Determination of atomic scattering lengths from measurements of molecular binding energies near Feshbach resonances

We present an analytic model to calculate the atomic scattering length near a Feshbach resonance from data on the molecular binding energy. Our approach considers finite-range square-well potentials and can be applied near broad, narrow, or even overlapping Feshbach resonances. We test our model on ${\mathrm{Cs}}_2$ Feshbach molecules. We measure the binding energy using magnetic-field modulation spectroscopy in a range where one broad and two narrow Feshbach resonances overlap. From the data we accurately determine the Cs atomic scattering length and the positions and widths of two particular resonances.

Figure 1

(Color online) Two-channel square-well model of Feshbach resonances. Scattering states are initially prepared in the entrance channel $\mid e⟩$; the closed channel $\mid c⟩$ supports a bare bound state at energy $E_c$. Feshbach coupling is modeled by the mixing of the two channels, which is resonantly enhanced when the bound state is near the scattering state. The interaction range is set to be the mean scattering length of the atomic van der Waals potential; see text.

Figure 2

(Color online) Example of a loss resonance at $48.2\mathrm{G}$. The losses are due to the coupling between atom pairs and $d$-wave molecules. The solid line is a Gaussian fit to the data. We find the resonant frequency to be $\nu =93.0\left(3\right)\mathrm{kHz}$ and the width $5.5\left(6\right)\mathrm{kHz}$. The inset shows the typical magnetic-field sequence used to associate ${\mathrm{Cs}}_2$ molecules with an oscillating magnetic field.

Figure 3

(Color online) Binding energy of cesium molecules near three Feshbach resonances as a function of the magnetic field. Zero energy corresponds to two Cs atoms in the absolute hyperfine ground-state sublevel ∣$F=3$, $m_F=3$⟩. The measurements are shown as open circles. The fit (solid line) is based on Eq. (13); see text. The inset shows an expanded view in the region of the two $d$- and $g$-wave narrow resonances. The error bars refer to the statistical uncertainties.

Figure 4

(Color online) Scattering length of ∣$F=3$, $m_F=3$⟩ cesium atoms in the magnetic-field range where three Feshbach resonances overlap. The solid curve shows the result of this work, while the dashed curve represents the prediction from a previous multichannel calculation [17].