Accurate prediction and measurement of vibronic branching ratios for laser cooling linear polyatomic molecules

Recent years have seen rapid progress in producing cold molecules and in using them to study molecular interactions and reactions,^1–3 quantum information processing and computation,^4–10 and precision measurement in search of new physics beyond the standard model (BSM).^11–14 Direct laser cooling and trapping has emerged as a particularly promising approach^5,15–30 to make molecules ultracold, as required by much of the current and future research in these areas. While molecules, compared to atoms, have wider applicability and often have higher sensitivity for BSM searches, their complex internal motions continue to pose challenges to applying the most powerful laser cooling techniques. The use of magneto-optical traps (MOTs) and very high-fidelity optical readout of internal quantum states typically rely on scattering more than 10⁴ photons;^15,16 to achieve this, a transition manifold with the overall fractional loss less than 10⁻⁴ is required.

Several diatomic molecules have been optically cycled well above the required 10⁴ photon threshold, leading to successful laser cooling^17,18 and loading into optical traps.^20,23–26 Laser cooling of polyatomic molecules, however, is more challenging. To understand the difficulty, this key metric should be considered—the quasidiagonality of the Franck–Condon matrix for the electronic laser cooling transition.^31–36 In order to quantify this metric, and thus determine whether a molecular candidate can be successfully loaded and held in a MOT, knowledge of all decays with intensity above about 10⁻⁵ is needed to ensure that a sufficient number of photons may be scattered by each molecule. Recent observations of nominally symmetry-forbidden decays^37,38 confirm that very weak decays, which can only be observed with considerable effort, can pose major obstacles to achieving an optical cycle capable of scattering >10⁴ photons per molecule. While experimental studies using high resolution laser spectroscopy^{30,37,39–43} have the potential to provide accurate branching ratios, it is a formidable endeavor to measure branching ratios smaller than 10⁻³ with current methods. The capability to calculate and measure the vibrational branching ratios accurately is thus a critical tool for developing laser cooling strategies for specific molecules.

In this Communication, we present generally applicable methods to calculate and measure vibronic branching ratios for all transitions with intensity above 10⁻⁵ in laser-coolable linear polyatomic molecules. Spin-vibronic perturbations^44–46 contribute significantly to vibronic decay pathways so that modeling their effects on laser cooling is of utmost importance. The present computational scheme performs coupled-cluster (CC)^47–50 calculations to parameterize a Köppel–Domcke–Cederbaum (KDC) multi-state quasidiabatic Hamiltonian⁵¹ including all relevant spin-vibronic perturbations and obtains vibronic levels and wave functions from discrete variable representation (DVR)^52–54 calculations. We compare the computational results to new measurements of branching ratios in CaOH and YbOH, with ∼10⁻⁵ relative intensity sensitivity. These measurements, which rely crucially on strong optical excitation, achieve a level of sensitivity with direct experimental access to the perturbations probed by our calculations. The good agreement between experiment and theory demonstrated here validates both methodologies for guiding the selection of candidate molecules for direct laser cooling and for understanding the role that weak perturbations play in the optical cycling process. It also provides clear pathways for deep laser cooling of the two studied species, for use in quantum computation and in the search of physics BSM. These methods, in general, provide an approach for assessing and understanding laser cooling of polyatomic molecules.

As shown in Fig. 1, the most commonly used optical cycle in a linear triatomic molecule of the type M-A-B, e.g., a metal hydroxide, involves the transitions from A²Π_1/2(000), the vibrational ground state of a low-lying excited electronic state, to the vibrational states of the ground electronic state X²Σ_1/2, i.e., X²Σ_1/2(000), X²Σ_1/2(100), X²Σ_1/2(010), etc. Here, $(v1v2ℓv3)$ denotes a vibrational state with v₁ quanta of excitation in the M-A stretching mode, v₂ quanta of bending excitation, and v₃ quanta of A-B stretching excitation. The superscript ℓ denotes a vibrational angular momentum due to bending excitations. X²Σ_1/2 is well separated from other electronic states. A variational vibrational calculation within the Born–Oppenheimer approximation thus can produce its vibrational levels and wave functions accurately.

In contrast, a number of small perturbations^44–46 contributes to the A²Π_1/2(000) vibronic wave function and hence to the targeted branching ratios. The Renner–Teller (RT) effects⁵⁵ and spin–orbit coupling (SOC) are well known to play important roles in the description of vibronic levels and wave functions for the A²Π states in a linear triatomic molecule.^56,57 In addition, importantly, linear vibronic coupling (LVC) and SOC between A²Π and B²Σ introduce A²Π(010) and B²Σ(010) components into the wave function of A²Π_1/2(000) and thus make important contributions to nominally symmetry-forbidden transitions to X²Σ_1/2(010)^58,59 with small but non-negligible branching factors. Therefore, aiming to obtain accurate branching factors higher than 10⁻⁵, it is necessary to calculate the A²Π_1/2(000) vibronic wave function using a multi-state Hamiltonian with all of these relevant perturbations taken into account. We thus adopt the KDC Hamiltonian widely used to account for spin–vibronic coupling in spectrum simulation.^60–70 We mention that LVC between A²Π and B²Σ borrows intensity for the B²Σ(000) → X²Σ(010) transition in a similar way and thus will be relevant when studying the B²Σ → X²Σ cycling.

