Mass scaling and nonadiabatic effects in photoassociation spectroscopy of ultracold strontium atoms

We report photoassociation spectroscopy of ultracold ${}^{86}\mathrm{Sr}$ atoms near the intercombination line and provide theoretical models to describe the obtained bound-state energies. We show that using only the molecular states correlating with the ${}^1{S}_0+{}^3{P}_1$ asymptote is insufficient to provide a mass-scaled theoretical model that would reproduce the bound-state energies for all isotopes investigated to date: ${}^{84}\mathrm{Sr},{}^{86}\mathrm{Sr}$, and ${}^{88}\mathrm{Sr}$. We attribute that to the recently discovered avoided crossing between the ${}^1{S}_0+{}^3{P}_1$ $0_u^{+}$ $\left({}^3{\Pi }_u\right)$ and ${}^1{S}_0+{}^1{D}_2$ $0_u^{+}$ $\left({}^1{\Sigma }_u^{+}\right)$ potential curves at short range and we build a mass-scaled interaction model that quantitatively reproduces the available $0_u^{+}$ and $1_u$ bound-state energies for the three stable bosonic isotopes. We also provide isotope-specific two-channel models that incorporate the rotational (Coriolis) mixing between the $0_u^{+}$ and $1_u$ curves which, while not mass scaled, are capable of quantitatively describing the vibrational splittings observed in experiment. We find that the use of state-of-the-art ab initio potential curves significantly improves the quantitative description of the Coriolis mixing between the two $-8$-GHz bound states in ${}^{88}\mathrm{Sr}$ over the previously used model potentials. We show that one of the recently reported energy levels in ${}^{84}\mathrm{Sr}$ does not follow the long-range bound-state series and theorize on the possible causes. Finally, we give the Coriolis-mixing angles and linear Zeeman coefficients for all of the photoassociation lines. The long-range van der Waals coefficients $C_6\left(0_u^{+}\right)=3868\left(50\right)$ a.u. and $C_6\left(1_u\right)=4085\left(50\right)$ a.u. are reported.

Figure 1

Potential curves correlating to the ${}^1{S}_0+{}^3{P}_1$ asymptote used in the calculation of the theoretical binding energies. The dashed curves represent the original ab initio Hund's case (a) potentials of Skomorowski et al. [44]. The solid lines are the potential curves fitted to the experimental data from photoassociation experiments. The top panel shows the potential curves in the Hund's case (a) representation, while the bottom panel shows them in the Hund's case (c) basis actually used in the calculation.

Figure 2

Coriolis mixing between the $0_u^{+}$ and $1_u$ series. The upper two plots show the two-channel wave functions of the strongly mixed ${}^{88}\mathrm{Sr}$ $J=1$ levels at $-8430$ and $-8200$ MHz $(\theta ={24}^{\circ }$ and ${114}^{\circ }$, respectively). In the case of the $-8430$-MHz level, the $0_u^{+}$ component (dark blue lines) constitutes the majority of the two-channel wave function, while the $-8200$-MHz state is predominantly of the $1_u$ symmetry (orange lines). For comparison, in the lower part of the figure we show two relatively pure $(\theta =5^{\circ }$ and ${95}^{\circ })$ $0_u^{+}$ and $1_u$ states at $-3463$ and $-2684$ MHz, respectively.

Figure 3

Theoretical $0_u^{+}$ energy levels as a function of the reduced mass $\mu$. The blue dashed lines are calculated using the ${}^3{\Pi }_u$ potential curve alone which is only capable of mass scaling between two isotopes at a time: ${}^{86}\mathrm{Sr}$ and ${}^{88}\mathrm{Sr}$ in this case. The solid lines show the results for a model that also includes the short-range crossing with a ${}^1{S}_0+{}^1{D}_2$ ${}^1{\Sigma }_u^{+}$ potential, as shown in Fig. 4. The latter model is capable of reproducing the experimental bound-state energies (marked as red crosses) for all isotopes. In this model, a short-range ${}^1{\Sigma }_u^{+}$ perturbing state crosses the ${}^1{S}_0+{}^3{P}_1$ dissociation limit at $2\mu \approx 84$ amu causing a sudden departure from the usual mass scaling behavior.

Figure 4

Avoided crossing between the ${}^1{S}_0+{}^3{P}_1{}^3{\Pi }_u$ and ${}^1{S}_0+{}^1{D}_2$ ${}^1{\Sigma }_u^{+}$ potential curves. The strong spin-orbit coupling between these two potential curves causes anomalies in the mass-scaling behavior of the photoassociation spectra near strontium intercombination line. In fact, a model that only includes the potential curve directly supporting the $0_u^{+}$ bound states close to the ${}^1{S}_0+{}^3{P}_1$ dissociation limit is incapable of properly describing mass scaling between the three bosonic isotopes in strontium.

Figure 5

Spectroscopy of the $0_u^{+}(\nu =-2)$ PA line of ${}^{86}\mathrm{Sr}$. (a) Atom number versus laser detuning from the one-photon ${}^1{S}_0-{}^3{P}_1$ atomic transition for interaction time $t=59.4$ ms. (b) The effective collision event rate constant derived from the atom loss using Eq. (A4). (c) Temperature versus detuning shows little variation, supporting the assumption of constant sample temperature.