Heteronuclear molecules in an optical lattice: Theory and experiment

We study properties of two different atoms at a single optical lattice site at a heteronuclear atomic Feshbach resonance. We calculate the energy spectrum, the efficiency of rf association, and the lifetime as a function of magnetic field and compare the results with the experimental data obtained for ${}^{40}\text{K}$ and ${}^{87}\text{R}\text{b}$ [C. Ospelkaus et al., Phys. Rev. Lett. 97, 120402 (2006)]. We treat the interaction in terms of a regularized $\delta$ function pseudopotential and consider the general case of particles with different trap frequencies, where the usual approach of separating center-of-mass and relative motion fails. We develop an exact diagonalization approach to the coupling between center-of-mass and relative motion and numerically determine the spectrum of the system. At the same time, our approach allows us to treat the anharmonicity of the lattice potential exactly. Within the pseudopotential model, the center of the Feshbach resonance can be precisely determined from the experimental data.

Figure 1

(Color online) Energy eigenvalues of ${}^{40}\text{K}$ and ${}^{87}\text{R}\text{b}$ as a function of scattering length without (black dashed line) and with coupling terms (blue solid line) due to anharmonicity and unequal trap frequencies in the lattice for parameters: $V_{\text{Rb}}=40.5E_{r,\text{Rb}}$, $V_{\text{K}}=0.86V_{\text{Rb}}$, and $\lambda =1030\text{nm}$. The deviation between the idealized model and the full solution is substantial in particular for the upper branch.

Figure 2

(Color online) Low-energy spectrum of states with center-of-mass energy $3/2\hslash {\omega }_{\text{c}\text{.m}\text{.}}$ for ${}^6\text{L}\text{i}$ and ${}^{133}\text{C}\text{s}$ and lattice parameters $V_{\text{Li}}=V_{\text{Cs}}=10{\hslash }^2{k}^2/2\mu$ and $\lambda =1\mu \text{m}$. The energy is much more lowered compared to the case of ${}^{40}\text{K}$ and ${}^{87}\text{R}\text{b}$. This is due to the large mass ratio of the ${}^6\text{L}\text{i}$ and ${}^{133}\text{C}\text{s}$ atoms.

Figure 3

(Color online) rf spectroscopy of ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ in a 3D optical lattice on the ${}^{40}\text{K}|9/2,-7/2⟩\to |9/2,-9/2⟩$ transition (see inset) at a lattice depth of $V_{\text{Rb}}=27.5E_{r,\text{Rb}}$ and a magnetic field of 547.13 G, where the interaction is attractive. The spectrum is plotted as a function of detuning from the undisturbed atomic resonance frequency and clearly shows the large atomic peak at zero detuning. The peak at 13.9 kHz is due to association of $|1,1⟩\otimes |9/2,-7/2⟩$ atom pairs into a bound state (figure from Ref. [9]).

Figure 4

(Color online) Experimentally observed energy spectrum together with theory without free parameters (black dashed line) and a least-squares fit for the resonance parameters $B_0$ and $\Delta B$ (red solid line). Experimental data from Ref. [9].

Figure 5

(Color online) Transfer efficiency of rf association as observed in the experiment and estimated from a Rabi model, both for (a) attractively interacting atoms and molecules and for (b) repulsively interacting pairs. The experimental data contain a global factor which has been chosen such that the value of the outermost right (a) [left (b)] data point is one (see text). Experimental data of (a) from Ref. [9].

Figure 6

(Color online) Sketch of expected lattice occupation. The arrows illustrate tunneling of remaining fermionic ${}^{40}\text{K}$ atoms to the “molecular” shell where they can undergo inelastic three-body collisions with a ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ molecule.

Figure 7

(Color online) Lifetime of heteronuclear ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ molecules in the optical lattice. The Lifetime is limited due to residual atoms which can tunnel to lattice sites with molecules and provoke inelastic three-body loss. The theoretical prediction uses the pseudopotential wave function and contains a global factor which was adjusted to the experimental data of Ref. [9].