Interaction spectroscopy of a two-component Mott insulator

We prepare and study a two-component Mott insulator of bosonic atoms with two particles per site. The mapping of this system to a magnetic spin model and the subsequent study of its quantum phases require a detailed knowledge of the interaction strengths of the two components. In this work we use radio-frequency transitions and an on-site interaction blockade for precise empirical determination of the interaction strengths of different combinations of hyperfine states on a single lattice site. We create a map of the interactions of the two lowest hyperfine states of ${}^7\mathrm{Li}$ as a function of magnetic field, including measurements of several Feshbach resonances with exceptional sensitivity, and we identify promising regions for the realization of magnetic spin models.

Figure 1

The three different combinations of two hyperfine states on a lattice site have interaction energies $U_{aa}$, $U_{ab}$, and $U_{bb}$ which can be tuned via Feshbach resonances. At a given field $B$, the splittings will in general be unequal, giving rise to an interaction blockade: The two possible transitions can be individually addressed by an rf drive with frequency $\omega ={\omega }_{\text{Zeeman}}+\mathrm{\Delta }U/\hslash$.

Figure 2

Fits to representative spectra of the transition between $b$ and $a$, taken between the two $bb$ Feshbach resonances. The spectra have been plotted so that the Zeeman-shifted peaks of the $n=1$ transitions overlap. The inset shows an example spectrum in which each point is an average of four measurements and the fit is a sum of two Gaussians. The crossing of the frequencies of the two peaks corresponds to $U_{ab}-U_{bb}=0$.

Figure 3

(a) Relative interactions $U_{ab}-U_{aa}$ and $U_{bb}-U_{ab}$ plotted as a function of magnetic field, measured via rf interaction spectroscopy in a $35E_R\times 35E_R\times 35E_R$ optical lattice. (b) Scattering lengths of the $aa$ and $bb$ interactions plotted as a function of magnetic field, measured using lattice AM and shown in units of the Bohr radius $a_0$. Also shown as a bold dashed line is the $ab$ scattering length, obtained using simultaneous hyperbolic fits to the rf spectroscopy and lattice AM data sets. The fits are shown as solid black lines. Dotted vertical lines indicate the position of a resonance.

Figure 4

Rabi oscillations of doublons (a) exactly at and (b) far away from the magnetic field at which the interactions are degenerate, as in (3). The measured atom number is normalized to the total number of $n=2$ sites. The system is initially prepared in state $b$ and we detect the total number of atoms in state $a$ after applying a resonant rf drive. Decaying sinusoidal fits determine the Rabi frequencies to be 1.68(4) and 2.30(1) kHz, respectively, consistent with the expected $\sqrt{2}$ ratio in Rabi frequency between resonant three-level and two-level systems. The time constant for the decoherence away from degeneracy is 10.2 ms. The oscillations in (a) seem to decay faster because these data were taken at the closest magnetic field to the degeneracy point which is permitted by the present resolution limit of our magnetic-field set point, so we ultimately observed beating between two nearly equal but off-resonant Rabi frequencies.