribution images of the BaH molecular beam because this wavelength matches the peak sensitivity of EMCCD and PMT detectors used to detect these molecules.
For this transition we also use two lasers, with the J = 3/2 hyperfine splitting addressed with a 40 MHz AOM. Both E 2 Π 1/2 lasers are broadened with a 3 MHz EOM, to ensure that they interact with all velocity classes and that the fluorescence signal accurately represents the total v = 0 population regardless of the specific hyperfine distribution of the molecular ensemble when it reaches the detection region.

IV. OPTICAL CYCLING
The first step in achieving laser control and cooling of molecular motion is to establish repeated scattering of photons (optical cycling) and characterize dominant loss channels.
As discussed in Sec. III, our detection scheme relies on a non-cycling transition at 683 nm, necessitating a different approach to characterizing the photon scattering dynamics for the main laser cooling transition at 1060 nm. The experimental setup used for characterizing repumping light (orange) was blocked, and the number of passes of the (0, 0) X 2 Σ + → A 2 Π 1/2 laser (blue) was varied. The 1060 nm laser beam was alternated between "on" and "off" to account for any drift in the molecular beam yield, and each data point is the average of 200 molecular beam pulses. Accounting for the beam forward velocity (160 ± 40 m/s [33]) and the measured diameter of the laser beam (1.5 ± 0.1 mm), we can convert the number of passes to the molecule-light interaction time. As seen from the error bars in Fig. 2(c,d), this conversion is the dominant source of uncertainty, primarily because of the substantial spread in the beam forward velocity. We can estimate the number of photons scattered (N scat ) based on the fraction of molecules that remain in ground vibrational state (P v=0 ), and the known diagonal FCF F 00 from previous measurements [47,53]. Figure   2(b) provides a representative ToF data for an unperturbed BaH beam (blue) and with the (0, 0) X 2 Σ + → A 2 Π 1/2 cycling laser on (orange) resulting in 15% of the molecules remaining in v = 0 at the detection region. We found no significant dependence of the scattering rate on an applied magnetic field used to destabilize the dark states, most likely because residual field in the interaction region of a few Gauss was sufficient to cause a dark state precession rate comparable to the excitation rate (∼ 10 6 s -1 ).
Following Di Rosa [12], we model the repeated spontaneous emission events by a molecule as a Bernoulli sequence with probability p = 1 − F 00 that decay will result in populating an excited vibrational level v > 0. The probability that a molecule initially in the vibrational ground state will still be in v = 0 after scattering N scat photons is given by: Therefore, we can convert the fraction P v =0 into the number of scattered photons for the remaining molecules: The expectation value of N scat for a molecular ensemble can be estimated by modeling the photon scattering process before the molecule is optically pumped into v = 1 as a geometric distribution with the expected value of and the standard deviation in N scat of where τ sp = 136.5 ns [47] is the spontaneous excited state lifetime and n g (n e ) is the number of ground (excited) m F magnetic sublevels. While Eq. (5) provides a useful way to approximate the maximum possible scattering rate for molecules, our estimates for the achievable scattering rate in the experiment using both a multilevel rate equation model (Fig. 11) as well as an optimized numerical simulation of the full system using the Lindblad master equation 3 predict R OBE ≈ 1.4 × 10 6 s -1 . The data presented in Fig. 2(c) shows that we achieve R scat ≈ 0.96R OBE in this experimental configuration with maximum interaction time t int ≈ 900τ sp . Our measurement of the scattering rate relies on (1, 0) X 2 Σ + ← A 2 Π 1/2 being the dominant loss mechanism out of the quasi-cycling transition. As shown in Fig. 2(b), with an addition of the (0, 1) X 2 Σ + → B 2 Σ + repumping laser in the "clean-up" region, we return most of the molecules (green curve) back into the ground vibrational level. The B 2 Σ + (v = 0) state has a good Franck-Condon overlap with X 2 Σ + (v = 0) (F 00 = 0.953 [47]) so molecules excited to this state decay to the desired ground state |v = 0, N = 1 with a 95% probability.
