Ultracold $^{88}\rm{Sr}_2$ molecules in the absolute ground state

We report efficient all-optical creation of an ultracold gas of alkaline-earth-metal dimers, $^{88}\rm{Sr}_2$, in their absolute ground state. Starting with weakly bound singlet molecules formed by narrow-line photoassociation in an optical lattice, followed by stimulated Raman adiabatic passage (STIRAP) via a singlet-dominant channel in the $(1)0_u^+$ excited potential, we prepare pure samples of more than 5500 molecules in $X^1\Sigma_g^+(v=0,J=0)$. We observe two-body collisional loss rates close to the universal limit for both the least bound and most bound vibrational states in $X^1\Sigma_g^+$. We demonstrate the enhancement of STIRAP efficiency in a magic-wavelength optical lattice where thermal decoherence is eliminated. Our results pave the way for the use of alkaline-earth-metal dimers for high-precision spectroscopy, and indicate favorable prospects for robust quantum state preparation of ultracold molecules involving closed-shell atoms, as well as molecule assembly in deep optical traps tuned to a magic wavelength.


Introduction
Molecules serve as natural test beds for molecular quantum electrodynamics [1][2][3][4], in addition to tests of beyond-Standard-Model physics such as searches for T -symmetry violation [5][6][7][8][9][10], dark matter [11][12][13], time variation of fundamental constants [14][15][16][17][18], and non-Newtonian gravity [19]. In this regard, alkaline-earth-metal molecules represent an exciting frontier since their closed-shell structure lead to 1 Σ ground potentials that are naturally insensitive to external perturbations. Furthermore, homonuclear combinations possess high-Q subradiant transitions to the excited potentials that can be utilized as in situ magnetometers for systematic calibrations. These qualities present a pristine environment for precise molecular spectroscopy as a means to probe new physics. In addition, molecular gases at cold or ultracold temperatures provide experimental benchmarks for understanding collisions and chemical reactions, where accurate theoretical descriptions that include quantum effects remain a significant computational challenge [20][21][22][23][24][25][26][27][28][29]. In these applications, often it is necessary to perform tailored state preparation and measurement of the molecules in specific rovibronic and hyperfine states.
On another front, the ability to prepare molecular ensembles with increasingly greater phase-space densities has been driven by efforts to observe novel many-body physics [30][31][32][33][34] and to realize platforms for quantum information processing [35][36][37][38][39][40][41][42][43] that take advantage of the rich rovibronic structure of molecules. In one approach to the creation of ultracold molecular ensembles, molecules are associated via optical or magnetic Feshbach resonances from laser-cooled ultracold atoms. By preserving the original atomic phase-space density, this has led to the creation of degenerate Bose [44][45][46] and Fermi [47] alkali-metal diatomic molecular gases. However, in many practical applications, it is advantageous to initialize the molecules in a deeply bound state. For example, in bi-alkalis [48][49][50][51][52][53][54][55][56][57][58][59], a common step involves de-exciting the weakly bound Feshbach molecules to the absolute ground state using stimulated Raman adiabatic passage (STIRAP) in order to access the large molecule-frame dipole moments. Despite these technical successes [60] and prior work on weakly-bound molecules of Sr 2 [61][62][63] and Yb 2 [64], the production of ultracold molecules consisting of closed-shell atoms in the absolute ground state has yet to be demonstrated.
In this paper, we extend the robust state control offered by STIRAP to the entire ground potential of 88 Sr 2 , a homonuclear alkaline-earth-metal molecule. This paper is organized into three main parts. First, we describe the spectroscopy and transition strength measurements of the (1)0 + u and X 1 Σ + g potentials in order to identify a feasible STIRAP pathway for adiabatic transfer within X 1 Σ + g . Next, as proof of concept, we experimentally perform STIRAP to create absolute-ground-state molecules and investigate factors that limit the transfer efficiencies both in free flight and in a magicwavelength optical lattice. Finally, we study the lifetime of the absolute-ground-state molecules and measure the two-body inelastic loss rates to extend our understanding of ultracold collisions, which may potentially inform the feasibility of producing stable molecular Bose-Einstein condensates [65][66][67]. We expect that the techniques described herein can be generalized to molecules consisting of at least one closed-shell atom [68][69][70][71][72][73], and to other molecular species in deep optical traps such as tweezers and lattices [74][75][76][77][78][79][80].
