Magnetic control of ultra-cold ⁶Li and ¹⁷⁴Yb(³P₂) atom mixtures with Feshbach resonances

For the determination of collisional properties of Li(²S$_{1/2}$) + Yb(³P₂) system we have (i) determined the short-range and long-range electronic interaction potentials $V(\vec{R},\tau )$ and (ii) setup a close-coupling calculation that treats the hyperfine and Zeeman interaction, molecular rotation, magnetic dipole–dipole (MDD) interaction, and the electronic interactions on equal footing. Here, $\vec{R}$ describes the interatomic separation and orientation and τ labels other quantum numbers that uniquely label the electronic potentials. We have calculated the non-relativistic doublet $^{2}{{\Sigma }^{+}}$ and $^{2}\Pi$ and quartet $^{4}{{\Sigma }^{+}}$ and $^{4}\Pi$ potentials, shown in figure 1, using the configuration-interaction method within the MolPro package [38].

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We used the aug-cc-pCVTZ basis set for Li [39] and chose a basis set constructed from the (15s 14p 12d 11f 8g)/(8s 8p 7d 7f 5g) wave functions of Dolg and Cao [40, 41] for Yb. The ytterbium basis relies on a relativistic pseudopotential that describes the inner orbitals up to the 3d¹⁰ shell. Only, the 2s valence electrons of Li and 4f¹⁰ and 6s² valence electrons of Yb are correlated in the ab initio calculation. Our potentials agree well with the calculations of [42]. However, the potentials are not spectroscopically accurate and we have modified their shape to fit to the experimental loss rate spectrum. For $R\gt 27{{a}_{0}}$, beyond the Le Roy radius where the electron clouds of the atoms have negligible overlap, these electronic potentials are smoothly connected to the long-range isotropic and anisotropic van-der-Waals potentials determined from atomic polarizabilities [9]. In addition, we included the MDD interaction.

Next, we find it instructive to analyze the relative strength of the long-range atom–atom interactions with the Zeeman, hyperfine, and relative rotational interactions. Figure 2 shows the two major anisotropic interactions, the MDD ${{C}_{3}}/{{R}^{3}}$ and the anisotropic dispersion $\Delta {{C}_{6}}/{{R}^{6}}$ (AD) interaction, which can lead to the reorientation of the Li and Yb$^{*}$ angular momenta during a collision, as a function of R. These potentials as well as the isotropic dispersion ${{C}_{6}}/{{R}^{6}}$ (ID) interaction were drawn, assuming weighted-average values of C₃ and C₆, and $\Delta {{C}_{6}}$, based on our previous calculations [9]. In fact, ${{C}_{6}}=({{C}_{6\Sigma }}+2{{C}_{6\Pi }})/3$, where ${{C}_{6\Sigma }}$ and ${{C}_{6\Sigma }}$ are the dispersion coefficient of the Σ and Π potentials, respectively. The $\Delta {{C}_{6}}$ coefficient is found from tensor analyses of the parallel and perpendicular components of the dynamic polarizability, as described in section 2 of [43]. In addition, the molecule can rotate, here estimated by ${{\hbar }^{2}}/(2{{\mu }_{r}}{{R}^{2}})$ with reduced mass ${{\mu }_{r}}$. Finally, hyperfine splitting (hs) of the ⁶Li atom, and the Zeeman interaction at magnetic field strengths B = 100 and 400 G are shown. At much shorter separation and larger energies, not shown in the figure, the exchange interaction due to the exponentially-increasing electron cloud overlap lifts the degeneracy of the double and quartet potentials and mixes the spin–orbit states of the Yb(³P_j) levels.