In order to determine the photon scattering rate for the (0, 1) X 2 Σ + → B 2 Σ + repumping transition, we begin by depleting v = 0 on the main cycling transition (0, 0) X 2 Σ + → A 2 Π 1/2 at the maximum interaction time. Then in a separate region ( Fig. 2(a)) we apply the repumping light, while varying the number of passes through the molecular beam.
Because F 00 ≈ 1 for the (0, 0) X 2 Σ + ← B 2 Σ + transition, it will only take 1/F 00 ≈1 photon scattered from the repumping laser to optically pump a molecule back to v = 0. As shown in Fig. 2(d), we model the interaction time required for scattering one photon in the CU region as an exponential distribution with a cumulative distribution function given as us to extract a scattering rate of 1.3(2) × 10 5 s -1 . Since R scat ∝ σ abs I 0 and the resonant absorption cross section depends on the corresponding FCF σ abs ∝ F v v , the scattering rate will be lower for the off-diagonal transition for a given laser intensity I 0 . However, using the experimentally measured R scat for the main cycling (0, 0) together with the estimate of the off-diagonal FCF F 01 ≈ 0.012, we determine that the rate of optical pumping into the excited vibrational level v = 1 in the optical cycling region (∼ 1.7 × 10 4 s −1 ) is a factor of 7 less than our measured repumping rate, indicating that there is sufficient repumping laser intensity to rapidly return the molecules into the optical cycle.

V. RADIATIVE DEFLECTION
The depletion-based scattering measurements described in Sec. IV provide strong evidence that we maintain a sufficiently high scattering rate in the optical cycling region to pump most of the molecules from the v = 0 vibrational manifold into v = 1. Moreover, we achieve a repumping rate that is significantly higher than the rate of optical pumping into v = 1. Therefore, by merging both (0, 0) repumping lasers ( Fig. 3(a)) we expect to deflect the BaH molecular beam using the radiation pressure force. In this experiment, the deflection laser passes through the vacuum chamber perpendicular to the molecular beam and strikes a 90 • mirror prism (a hollow rooftop mirror, HRM) which reflects the light back through the vacuum chamber but displaced downward by ∼ 2 cm, thus traversing below the molecular beam. The light then strikes another 90 • mirror prism that translates the beam upward and redirects it back through the molecules, deflecting the molecular beam in the same direction as the first pass ( Fig. 3(b)). The process is repeated 10 times to increase the interaction time while maximizing the laser intensity. Since the number of mirror bounces is doubled in order to propagate the light from a single direction and because the reflectivity r HRM < r mirror , the effective molecule-light interaction time is shorter than in Sec. IV.  not see an appreciable increase in the width of the molecular beam (Fig. 3(d)), indicating consistent scattering for each detected molecule. The scattering rate of 8 × 10 5 s -1 extracted from the deflection of the molecular beam accurately represents the scattering rate that we can achieve and maintain for the entire ensemble of molecules.

VI. TRANSVERSE LASER COOLING
While laser deflection results presented in Sec. V provide a valuable benchmark for the development of radiative slowing of BaH molecules, the data does not demonstrate a decrease in the entropy of the molecular ensemble (as can be seen from the beam widths in Fig. 3(d)). To achieve a reduction in transverse velocity spread for the molecular beam, we establish a 1D standing light wave intersecting the molecular beam ( Fig. 4(a)). Figure   4(b) demonstrates effective transverse temperature of the molecular beam as a function of the common detuning for the (0, 0) X 2 Σ + → A 2 Π 1/2 cooling laser with the repumping (0, 1) X 2 Σ + → B 2 Σ + laser fixed on resonance. We observe broadening of the molecular beam for red-detuned laser frequencies and narrowing for blue-detuned frequencies, consistent with Sisyphus laser heating and cooling of the ensemble, respectively [35,37,38,54].
