Optical cycling, radiative deflection and laser cooling of barium monohydride (¹³⁸Ba¹H)

Various approaches have been developed for controlling atomic motion [1–3]. 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 desirable for repeated photon scattering [10].

In order to extend laser cooling to molecules, the use of quasi-closed cycling vibrational 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 continuous-wave 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 [14–17]. 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) characterization of optical cycling (∼10–80 photons) using ground vibrational state depletion, (ii) deflection of molecules with radiation pressure force (∼80 photons), and (iii) laser compression of a molecular beam along one transverse dimension (≳100 photons). Our observations clearly show spatial manipulation of the BaH molecular beam, and demonstrate understanding of the optical cycling process out to several hundred 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 × 10⁶ s⁻¹), a molecular class frequently encountered in fundamental physics applications [27, 30, 31].

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⁻¹) together with a low possible scattering rate (R_scat ≲ 1 × 10⁶ s⁻¹) require a use of a slow cryogenic buffer-gas beam [32] in order to provide enough molecule-laser 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 Δv_rad ∼Γ_sp/k ≈ 1 m s⁻¹ 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 Δv_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 (figure 1) broadband anti-reflection coated for a reflectivity of r_window < 0.5% per surface and out-of-vacuum mirrors with broadband dielectric high-reflectivity coatings with r_mirror > 99.8% for 600 nm–700 nm and 1000–1100 nm spectral regions, enough to cover all wavelengths needed for optical cycling. 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. However, for the experiments presented here, the laser intensity stays high enough to not limit the scattering rate. This is confirmed experimentally because we do not see a deviation from constant scattering rate, even after many passes as shown in figure 3(c). 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.

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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 alkaline-earth-metal like atom (Ca, Sr or Yb) ionically and monovalently bonded to an electronegative fluorine (F) [35–37] or hydroxyl (OH) [38–40] ligand. Such ligands with a moderate electron-withdrawing 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 (MCH₃) 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) X²Σ⁺ ↔ A²Π_1/2 at 1061 nm (τ_sp ≈ 140 ns, Doppler cooling limit T_D ≈ 30  μK), (ii) X²Σ⁺ ↔ B²Σ⁺ at 905 nm (τ_sp ≈ 130 ns, T_D ≈ 30  μK), and (iii) X²Σ⁺ ↔ H²Δ_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 (${\mathcal{F}}_{{v}^{{\prime\prime}}{v}^{\prime }}$) for the X²Σ⁺ ← A²Π_1/2 electronic decay³ (${\mathcal{F}}_{00}{ >}0.987$ [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 B²Σ⁺ excited electronic state (figure 2(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.

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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 ΔN = −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 ²Σ⁺ electronic states scale as ${\omega }_{\text{vib}}\propto {\mu }_{\text{red}}^{-1/2}$, ${B}_{\text{rot}}\propto {\mu }_{\text{red}}^{-1}$ and ${\gamma }_{\text{SR}}\propto {\mu }_{\text{red}}^{-1}$ 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 (figures 2(b) and (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 $\left(0,0\right){X}^{2}{{\Sigma}}^{+}\to {A}^{2}{{\Pi}}_{1/2}$ transition⁴, 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 (figure 2(b)). For the $\left(0,1\right){X}^{2}{{\Sigma}}^{+}\to {B}^{2}{{\Sigma}}^{+}$ 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 X²Σ⁺ → E²Π_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 E²Π_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.

