Certifying the Adiabatic Preparation of Ultracold Lattice Bosons in the Vicinity of the Mott Transition

We present a joint experimental and theoretical analysis to assess the adiabatic experimental preparation of ultracold bosons in optical lattices aimed at simulating the three-dimensional Bose-Hubbard model. Thermometry of lattice gases is realized from the superfluid to the Mott regime by combining the measurement of three-dimensional momentum-space densities with ab initio quantum Monte Carlo (QMC) calculations of the same quantity. The measured temperatures are in agreement with isentropic lines reconstructed via QMC for the experimental parameters of interest, with a conserved entropy per particle of $S/N=0.8(1)k_B$. In addition, the Fisher information associated with this thermometry method shows that the latter is most accurate in the critical regime close to the Mott transition, as confirmed in the experiment. These results prove that equilibrium states of the Bose-Hubbard model—including those in the quantum-critical regime above the Mott transition—can be adiabatically prepared in cold-atom apparatus.

Figure 1

Experimental reduced temperatures $T_J=k_{B}T/J$ (open dots) obtained from the momentum-space-density thermometry (see text) plotted as a function of the ratio $u=U/J$, and compared to isentropic lines. The underlying false-color plot shows the theoretical map of the entropy per particle $S/N{k}_B$ of the trapped 3D BH model, with the experimental parameters. The white dashed curve is the isentropic line at $S/N=0.8k_B$. The black dashed line represents the line of critical temperatures for the uniform 3D BH model at unit filling (from Ref. [22]).

Figure 2

(a) Time sequence for the loading of the ${{}^4\mathrm{He}}^{*}$ BECs from the optical dipole trap in the 3D optical lattice. The intensity of the 3D lattice is increased linearly with time at a rate of $0.3E_r{\mathrm{ms}}^{-1}$. For two different final values of the lattice intensity, the two ramps coincide up to reaching the lowest of the two values. The linear increase of the lattice intensity corresponds to an approximately exponential increase of the ratio $U/J$ over time. (b)–(c) 1D cut $\rho [\mathit{k}=(k,0,0)]$ along the ${\overrightarrow{u}}_x$ axis through the 3D momentum-space densities measured at $u=5$ and $u=92$.

Figure 3

(a) Plot of the momentum-space density ${\tilde{\rho }}_{\exp }(k)$ measured in the experiment (restricted to the first Brillouin zone) and of the theoretical one ${\tilde{\rho }}_{\mathrm{QMC}}(k;T)$ at various temperatures $T_J$. The comparison is shown for a ratio $u=30$ corresponding to the location of the quantum critical point in the ground state. (b) Measured temperature $T$ in recoil units $E_r$ as a function of $u$. The dashed line corresponds to an energy $2.2J$ (expressed in units of $E_r$) which fits best the data in the range $u=5–20$. (c) Fisher information $I(T)$ for the temperature estimation from $\tilde{\rho }$ obtained from the QMC data; the experimental temperatures are reproduced for reference.