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The figure is most easily interpreted in the coordinate system and angular-momentum basis set with projection quantum numbers defined along the external magnetic field direction. In this coordinate system the rotational, spin–orbit, and hyperfine, and Zeeman interactions as well as the ID potential shift molecular levels, whereas the exchange interaction, the MDD interaction, and the anisotropic component of the dispersion potential lead to coupling between rotational, hyperfine, and Zeeman components. For different interatomic separations different forces dominate. For example, when the curves for the magnetic dipole or AD interaction cross the hyperfine, Zeeman and/or rotational energies spin flips combined with changes in the rotational state can occur. In fact, the AD curve crosses the B = 100 and 400 G Zeeman curves near $R=50{{a}_{0}}$ and $40{{a}_{0}}$, respectively, whereas the MDD curve crosses ${{\delta }_{{\rm hs}}}$ and ${{\Delta }_{Z}}$ at much shorter R, where chemical bonding or exchange interaction will also play an important role.

We have setup a close-coupling model for the scattering of fermionic ⁶Li and the metastable bosonic ¹⁷⁴Yb isotope. The lithium atom is uniquely specified by electron spin ${{s}_{{\rm Li}}}=1/2$ and nuclear spin ${{i}_{{\rm Li}}}=1$. The metastable Yb$^{*}$ is specified by electron spin ${{s}_{{\rm Yb}}}=1$ and electron orbital angular momentum ${{l}_{{\rm Yb}}}=1$. It is also convenient to define total atomic angular momenta ${{\vec{f}}_{{\rm Li}}}={{\vec{s}}_{{\rm Li}}}+{{\vec{\imath }}_{{\rm Li}}}$ and ${{\vec{j}}_{{\rm Yb}}}={{\vec{s}}_{{\rm Yb}}}+{{\vec{l}}_{{\rm Yb}}}$ for Li and Yb$^{*}$, respectively. Their projections along the B-field direction are ${{m}_{{\rm Li}}}$ and ${{m}_{{\rm Yb}}}$. The Hamiltonian for the relative motion of the two such atoms in a magnetic field B along the $\hat{z}$ direction is

$$\begin{eqnarray}&&H=-\frac{{{\hbar }^{2}}}{2{{\mu }_{r}}}\frac{{{{\rm d}}^{2}}}{{\rm d}{{R}^{2}}}+\frac{{{{\vec{\ell }}}^{2}}}{2{{\mu }_{r}}{{R}^{2}}}+{{H}_{{\rm SO}}}+{{H}_{{\rm hf}}}+{{H}_{Z}}+U(\vec{R}),\end{eqnarray} \tag{ 1 }$$

where the rotational Hamiltonian is ${{\hbar }^{2}}{{\vec{\ell }}^{2}}/(2\mu {{R}^{2}})$, the spin–orbit interaction of Yb$^{*}$ is ${{H}_{{\rm SO}}}={{a}_{{\rm SO}}}{{\vec{l}}_{{\rm Yb}}}\cdot {{\vec{s}}_{{\rm Yb}}}$, the hyperfine interaction of ⁶Li is ${{H}_{{\rm hf}}}$ $\;=\;{{a}_{{\rm hf}}}{{\vec{s}}_{{\rm Li}}}\cdot {{\vec{\imath }}_{{\rm Li}}}$, and the Zeeman interaction ${{H}_{Z}}=({{g}_{e,{\rm Li}}}{{s}_{{\rm Li},z}}+{{g}_{N,{\rm Li}}}{{i}_{{\rm Li},z}}+{{l}_{{\rm Yb},z}}+{{g}_{e,{\rm Yb}}}{{s}_{{\rm Yb},z}}){{\mu }_{B}}B$, with Bohr magneton ${{\mu }_{B}}$ and projection operators ${{s}_{{\rm Li},z}}$ etc. The strengths ${{a}_{{\rm SO}}}$ and ${{a}_{{\rm hf}}}$ are taken from [44, 45]. The ${{g}_{e,{\rm Li}}}$, ${{g}_{e,{\rm Yb}}}$, and ${{g}_{N,{\rm Li}}}$ are the electronic and nuclear g-factors of Li and Yb* from [45, 46]. Finally, $U(\vec{R})$ describes the MDD interaction and the non-relativistic electronic potentials, which are reexpressed in terms of (tensor) coupling operators between electron spins ${{\vec{s}}_{{\rm Li}}}$ and ${{\vec{s}}_{{\rm Yb}}}$, and the angular momenta ${{\vec{l}}_{{\rm Yb}}}$ and $\vec{\ell }$. For $R\to \infty$ the interaction $U(\vec{R})\to 0$.