This prepares the atoms in the 1 S 0 electronic ground state at approximately 2(1) µK as inferred from a time-of-flight ballistic expansion. During the narrow-line MOT cooling stage, the atoms are overlapped with a one-dimensional optical lattice with a typical trap depth of 50 µK where they remain optically trapped after the MOT coils are ramped off. In this work, the light for the lattice is derived from a Ti:sapphire laser. To produce weakly bound molecules in the electronic ground potential, X 1 Σ + g , we photoassociate the atomic sample with a 1.5 ms laser pulse to (1)0 + u (v = −4, J = 1) where a sizable fraction subsequently decays down to the least bound vibrational states X(62, 0) and X(62, 2) [63]. In a few instances, we alternatively photoassociate to (1)0 + u (−5, 1) to produce X(61, 0) and X(61, 2).
The nuclear spin of 88 Sr is I = 0, resulting in the absence of hyperfine structure. Spin statistics forces the molecular wavefunction to be even upon the exchange of the bosonic nuclei in 88 Sr 2 . This implies that only even J (total angular momentum of the molecule) are allowed in the gerade X 1 Σ + g , and only odd J are allowed in the ungerade (1)0 + u . Since the total number of bound states in a given potential is not always known a priori, in some cases we use negative values for v (the vibrational quantum number) that count down from the dissociation threshold of the respective potentials.
While we typically work with nearly equal mixtures of J = 0, 2 ground state molecules, we can perform a purification step that clears away J = 2 molecules. This is done by photodissociating J = 2 molecules to the 1 S 0 + 3 P 1 threshold with an additional laser for 1 ms, and blasting the resulting atoms out of the trap with 461 nm laser light near-resonant with 1 S 0 -1 P 1 . In either case, we prepare ∼ 10 4 molecules in J = 0 to serve as our spectroscopic signal. We absorption-image the atomic fragments on 1 S 0 -1 P 1 after photodissociating J = 0 molecules above the 1 S 0 + 3 P 1 threshold with a 120 µs pulse. While we usually perform absorption imaging at a slight grazing angle along the lattice direction to maximize the optical depth (and hence the signal-to-noise ratio), for calibration purposes we also take images perpendicular to the lattice direction from which we deduce that the molecules typically fill ∼570 lattice sites.
We use diode lasers for both the pump and anti-Stokes STIRAP lasers. The pump laser is stabilized using the Pound-Drever-Hall technique to a high-finesse cavity (F > 10 5 , ultralow expansion glass). This phase stability is transferred to the frequency comb spectrum of a low-noise erbium-doped femtosecond fiber laser (FC1500-250-ULN, Menlo Systems GmbH) by referencing the repetition rate to the pump laser. By phase locking the anti-Stokes to the comb, the relative frequency stability of the pump and anti-Stokes is maintained to < 1 kHz (estimated from initial scans of the Raman clock transition X(62, 2) → X(0, 0) using these two lasers; the true value is expected to be at the Hz level). The STIRAP lasers pass through a common Glan-Thompson polarizer with a 100,000:1 extinction ratio to drive π-transitions, and co-propagate along the lattice to minimize momentum transfer. Binding energy (cm -1 ) Figure 1. One-photon spectroscopy of (1)0 + u . (a) The spin-orbit interaction couples A 1 Σ + u and c 3 Π u (dashed and dotted-dashed respectively) in the manifold of excited potentials resulting in the admixed relativistic potentials (1)0 + u (solid red) and (2)0 + u (solid blue). Low-lying states of (1)0 + u present a pathway for adiabatic transfer in the ground potential X 1 Σ + g (solid black). Ω 1 and Ω 2 are the angular Rabi frequencies of the pump and anti-Stokes lasers respectively. For the spectroscopy of (1)0 + u , only the pump laser is present. (b) Binding energies of the first 37 vibrational levels belonging to (1)0 + u with J = 1. The marked change in trend at approximately −1500 cm −1 can be attributed to the avoided crossing of the non-relativistic potentials. (c) Transition strengths S to (1)0 + u (v = 9 − 20, 1) from X(62, 0) (black squares) and X(61, 0) (blue triangles). (d) Measured linewidths of the (1)0 + u states (black squares). Yellow diamonds are the theory predictions of S and Γ using the Morse/long-range (1)0 + u potential described in reference [81]. All error bars represent 1σ uncertainties. The binding energies and transition strength values are listed in table A1 in the Appendix.