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 section 3, 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 the $\left(0,0\right){X}^{2}{{\Sigma}}^{+}{\leftrightarrow}{A}^{2}{{\Pi}}_{1/2}$ scattering rate is shown in figure 3(a): the $\left(0,1\right){X}^{2}{{\Sigma}}^{+}\to {B}^{2}{{\Sigma}}^{+}$ repumping light (orange) was blocked, and the number of passes of the $\left(0,0\right){X}^{2}{{\Sigma}}^{+}\to {A}^{2}{{\Pi}}_{1/2}$ laser (blue) was varied from zero to ∼20 passes. 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 and we integrate the signal for all arrival times to estimate the total X²Σ⁺(v'' = 0) population. These depletion measurements were performed with a total power of 100 mW (30 mW for repumping) divided evenly between each hyperfine component. Accounting for the beam forward velocity (160 ± 40 m s⁻¹ [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 figures 3(c) and (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 $\left({N}_{\text{scat}}\right)$ based on the fraction of molecules that remain in ground vibrational state (P_v=0), and the known diagonal FCF ${\mathcal{F}}_{00}$ from previous measurements [47, 53]. Figure 3(b) provides a representative ToF data for an unperturbed BaH beam (blue) and with the $\left(0,0\right){X}^{2}{{\Sigma}}^{+}\to {A}^{2}{{\Pi}}_{1/2}$ 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 (∼10⁶ s⁻¹).

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Following Di Rosa [12], we model the repeated spontaneous emission events by a molecule as a Bernoulli sequence with probability $p=1-{\mathcal{F}}_{00}$ 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:

$$\begin{equation}{P}_{{v}^{{\prime\prime}}=0}={\left({\mathcal{F}}_{00}\right)}^{{N}_{\text{scat}}}.\end{equation} \tag{ 1 }$$

Therefore, we can convert the fraction P_v''=0 into the number of scattered photons for the remaining molecules:

$$\begin{equation}{N}_{\text{scat}}=\frac{\mathrm{log}\enspace {P}_{{v}^{{\prime\prime}}=0}}{\mathrm{log}\enspace {\mathcal{F}}_{00}}.\end{equation} \tag{ 2 }$$

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

$$\begin{equation}\langle {N}_{\text{scat}}\rangle =\frac{1}{1-{\mathcal{F}}_{00}}\approx 80,\end{equation} \tag{ 3 }$$

and the standard deviation in N_scat of

$$\begin{equation}{\sigma }_{{N}_{\text{scat}}}=\frac{\sqrt{{\mathcal{F}}_{00}}}{1-{\mathcal{F}}_{00}}\approx 80.\end{equation} \tag{ 4 }$$

Equation (2) allows us to estimate the photon scattering rate for the molecules remaining in v'' = 0 as a function of the laser interaction time, N_scat = R_scatt_int. However, as shown in equations (3) and (4), for any specific molecule there is a wide range of actual number of scattering events, meaning this setup would not be efficient for optical manipulation, as a wide range in transverse momentum would be imparted. As shown in figure 3(c), a constant scattering rate of 1.4(1) × 10⁶/s can be maintained, which is ∼80% of the expected maximum scattering rate R_scat,max based on the ground and excited state multiplicities,

$$\begin{equation}{R}_{\text{scat,max}}=\frac{1}{{\tau }_{\text{sp}}}\frac{{n}_{\text{e}}}{{n}_{\text{e}}+{n}_{\text{g}}}=1.8{\times}1{0}^{6}/\mathrm{s},\end{equation} \tag{ 5 }$$