There are four contributions to $U(\vec{R})$: a spin-independent isotropic potential ${{V}_{{\rm iso}}}(R)$, which is proportional to an attractive $1/{{R}^{6}}$ potential for large separations, a short-range isotropic exchange interaction $V_{{\rm exc}}^{\Sigma ,\Pi }(R){{[{{\vec{s}}_{{\rm Li}}}\;\otimes \;{{\vec{s}}_{{\rm Yb}}}]}_{00}}=-V_{{\rm exc}}^{\Sigma ,\Pi }(R){{\vec{s}}_{{\rm Li}}}\cdot {{\vec{s}}_{{\rm Yb}}}/\sqrt{3}$, which splits doublet from quartet Σ and Π potentials and falls of exponentially for large R, and an anisotropic quadrupole-like interaction ${{V}_{{\rm ani}}}(R){{[{{\hat{C}}_{2}}(\hat{R})\;\otimes \;{{[{{\vec{l}}_{{\rm Yb}}}\;\otimes \;{{\vec{l}}_{{\rm Yb}}}]}_{2}}]}_{00}}$, which lifts the degeneracy of the Σ and Π potentials and ${{V}_{{\rm ani}}}(R)\propto 1/{{R}^{6}}$ for large R. Here, ${{\hat{C}}_{kq}}(\hat{R})$ is a spherical harmonic. The four interaction strengths ${{V}_{{\rm iso}}}(R)$, $V_{{\rm exc}}^{\Sigma ,\Pi }(R)$, and ${{V}_{{\rm ani}}}(R)$ are constructed such that in the body-fixed frame with projections along the internuclear axis the sum of the interactions closely reproduces our four non-relativistic potentials.

The Hamiltonian is constructed in the atomic basis or channels ${{Y}_{\ell {{m}_{\ell }}}}(\theta ,\phi )|{{\alpha }_{{\rm Li}}}\rangle |{{\alpha }_{{\rm Yb}}}\rangle$, where ${{Y}_{\ell {{m}_{\ell }}}}(\theta ,\phi )$ is a spherical harmonic and angles θ and ϕ give the orientation of the internuclear axis relative to the magnetic field direction. Hence, the rotational Hamiltonian is diagonal in this basis. The kets $|{{\alpha }_{{\rm Li}}}\rangle$ are eigen states of the atomic Zeeman plus hyperfine Hamiltonian of ⁶Li. For fields $B\gt 100$ G, of interest in this paper and where the Paschen–Back limit holds, we denote states by ${{\alpha }_{{\rm Li}}}={{m}_{s}},{{m}_{i}}$, where m_s and m_i are the projections of ${{\vec{s}}_{{\rm Li}}}$ and ${{\vec{\imath }}_{{\rm Li}}}$, respectively (and ${{m}_{{\rm Li}}}={{m}_{s}}+{{m}_{i}}$). The kets $|{{\alpha }_{{\rm Yb}}}\rangle$ are eigen states of the atomic spin–orbit and Zeeman Hamiltonian of ¹⁷⁴Yb in the ³P term. As the spin–orbit interaction is orders of magnitude larger than the Zeeman interaction we denote the states by ${{\alpha }_{{\rm Yb}}}={{j}_{{\rm Yb}}}{{m}_{{\rm Yb}}}$. Coupling between the basis states is due to $U(\vec{R},\tau )$. The Hamiltonian conserves ${{M}_{{\rm tot}}}={{m}_{{\rm Li}}}+{{m}_{{\rm Yb}}}+{{m}_{\ell }}$ and only even (odd) ℓ are coupled.