High-resolution spectroscopy
Just as for atomic Sr, due to the heavy mass of the strontium nucleus the singlet and triplet electronic states of Sr 2 are strongly mixed by the spin-orbit interaction [82]. This effect is manifest in the relativistic potential (1)0 + u that asymptotes to the 1 S 0 + 3 P 1 threshold, which can be identified as the lower branch of the avoided crossing between the A 1 Σ + u and c 3 Π u non-relativistic potentials ( figure 1(a)). We spectroscopically locate the first 37 vibrational levels of (1)0 + u with J = 1 via one-photon excitation of X(62, 0). Previous studies have hitherto only probed weakly bound states near the intercombination [83][84][85][86][87][88][89]. As shown in figure 1(b), the effect of the spin-orbit perturbation is markedly noticeable from the change in the trend of the binding energies near −1500 cm −1 with respect to the 1 S 0 + 3 P 1 threshold. Having found the rovibrational ground state of (1)0 + u , we fit the binding energies of the first 11 states to the energies Table 1. Extracted spectroscopic constants of (1)0 + u in units of cm −1 . These were obtained from fitting the 11 lowest bound states to the energies of a vibrating rotor. Since we only measured states of the same angular momentum (J = 1), we cannot separate out the vibrational and rotational constants.
Spectroscopic constant This work Reference [90] −D e + 2B e -2791. 21 where ω e , x e , B e , α e , and D e are the vibrational, anharmonicity, rotational, and vibration-rotation coupling spectroscopic constants and the potential depth, respectively. Our results, listed in table 1, compares favorably with those quoted for A 1 Σ + u inferred from the observed spectra of several isotopic combinations in reference [90].
For coherent transfer within the singlet X 1 Σ + g ground potential, we ideally require intermediate states with marginal triplet admixture, favorable Frank-Condon overlap with the initial and final states, and a narrow linewidth. In an earlier work [91], we predict that the first two criteria can be satisfied in the vicinity of the A 1 Σ + u -c 3 Π u avoided crossing, where transitions strengths with mid-to-low-lying states of X 1 Σ + g are expected to be as large as 10 −2 (ea 0 ) 2 , while simultaneously maintaining reasonable transition strengths with photoassociated weakly bound molecules. As shown in figures 1(c) and (d), we verify this by measuring the transition strengths of X(62, 0) and X(61, 0) to (1)0 + u states on both sides of the avoided crossing, as well as their linewidths, Γ. To do so, we follow references [92,93] and measure the number of remaining molecules after a depletion pulse, N (δ 1 , t), as a function of the pulse time t and angular frequency detuning δ 1 of the pump laser with respect to the X → 0 + u transition. Straightforward rate equations imply that where N 0 is the initial molecule number and Ω 1 is the angular Rabi frequency of the X → 0 + u pump transition. By simultaneously fitting the depletion curves versus t and δ 1 , we extract Ω 1 and Γ. We then convert Ω 1 into transition strength, S, from conservative estimates of the pump laser beam waist and power.
Our data suggests that (1)0 + u (11, 1) offers one of the strongest pump couplings. Moreover, the required laser wavelengths of 793 nm and 732 nm (for the pump and  anti-Stokes respectively) to address the entire depth of X 1 Σ + g are within the operating range of commercially available AR-coated laser diodes and our frequency comb. While transitions starting from X(62, 0) are generally weaker than those from X(61, 0), this is outweighed by the fact that we create larger samples of the former and detect them with better efficiency. Thus, in this study we choose X(62, 0) as our initial state and the singlet-dominant (1)0 + u (11, 1) as the intermediate state for the STIRAP transfer. We note that a similar singlet pathway has been recently shown to be favorable for bi-alkali molecules [94].
