Observable Resonances in Efimov-unfavored Systems

TL;DR

The paper addresses how the intraspecies scattering length $a_ ext{BB}$ shifts Efimov resonances and the three-body parameter $a_-^{(0)}$ in Efimov-unfavored LLH systems. It introduces a finite-range, separable-potential three-body framework and compares with zero-range hyperspherical results to study ^{23}Na$_2$^{40}$K, revealing two distinct mechanisms dictated by the sign of $a_ ext{BB}$ and the large disparity between scaling parameters $s_0$ and $s_0^*$. For $a_ ext{BB}<0$, a transition region between two Efimov scalings causes strong $|a_ ext{BB}|$-dependence of the spectrum and $a_-^{(n)}$, while for $a_ ext{BB}>0$ the spectrum splits into two well-separated ladders, pushing the first resonance $a_-^{(0)}$ to very large values. Theoretical results show good agreement with experimental Na$_2$K trimer binding energies, validating the mechanisms and highlighting that observable 3BP values arise in LLH systems when $a_ ext{BB}<0$.

Abstract

Three-body loss resonances associated with heavy-heavy-light Efimov states have been observed for over a decade in ultracold mixtures tuned near interspecies Feshbach resonances. For light-light-heavy systems, observing such resonances has been far more challenging due to the substantially large Efimov spacing. In these Efimov-unfavored systems, the intraspecies scattering length $a_\text{BB}$ has been shown to significantly affect the overall Efimov scenario, namely, the positions of the Efimov resonances $a_{-}^{(n)}$ and the three-body parameter (3BP) $a_{-}^{(0)}$. The present article explains the origin behind this influence by highlighting two primary mechanisms via which both the magnitude and sign of $a_\text{BB}$ govern the Efimov spectrum and set the resulting 3BP $a_{-}^{(0)}$. By employing van der Waals interactions for $^{23}$Na$_2{}^{40}$K, we attribute the vital role of $a_\text{BB}$ in Efimov-unfavored systems to the large difference between the Efimov scaling parameters for two and three resonant interactions, $s_0$ and $s_0^*$. In particular, we account for the unusually large $a_{-}^{(0)}$ obtained in light-light-heavy systems with $a_\text{BB}>0$ (e.g., $^{41}$K$_2{}^{87}$Rb), and show that the first Efimov resonance can still occur at an experimentally accessible value when $a_\text{BB}<0$.

Figures (8)

• Figure 1: The Efimov scaling parameters for two (solid) and three (dashed) resonant pairs $s_0$ and $s_0^*$ plotted vs. the system mass imbalance $m_\text{B}/m_\text{X}$
• Figure 2: The Efimov spectrum for ^23Na2^40K at heteronuclear unitarity ($\abs{a_\text{BX}}\rightarrow\infty$) with $\kappa_n=\sqrt{-m_\text{B}E_n}/\hbar$. Each plotted quantity is computed from both a finite-range and a zero-range calculation. The dimer threshold (green and orange) divides the spectrum into lower trimers (red and blue) and upper resonances (cyan and black).
• Figure 3: The zero-range hyperspherical potentials $U_n(R)$ at heteronuclear unitarity ($\abs{a_\text{BX}}\rightarrow\infty$), shown for $a_\text{BB}<0$ (a) and $a_\text{BB}>0$ (b). The insets display the corresponding hyperangular eigenvalues $\lambda_n(R)$ defined in Eq. (\ref{['ZRpot']}). In the right panel, the upper potential $U_2(R)$ (cyan) is magnified by a factor of 100 for better visibility.
• Figure 4: The variable Efimov scaling parameter, given through the ratios $E_n/E_{n+1}$ at $\abs{a_\text{BX}} \rightarrow\infty$, graphed for the lowest three pairs of consecutive trimer states in Fig. \ref{['fig1spectrum']}. Solid curves correspond to FR calculations, while dashed curves correspond to ZR calculations. The horizontal dashed lines represent the universal Efimov scaling parameters: $s_0^*=1.018$ (upper) at $1/a_\text{BB}=0$ and $s_0=0.285$ (lower) at $1/a_\text{BB}\rightarrow-\infty$, corresponding to three and two resonant pairs respectively.
• Figure 5: The atom-dimer (K-Na2) scattering length $a_\text{AD}$ at $\abs{a_\text{BX}}\rightarrow\infty$, calculated for the lower ZR potential $U_1(R)$ with $a_\text{BB}>0$ (Fig. \ref{['fig2potcurves']}(b)). The vertical dashed lines mark the values of $a_\text{BB}$ at which the lower spectrum trimers intersect the Na2 + K threshold, as determined from the ZR data in Fig. \ref{['fig1spectrum']}. Recall that this ZR calculation is equipped with a log-derivative that ties it with the FR model.