Microscopy of an ultranarrow Feshbach resonance using a laser-based atom collider: A quantum defect theory analysis

We employ a quantum defect theory framework to provide a detailed analysis of the interplay between a magnetic Feshbach resonance and a shape resonance in cold collisions of ultracold ${}^{87}\mathrm{Rb}$ atoms as captured in recent experiments using a laser-based collider [M. Chilcott et al., Phys. Rev. Research 3, 033209 (2021)]. By exerting control over a parameter space spanned by both collision energy and magnetic field, the width of a Feshbach resonance can be tuned over several orders of magnitude. We apply a quantum defect theory specialized for ultracold atomic collisions to fully describe of the experimental observations. While the width of a Feshbach resonance generally increases with collision energy, its coincidence with a shape resonance leads to a significant additional boost. By conducting experiments at a collision energy matching the shape resonance and using the shape resonance as a magnifying lens, we demonstrate a feature broadening to a magnetic width of 8 G compared to a predicted Feshbach resonance width much less than $0.1$ mG.

Figure 1

Representative interaction potentials. The $s$-wave channel is shown by the blue solid curve. The addition of $d$-wave angular momentum gives rise to a potential (orange solid curve), with a barrier hosting a quasibound state (dashed orange line). The centrifugal barrier is highly exaggerated to be visible; in reality its height is orders of magnitude smaller than the depth of the well. We also show the potential of a closed-channel potential in red, also with $d$-wave character and hosting a bound state (dashed red line). The relative position of the closed channel can be tuned with a magnetic field, and the bound state associated with the Feshbach resonance can be pulled through the shape resonance.

Figure 2

Fractions of scattered atoms from postcollision absorption images. (a) Absorption images at a range of magnetic fields over the resonance for a collision energy of $E/k_{\mathrm{B}}=270\mu \mathrm{K}$. The collision axis is horizontal. (b) Close-up of a collision near 940 G. The scattering here is strongly anisotropic and has the signature of $d$-wave scattering [42]. (c) Integrated column density [blue (top) curve], to which our model is fitted in order to extract the parameters of the collision. From the fit the components of unscattered (orange curve) and the $s$-wave (red curve) and $d$-wave (green curve) scattered atoms are extracted.

Figure 3

Examples of Fano profiles of differing shape parameters $q$ as encountered in our experiments. Points show the measured scattered fraction as a function of magnetic field for four different collision energies $E$. The solid lines are curve fits based on the Beutler-Fano model outlined in Sec. 2c. The fitted parameters $B_{\mathrm{res}}$ and ${\mathrm{\Gamma }}_B$ [cf. Eq. (3)] are indicated as a dashed line and a shaded area, respectively. The experimental data illustrating the characteristic Fano profiles were previously published in [35].

Figure 4

Different length and energy scales of the scattering problem in the $\ell =2$ entrance channel. (a) Triplet potential of the entrance channel $V$ from Ref. [50]. At the long range, $V$ is well approximated by the van der Waals potential $V_{\text{eff}}$. (b) At threshold and over a wider range of separation, we show two scattering wave functions derived from coupled-channel calculations, offset vertically by their collision energies. An arrow labels the length scale of the van der Waals potential $\beta$. (c) At short range, the shape of the wave function is independent of energy apart from variations of the amplitude. The increase in amplitude demonstrated at $300\mu \mathrm{K}$ is associated with the nearby shape resonance. (d) Scattering wave function $u$ at long range along with the free-particle wave function $u_{\text{free}}$. The difference between the two is captured by a phase shift $\eta$.

Figure 5

Solution pairs inside the van der Waals radius: the zero-energy analytic solutions $u_{\pm }$, the WKB solutions $w_s$ and $w_c$ for ${\varphi }_1=0$ (computed using $R_0=0.1$) at $E/k_{\mathrm{B}}=300\mu \mathrm{K}$, and the corresponding reference functions ${\widehat{f}}_1$ and ${\widehat{g}}_1$.

Figure 6

QDT parameters (a) $\xi$, (b) $tan\lambda$, and (c) $C^{-2}$ as a function of energy. The blue lines show numerically calculated parameters using QDT reference functions based on the phase ${\varphi }_1=0$ [cf. Eq. (22)]. The orange lines show the QDT parameters transformed to ${\varphi }_1=0.590\pi$ using the analytic rotation formulas (B12).

Figure 7

Properties of the Feshbach resonance as measured at a number of collision energies: (a) the background phase shift and corresponding $q$ parameters, (b) the magnetic field of the resonance, and (c) the width of the resonance. We include predictions from the multichannel quantum defect theory model. The dashed line in (b) indicates the expected position of the Feshbach resonance in the absence of the open-channel effects (shape resonance).

Figure 8

MQDT calculations in both collision energy and magnetic field: (a) ${sin}^2{\delta }_d$, which is proportional to the scattering cross section, and (b) the scattering phase shift. In both panels, the position of the Fano resonance in the magnetic field is shown in red, while in (b) the resonance positions in energy are shown as orange solid lines and the noninteracting resonance positions are shown as orange dashed lines.

Figure 9

Representative measurement of the atom loss fitted by the described model. The extracted resonance position is shown in red.