Observation of Broad $d$-Wave Feshbach Resonances with a Triplet Structure

High partial-wave ($l\ge 2$) Feshbach resonance (FR) in an ultracold mixture of ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ atoms is investigated experimentally aided by a partial-wave insensitive analytic multichannel quantum-defect theory. Two “broad” resonances from coupling between $d$ waves in both the open and closed channels are observed and characterized. One of them shows a fully resolved triplet structure with a splitting ratio well explained by the perturbation to the closed channel due to interatomic spin-spin interaction. These tunable “broad” $d$-wave resonances, especially the one in the lowest-energy open channel, could find important applications in simulating $d$-wave coupling dominated many-body systems. In addition, we find that there is generally a time and temperature requirement, associated with tunneling through the angular momentum barrier, to establish and observe resonant coupling in nonzero partial waves.

Figure 1

The $d$-wave Feshbach resonance. (a) A simplified two-channel illustration for the origin of the triplet structure in $d$-wave FRs. The inset shows the angular part of the molecular wave function (spherical harmonic $|Y_2^{m_l^{\prime }}{|}^2$), with arrows illustrating the spins of the valence electrons aligned with the bias magnetic field (for the spin-triplet closed channel). In this example, the $m_l^{\prime }=0$ ($|m_l^{\prime }|=2$) state has lower (higher) energy because the magnetic dipole-dipole interaction is dominated by the head-to-tail attractive (side-by-side repulsive) configuration. The splitting of the bound states gives rise to a triplet FR structure when the open channel is also $d$ wave. Panels (b)–(d) show the remaining fraction of ${}^{85}\mathrm{Rb}$ (normalized to the baseline value in off-resonance regions) after coexisting with ${}^{87}\mathrm{Rb}$ atoms (b) for 1 s at $16\mu \mathrm{K}$, (c) for 1 s at $1.2\mu \mathrm{K}$, and (d) for 1.6 s at 400 nK. The number of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ atoms in the off-resonance regions are, respectively, about $2\times {10}^5$ and $4\times {10}^5$ for (b), $1.2\times {10}^4$ and $1.2\times {10}^5$ for (c), and $3.5\times {10}^3$ and $5.5\times {10}^4$ for (d). The characteristic triplet structure for $l=l^{\prime }=2$ is clearly visible in (d). The shaded gray region indicates the position at which the $|m_l^{\prime }|=2$ bound states become degenerate with the threshold of the open channel. All data points are averages over five measurements, and the error bars show the typical standard deviations. The eye-guiding black solid lines in (b) [(c),(d)] are fits to the data using an asymmetric double sigmoidal [multiple-peak] function.

Figure 2

Reduced generalized scattering lengths ${\tilde{a}}_l(B)/{\overline{a}}_l$ in the (a) ${}^{85}\mathrm{Rb}|2,-2⟩+{}^{87}\mathrm{Rb}|1,-1⟩$ and (b) ${}^{85}\mathrm{Rb}|2,+2⟩+{}^{87}\mathrm{Rb}|1,+1⟩$ scattering channels, for partial waves $l=0$ through $l=3$. The blue filled (empty) circles denote experimentally observed positions for the “broad” (“narrow”) $d$-wave FRs. The arrows in (b) link the observed FRs to their corresponding predicted positions. The identification is done through coupled-channel calculations.

Figure 3

Tunneling time through the angular momentum barrier, ${\tau }_l$, for the ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ system, as a function of energy $\epsilon$ in units of $\epsilon /k_B$.