To estimate the temperature of the beam we performed Monte Carlo simulations for various transverse temperature distributions, then selected the temperature that matches the observed expansion of the beam after a 5 mm aperture at the entrance to the interaction region. This relates the width of the unperturbed molecular beam as imaged on the camera to an effective transverse temperature. Sisyphus cooling applied over a 3 cm long interaction region reduced the effective transverse temperature by ∼ 25%, from 20 mK to 15 mK, as shown in Fig. 4(b). We benchmark the strength of the Sisyphus cooling force by comparing a Monte Carlo simulation with only Doppler cooling to one with both Doppler and Sisyphus cooling. We find that achieved Sisyphus force is ∼ 5 times greater than Doppler, explaining why the Sisyphus cooling signature was more pronounced than pure Doppler cooling. (Γ nat /2π ≈ 1.2 MHz) leads to an acute dependence of the cycling rate on the alignment and detuning of the cooling laser relative to the molecular beam ( Fig. 5(a)). The asymmetric Doppler shifts lead to a unidirectional deflection of the molecular beam in the cooling configuration ( Fig. 5(b)), where the direction of the deflection depends on the alignment angle as shown in the data in Fig. 5(c). While such shifts were not important for transverse beam cooling of molecules with larger natural linewidth like SrF [35] and SrOH [38], a pronounced effect for misalignment of < 1 • observed in our work indicates that a careful geometry optimization will be required for performing precision spectroscopy for molecular beams of laser-coolable molecules with Γ nat /2π ≈ 1 MHz (e.g. TlF [30] or TlCN [31]). For a two level system, we can provide a simplified model of this effect as shown in Fig. 5(d). This model uses a realistic Rabi rate for our experiment, and realistic distribution of forward velocity.
We see that even small angular misalignment can lead to large imbalance in the maximum force pushing the beam to either direction. To study the scattering rate, we monitored the instantaneous X 2 Σ + (v = 0) population at two points along the molecular beam propagation direction as a function of the A 2 Π 1/2 light power, alternating the A 2 Π 1/2 light on and off every shot. By taking the ratio of consecutive shots we reduce our sensitivity to molecular beam fluctuations and can isolate the effect of the A 2 Π 1/2 light ( Fig. 6(a)). Because we detect molecules with the excitation to the E 2 Π 1/2 state, while simultaneously cycling on the (0, 0) X 2 Σ + ↔ A 2 Π 1/2 transition, we can observe the instantaneous vibrational ground state fraction 75 cm and 150 cm from the beam source; the presence of optical cycling will manifest as a reduction in the fraction of molecules residing in v = 0. While for laser powers below 90 mW the measured v = 0 population fraction is the same for both regions, for high A 2 Π 1/2 light powers (>90 mW) we observe a v = 0 population of 49 ± 3% in the near region and 37 ± 4% in the far region ( Fig. 6(b)). Using ab initio calculations that utilized spectroscopically accurate molecular potentials for BaH [47], we attribute this population decrease to a combined loss into the X 2 Σ + (v = 2) excited vibrational state and H 2 ∆ 3/2 metastable electronic state. We experimentally confirmed there is no dependence of the signal in the far region, on the presence of the E 2 Π 1/2 laser in the close detection region. Based on the previous measurements and calculations of the BaH vibrational branching ratios [47], this v = 0 population reduction allows us to estimate ∼ 4, 500 scattering events between the near and far regions or ∼ 8, 500 total photon cycles. Given the time it takes the average molecule to reach the far detection region (11 ms), this gives a rate of R scat ∼ 8 × 10 5 photons/s, consistent with what we measured using transverse beam deflection in Sec. V. We see equal depletion for the full ToF beam profile, which indicates that we are able to maintain this high scattering rate for all forward velocities despite the additional complexity of achieving rapid photon cycing in the slowing configuration.
