Characterization of Feshbach resonances in $^6\mathrm{Li}{-}^7\mathrm{Li}$ using improved interaction potentials

Abstract

We characterize Feshbach resonances in all isotopologues of the $\mathrm{Li}{-}\mathrm{Li}$ system with improved interaction potentials. Starting from spectroscopically accurate Morse/long-range (MLR) potential-energy curves for the singlet ($X^{1}Σ^{+}$) and triplet ($a^{3}Σ^{+}$) electronic states of $\mathrm{Li}_2$, we apply small phenomenological inner-wall adjustments (following Julienne and Hutson, Phys. Rev. A 89, 052715 (2014), arXiv:1404.2623v3) and fit the resulting potentials to threshold measurements for the $^{6}\mathrm{Li}{-}^{6}\mathrm{Li}$ and $^{7}\mathrm{Li}{-}^{7}\mathrm{Li}$ isotopologues, including binding energies, scattering lengths, and Feshbach resonance positions. Using the optimized potentials in coupled-channels scattering calculations, we predict and characterize s-wave Feshbach resonances in the $^{6}\mathrm{Li}{-}^{7}\mathrm{Li}$ isotopologue. In its lowest-energy hyperfine channel, all resonances are narrow ($\sim 0.01{-}0.1$ G), strongly closed-channel dominated, and predominantly triplet in electronic spin character, in marked contrast to the homonuclear systems. These results provide a foundation for designing Raman optical-transfer pathways to produce ultracold $\mathrm{Li}_2$ molecules in deeply bound rovibrational levels of both the $X^1Σ^{+}$ and $a^3Σ^{+}$ potentials across all three isotopologues.

Figures (6)

• Figure 1: Zeeman shifts of the hyperfine levels in the electronic ground state ($^{2}S_{1/2}$) of ${}^{6}\mathrm{Li}$ (red) and ${}^{7}\mathrm{Li}$ (blue), labeled by the corresponding quantum numbers $f$ and $m_f$. The zero of energy is defined as the energy in the absence of hyperfine and Zeeman interactions. The numbers on the right label the atomic hyperfine states used to specify each atom in a scattering channel.
• Figure 2: Adiabatic Born--Oppenheimer (ABO) potential energy curves for the ground singlet $X^1\Sigma^+$ and triplet $a^3\Sigma^+$ electronic states of Li$_2$ isotopologues near the dissociation threshold at $B = 0$. Energies are shown in units of GHz, and $R$ represents the internuclear separation (in units of Å). The solid horizontal lines represent energy levels of the separated atom hyperfine states which have the same $M_F$ as the lowest energy hyperfine state (${}^{6}$Li$_2$: $M_F = 0$; ${}^{7}$Li$_2$: $M_F = 2$; ${}^{6}$Li${}^{7}$Li: $M_F = 3/2$). The dash-dot horizontal lines represent the last-bound vibrational levels supported by the ABO potentials, which do not account for hyperfine interactions. The dotted horizontal lines represent molecular bound states from the coupled-channel calculations, labeled by the approximately good quantum numbers $(v, I, M_I)$.
• Figure 3: (a) Zeeman shifts of the lowest hyperfine thresholds (solid curves) and $s$-wave molecular bound states (dotted curves) for ${}^{6}$Li--${}^{6}$Li (red), ${}^{7}$Li--${}^{7}$Li (blue), and ${}^{6}$Li--${}^{7}$Li (green). The bound state curves are labeled by appropriate vibrational and spin quantum numbers. Crossing points between the bound states and the hyperfine thresholds correspond to Feshbach resonances, and are indicated by arrows. (b) Singlet fractions of the molecular bound states as a function of $B$.
• Figure 4: Closed-channel fraction for the lowest-energy hyperfine-channel Feshbach molecules. $(B-B_0)/|\Delta|$ represents the scaled detuning. Note that $Z_{\mathrm{closed}}=0$ at $B = B_0$ according to the two-channel model definition of the open/closed-channel fractions in Ref. Chin2010Rev.Mod.Phys..
• Figure 5: Vibrational-level energy shifts produced by adding the short-range shift term to the Li$_2$ interaction potentials. The ordinate shows $\Delta E(v)=E_{\rm shifted}(v)-E_{\rm MLR,2013}(v)$ (MHz), referenced to the unshifted 2013 MLR potentialsGunton2013Phys.Rev.ASemczuk2013Phys.Rev.A; the horizontal dashed line marks $\Delta E=0$. (a) Ground singlet $X^{1}\Sigma_{g}^{+}$ manifold for $^{7}$Li$_2$ and $^{6}$Li$_2$ over the vibrational range shown. (b) Triplet $a^{3}\Sigma_{u}^{+}$ manifold. Curves compare the shifted MLR potential optimized in this work (MLR$+$shift) and the shifted 2014/NIST potentialsJulienne2014Phys.Rev.A (NIST$+$shift). Positive (negative) $\Delta E$ corresponds to a lower (higher) bound level relative to the 2013 MLR reference.