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 equation (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 (figure C.1) as well as an optimized numerical simulation of the full system using the Lindblad master equation⁵ predict R_OBE ≈ 1.4 × 10⁶ s⁻¹. The data presented in figure 3(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 $\left(1,0\right){X}^{2}{{\Sigma}}^{+}{\leftarrow}{A}^{2}{{\Pi}}_{1/2}$ being the dominant loss mechanism out of the quasi-cycling transition. As shown in figure 3(b), with an addition of the $\left(0,1\right){X}^{2}{{\Sigma}}^{+}\to {B}^{2}{{\Sigma}}^{+}$ repumping laser in the 'clean-up' region, we return most of the molecules (green curve) back into the ground vibrational level. The ${B}^{2}{{\Sigma}}^{+}\left({v}^{\prime }=0\right)$ state has a good Franck–Condon overlap with ${X}^{2}{{\Sigma}}^{+}\left({v}^{{\prime\prime}}=0\right)$ (${\mathcal{F}}_{00}$ = 0.953 [47]) so molecules excited to this state decay to the desired ground state $\left\vert {v}^{{\prime\prime}}=0,{N}^{{\prime\prime}}=1\right\rangle$ with a 95% probability. In order to determine the photon scattering rate for the $\left(0,1\right){X}^{2}{{\Sigma}}^{+}\to {B}^{2}{{\Sigma}}^{+}$ repumping transition, we begin by depleting v'' = 0 on the main cycling transition $\left(0,0\right){X}^{2}{{\Sigma}}^{+}\to {A}^{2}{{\Pi}}_{1/2}$ at the maximum interaction time. Then in a separate region (figure 3(a)) we apply the repumping light, while varying the number of passes through the molecular beam. Because ${\mathcal{F}}_{00}\approx 1$ for the $\left(0,0\right){X}^{2}{{\Sigma}}^{+}{\leftarrow}{B}^{2}{{\Sigma}}^{+}$ transition, it will only take $1/{\mathcal{F}}_{00}\approx 1$ photon scattered from the repumping laser to optically pump a molecule back to v'' = 0. As shown in figure 3(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 ${N}_{\text{scat}}={N}_{\text{offset}}\left(1-\mathrm{exp}\left\{-{R}_{\text{scat}}{t}_{\mathrm{int}}\right\}\right)$, which becomes N_scat ≈ N_offsetR_scatt_int for t_int ≪ 1/R_scat and allows us to extract a scattering rate of 1.3(2) × 10⁵ s⁻¹. Note that an offset (N_offset) is included in the fit to account for imperfect alignment of the CU laser and the depletion laser in figure 3(d), resulting in incomplete vibrational repumping. For perfect alignment, we would expect N_offset = 1, and for the fit shown, we obtain N_offset = 0.9 ± 0.1, indicating that we repump almost all molecules. Since R_scat ∝ σ_absI₀ and the resonant absorption cross section depends on the corresponding FCF ${\sigma }_{\text{abs}}\propto {\mathcal{F}}_{{v}^{{\prime\prime}}{v}^{\prime }}$, the scattering rate will be lower for the off-diagonal transition for a given laser intensity I₀. However, using the experimentally measured R_scat for the main cycling $\left(0,0\right){X}^{2}{{\Sigma}}^{+}\to {A}^{2}{{\Pi}}_{1/2}$ excitation, together with the estimate of the off-diagonal FCF ${\mathcal{F}}_{01}\approx 0.012$, we determine that the rate of optical pumping into the excited vibrational level v'' = 1 in the optical cycling region (∼1.7 × 10⁴ 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.

The depletion-based scattering measurements described in section 4 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 $\left(0,0\right){X}^{2}{{\Sigma}}^{+}\to {A}^{2}{{\Pi}}_{1/2}$ main and $\left(0,1\right){X}^{2}{{\Sigma}}^{+}\to {B}^{2}{{\Sigma}}^{+}$ repumping lasers (figure 4(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 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 (figure 4(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 section 4. This experiment was performed with the same beam size and power as the depletion experiments in section 4.

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Figures 4(c) and (d) summarize the radiative deflection experiments for BaH. The magnitude of the displacement is determined by fitting the molecular beam images to a 2D Gaussian distribution and monitoring the fitted center of the molecular beam with and without the deflection light. Quoted error bars are determined based on the statistical fit uncertainty. In figure 4(d) we observe that the deflected beam appears ∼10% broader, potentially indicating that a fraction of detected molecules scatter a higher number of photons. By using the change in the fitted center of the whole molecular beam we provide a conservative lower bound on the number of scattered photons per molecule. The measured center-of-mass molecular beam deflection is ∼1 mm, which is consistent with each molecule scattering ∼80 photons at the average rate of 8 × 10⁵ photons/s. A number of experimental differences could explain why this scattering rate is 40% slower than that measured via depletion experiments: (i) slower than anticipated repumping in the main region, (ii) the presence of transverse Doppler shifts, and (iii) the presence of imperfectly remixed 'dark' states which do not play a dominant role in depletion experiments. Importantly, we do not see a significant increase in the width of the molecular beam (figure 4(d)), indicating consistent scattering for each detected molecule. The scattering rate of 8 × 10⁵ 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.

