Finite-range corrections near a Feshbach resonance and their role in the Efimov effect

We have measured the binding energy of ${}^7$Li Feshbach molecules deep into the nonuniversal regime by associating atoms in a Bose-Einstein condensate with a modulated magnetic field. We extract the scattering length from these measurements, correcting for nonuniversal short-range effects using the field-dependent effective range. With this more precise determination of the Feshbach resonance parameters we reanalyze our previous data on the location of atom loss features produced by the Efimov effect [Pollack et al., Science 326, 1683 (2009)]. We find the measured locations of the three- and four-body Efimov features to be consistent with universal theory at the 20%–30% level.

Figure 1

Magneto-association induced loss at $B=734.5$G, where $a\simeq 1100a_0$. The main plot shows loss spectra for thermal gases with the following modulation amplitudes and approximate temperatures: green (triangles) 0.57 G/$10\mu \text{K}$; blue (squares) 0.14 G/$3\mu \text{K}$; and red (circles) 0.57 G/$3\mu \text{K}$. The solid curves are fits to Lorentzians convolved with thermal Boltzmann distributions. For the 0.57 G/$3\mu \text{K}$ data the lower frequency resonance is a subharmonic response, while the primary resonance is thermally broadened by strong modulation. The inset (black diamonds) corresponds to a BEC with a modulation amplitude of 0.14 G. The solid black line is a Lorentzian fit to the condensate resonance and the vertical dashed line in the main figure is the resonance location $E_{\mathrm{b}}/h=450$kHz found from this fit.

Figure 2

Results of modulation spectroscopy using condensates (red circles) and $\sim$$3\mu \text{K}$ thermal clouds (black squares). (a) and (b) $E_{\mathrm{b}}$ vs $B$; and (c) and (d), the same data plotted as $\gamma \equiv {(m{E}_{\mathrm{b}}/{\hslash }^2)}^{1/2}$ vs $B$. The vertical error bars correspond to uncertainty in fitting to the binding energy resonances, while the horizontal error bars are the statistical uncertainties due to shot-to-shot variations of the magnetic field. The relatively large error bars below 725 G arise from the broadening of the resonance from the strong modulation required to produce a detectable signal. The lines are fits of the measurements to Eq. (1) using the various models. Solid (black), universal model, $E_{\mathrm{b}}={\hslash }^2/m{a}^2$; dashed (green), simple two-channel model, Eq. (2), with the parameters given in the text; dot-dashed (blue), complex two-channel model given in Ref. [11] [Eq. (26)], again with parameters as given in the text (the two-channel model of Ref. [10] gives similar results); and dotted (red), Eq. (4) using $R_{\mathrm{e}}$($B$) from Fig. 3. The fits exclude data below 725 G, where $a\simeq 250a_0$ is no longer much greater than $R^{*}$, and the validity of the nonuniversal corrections becomes questionable. While the nominal fitting parameters are $a_{\mathrm{bg}}$, $\Delta$, and $B_{\infty }$, $\Delta$ is fixed at $-$174 G. The fits are weighted by the inverse uncertainties. The resulting Feshbach resonance parameters are given in Table .

Figure 3

Coupled-channels calculation of $a$ and $R_{\mathrm{e}}$ for the $F=1,m_F=1$ Feshbach resonance in ${}^7$Li. The horizontal and vertical dashed (blue) lines indicate $a=0$ and $B_{\infty }$, respectively. $R_{\mathrm{e}}$ was fit to a polynomial expansion in the scaled field $\beta =(B-737.7$ G)/G to obtain $R_{\mathrm{e}}\left(\beta \right)/a_0=-6.2+3.50\beta -9.2\times {10}^{-3}{\beta }^2-6.5\times {10}^{-5}{\beta }^3+5.7\times {10}^{-7}{\beta }^4$. This polynomial fit is used to obtain $R_{\mathrm{e}}$ in Eq. (4). Similar calculations for the $F=1,m_F=0$ [27] and $F=1,m_F=1$ [44] resonances have been previously presented. In the latter case, the calculation of $R_{\mathrm{e}}$ agrees well with our results.

Figure 4

Three-body loss rate coefficient $L_3$ vs $a$. The points are values extracted from the measured trap loss, with the green, blue, and red points corresponding to condensates with different trap frequencies, and the purple to a thermal gas ($T\approx$ 1–3 $\mu$K), as reported in Ref. [5]. The solid line shows universal scaling in $a$, where the positions of the features are determined by the single feature $a_2^{+}$. The only additional fitted parameters are the widths ${\eta }_{+}=0.075$ and ${\eta }_{-}=0.17$, for the $a>0$ and $a<0$ sides of the resonance, respectively, and an overall scale factor of 5.5 on the $a>0$ side of resonance. The need for scaling the universal theory for positive $a$ is unknown. We ascribe the deviation from the universal curve for small, negative $a$ to the presence of the four-body feature $a_{1,1}^T$. The dashed lines are guides to the eye, showing $a^4$ scaling.