To locate the bound states of X 1 Σ + g , we perform dark-resonance spectroscopy. We lock the pump laser on resonance with X(62, 0) → (1)0 + u (11, 1) and adjust the laser power and pulse time such that we achieve nearly full depletion without bleaching the spectroscopic signal. For this part of the study, we use a tunable Ti:Sapphire laser as the anti-Stokes (∼500 W cm −2 ). On resonance, (1)0 + u (11, 1) is significantly Stark shifted by the anti-Stokes and the spectroscopic signal is no longer depleted by the pump. In this way, we find all 63 vibrational states in X 1 Σ + g with J = 0 and 2, as shown in figures 2(a) and (b). To compare our spectroscopy with Fourier transform spectra in a heatpipe Table 2. Spectroscopic constants for X 1 Σ + g in units of cm −1 . The binding energies were determined to an accuracy of 0.002 cm −1 , limited by that of the wavelength meter (High Finesse WS7-60) used to determine the laser frequencies. The rotational splittings have tighter uncertainties because the relative precision of the wavemeter is approximately 1 × 10 −4 cm −1 . Only the first 5 vibrational levels were used, and the reduced χ 2 of the fits to the binding energies and rotational splittings are 0. 96 By simultaneously fitting the binding energies and the J = 0, 2 rotational splittings of the first 5 vibrational levels to equations 1 and 3 respectively, we extract the spectroscopic constants (see table 2) and find excellent agreement with the values reported in references [95,96] that are weighted across several isotopologues. To verify that the addressed states form a Λ-system, we lock the anti-Stokes laser on resonance with the absolute rovibrational ground state of the 88 Sr 2 dimer (i.e. X(0, 0) → (1)0 + u (11, 1)) and reduce its laser power. Sweeping the pump frequency reveals a narrow electromagnetically induced transparency (EIT) peak within the broad single-photon depletion spectrum (see figure 2(c)), heralding the formation of a dark state necessary for STIRAP transfer. At bias magnetic fields of < 1 G the Zeeman sub-levels of (1)0 + u (11, 1) are essentially unresolved due to the ∼5 MHz excited state linewidth, consistent with the expected magnitude of the Zeeman shift [97]. From measurements of Autler-Townes splittings we obtain Ω 2 , the angular Rabi frequency of the anti-Stokes coupling for various anti-Stokes laser powers. From a conservative estimate of the anti-Stokes laser beam waist, we find a transition strength of 8.6(9) × 10 −2 (ea 0 ) 2 for X(0, 0) → (1)0 + u (11,1). This is one of the strongest molecular transitions in 88 Sr 2 , with a value approaching that of a typical atomic transition.

Numerical model
We model the dynamics of the 5-level Λ-system (figure 2(d)) with the Lindblad master +1) respectively, and m J is the projection of the total angular momentum onto the lab-frame quantization axisẐ. The pump and anti-Stokes electric fields are where ω j are the laser angular frequencies and j = {1, 2} label the ground states. We pick the convention that the lasers propagate along the positiveŶ direction and write the polarization vectors as where the inclination θ with respect toẐ and the phase φ are angles that parametrize the polarization state. The angular Rabi frequencies coupling the ground states to the excited Zeeman sub-levels ( Selection rules force q = −1, 0, +1 for state labels k = 3, 4, 5 respectively. This is true for both j = 1, 2 since we have a Λ-system. To make contact with the experiment, we note that the excitation rates and Autler-Townes splittings are both proportional to q |Ω j (j),q | 2 = Ω 2 j which are in turn proportional to the respective laser intensities, so the quantity that we measure in the preceding section is Ω j even if the polarization is elliptical. Physically, this quantity is equivalent to the Rabi angular frequency for the ideal case of a π-transition (J = 0 ↔ 1, ∆m J = 0) driven with pure linear polarization exactly parallel to the quantization axis.