We use a KDC Hamiltonian of a form

including six-scalar quasidiabatic electronic states, A²Π_x(α), A²Π_y(α), B²Σ(α), A²Π_x(β), A²Π_y(β), and B²Σ(β), as the electronic basis. Here, α and β refer to S_z = 1/2 and −1/2. $EA2Πxqd$⁠, $EA2Πyqd$⁠, and $EB2Σqd$ denote potential energies of these quasidiabatic states. V^RT’s represent RT coupling, V^LVC’s represent LVC, and h^SO’s denote SOC matrix elements. This Hamiltonian includes all the perturbations pertinent to the calculations of branching ratios for the A²Π_1/2 → X²Σ_1/2 optical cycle.

We first calculate the adiabatic potential energy surfaces (PESs) for A²Π₊ and A²Π₋, the two scalar adiabatic states of A²Π, as well as for B²Σ,

Then, we apply the following adiabatic to diabatic transformation:

where Q_x and Q_y represent the normal coordinates of the two bending modes. This transformation was derived for harmonic potentials. The application to potentials with anharmonic contributions corresponds to a quasidiabatization process. This transforms A²Π₊ and A²Π₋ into two quasidiabatic states, hereafter referred to as A²Π_x and A²Π_y, and the adiabatic potentials in Eq. (2) into the quasidiabatic potentials

Next, we include LVC using the linear diabatic coupling constants⁷¹ between A²Π_x(y) and B²Σ. This also introduces corrections to $Ẽqd$⁠, $ṼRT$⁠, and $EB2Σad$⁠,⁷¹ leading to

Finally, we augment Eq. (5) with SOC and obtain Eq. (1), hereby completing the construction of the quasidiabatic Hamiltonian.

We used the CFOUR program^49,72–75 to calculate all parameters in the KDC Hamiltonian including the adiabatic PESs, linear diabatic coupling constants,⁷¹ and SOC matrix elements.^76–78 The present study used the equation-of-motion (EOM) electron attachment CC singles and doubles (EOMEA-CCSD) method⁷⁹ and correlation-consistent basis sets^80–84 to provide accurate PESs around the equilibrium structures pertinent to the calculations of the low-lying vibronic states involved in laser cooling. We accounted for scalar-relativistic effects for SrOH and YbOH using the spin-free exact two-component theory in its one-electron variant (SFX2C-1e).^85,86 Having the quasidiabatic Hamiltonians expanded in “real space” basis sets, we carried out DVR⁵³ calculations in the representation of the four vibrational normal coordinates to obtain vibronic levels and wave functions. The normal coordinate DVR calculations previously produced accurate vibrational levels for the X²Σ state of YbOH.⁴³ We then combine the Franck–Condon overlap integrals with the EOMEA-CCSD electronic transition dipole moments to obtain the branching factors. We refer the readers to the supplementary material for more details about the computations.

Branching ratios for CaOH and YbOH were recorded using dispersed laser-induced fluorescence (DLIF) measurements. The sensitivity achieved was improved by approximately two orders of magnitude over previous measurements in similar species,^41–43,92 primarily due to the use of bright cryogenic molecular beams and photon cycling described below and in the supplementary material.

In brief, cryogenic buffer gas beams of CaOH and YbOH were produced using the beam source described in Ref. 14. Laser-excitation-enhanced chemical reactions between metallic Yb or Ca and H₂O vapor^93–95 increased the molecular beam flux by a factor of ∼10 compared to previous reports.^14,27 The apparatus was modified with a spectrometer similar to that used in Ref. 38. After propagating ∼30 cm downstream from the source, the molecules interacted with laser beams addressing the rotationally resolved $Ã2Π1/2(000)←X̃2Σ+(000)$ and $Ã2Π1/2(000)←X̃2Σ+(100)$ transitions used for optical cycling in laser cooling experiments.^14,27 In this detection region, each molecule scattered, on average, 50–100 photons. Using the optical cycling transitions therefore directly increased the number of photons collected and thus the sensitivity, compared to typical DLIF measurements where molecules scatter, on average, only a single photon.⁴¹ 

The molecular fluorescence was collected and dispersed on a 0.67 m Czerny–Turner style spectrometer, before being detected with an electron-multiplying charged-coupled device (EMCCD). The wavelength and intensity axes were carefully calibrated in separate measurements (see the supplementary material). See the supplementary material for more details on the experimental apparatus, calibration, and data analysis. A representative DLIF spectrum recorded for YbOH is shown in Fig. 2, demonstrating the sensitivity achieved by this method.

