Orbital Feshbach resonances with a small energy gap between open and closed channels

Recently, a new type of Feshbach resonance, i.e., orbital Feshbach resonance (OFR), was proposed for the ultracold alkaline-earth-metal–like atoms and was experimentally observed in the ultracold gases of ${}^{173}\mathrm{Yb}$ atoms. Unlike most of the magnetic Feshbach resonances of ultracold alkali atoms, when the OFR of ${}^{173}\mathrm{Yb}$ atoms appears, the energy gap between the thresholds of the open channel (OC) and the closed channel (CC) is much smaller than the characteristic energy of the interatomic interaction, i.e., the van der Waals energy. In this paper we study the OFR in systems with a small CC-OC threshold gap. We show that in these systems the OFR can be induced by the coupling between the OC and either an isolated bound state of the CC or the scattering states of the CC. Moreover, we also show that in each case the two-channel Huang-Yang pesudopotential is always applicable for the approximate calculation of the low-energy scattering amplitude. Our results imply that in the two-channel theoretical calculations for these systems it is appropriate to take into account the contributions from the scattering states of the CC.

Figure 1

Energy-level diagram of a single atom. Here ${\delta }_e$ is the Zeeman energy difference between states ${|e\rangle }^{\left(\mathrm{E}\right)}{|\downarrow \rangle }^{\left(\mathrm{N}\right)}$ and ${|e\rangle }^{\left(\mathrm{E}\right)}{|\uparrow \rangle }^{\left(\mathrm{N}\right)}$, and ${\delta }_g$ is the one between states ${|g\rangle }^{\left(\mathrm{E}\right)}{|\downarrow \rangle }^{\left(\mathrm{N}\right)}$ and ${|g\rangle }^{\left(\mathrm{E}\right)}{|\uparrow \rangle }^{\left(\mathrm{N}\right)}$. The Zeeman energy difference $\delta$ in Eq. (3) can be expressed as $\delta ={\delta }_e-{\delta }_g$. In our system with $\delta >0$, the open channel is the state $|o\rangle$ with one atom in ${|e\rangle }^{\left(\mathrm{E}\right)}{|\uparrow \rangle }^{\left(\mathrm{N}\right)}$ and one atom in ${|g\rangle }^{\left(\mathrm{E}\right)}{|\downarrow \rangle }^{\left(\mathrm{N}\right)}$.

Figure 2

Schematic diagram of the Feshbach resonances. Here $r$ is the interatomic distance, the black solid lines are the potential curves of the OC and the CC, and the black dotted lines is the threshold of the OC. (a) The Feshbach resonance induced by the coupling between the OC and an isolated bound state (red dashed line) of the CC. (b) The Feshbach resonance induced by the coupling between the OC and the scattering states (blue dash-dotted lines) of the CC. In this case all the bound states (red dashed line) of the CC are far off resonant from the threshold of the OC.

Figure 3

The square-well model for the potentials $U^{(+)}\left(r\right)$ and $U^{(-)}\left(r\right)$.

Figure 4

The scattering length $a_{\mathrm{s}}$ of the square-well model. We show the results given by exact numerical calculation (red circles) and the two-channel Huang-Yang pseudopotential (blue solid lines). In (a) and (c) we also show the single-pole approximation (black squares). Here we consider the cases with (a) $a^{(+)}=1000b\left(u^{(+)}\approx 2.47b^{-2}\right)$ and $a^{(-)}=0.5b\left(u^{(-)}\approx -3.67b^{-2}\right)$, (b) $a^{(+)}=0.15b\left(u^{(+)}\approx 19.8b^{-2}\right)$ and $a^{(-)}=1000b\left(u^{(-)}\approx 22.2b^{-2}\right)$, and (c) $a^{(+)}=870b\left(u^{(+)}\approx 2.4697b^{-2}\right)$ and $a^{(-)}=910b\left(u^{(-)}\approx 2.4696b^{-2}\right)$.

Figure 5

The scattering length $a_{\mathrm{s}}$ of the square-well model. We show the results given by exact numerical calculation (red circles) with the approach in Appendix pp1 and the two-channel Huang-Yang pseudopotential (blue solid lines). Here we consider the cases with $a^{(+)}=22.4b\left(u^{(+)}\approx 2.56b^{-2}\right)$ and $a^{(-)}=2.34b\left(u^{(-)}\approx 3.76b^{-2}\right)$.