While laser deflection results presented in section 5 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 figure 4(d)). To achieve a reduction in transverse velocity spread for the molecular beam, we establish a 1D standing light wave intersecting the molecular beam (figure 5(a)). Figure 5(b) demonstrates effective transverse temperature of the molecular beam as a function of the common detuning for the $\left(0,0\right){X}^{2}{{\Sigma}}^{+}\to {A}^{2}{{\Pi}}_{1/2}$ cooling laser with the repumping $\left(0,1\right){X}^{2}{{\Sigma}}^{+}\to {B}^{2}{{\Sigma}}^{+}$ 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].

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To estimate the temperature of the beam we performed Monte Carlo simulations of the molecular beams spatial propagation. We begin with molecules uniformly sampled across a 5 mm radius circle (the size of the collimation aperture in the experiment). We then give each molecules a transverse velocity based on a fixed temperature, and propagate them from the collimation aperture, through the interaction region where they feel the optical force, and into the detection region at a typical forward velocity. Based on the width of the distribution at the detection region, relative to the 5 mm width defined at the aperture, we can extract the effective transverse temperature. For example, if the transverse temperature is effectively zero, the beam will not expand at all, and will remain 5 mm wide. This allows us to map a relationship between a measured width, and 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 figure 5(b). This experiment and simulations were performed with the same beam size and power as the depletion experiments in section 4. 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 the at the experimentally determined optimal detuning of 5 MHz the magnitude of the damping coefficient for the Sisyphus force is ∼5.6 ± 1.7 times greater than Doppler, with error dominated by fit uncertainty. The experimental power and optimal detuning corresponds to a damping coefficient with a magnitude of 330 ± 100 s⁻¹ for Sisyphus cooling.

While attempting to observe Doppler cooling of the BaH beam by reducing the laser intensity and increasing the collimating aperture size to allow for a broader transverse velocity class to enter the cooling region, we have observed a systematic shift in the molecular beam position as shown in figure 6. A small natural linewidth of the cooling transition (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 (figure 6(a)). The asymmetric Doppler shifts lead to a unidirectional deflection of the molecular beam in the cooling configuration (figure 6(b)), where the direction of the deflection depends on the alignment angle as shown in the data in figure 6(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 figure 6(d). This model uses a realistic Rabi rate for our experiment, and realistic distribution of forward velocity. Force imbalance (ΔF) is based on the detuning dependent force from the left F_l(δ) (right F_r(δ)) propagating laser beam:

$$\begin{equation}{\Delta}F=\frac{\mathrm{max}\left({F}_{\text{l}}\left(\delta \right)\right)-\mathrm{max}\left({F}_{\text{r}}\left(\delta \right)\right)}{\mathrm{max}\left({F}_{\text{l}}\left(\delta \right),{F}_{\text{r}}\left(\delta \right)\right)}.\end{equation} \tag{ 6 }$$

We see that even small angular misalignment can lead to large imbalance in the maximum force pushing the beam to either direction. As shown in figure 6(d) an imbalance as large as 60% is possible for even a small misalignment of 1 degree. This is consistent with the data showing a 0.6 mm deflection in figure 6(c), and the maximum deflection of 1 mm we observed in figure 4.

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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 reference [56] and later used to model the MgF MOT properties [57]. The simulation included n_l = 12 magnetic sublevels of the N'' = 1 rotational manifold in the vibronic ground state (figure 2(b)), n_u = 4 magnetic sublevels of the J' = 1/2 manifold (F' = 0, 1) of the v' = 0 vibrational level of the excited A²Π_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 detunings⁶ [56]. 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 [58].