In the experiment of [1] the ⁶Li atoms are prepared in the energetically-lowest hyperfine state with ${{m}_{{\rm Li}}}=1/2$ while the Yb atoms are prepared in the ³P₂ ($|{{j}_{{\rm Yb}}}{{m}_{{\rm Yb}}}\rangle =|2,-1\rangle$) sublevel. For ultra-cold gasses with temperatures near 1 μK we can focus on so-called s-wave or $\ell =0$ scattering. Theoretically, this corresponds to calculations including channels with ${{M}_{{\rm tot}}}=-1/2$ and even partial waves ℓ. We will also present results for ultra-cold scattering with Yb$^{*}$ prepared in the state $|{{j}_{{\rm Yb}}}{{m}_{{\rm Yb}}}\rangle =|2,-2\rangle$. This corresponds to calculations for ${{M}_{{\rm tot}}}=-3/2$.

Figure 3 shows a sample of the medium- to long-range adiabatic potentials, obtained by diagonalizing H excluding the radial kinetic energy operator, as a function of interatomic separation R near the ${\rm Y}{{{\rm b}}^{*}}$(³P₂) dissociation limit at a magnetic field of B = 300 G. Only potentials dissociating to the ${{f}_{{\rm Li}}}=1/2$ or $3/2$ and ${{j}_{{\rm Yb}}}=2$, ${{m}_{{\rm Yb}}}=0,-1,-2$ limits are visible in the figure. Inelastic collisional losses to energetically lower-lying ${\rm Y}{{{\rm b}}^{*}}$(³P, ${{j}_{{\rm Yb}}}=0,1$) states do nevertheless exist. Feshbach resonances occur in channels with a dissociation energy above that of the entrance channel.

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Figure 4 shows the experimental inelastic rate coefficient [1] at a temperature of $1.6\;\mu {\rm K}$ in the collision between ⁶Li and ¹⁷⁴ ${\rm Y}{{{\rm b}}^{*}}$ atoms and our best-fit theoretical inelastic rate coefficient at a collision energy of $E/k=1.6\;\mu {\rm K}$. We include channels with ${{M}_{{\rm tot}}}=-1/2$ and even ℓ up to 8 leading to a close-coupling calculation with 99 channels. A thermal average of the theoretical inelastic rate coefficient is not required as we are in the Wigner threshold limit where this rate is independent of collision energy. We observe at least one clear resonance at B = 450 G and possibly three weaker resonances. From calculations that include fewer partial waves we find that the resonances can not be labeled by a single partial wave. Their locations shift significantly and only converge to within a few Gauss when $\ell =8$ channels are included. Including channels with $\ell \leqslant 10$ shifts resonance by no more than 0.1 G. This is well within the width of the resonances for both figures 4 and 5. Consequently, we conclude that it is sufficient to include channels up to $\ell \leqslant 8$.

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Our bound-state calculation allows us to identify the dominant closed channel that couples to the s-wave entrance channel at B = 450 G. We identify this channel with the ℓ = 4 partial wave and projection ${{m}_{\ell }}\;=\;1$. This coupling is due to anisotropic interactions, such as the AD forces at large separations. Moreover, this channel dissociates to the atomic limit ⁶Li(m_s = 1/2, m_i = −1) + ¹⁷⁴Yb(³P, j = 2, m = −1), which lies above the entrance channel (see figure 3).

Figure 4 also provides evidence that the inelastic rate of the strong resonance is close to the universal rate, as shown by the dashed line in figure 4. This is a clear indication that the collisional system is close to the universal regime for losses on the dominant resonance and lies well below for other magnetic fields [43, 47].

