Precision Measurements on a Tunable Mott Insulator of Ultracold Atoms

We perform precision measurements on a Mott-insulator quantum state of ultracold atoms with tunable interactions. We probe the dependence of the superfluid-to-Mott-insulator transition on the interaction strength and explore the limits of the standard Bose-Hubbard model description. By tuning the on-site interaction energies to values comparable to the interband separation, we are able to quantitatively measure number-dependent shifts in the excitation spectrum caused by effective multibody interactions.

Figure 1

(a) Scattering length $a_S$ for Cs atoms in the lowest hyperfine ground state as a function of the magnetic field $B$ [32]. (b) Example of an integrated density profile of the BEC for 50 ms time of flight. The arrows indicate the FWHM $d$. (c) Center peak FWHM $d$ as a function of lattice depth $V_0$ for $427a_0$ (circles), $320a_0$ (squares), and $212a_0$ (diamonds). The solid lines are fits [34] from which the critical lattice depth $V_C$ is determined. (d) Critical depth $V_C$ as a function of $a_S$. The solid (dashed) line corresponds to the transition points for the SF-to-MI transition given by the quantum Monte Carlo (QMC) (mean-field) calculation $(U/J{)}_c=29.3$ (34.8) [19, 25], and the shaded area indicates our uncertainty for $V_0$ for the QMC calculation.

Figure 2

(a) Excitation spectrum in the MI phase at $V_0=20E_R$ and $a_S=212a_0$. The solid line is a triple Gaussian fit. We take the resonance positions as the centers of the Gaussian peaks. (b) Frequency of the $U$ resonance (circles) and the $2U$ resonance (diamonds) for various values of $V_0$ for $a_S=212a_0$. The solid lines are the calculated frequencies corresponding to $U(1)$ and $2U(1)$. At this value for $a_S$ the transition point occurs at $\approx 12E_R$. (c) Excitation spectrum in the MI regime at $20E_R$ and $a_S=320a_0$. The solid line is a five-peak Gaussian fit.

Figure 3

(a) Detailed structure of the resonance near $U$ for different values of the initial atom density at $a_S=427a_0$ [low density, no double occupancy (circles), intermediate density, some double occupancy (squares), high density, high double occupancy (diamonds); for details see text]. Every data point is the statistical average of five measurements, the error bars are the standard deviation. (b) Remaining atom number $n_A$ after AM at the same interaction strength as in (a), normalized to $n_A$ without AM. The solid line is a Gaussian fit. (c) Remaining molecule number $n_M$ after AM at the same interaction strength as in (a), normalized to $n_M$ without AM. The solid line is a double Gaussian fit.

Figure 4

(a) Frequency of the upper ($R_2$, circles) and lower ($R_1$, squares) resonance around $U$ and of the resonance around $2U$ (diamonds) as a function of $a_S$ for a lattice depth $V_0=20E_R$. The dotted lines are the calculated $U(1)/h$ and $2U(1)/h$ values from the standard BH model. The solid lines are the result of the more elaborate calculations for $U(2)/h$ and $3U(3)/h-2U(2)/h$ (see text). The dashed line is the calculated $3U(3)/h-2U(2)/h$ using the renormalized perturbation theory [23]. (b) Same as in (a), but for $V_0=25E_R$.