Following earlier work [56–58], we present the strength of the achievable BaH magneto-optical trapping configurations by plotting molecular acceleration as a function of (a) the distance to the trap center for the v = 0 velocity class, and (b) as a function of the velocity for a fixed position slightly displaced from the trap center (d = 0.1 mm), in order to have a well defined quantization axis set by the direction of the magnetic field.

Figure 7 provides a summary of the predicted acceleration profiles as a function of distance to the MOT center (for v = 0 m s⁻¹) and molecular velocity (for d = 0.1 mm) under experimental conditions approximating those in the experiments above (P_tot = 200 mW and 1-inch 1/e²-diameter laser beams) and previously achieved in molecular MOT experiments (15 G cm⁻¹ magnetic field gradient [16]). By comparing different MOT operation parameters (varying the number of laser frequencies and polarization settings) and configurations (AC vs DC vs DC dual-frequency), we conclude that the AC MOT has the highest potential for future trapping of BaH molecules providing both the highest peak deceleration (∼2 km s⁻²) and the largest velocity range affected by the MOT potential (up to ∼15 m s⁻¹). We determine that the optimal polarization setting for the AC MOT is the same as that used for capturing CaF molecules [16].

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A unique property of BaH that distinguishes it from other molecules to which magneto-optical 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 A²Π_1/2 state [33]), arising from a strong mixing with the adjacent B²Σ⁺ 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 reference [56], it was anticipated that the DC configuration will lead to strong MOT confining forces for BaH molecules [33]. However, as can be seen from figure 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 with a repulsive trap center and a small effective velocity range (figure 7(b)). It was previously shown that for molecules with small Zeeman shifts in the excited state (like CaF [56] and MgF [57]), 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 figure 7, using the dual-frequency method outlined in reference [58] 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⁻¹) [59] and SrF (v_cap ≈ 5 m s⁻¹) molecules [60] 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 [61, 62]. 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⁻¹, leading to the MOT capture velocity ∼3.5 m s⁻¹.⁷

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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 × 10⁵ s⁻¹, enabling radiative deflection and Sisyphus laser cooling of the BaH molecular beam. These results validate the feasibility of laser cooling and trapping heavy molecules with relatively long-lived excited electronic states, despite previously unexplored technical challenges. In order to guide future work on laser slowing and magneto-optical trapping, we have carried out realistic estimations of MOT forces, pinned to the experimental benchmarks that we have already achieved. While our results demonstrate that a large Zeeman shift in the excited electronic state does not lead to a strong confining potential, an AC configuration of a magneto-optical trap has potential to capture BaH molecules with velocities ≲3 m s⁻¹.

We would like to acknowledge helpful discussions with B L Augenbraun and Z Lasner Additionally, we thank Qi Sun for experimental assistance. This work was supported by the ONR Grant N00014-17-12246 and the W M Keck Foundation. RLM and KW gratefully acknowledge support by the NSF IGERT Grant No. DGE-1069240. IK has been supported by the Simons Junior Fellow Award.

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,

$$\begin{equation}\frac{\mathrm{d}\rho \left(t\right)}{\mathrm{d}t}=\mathcal{L}\rho \left(t\right),\end{equation} \tag{ A1 }$$

with $\mathcal{L}$ being the Lindblad superoperator of the form

$$\begin{equation}\mathcal{L}\rho \left(t\right)=-i\hslash \left[H,\rho \right]+\sum _{i=1}^{{N}^{2}-1}{\gamma }_{i}\left({C}_{i}\rho {C}_{i}^{{\dagger}}-\frac{1}{2}\left\{{C}_{i}^{{\dagger}}{C}_{i},\rho \right\}\right),\end{equation} \tag{ A2 }$$

where C_i belong to a set of orthonormal operators with eigenvalues γ_i and N is number of states included [64, 65]. We used jump operators as our orthonormal set [66], and in our case the dissipative part of the superoperator included only effects of spontaneous emission: for a decay from state $\left\vert i\right\rangle$ to $\left\vert f\right\rangle$ with rate Γ_i→f, we used ${G}_{i\to f}=\sqrt{{{\Gamma}}_{i\to f}}{C}_{i\to f}=\sqrt{{{\Gamma}}_{i\to f}}\left\vert f\right\rangle \left\langle i\right\vert$.

