Analyzing Feshbach resonances: A ${}^6\mathrm{Li}-{}^{133}\mathrm{Cs}$ case study

We provide a comprehensive comparison of a coupled channel calculation, the asymptotic bound-state model (ABM), and the multichannel quantum defect theory (MQDT). Quantitative results for ${}^6\text{Li}-{}^{133}\text{Cs}$are presented and compared to previously measured ${}^6\text{Li}-{}^{133}\text{Cs}$Feshbach resonances (FRs) [Repp et al., Phys. Rev. A 87, 010701(R) (2013)]. We demonstrate how the accuracy of the ABM can be stepwise improved by including magnetic dipole-dipole interactions and coupling to a nondominant virtual state. We present a MQDT calculation, where magnetic dipole-dipole and second-order spin-orbit interactions are included. A frame transformation formalism is introduced, which allows the assignment of measured FRs with only three parameters. All three models achieve a total rms error of $<1\mathrm{G}$ on the observed FRs. We critically compare the different models in view of the accuracy for the description of FRs and the required input parameters for the calculations.

Figure 1

Molecular energy levels for the ${}^6\mathrm{Li}|F=1/2,m_F=-1/2\rangle {\oplus }^{133}\mathrm{Cs}|3,3\rangle$ channel for $l=0$, $s$ waves. The energies with respect to the open-channel asymptote for the bare ABM (blue line), dressed ABM (black line), MQDT (red dashed-dotted line), and CC (green dashed line) are depicted. The horizontal line at zero energy represents the continuum threshold. The crossings of the molecular channels with the threshold mark the positions of the FRs. Calculated quasibound levels and box states from the CC model are removed for clarity. The inset shows a zoom into the region of the resonance at $\sim 889$ G, where the differences between the models are clearly visible. For example, the energy level of the bABM is not shifted, as it neglects the coupling to the continuum.

Figure 2

These graphs show the $l=1$ resonances near (a) 658.2 G and (b) 773.1 G. The (red) curves are the best fit form of Eq. (A1). The (blue) dots are the MQDT calculation of $a_1^3\left(B\right)$. The (black) dashed line is the function $a_{1,\mathrm{bg}}^3\left(B\right)$.

Figure 3

This graph shows the $l=1$ resonance near 760.4 G. The (red) curve is the best fit $a_1^3\left(B\right)$ from Eq. (A2). The (blue) dots are the MQDT calculation of $a_1^3\left(B\right)$. The (black) dashed line is the high-order polynomial $a_{1,\mathrm{bg}}^3\left(B\right)$. This fit excludes data around the narrow resonance near 773.1 G.