The data in Fig. 6(b) suggests that the scattering rate bottleneck in this measurement is the repumping rate out of the v = 1 state. If the repumping light power was not limiting the overall scattering rate, there would be very little population in the X 2 Σ + (v = 1) state as we are decoupling the cycling and repumping lasers using two different electronic states. The main cycling X − A laser light couples 12 ground state to 4 excited state sublevels, making 75% an expected population fraction residing in X 2 Σ + (v = 0) in a steady-state cycling configuration. Combining the X 2 Σ + (v = 0) population measurement with the measured scattering rate from Sec. IV, we can estimate state population of 50%, 37.5% and 12.5% for the X 2 Σ + (v = 0), X 2 Σ + (v = 1) and A 2 Π 1/2 (v = 0) states, respectively. This reduction of the excited state population from the maximum attainable value of 25% to 12.5% leads to a reduction in the scattering rate to a value of R scat ∼ 9 × 10 5 photons/s, consistent with the estimate based on the molecule loss to X 2 Σ + (v = 2) and H 2 ∆ 3/2 . This high number of observed photon scattering events implies a reduction in the average beam velocity of ∼ 25 m/s, or ∼ 15%. Unfortunately, the current apparatus is not capable of Doppler-sensitive forward velocity measurements since the narrow linewidth of the cooling transition and the large spread in forward velocities would reduce the signal-to-noise ratio by a factor of ∼ 20. A planned upgrade using two-photon detection via a higher-lying electronic state, combined with a high-solid-angle detection system, should make this possible.

VIII. PROSPECTS FOR MAGNETO-OPTICAL TRAPPING OF BAH
In order to characterize the feasibility for magneto-optical trapping for BaH molecules, we perform studies of confining forces using numerical solutions of the multilevel rate equation model following the framework presented in Ref. [58] and later used to model the MgF MOT properties [59]. The simulation included n l = 12 magnetic sublevels of the N = 1 rotational manifold in the vibronic ground state ( Fig. 1(b)), n u = 4 magnetic sublevels of the J = 1/2 manifold (F = 0, 1) of the v = 0 vibrational level of the excited A 2 Π 1/2 electronic state, and between three and six light frequency components from each direction. Because of the complex interplay between the ground and excited state g-factors as well as the specific nature of spacings between the hyperfine components in the ground vibronic state, a detailed numerical study is necessary in order to identify the optimal laser polarization structure and detunings 4 [58]. Depending on whether the current in magnetic field coils used for MOT operation is static (DC) or alternating (AC), there are two types of molecular MOT operating regimes, correspondingly [10,13]. Moreover, in order to enhance the confining force in the DC MOT configuration, both "blue" and "red" detuned laser beam components can be applied resulting in a dual-frequency DC MOT [60].  km/s 2 ) and the largest velocity range affected by the MOT potential (up to ∼ 15 m/s). We determine that the optimal polarization setting for the AC MOT is the same as that used for capturing CaF molecules [16].
A unique property of BaH that distinguishes it from other molecules to which magnetooptical forces have been applied (SrF [14], CaF [16], YO [17,34] and CaOH [40]) is that a large excited state g-factor (g eff ≈ −0.51 for the A 2 Π 1/2 state [33]), arising from a strong mixing with the adjacent B 2 Σ + electronic state, is approximately the same as that of the ground state (g eff ≈ +0.56 for J = 3/2 [33]). Based on the model proposed in Ref. [58], it was anticipated that the DC configuration will lead to strong MOT confining forces for BaH molecules [33]. However, as can be seen from Fig. 7(a), a complex interplay between the Zeeman shifts for the ground and excited magnetic sublevels contributes to a relatively weak confining force with an undesirable spatial structure and a small effective velocity range ( Fig. 7(b)). It was previously shown that for molecules with small Zeeman shifts in the excited state (like CaF [58] and MgF [59]), the "dual-frequency" contribution to the MOT forces far outweighs the confining effects arising from non-zero g-factors in the excited state. As can be seen from Fig. 7, using the dual-frequency method outlined in Ref. [60] we can significantly improve the BaH MOT properties. However, in order to obtain a large velocity capture range, the use of the AC MOT configuration is necessary.
