Dissociation of Feshbach molecules into different partial waves

Ultracold molecules can be associated from ultracold atoms by ramping the magnetic field through a Feshbach resonance. A reverse ramp dissociates the molecules. Under suitable conditions, more than one outgoing partial wave can be populated. A theoretical model for this process is discussed here in detail. The model reveals the connection between the dissociation and the theory of multichannel scattering resonances. In particular, the decay rate, the branching ratio, and the relative phase between the partial waves can be predicted from theory or extracted from experiment. The results are applicable to our recent experiment in ${}^{87}\mathrm{Rb}$, which has a $d$-wave shape resonance.

Figure 1

A typical scattering resonance. (a) Scattering phase calculated from Eqs. (17, 18) for $q=-\cot {\delta }_l^{\mathrm{bg}}=2$ as a function of the dimensionless energy $ϵ$. The horizontal lines show ${\delta }_l^{\mathrm{bg}}$ and ${\delta }_l^{\mathrm{bg}}+\pi$, which are reached for $ϵ\to \pm \infty$. (b) Beutler-Fano profile for the cross section calculated from Eq. (19) for the same resonance. The horizontal line shows the background value.

Figure 2

A quasibound state in the collision of two ${}^{87}\mathrm{Rb}$ atoms. Potentials for the $s$-wave (dotted line) and the $d$-wave (solid line) are shown as a function of radius. The centrifugal barrier for the $d$-wave is clearly visible. In addition, the highest-lying $s$-wave bound state (horizontal dotted line) and the corresponding $d$-wave state (horizontal solid line) are shown. The $d$-wave state lies above threshold (i.e., $E>0$), so that it is only quasibound and gives rise to a scattering resonance.

Figure 3

Shape resonance for the scattering of two ${}^{87}\mathrm{Rb}$ atoms in the hyperfine state ∣1,1⟩ at $B=0$. (a) Scattering phases and (b) cross sections are shown for the $s$-wave (dotted lines) and the $d$-wave (solid lines). The $d$-wave shape resonance is clearly visible. Other partial waves yield no noticeable scattering in this energy range.

Figure 4

Scheme of a Feshbach resonance. The interaction Hamiltonian can cause transitions between the closed channel and the incoming flux in the open channel. A Feshbach resonance occurs when the energy of the incoming flux matches the energy of a closed-channel quasibound state.

Figure 5

Combination of a shape resonance and a Feshbach resonance for scattering of ${}^{87}\mathrm{Rb}$ in state ∣1, 1⟩. The partial-wave components ${\sigma }_l$ of the total cross section are shown for the $s$-wave (dotted line) and the $d$-wave (solid line). The magnetic field $B$ is held at various values above the Feshbach resonance at $B_{\mathrm{res}}\sim 632\mathrm{G}$. The Feshbach resonance is much narrower than the shape resonance. By changing $B$, the position of the Feshbach resonance can be tuned through the shape resonance.

Figure 6

Decay rates of the molecular state for the $632\mathrm{G}$ Feshbach resonance in ${}^{87}\mathrm{Rb}$. The partial decay rates into the $s$-wave (dotted line) and $d$-wave (solid line) are shown. The $d$-wave shape resonance obviously has a drastic effect on the $d$-wave decay rate.

Figure 7

Fitting to the $S$-matrix elements. Results from the coupled-channels calculation (circles) are shown versus magnetic field $B$ for a fixed energy of $k_B\times 255\mu \mathrm{K}$. The fit curves (solid lines) are hardly visible, because they agree so well with the coupled-channels results.