In the electric dipole and rotating-wave approximation, the Hamiltonian governing the unitary evolution is [98] where Z is the Zeeman splitting of the excited state, δ j ≡ ω j − ω 0 are the angular frequency detunings from the m J = 0 sub-level with transition frequency ω 0 , and we include an additional auxiliary state |6 that does not participate in the coherent dynamics but merely functions to collect population decay from |k .
To include the relaxation dynamics, we compute where {,} is the anti-commutator, G k ≡ √ Γ|k 6| are the so called jump operators, Γ is the excited state linewidth, and ρ is the 6 × 6 density matrix for the whole system. The effect of L k is to generate decay terms proportional to −Γ in the diagonals of |k and −Γ/2 in the off-diagonals between |k and |6 .
In some cases it is useful to include a phenomelogical decoherence rate Γ eff which can be interpreted as the relative linewidth between the pump and anti-Stokes. To do so we compute where P j ≡ |j j| are projection operators onto the corresponding diagonal element. The effect of D 1,2 is to generate decay terms proportional to −Γ eff /2 in the off-diagonals between |1 and |2 . The Lindblad master equation for our system is thus where [,] is the commutator. We numerically solve the time evolution of equation 9 for the input parameters Ω j , δ j , θ j , φ j , Z, Γ, and Γ eff , with the initial condition ρ 11 (t = 0) = 1 and zero for all other entries. For later convenience, we define the onephoton (common) detuning ∆ ≡ δ 2 , and the two-photon (Raman) detuning δ ≡ δ 1 − δ 2 . In the weak probe limit (Ω 1 Ω 2 ) such that ρ 11 ≈ 1, ρ 22 ≈ ρ 33 ≈ 0, we can derive a general analytical expression for the excitation or EIT lineshape. Setting Z ≈ 0 (unresolved excited states) and discarding negligible terms O(Ω 1 /Ω 2 ), we obtaiṅ From the first equation, we see that the rate of excitation out of |1 is R = k Im Ω * 1,k ρ 1k , where Im() denotes the imaginary part. To find the excitation rate approaching steady-state conditions, we setρ 12 ≈ρ 1k ≈ 0 to solve for ρ 1k and substituting back into the expression for R, we find The general EIT lineshape for non-cycling transitions is then Finally, for the ideal case where the Raman lasers have exactly the same polarization (i.e. θ 1 = θ 2 , φ 1 = φ 2 ), we have k Ω * 1,k Ω 2,k = k Ω 1,k Ω * 2,k = Ω 1 Ω 2 since both ground states have J = 0, and equations 11 and 12 further simplify to where the single-photon and two-photon effects are separated into distinct terms; e.g.
for Ω 2 = 0, we immediately recover equation 2. Equation 13 can be shown to be in exact agreement with reference [93], and is also valid in the case where the excited state structure is very well separated such that one effectively addresses a three-level Λsystem. As can be seen in figure 2(c), the analytical form of equation 13 is an excellent fit to the experimental data.  N 3 are the initial and final molecule numbers respectively, and N 2 is the remaining molecule number after the forward transfer. For our Rabi frequencies, we typically achieve full extinction such that N 2 = 0, except in a few extreme cases where ∆ is very large resulting in reduced adiabaticity. Assuming equal efficiencies for the forward and reverse transfer, we routinely achieve one-way transfer efficiencies of η = 85(3)%. Due to the unresolved Zeeman structure of (1)0 + u (11, 1), a finite relative angle between the polarizations of the Raman (pump and anti-Stokes) lasers dilutes the Rabi couplings and destabilizes the dark state. This leads to increased near-resonant scattering which diminishes the overall transfer efficiencies. In the current study, we measure a relative polarization angle of 16(2) degrees between the pump and anti-Stokes. The cause was traced to a dichroic mirror combining the lasers with the lattice. While we could solve the issue by placing a Glan-Thompson polarizer after this dichroic mirror and immediately before the chamber viewport, geometric constrains in our current setup prohibit this. Nevertheless, for J = 0 ↔ 1 transitions, we can circumvent this by either lifting the degeneracy of Zeeman sublevels, or performing STIRAP at ∆ larger than Γ, the linewidth of the X → 0 + u transition, as demonstrated in figure 3(b). Numerical simulations using the measured relative polarization angle and expected Γ eff show good agreement. We note that detuned STIRAP has been reported to mitigate other technical imperfections such as laser phase noise and stray reflections [57,99,100]. When choosing an operational common detuning, it is prudent to carefully survey the molecular structure. For instance, in 88 Sr 2 , the rotational splitting of X(62, 2) and X(62, 0) is approximately 70 MHz. Therefore, blue detuning is preferred so as to avoid accidental perturbation of the excited state by the pump laser should it be tuned close to 0 + u (11, 1) → X (62,2). Similarly, technical leakage light through the acoustooptic modulators used to modulate the laser intensities can diminish transfer efficiencies should the residual diffraction orders accidentally address a X → 0 + u resonance. These technical effects are non-negligible since the laser intensities used in STIRAP are large.