Estimation of escape and capture velocities (v esc , v cap ) are experimentally relevant ways to characterize the magneto-optical trapping potential. 3D MOT capture velocities have been measured for CaF (v cap ≈ 11 m/s) [61] and SrF (v cap ≈ 5 m/s) molecules [62] and provide useful benchmarks for our calculations. Previously it has been experimentally observed that an approximately linear relationship can be established between the MOT capture and escape velocities, v cap = bv esc , with a proportionality coefficient b 1 [63,64]. Figure 8 presents the simulated trajectories of BaH molecules that start at the geometric center of the MOT, for different initial velocities. As shown in the plotted curves, we estimate v esc for the BaH AC MOT to be ∼ 3 m/s, leading to the MOT capture velocity ∼ 3.5 m/s. 5

IX. CONCLUSION
Using a number of different methods, we have experimentally characterized optical cycling dynamics in BaH molecules. Specifically, we achieved a photon cycling rate of 8 × 10 5 s -1 , enabling radiative deflection and Sisyphus laser cooling of the BaH molecular beam.  Fig. 7(b). The time evolution for the trapped and untrapped BaH trajectories is presented in the Appendix (Fig. 12 To obtain accurate theoretical estimates for the photon scattering rate we numerically solved the master equation for time evolution of the density matrix ρ in the Lindblad form, with L being the Lindblad superoperator of the form where C i belong to a set of orthonormal operators with eigenvalues γ i and N is number of states included [66,67]. We used jump operators as our orthonormal set [68], and in our case the dissipative part of the superoperator included only effects of spontaneous emission: for a decay from state |i to |f with rate Γ i→f , we used G i→f = Γ i→f C i→f = Γ i→f |f i|. In our calculations we included both spin-rotational manifolds J = 1/2 and J = 3/2 in the v = 0 vibrational state of the X 2 Σ + ground electronic state with all the hyperfine levels and the J = 1/2 rotational manifold of the A 2 Π 1/2 excited electronic state. We have also assumed no decay into higher vibrational states (F 00 ≈ 1). Having set up the equations, we performed an optimization of the average scattering rate over the experimental interaction time T , where the sum is over all excited states decaying with the same rate Γ. The optimization was performed with respect to the Rabi rate for the |X 2 Σ + ; J = 1/2 to A 2 Π 1/2 ; J = 1/2 transition, Rabi rate for the |X 2 Σ + ; J = 3/2 to A 2 Π 1/2 ; J = 1/2 transition, detunings of both transitions, polarization of the light fields, and the background magnetic field responsible for dark state remixing. Given the experimental constraints, we found the maximum achievable average scattering rate of Γ ≈ Γ/5.21, which agrees well with the highest scattering rate we achieved in the experiment (Sec. V).
In Fig. 9 we show the average scattering rate obtained in the simulations as a function of the Rabi rate and detuning of the J = 3/2 laser. We observe that the scattering rate is highest for relatively large Rabi rates, which can be expected since the excitation rates have to match remixing rates in order to reach optimal values [69]. We also see that, because of the nature of our coupling scheme where we effectively create a Λ-type system with many more ground states than excited states, having both lasers on resonance is detrimental to achieving high scattering rates.  generally using the following Hamiltonian, with the spin-rotation constant γ SR and hyperfine constants b, c taken from our previous measurements [33]. Figure 10 shows the results of the Hamiltonian in Eq. B1 diagonalized for a range of fields between 0 and 100 G. As can be seen from Fig. 10(a), a linear approximation for the m F Zeeman sublevels is valid for fields up to ∼ 30 G for the |N = 1, J = 3/2 manifold, while because of the small hyperfine splitting of the |N = 1, J = 1/2 sublevels, we use a linear approximation for the m J Zeeman sublevels.  Fig. 11 can be compared to experimental observations in Fig. 5(b). Notice the qualitative agreement between the blue curves in both figures, as we observe peak beam center shift at detuning of 0, ±20 MHz.
We also detected beam width change around the -20 MHz detuning, consistent with the calculated acceleration curve. However, we do not observe consistent width change arising from the Doppler forces (modelled by the rate equations) around zero detuning, perhaps due to cancelling out between the Doppler and Sisyphus effects. ms [47]. In order to understand the time evolution of BaH trajectories inside a magnetooptical potential for different initial velocities, we plotted both trapped and untrapped BaH trajectories in Fig. 12 with the time information provided as a color gradient. In order to minimize the loss of molecules to dark rotational sublevels due to spontaneous vibrational decay, one would ideally transfer BaH molecules from a MOT into a conservative magnetic Color gradient scale provides time information not available in Fig. 8.