In our calculations we included both spin-rotational manifolds J = 1/2 and J = 3/2 in the v'' = 0 vibrational state of the X²Σ⁺ ground electronic state with all the hyperfine levels and the J' = 1/2 rotational manifold of the A²Π_1/2 excited electronic state. We have also assumed no decay into higher vibrational states (${\mathcal{F}}_{00}\approx 1$). Having set up the equations, we performed an optimization of the average scattering rate over the experimental interaction time T,

$$\begin{equation}\overline{{\Gamma}}=\sum _{i}\frac{1}{T}{\int }_{0}^{T}{\rho }_{{e}_{i}{e}_{i}}\left(t\right){\Gamma}\enspace \mathrm{d}t,\end{equation} \tag{ A3 }$$

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 $\left\vert {X}^{2}{{\Sigma}}^{+};J=1/2\right\rangle$ to $\left\vert {A}^{2}{{\Pi}}_{1/2};J=1/2\right\rangle$ transition, Rabi rate for the $\left\vert {X}^{2}{{\Sigma}}^{+};J=3/2\right\rangle$ to $\left\vert {A}^{2}{{\Pi}}_{1/2};J=1/2\right\rangle$ 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 $\overline{{\Gamma}}\approx {\Gamma}/5.21$, which agrees well with the highest scattering rate we achieved in the experiment (section 5).

In figure A.1 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 [67]. 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.

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In order to develop a proper theoretical model of the magneto-optical trapping forces for molecules, an accurate approximation of the molecular energy shifts with applied magnetic fields (Zeeman shifts) is necessary. Zeeman shift measurements for BaH have been previously reported [33] in the intermediate magnetic field range (50–100 G). Using the extracted hyperfine and spin-rotation parameters, we can estimate the Zeeman energy levels more generally using the following Hamiltonian,

$$\begin{equation}H={\gamma }_{\text{SR}}\mathbf{N}\cdot \mathbf{S}+b\mathbf{I}\cdot \mathbf{S}+c{I}_{Z}{S}_{Z}+{\mu }_{\text{B}}{g}_{S}\mathbf{B}\cdot \mathbf{S},\end{equation} \tag{ B1 }$$

with the spin-rotation constant γ_SR and hyperfine constants b, c taken from our previous measurements [33]. Figure B.1 shows the results of the Hamiltonian in equation (B1) diagonalized for a range of fields between 0 and 100 G. As can be seen from figure B.1(a), a linear approximation for the m_F Zeeman sublevels is valid for fields up to ∼30 G for the $\left\vert {N}^{{\prime\prime}}=1,\enspace {J}^{{\prime\prime}}=3/2\right\rangle$ manifold, while because of the small hyperfine splitting of the $\left\vert {N}^{{\prime\prime}}=1,\enspace {J}^{{\prime\prime}}=1/2\right\rangle$ sublevels, we use a linear approximation for the m_J Zeeman sublevels.

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In order to better understand the laser cooling process accompanying the observed beam deflection signature described in section 6, we solved the rate equation model with the three EOM sidebands spaced by 20 MHz. Results presented in figure C.1 can be compared to experimental observations in figure 6(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.

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Large vibrational spacing for diatomic monohydrides (MH) compared to monofluorides (MF) and monohydroxides (MOH) will lead to enhanced vibrational decay from the v'' = 1 levels populated during the MOT loading process. Previous accurate ab initio calculations for BaH predict spontaneous vibrational lifetime for v'' = 1, N'' = 1 state of τ_vib = 12.6 ms [47]. In order to understand the time evolution of BaH trajectories inside a magneto-optical potential for different initial velocities, we plotted both trapped and untrapped BaH trajectories in figure D.1 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 an MOT into a conservative magnetic [68, 69] or optical [70] trap within the first ∼10 ms of capturing them in an MOT.

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