The agreement between the experimental [1] and our theoretical spectrum is only obtained by allowing the coefficients ${{V}_{{\rm iso}}}(R)$, $V_{{\rm exc}}^{\Sigma ,\Pi }(R)$, and ${{V}_{{\rm ani}}}(R)$ to vary such that the depth of the four non-relativistic potential curves is changed. We can not exclude the existence of other shapes of potentials which will lead to a loss rates that is consistent with the experimental data.

The lower panel of figure 4 shows the scattering length a as a function of the magnetic field strength for the collisions between ⁶Li(${{m}_{s}}=-1/2,{{m}_{i}}=1$) and ¹⁷⁴Yb($^{3}{{{\rm P}}_{2}},{{j}_{{\rm Yb}}}=2,{{m}_{{\rm Yb}}}=-1$) atoms at a collisional energy of 1.6 μK. Here the scattering length is defined through ${{S}_{{\rm elastic}}}={\rm exp} (-2{\rm i}k[a-{\rm i}b])$, where ${{S}_{{\rm elastic}}}$ is the elastic S-matrix element for the s-wave scattering channel, wavenumber k is given by $E={{\hbar }^{2}}{{k}^{2}}/(2{{\mu }_{r}})$, and positive length b can be related to the inelastic loss rate coefficient. The resonance features of a correspond to those for the inelastic rate in figure 4. The Feshbach resonance centered at 450 G has a well resolved Fano profile with a background value close to $75{{a}_{0}}$.

We have also studied the role of the Yb spin–orbit interaction and anisotropy of the electronic potentials on the Feshbach resonance structure. Our analyses indicate that for most magnetic fields the resonance features are broadened by the losses energetically lower collision channels with non-zero orbital angular momentum ℓ. This confirmes our predictions about the importance of anisotropic coupling between the short-range potentials. Moreover, we also observe that, except for small fields, more than one half of the loss goes into ³P$_{2},j=2,m=-2$ channels. In addition, a significant fraction of the population ends up in the Yb(³P₁) spin–orbit state (see figure 4(b) in [1]).

We discussed advances in the understanding of scattering properties of high-spin open-shell atomic systems. In particular, our attention was directed towards collisions between fermionic lithium atoms in their absolute ground state with bosonic ytterbium in its meta-stable ³P₂ state. Both species, either with their bosonic or fermionic isotopes, have been successfully cooled to quantum degeneracy and confined in optical dipole traps or optical lattices, allowing the study of their collisions at the quantum level. We observed magnetic Feshbach resonances in collisions with the meta-stable ¹⁷⁴Yb atoms even though this atom was prepared in the ${{m}_{{\rm Yb}}}=-1$ magnetic sub level, which is not the energetically-lowest ³P₂ Zeeman state. With couple-channels calculations we have elucidated the role of anisotropies in the creation of these resonances and by optimizing the shape of the short-range potentials reproduced the measured loss rate for magnetic fields up to 500 G.

Initially, collisions between ground state Li and metastable Yb in the ³P₂ state seemed to be a good system in which to form strong Feshbach resonances, since the Yb life time is 15s and interactions with Li atoms is highly anisotropic. Our calculations, however, show that when Yb atoms are not prepared in the energetically-lowest magnetic sublevel the resonances have large inelastic losses (on the order of 10⁻¹⁰ cm³ s⁻¹). This large rate coefficient might be an obstacle to use them in stimulated Raman adiabatic passage schemes to create the ground state molecules. On the other hand, we also found that inelastic loss for the resonances in collisions with the energetically-lowest magnetic sublevel of Yb is an order of magnitude smaller. These resonances if experimentally confirmed can be promising candidates for Raman transfer. In figure 5 we present a prediction for the Feshbach spectrum for the collision between ⁶Li and ¹⁷⁴Yb ($^{3}{{{\rm P}}_{2}},j=2,m=-2$) based on the optimized potentials.

This work is supported by the AFOSR grant No. FA9550-14-1-0321, the ARO MURI grant No. W911NF-12-1-0476, and the NSF grant No. PHY-1308573.