STIRAP in a non-magic optical lattice
A hallmark of STIRAP that gives it an edge over other transfer schemes (e.g. a Raman π-pulse [74,101]) is its robustness against small perturbations (e.g. laser intensity stability, frequency drifts) as evident by the detuned-STIRAP resonance in figure 3(c), where the efficiency remains > 50% even as the relative Raman detuning is scanned over > 100 kHz. This range can be made wider with larger laser intensities. The detuned-STIRAP lineshape also shows the expected asymmetry with the sharper edge facing the one-photon resonance location.

STIRAP in a magic lattice
We now explore performing STIRAP in a deep optical lattice, at a trap depth of U 0 = 1009(44) E rec . Unlike in the previous subsection, here we leave the trap light on throughout the sequence. Our current strategy for engineering magic lattices involves a lattice blue-detuned from a transition connecting the deeply bound ground state with a narrow rovibronic state in the (1)1 u potential (ungerade, Ω = 1) that asymptotes to the 1 S 0 + 3 P 1 threshold [81,101], as illustrated in figure 4(a). At the magic wavelength, the polarizabilities of the two J = 0 vibrational ground states are matched (α /α = 1), resulting in equal trap depths. This effectively removes lattice-induced thermal decoherence and precludes the excitation of breathing modes during the state transfer.
We observe enhancement of STIRAP efficiency in a magic wavelength lattice as shown in figure 4(b), accompanied by lattice-induced Stark shift of the peak STIRAP efficiency ( figure 4(c)). To understand this, we measure the polarizability ratio of the ground states at various lattice wavelengths near the X → 1 u transition (figure 4(d)), where α , α are the scalar polarizabilities of X(0, 0) and X(62, 0) respectively, L 0 is the differential lattice-induced lightshift, f ax is the axial trap frequency (section 5), and f rec ≡ h/(2M λ 2 ) = E rec /h is the standard expression for the recoil frequency. Since L 0 and f 2 ax both depend linearly on lattice intensity, this calculation bypasses the need to determine any geometric parameters (e.g. lattice beam waist).
For X(0, 0), the optimal magic wavelength [81] with the greatest magic detuning (∆ m = 4.456(3) GHz) occurs at 1004.7723(1) nm, blue detuned from X(0, 0) → (1)1 u (9, 1) with a measured transition strength of 1.33(15) × 10 −4 (ea 0 ) 2 using an allfrequency method following reference [81]. This is theoretically predicted to be the strongest X → 1 u transition that is below the (1)0 + u potential minimum in 88 Sr 2 . In the Lamb-Dicke regime, the first-order Doppler effect is suppressed (Γ eff = 0). In general, the lattice-induced differential lightshift on a transition in a non-magic trap becomes compounded for higher trap motional states. For an experimentally set (bare) Raman detuning of the pump and anti-Stokes in the motional ground state, the thermal distribution of the molecules occupying various trap motional states maps to a probability density, p, for the molecule to experience a Raman detuning additionally  to X(62, 0) around the magic wavelength, determined from the differential lattice lightshifts. In both (c) and (d), the red solid line is the fit to the data in the form a/(x − x 0 ) + b where a and b are free parameters and x 0 is fixed to the independently measured X(0, 0) → (1)1 u (9, 1) resonance wavelength. All error bars represent 1σ of statistical error.