The computed vibronic energy levels for SrOH and CaOH, as summarized in Table I, show excellent agreement with measured values, with discrepancies lower than 10 cm⁻¹. The calculations also accurately reproduced the splittings between (02²0) and (02⁰0) and among the four A²Π(010) states. The present KDC Hamiltonian calculations thus accurately describe the vibronic wave functions. We provide a complete list of computed vibronic levels for the X²Σ, A²Π, and B²Σ states in the supplementary material.

Tables II and III summarize the calculated and measured vibrational branching ratios for decay from A²Π_1/2(000) to the X²Σ state in CaOH and YbOH. All branching ratios above 10⁻⁵ are shown; computed branching to the O–H stretch mode is below 10⁻⁶ and therefore negligible. The present level of consistency between computations and measurements for CaOH and YbOH, i.e., excellent agreement for stronger transitions and qualitative agreement for weaker transitions below 0.1%, is very promising. In CaOH, although branching ratios for the origin transitions are as large as ∼95%, one has to take into account up to (300) to saturate the Ca–O stretch branching ratios to below 10⁻⁵. In YbOH, the branching ratios for the Yb–O stretch are less diagonal and (400) is relevant.

The nominally symmetry-forbidden $Δv2=1,3,5,…$ transitions borrow intensities through vibronic and/or spin–orbit coupling.^37,58,59 As shown in Fig. 3, the “direct vibronic coupling” (DVC) mechanism mixes B²Σ(010) into the A²Π_1/2(000) wave function through LVC. The “spin–orbit-vibronic coupling” (SOVC) mechanism couples B²Σ(000) with A²Π(000) through SOC and then mixes A²Π_3/2(010) into the A²Π_1/2(000) wave function through LVC. A perturbative analysis gives the following ratio between the SOVC and DVC contributions to the intensity:

where $hAXdip$ and $hBXdip$ are the A-X and B-X electronic transition dipole moments. The percentages of the DVC contributions obtained using the B²Σ(010) and A²Π(010) components in the computed A²Π_1/2(000) wave function amount to 98%, 89%, and 70% for CaOH, SrOH, and YbOH, respectively, which are consistent with the values of 99%, 92%, and 76% obtained from the perturbative analysis using Eq. (6). The DVC channel dominates this intensity borrowing process even in molecules containing heavy atoms since the effects of large $hBASO$ are offset by large $hAASO$ in the denominator.

It is desirable to minimize the coupling between X²Σ(100) and X²Σ(020) in designing laser-coolable linear triatomic molecules. As shown in Table II, the intensities of the transitions to X²Σ(020) in SrOH are significantly lower than in CaOH. This is due to the larger energy separation between the X²Σ(100) and X²Σ(020) levels in SrOH (534 and 702 cm⁻¹) compared to CaOH (611 and 694 cm⁻¹), and hence weaker coupling. Similarly, there is non-negligible vibrational branching to X²Σ(120) and X²Σ(220) of CaOH; in contrast, the branching ratio to X²Σ(120) in SrOH is lower than 10⁻⁶.

We have presented a computational and experimental scheme to determine vibronic branching ratios pertinent to scattering around 100 000 photons in linear polyatomic molecules. The computational methodology is generally applicable to the study of optical cycles for linear triatomic molecules, such as metal hydroxides and isocyanides proposed for laser cooling. We have found good agreement with new experimental measurements that used optical cycling to achieve relative intensity sensitivities at the 10⁻⁵ level. The agreement supports the accuracy of both theoretical and experimental techniques.

Further work for the future includes the study of the accuracy for the parameterization of the quasidiabatic Hamiltonian, including the higher-order correlation effects on PESs, the effects of quadratic vibronic coupling, and rovibrational coupling, aiming to obtain quantitative accuracy for weak transitions. It is also of interest to study the transitions involving dark electronic states^26,28,96,97 in linear triatomic molecules, such as BaOH. The generalization to calculations of linear tetraatomic molecules, e.g., metal monoacetylides, should also provide very interesting results. The calculation of nonlinear molecules is the next frontier, and we assess that we can follow the same generic scheme although the perturbations relevant to the calculations will be molecule specific. The computational framework presented here thus forms a basis for the calculation of vibronic levels and branching ratios that are sufficiently accurate and complete to facilitate and guide laser cooling experiments for even highly complex polyatomic molecules.

The supplementary material contains details of experimental measurements and computations for the sake of self-containedness and a complete account of computed vibronic levels.

C.Z. thanks Yifan Shen (Baltimore) for stimulating discussions. L.C. is grateful to Paul Dagdigian (Baltimore) and John Stanton (Gainesville) for reading the manuscript and for helpful comments. This computational work at the Johns Hopkins University was supported by the National Science Foundation, under Grant No. PHY-2011794. All calculations were carried out at the Maryland Advanced Research Computing Center (MARCC). CaOH measurements were supported by the AFOSR and NSF. YbOH measurements were supported by the Heising-Simons Foundation, the Gordon and Betty Moore Foundation, and the Alfred P. Sloan Foundation. N.B.V. acknowledges support from the NDSEG fellowship.

The data that support the findings of this study are available within this article and its supplementary material.