Multichannel effects in the Efimov regime from broad to narrow Feshbach resonances

We study Efimov physics of three identical bosons with pairwise multichannel interactions for Feshbach resonances of adjustable width in magnetic field. We find that the two-body multichannel realization of the interaction can affect the universal Efimov spectrum, especially for resonances of intermediate width. We analyze two scenarios that are equivalent on the two-body level but differ in their realization in three-body multichannel spin space. The deviations in the Efimov spectra of these scenarios are caused by trimer states in the closed interaction channels whose binding energy and coupling strength to the Efimov states depend on the spin realization. However, in the narrow resonance limit the Efimov spectrum is set by the resonance width parameter $r^{*}$ only and does not depend on the realization in spin space. We find this limit to be even independent of the interaction potential considered. In the broad resonance limit all excited Efimov trimer energies approach the ones from the corresponding single-channel system for the scenarios investigated.

Figure 1

We compare the three-body spectra of the systems we have analyzed. The red solid lines correspond to the $Q=\underline{bb}$ realization, whereas the blue dashed lines correspond to the $Q=\underline{ab}$ realization of the system. In both cases the resonance positions are fixed at a threshold difference of ${\overline{\epsilon }}_0=1.5$ and the background scattering length is fixed to ${\overline{a}}_{bg}=-0.2$. In (a) we show the inverse dissociation scattering lengths $1/a_{-}^{\left(n\right)}$ of the three most deeply bound trimer states rescaled with the lowest (in absolute value) single-channel dissociation scattering length $a_{-}^{sc}$ as a function of the resonance width ${\overline{r}}^{*}$. The open circles and diamonds correspond to experimental data [22, 23, 25, 34, 37, 38, 39, 47], where we rescaled $a_{-}^{\left(0\right)}$ by the universal value of $-9.73r_{\mathrm{vdW}}$ [26] and set ${\overline{r}}^{*}=r^{*}/r_{\mathrm{vdW}}$. The diamonds correspond to very small values of $r^{*}$ and have been shifted by ${\overline{r}}^{*}=0.01$ to fit on the plot. The red band indicates the van der Waals universal region up to $\pm 15\%$. The thin black dashed line indicates $a_{-}^{\mathrm{sc}}/a_{\mathrm{bg}}$, while the light gray horizontal lines correspond to the trimer positions of the single-channel system. The vertical gray lines indicate the resonance widths at which we obtained full Efimov spectra shown in (b)–(d) for ${\overline{r}}^{*}=0.01,1,100$, respectively. In (b)–(d) we show the binding energies of the lowest three trimer states (red solid and blue dashed lines) as a function of the inverse scattering length $1/\overline{a}$. For better visibility both axes are rescaled as indicated by the axes labels. The thick black line indicates the dimer binding energy which agrees for both realizations since the $t$ operator is identical. The thin gray lines correspond to the trimer spectrum of the single-channel model, while the thick gray line corresponds to the single-channel dimer binding energy. In (b) the trimer and dimer lines stop at some positive inverse scattering length close to $1/{\overline{a}}^{1/4}=0.4$, since the threshold difference related to this value of the scattering length is zero and we enter a regime irrelevant for this investigation beyond this point.

Figure 2

Efimov spectrum in the narrow resonance limit. The purple (gray) lines indicate the five lowest Efimov trimer binding energies, whereas the thick black line indicates the ground-state dimer binding energy. The binding energies are shown for varying values of inverse scattering length. All quantities are given in units related to $r^{*}$ (see Sec. 4) and are rescaled as indicated by the axes labels for better visibility.

Figure 3

Narrow resonance limit of the closed-channel ($\underline{ab}a$ and $\underline{aa}b$) trimer binding energies relative to the closed-channel dimer binding energy with respect to the closed-channel threshold, which in the narrow resonance limit coincides with the resonance position ${\overline{\epsilon }}_0$. The gray line indicates the value ${\overline{\epsilon }}_0=1.5$ that we set the resonance position to throughout our analysis of the multichannel system with varying values of ${\overline{r}}^{*}$.