shifted by −δ is a factor that depends on the temperature and polarizability mismatch [102]. This probability density p(−δ ) peaks at δ max = 2/B. To model the transfer efficiencies, we first simulate ideal STIRAP efficiencies, η(δ), for the measured parameters as a function of δ. The function η(δ) peaks at approximately δ = 0. Thus, conceptually the overall efficiency involves an overlap integral between η and p, which we maximize in the actual experiment by manually adjusting the bare Raman detuning (as in figure 4(c)). We can account for this experimental detail to some extent in the model by shifting δ → δ − δ max such that the peaks of the distributions p and η line up. The overall efficiency is then calculated as the thermal average of η (i.e. a convolution) and the lattice wavelength dependence enters implicitly through α /α. The solid line in figure 4(b) shows the result of the simulation which reproduces the salient features of the measurement fairly well.

Two-body ultracold reactive collisions
With the creation of large numbers of molecules in X 1 Σ + g (0, 0), we are in a good position to study state-specific losses of an ultracold gas of alkaline-earth-metal dimers. To this end, we prepare a purified sample of J = 0 molecules by wiping away J = 2 from the initial photoassociated mixture (see section 2). We hold the v = 0 ground state molecules after a forward STIRAP sequence in a non-magic optical lattice (λ = 914.0(1) nm) for a variable amount of time, and then reverse the STIRAP sequence to recover weakly bound molecules which we detect. Figure 5 shows the decay of the v = 0 trapped molecule number over time. Fitted curves to the rate equationṄ = −kN γ (where k is a free parameter) strongly suggest two-body collisions (γ = 2) to be the dominant loss channel.
We can extract the two-body loss parameter, β, with a density calibration. Following reference [29], the rate equation can be written aṡ where N is the number of molecules per lattice 'pancake', A ≡ (ω 2 M/4πk B ) 3/2 , and ω ≡ 2π(f ax f 2 rad ) 1/3 . To simplify the analysis, in this study we assume that the molecules remain at the same temperature, T , throughout the hold duration.
Since the molecules are tightly trapped in the Lamb-Dicke and resolved sideband regimes, we can spectroscopically access the axial and radial trap frequencies (f ax and f rad respectively) by probing the shallow-to-shallow Raman transition X(62, 0) → X(61, 0) via (1)0 + u (12, 1) as shown in figures 6(a) and (b). Here, we use counter-propagating probe beams to maximize the imparted momentum along the axial lattice direction so as to observe the axial sidebands. This Raman transition is nearly magic since the polarizability difference of adjacently bound near-threshold states is negligible ( figure 6(c)). Asω 2 is measured for X(62, 0), in the case of X(0, 0) we must further scaleω 2 by the polarizability ratio α /α. From measurements of differential lightshift for the shallow-to-deep Raman transition X(62, 0) → X(0, 0) (figure 7(a)) and equation 14, we find α /α = 1.5176(59) at λ = 914.0(1) nm.
The next step is to determine the temperature of the molecular ensemble. We do so using carrier thermometry [102] with the shallow-to-deep Raman transition X(62, 0) → X(0, 0) and co-propagating probes (see figure 7(b)). For the purposes of inferring temperature via thermally imprinted lattice lightshifts (i.e. equation 15), probing a transition with a large polarizability difference is advantageous and has better accuracy since the effect is exaggerated over other broadening sources (e.g. power broadening) [103]. In figure 7(c) we compare the temperatures extracted using carrier thermometry to the conventional method of taking the ratio of the integrated area under the sidebands and find good agreement.
