Ab initio calculation of the spectrum of Feshbach resonances in NaLi $+$ Na collisions

We present a combined experimental and theoretical study of the spectrum of magnetically tunable Feshbach resonances in NaLi $\left(a^3{\mathrm{\Sigma }}^{+}\right)+$ Na collisions. In the accompanying paper, J. J. Park, H. Son, Y.-K. Lu, T. Karman, M. Gronowski, M. Tomza, A. O. Jamison, and W. Ketterle, Phys. Rev. X 13, 031018 (2023), we observe experimentally 8 and 17 resonances occurring between $B=0$ and 1400 G in upper and lower spin-stretched states, respectively. Here, we perform ab initio calculations of the NaLi $+$ Na interaction potential and describe in detail the coupled-channel scattering calculations of the Feshbach resonance spectrum. The positions of the resonances cannot be predicted with realistic uncertainty in the state-of-the-art ab initio potential, but our calculations yield a typical number of resonances that is in near-quantitative agreement with experiment. We show that the main coupling mechanism results from spin-rotation and spin-spin couplings in combination with the anisotropic atom-molecule interaction. The calculations furthermore explain the qualitative difference between the numbers of resonances in either spin state.

Figure 1

Interaction potentials for the NaLi $+$ Na collision complex. The potentials are shown as a function of $R$, the center-of-mass distance between the atom and molecule, and $\theta$ the Jacobi angle between the orientation of the molecule and the approach of the atom, with $\theta =0$ corresponding to Na-NaLi and $\theta ={180}^{\circ }$ to Na-LiNa. The molecular bond length is fixed at the triplet equilibrium bond length $r=8.8a_0$. (a) High-spin quartet $S=3/2$ potential. (b) Low-spin doublet $S=1/2$ potential. In principle, there are two doublet potentials that can have an avoided crossing. What is shown here is the doublet potential for a pure triplet NaLi molecule, $s=1$.

Figure 2

Convergence of the coupled-channels calculations with truncation of the basis set. (a) Calculation with fixed $N_{\max }=20$. Cross sections converge with $J_{\max }=2$. When the basis is truncated at $J_{\max }=1$, the cross sections scale quadratically with the spin-rotation coupling constant ${\gamma }_s$. This is demonstrated by agreement with the crosses which show one-quarter of the cross section obtained by scaling ${\gamma }_s$ up by a factor of 2. When the basis is truncated at $J_{\max }=2$, spin-spin coupling contributes and the cross section scales perturbatively with both coupling constants (crosses). Spin-spin coupling is typically dominant, but not by a large factor so both mechanisms contribute. Magnetic dipole-dipole coupling does not play an important role. (b) Calculation with fixed $J_{\max }=1$ The scattering cross section can be converged with $N_{\max }$, but requires an impractically large basis set. (c) Calculation with fixed $J_{\max }=2$ The scattering cross section is not converged with $N_{\max }$, but the density and typical width of resonances are not strongly dependent on $N_{\max }$.

Figure 3

Dependence of loss rates on scaling of three-body interactions. Loss rates as a function of $B$ field and scaling of the three-body interaction by a factor $1+\lambda$ for $N_{\max }=20$, shown for the upper stretched state. This shows that due to the uncertainty of $\lambda$ of at least several percent, the background loss and resonance positions are undetermined. Several $B$-independent resonances are observed, where the $\lambda$ scaling tunes the initial spin-stretched potential such that it supports a resonance, i.e., a bound state near zero energy. Hence, prediction of resonance positions requires knowledge of the interaction potentials to an accuracy that cannot realistically be achieved by ab initio calculations.

Figure 4

Spin dependence of three-body interactions. Loss rates are shown as a function of $B$ field and scalings of the interactions potential. Calculations are done with $N_{\max }=6$, which artificially reduces the number of resonances compared to Fig. 3. (a) The spin-independent three-body interaction is scaled by $1+\lambda$. In this case, all resonance positions depend on the scaling of the potential, the background loss rate varies strongly, and new $B$-independent resonances appear by scaling the initial-state potential such that supports a bound state near zero energy. (b) The spin splitting is scaled by $1+\lambda$, while the spin-stretched potential is kept fixed. Therefore, only the doublet potential is varied. The resonances near 150, 350, 1150, and 1250 G are independent of the scaling of the low-spin potentials, i.e., they are completely supported by the nonreactive spin-stretched potential. The feature just above 1000 G has a weak dependence on the scaling of the doublet potentials, and several broader features such as that around 500 G have a stronger dependence.

