Feshbach resonances of harmonically trapped atoms

Employing a short-range two-channel description, we derive an analytic model of atoms in isotropic and anisotropic harmonic traps at a Feshbach resonance. On this basis we obtain a parametrization of the energy-dependent scattering length that differs from the one previously employed. We validate the model by comparison to full numerical calculations for ${}^6\mathrm{Li}$-${}^{87}\mathrm{Rb}$ and explain quantitatively the experimental observation of a resonance shift and trap-induced molecules in exited bands. Finally, we analyze the bound state admixture and Landau-Zener transition probabilities.

Figure 1

(a) Energy spectrum of ${}^6\mathrm{Li}$-${}^{87}\mathrm{Rb}$ as a function of the magnetic field $B$ in a trap with $\omega =2\pi \times 200$ kHz. Dots indicate MC calculations, and lines indicate solutions of Eq. (6). The eigenenergies of Eq. (11) are shown as dashed lines. (b) Scattering length $a(E,B)/a_{\mathrm{ho}}=f(E)$ corresponding to an eigenenergy $E$ of the system. At $E=\hslash \omega (2n+1/2)$ with $n=\mathbb{N}$, resonances appear. (c) Contribution $A^2$ of the RBS for each energy level as a function of the magnetic field $B$. Dots indicate MC calculations, and lines indicate results of Eq. (9) with energies from solutions of Eq. (6).

Figure 2

Atom number vs magnetic field (dots) as measured in [9]. RBS admixture of eigenstates according to Eq. (9) with ${\gamma }_E={\gamma }_{\mathrm{bg}}$, $a^{*}\gg a_{\mathrm{bg}}$, and $a_{\mathrm{bg}}=a_0$ (solid line) and for an energy-dependent $a_{\mathrm{bg}}$ given in the text (dashed line).