The measured two-body loss rate coefficient β for X(0, 0) and X(62, 0) are 0.97(28) × 10 −10 cm 3 s −1 and 2.2(0.3) × 10 −10 cm 3 s −1 respectively, which are found to be unaffected by the presence of J = 2 vibrationally least bound molecules. These results are summarized in the inset to figure 5. An estimate of the universal inelastic loss rate via the dominant s-wave channel for X(0, 0) yield 1.22 × 10 −10 cm 3 s −1 , independent of the ensemble temperature [104]. This estimate relies on a coupled cluster computation for the isotropic van der Waals coefficient, C 6 , using the explicitly connected representation of the expectation value and polarization propagator [105,106] and the best approximation XCCSD4 method [107]. We find C 6 = 15685 a.u. for Sr 2 dimers at the equilibrium distance R e = 8.829 a 0 , consistent with an independent calculation from reference [108]. We had previously applied the same method and basis set to calculate the leading van der Waals coefficient for atomic strontium in the ground state and found excellent agreement with high-resolution Fourier transform spectra [82,96], thus we expect a similar level of accuracy for the absolute ground state strontium molecules in the present investigation. The closeness of our measured β for X(0, 0) molecules to the universal loss limit suggests that the molecules react with near unity probability following a collision at short range. For a homonuclear alkaline-earth dimer in the absolute ground state, one possible exoergic process that will manifest in such a loss is the formation of stable trimers [108]. Another possibility is the photoexcitation of "sticky" four-body complexes by the intense lattice light [109,110]. For the near-threshold X(62, 0) state, the slightly larger loss rate may indicate that vibrational relaxation effects are non-negligible. The results, however, do not limit the prospects of using these molecules for precision spectroscopy, since the implementation of a 3D lattice can strongly suppress collisional losses and enable long interrogation times [111].

Conclusion
In summary, we have created ultracold 88 Sr 2 molecules in the absolute ground state using STIRAP to coherently transfer photoassociated molecules. We achieve favorable transfer efficiencies of nearly 90%, both in free flight and in a deep optical lattice tuned to a magic wavelength, limited by the available pump laser power. Future experiments should benefit from a redesigned experimental setup such that the relative polarization of the pump and anti-Stokes beams can be made more parallel. We have mapped out all 63 vibrational states with J = 0, 2 in X 1 Σ + g to an accuracy of 0.002 cm −1 . We spectroscopically observed the effect of spin-orbit coupling on the bound states of (1)0 + u , verified its potential depth, and identified a favorable STIRAP pathway from the measured transition strengths. We investigated the lifetime of absolute ground state molecules in an optical trap and find two-body collisions near the universal loss rate.
The work presented here demonstrates the ability to access the full vibrational state space of X 1 Σ + g closed-shell molecules, and in doing so, opens up the variety of experiments where efficient state initialization in a deeply bound level is necessary. The X → 1 u transition described here presents one of the most favorable conditions for engineering a near-resonant magic lattice for Raman transitions within the ground potential, so as a natural next step we plan to utilize this to explore the stability and accuracy of the vibrational molecular clock. Large samples of ground state molecules should allow us to gain insight into the photon scattering process in a near-resonant magic lattice [101], and enable loading into 3D magic lattices at mid-IR wavelengths tuned far below the minima of the excited potentials where we expect the main loss processes to be diminished. With these, tests of fundamental physics with alkalineearth-metal molecules are now within reach.  Table A1 shows the binding energies for the lowest 37 vibrational states of (1)0 + u with J = 1 via laser excitation from X(62, 0), with respect to the atomic threshold 1 S 0 + 3 P 1 . We take the intercombination frequency from reference [112] and the binding energy of X(62, 0) from reference [89].
For v = 9-20 we use a frequency comb to determine the pump laser frequency with a high precision. The X → 0 + u resonance frequencies are linearly extrapolated to zero lattice intensity, and we report uncertainties inflated by the square root of the reduced χ 2 if it is greater than unity. The rest were determined using a wavelength meter to an uncertainty of 60 MHz.
In addition, we also list S X(62,0) and S X(61,0) , the measured X → 0 + u transition strengths from X(62, 0) and X(61, 0) respectively. The method of determining these transition strengths is described in the main text. Table A1. Binding energies for the lowest 37 vibrational states of (1)0 + u with J = 1 with respect to the atomic threshold 1 S 0 + 3 P 1 . Also shown are S X(62,0) and S X(61,0) , the measured X → 0 + u transition strengths from X(62, 0) and X(61, 0) respectively, in units of (10 −5 (ea 0 ) 2 ). These values are plotted in figure 1. All reported values are obtained in this work.