Contrasting the wide Feshbach resonances in ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$

We compare and contrast the wide Feshbach resonances and the corresponding weakly bound states in the lowest scattering channels of ultracold ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$. We use high-precision measurements of binding energies and scattering properties to determine interaction potentials that incorporate non-Born-Oppenheimer terms to account for the failure of mass scaling between ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$. Correction terms are needed for both the singlet and the triplet potential curves. The universal formula relating binding energy to scattering length is not accurate for either system. The ${}^6\mathrm{Li}$ resonance is open-channel-dominated and the van der Waals formula of Gao [J. Phys. B 37, 4273 (2004)] gives accurate results for the binding energies across much of the resonance width. The ${}^7\mathrm{Li}$ resonance, by contrast, is weakly closed-channel-dominated and a coupled-channel treatment of the binding energies is required. Plotting the binding energies in universal van der Waals form helps illustrate subtle differences between the experimental results and different theoretical forms near the resonance pole.

Figure 1

Energy levels of the ${}^2{S}_{1/2}$ ground state of the ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$ atoms, showing the different hyperfine structure of the two isotopes, which at zero field have upper and lower total angular momentum quantum numbers $f=\frac{1}{2}$ and $\frac{3}{2}$ for ${}^6\mathrm{Li}$ and $f=1$ and 2 for ${}^7\mathrm{Li}$. The energy zero is the energetic center of gravity of the hyperfine multiplet. The numbers indicate the state labels for all the states of ${}^7\mathrm{Li}$ and the lowest two states of ${}^6\mathrm{Li}$. The lowest- and highest-energy states at finite $B$ have $m_f=\frac{1}{2}$ and $\frac{3}{2}$ for ${}^6\mathrm{Li}$ and $m_f=1$ and 2 for ${}^7\mathrm{Li}$.

Figure 2

Born-Oppenheimer potential energy curves of the Li${}_2$ molecule correlating with two ground-state atoms. The inset shows the adiabatic potential curves of the five-channel potential matrix at $B=0$ for the states with partial wave $L=0$ ($s$ wave) and total spin projection $M_F=2$, using the notation of Fig. 1 to label the channels. The inset also shows the location of the last zero-field $s$-wave level of the ${}^7\mathrm{Li}$${}_2$ molecule.

Figure 3

Energies of the separated-atom channels for $s$-wave scattering with total spin projection $M_F=2$ for ${}^7\mathrm{Li}$${}_2$ (solid lines) and $M_F=0$ for ${}^6\mathrm{Li}$${}_2$ (dotted lines). The dashed lines show the energy of the last molecular bound state in each case. The arrows show the pole positions $B_0$ of the Feshbach resonances in the lowest-energy $s$-wave channel of each species.

Figure 4

Scattering length $a$ as a function of magnetic field $B$ for the (1,2) channel of ${}^6\mathrm{Li}$ and the (1,1) channel of ${}^7\mathrm{Li}$, showing the difference in pole strength for the two systems despite a superficial similarity in magnetic-field width $\Delta$. The feature near 543 G in ${}^6\mathrm{Li}$ is an additional narrow resonance.

Figure 5

Observed and calculated bound-state energies $E_{\mathrm{b}}$ in the (1,1) channel of ${}^7\mathrm{Li}$ as a function of magnetic field $B$: coupled-channel calculation (solid line), experimental results from Refs. [22] (open circles) and [57] (diamonds), and approximations from the universal (dotted), Gribakin-Flambaum (dot-dashed), and Gao (dashed) formulas.

Figure 6

Plots of $\epsilon r^2$ against $1/r^2$, using van der Waals units of energy and length. The solid lines show the coupled-channel calculation, the points give the experimental results mapped using the calculated $a\left(B\right)$ function evaluated at the measured $B$ values, and the dotted, dot-dashed, and dashed lines give the results of the universal, Gribakin-Flambaum, and Gao formulas, respectively. (a) Bound-state energies in the (1,2) channel of ${}^6\mathrm{Li}$, showing experimental results from Zürn et al. [13]. The inset shows an expanded view near $1/r^2=0.00018$, or $a\approx 2200a_0$. (b) Bound-state energies in the (1,1) channel of ${}^7\mathrm{Li}$, showing experimental results from Dyke et al. [22].

Figure 7

Plot of $\epsilon r^2$ against $1/r^2$ in the (1,1) channel of ${}^7\mathrm{Li}$, showing the experimental results of Dyke et al. [22], mapped with two different $a\left(B\right)$ functions calculated when the ${}^1{{\Sigma }_g}^{+}$ potential is adjusted slightly to give two different pole positions, 737.74 and 737.64 G, which differ by $\pm 0.05$ G from our fitted value of 737.69(2) G.