Figure 5

Cross section and rated as a function of $\lambda$ scaling. (a) The elastic collision cross section obtained by scaling the nonadditive three-body interaction by $1+\lambda$ at a fixed magnetic field $B=1500$ G. Different colors correspond to different spin states, $|s{m}_s\rangle |s_3{m}_3\rangle$, as indicated. (b) Loss rate coefficients as a function $\lambda$ scaling for fixed $B=1500\mathrm{G}$.

Figure 6

Probability distribution of collision rates. Cumulative probability distribution obtained by sorting cross sections and rate coefficients for different $\lambda$ scaling at fixed $B=1500$ G [66]. Different colors correspond to different states, $|s{m}_s\rangle |s_3{m}_3\rangle$, as indicated. (a) Distribution of the elastic cross section, (b) loss rate coefficient, and (c) the ratio of elastic-to-inelastic collisions. Solid (dashed) lines correspond to $N_{\max }=20$ (10). In panel (a) the black markers indicated the expected distribution of elastic cross sections for a van der Waals potential [68].

Figure 7

Correlation of elastic and inelastic cross sections obtained for various $\lambda$ scaling and magnetic field. For each spin state, there exists a positive correlation between the magnitudes of the elastic cross section and the loss rate coefficient.

Figure 8

Representative magnetic field scans for different basis set truncation. The figure shows representative magnetic-field scans of the collisional loss-rate coefficient. Different panels correspond to truncation of the basis set at $N_{\max }=2,$ 10, 30, as indicated. The left (right) hand column shows results for the upper (lower) spin-stretched state.

Figure 9

Number of resonances. The plot shows the expected number of resonances between 0 and 1500 G from (a) quasiclassical phase-integrals Eqs. (19) and (20) as a function of the maximum molecule-atom center-of-mass separation $R_{\max }$ up to which the integrals are evaluated and (b) quantum mechanical calculations of bound states on each adiabatic potential energy curve, as a function of $N_{\max }$ that truncates the channel basis set.

Figure 10

Representative magnetic field scans with noninitial Zeeman states removed. Shown are representative magnetic field scans obtained by excluding the noninitial Zeeman states for $N=0$ from the calculation. Different panels correspond to truncation of the basis set at $N_{\max }=2,$ 10, 30, as indicated. The left-hand (right) column shows results for the upper (lower) spin-stretched state. Excluding the lower-lying Zeeman levels for the upper spin-stretched state increases the typical number of resonances, whereas excluding excited Zeeman levels for the lower spin-stretched state does not reduce the typical number of resonances.

Figure 11

Representative magnetic field scans with the channel basis truncated with $J_{\max }=1$. Different panels correspond to truncation of the basis set at $N_{\max }=2$, 10, 30, as indicated. The left-hand (right) column shows results for the upper (lower) spin-stretched state. Truncation of the basis set with $J_{\max }=1$ reduces the number of resonances, and the qualitative difference in the number of visible resonances between the two spin states disappears.

Figure 12

Including hyperfine interactions in the calcuations. Representative magnetic field scans are obtained with and without hyperfine interactions. Panel (a) shows results for the upper spin-stretched state and panel (b) the lower spin-stretched state. Including hyperfine interactions increases the number of resonances somewhat, but not significantly, and it does not lead to clearly identifiable multiplet splittings.

Figure 13

Collisional loss rates at higher magnetic fields. Plotted are representative magnetic field scans up to higher magnetic fields than are probed experimentally. In the upper spin-stretched state, a strong feature appears which is caused by a resonance between the Zeeman relaxation energy and a rotational excitation. In the lower spin-stretched state, more resonances are typically visible, but the strong feature near 5000 G is missing as Zeeman relaxation is not possible in this spin state.

Figure 14

Perturbative analysis of calculated collision rates. Resonance spectra are displayed for the upper and lower spin-stretched states in orange and blue solid lines, respectively, calculated for $\lambda =-0.02$ and $N_{\max }=30$, also shown as Fig. 4 in the accompanying paper [36]. Markers show four times the loss rate obtained with the spin-spin and spin-rotation couplings halved, and agreement with the solid lines indicates these spin-dependent couplings act perturbatively. For the upper stretched state, the dashed-dotted line indicates the loss rate obtained with the interaction anisotropy turned off, which is entirely due to the magnetic dipole-dipole interaction. The much smaller loss rate in this case identifies a combination of the anisotropic interaction and the spin-spin and spin-rotation coupling as the dominant loss mechanism. The dotted line is obtained by halving the strength of anisotropic interactions and multiplying the resulting loss rate by a factor of 4. This completely changes the shape and resonance positions. The disagreement between the dotted and the full line indicates that the anisotropic interactions do not act perturbatively.