Simple model for predicting and analyzing magnetically induced Feshbach resonances

We present a simple theoretical model for describing ultracold-atomic collision dynamics and Feshbach resonances (FRs) without onerous numerical computation. An effective quantum-defect phase shift ${\delta }_{\mathrm{Y}}^{\mathrm{eff}}$ is derived and the reference potentials are shown acting as an assistant tool in the calculation. With the help of singlet and triplet scattering lengths $a^{\mathrm{S}}$ and $a^{\mathrm{T}}$, as well as an adjustable parameter ${\gamma }_{\mathrm{sr}}$, this model can be used to reliably predict FRs. An application example of ${}^6\mathrm{Li}-{}^{40}\mathrm{K}$ collision shows that the results given by the simple model agree quite well with both the relevant experimental data and the coupled-channel calculations.

Figure 1

Comparison of the calculated results $(◯)$ using Eq. (16) in Ref. [28] with those $(+)$ using Eq. (7) in the present work. (a), (c) $Y^{\mathrm{S}}$ and $Y^{\mathrm{T}}$ vs ${log}_{10}(a^{\mathrm{ref}}/\sigma )$. (b),(d) ${\delta }_{\mathrm{Y}}^{\mathrm{S}}$ and ${\delta }_{\mathrm{Y}}^{\mathrm{T}}$ vs ${\Delta }_{\mathrm{th}}^{\mathrm{ref}}$. The vertical dashed lines correspond to the point of $a^{\mathrm{ref}}=0$.

Figure 2

(a) The scattering phase shift ${sin}^2\delta$ for the energetically lowest $M_T=-4$ channel of ${}^6\mathrm{Li}-{}^{40}\mathrm{K}$ as a function of magnetic field B and relative momentum k. (b)–(d) ${sin}^2\delta$ in the $M_T=-4$ channel vs k at $B=148$, 158, and 168 G, respectively. Solid and dotted lines in (b)–(d) show the results calculated using Eqs. (10) and (11), respectively. The horizontal dashed line in (a) and vertical arrows in (b)–(d) indicate the experimental collision energy $E=12$ $\mu \mathrm{K}$ in Ref. [8]. A and B in (a) denote the resonance positions at $E=0$.