Mott transition for strongly interacting one-dimensional bosons in a shallow periodic potential

We investigate the superfluid-insulator transition of one-dimensional interacting bosons in both deep and shallow periodic potentials. We compare a theoretical analysis based on quantum Monte Carlo simulations in continuum space and Luttinger liquid approach with experiments on ultracold atoms with tunable interactions and optical lattice depth. Experiments and theory are in excellent agreement. Our study provides a quantitative determination of the critical parameters for the Mott transition and defines the regimes of validity of widely used approximate models, namely, the Bose-Hubbard and sine-Gordon models.

Figure 1

QMC results for the phase diagram in the $g\text{−}\mu$ plane for a fixed amplitude of the periodic potential $V=2E_{\text{r}}$. The black (dark) points joined by lines are determined from the $K=1$ criterion. The crossing points of the compressibility (red), the position of the compressibility cusps (green), and the crossing points of the Luttinger parameter (blue) are also shown and are practically indiscernible. The left-hand red crosses are the hard-core limits $[({\hslash }^2/ma)/g\to 0]$. The inside of the lobe (red shaded region) and the red dashed line correspond to a density of one particle per potential spacing. The blue dotted line is the prediction of the BH model and the right-hand blue cross is the corresponding tip of the lobe. Inset: The particle filling, $na$, the compressibility $\kappa$, the superfluid fraction $f_{\text{s}}$, and the Luttinger parameter $K$ are shown versus the chemical potential $\mu$, for a periodic potential amplitude $V=2E_{\text{r}}$ and an interaction strength $g=7{\hslash }^2/ma$. The various curves are the QMC results for different system sizes, $L/a=30$, 50, and 100 (blue, green, and red, respectively), showing a sharper transition as the size increases. The vertical dotted lines show the transition points determined from the criterion $K=1$.

Figure 2

Analysis of the correlation function $g_1\left(x\right)=\langle {\widehat{\psi }}^{†}\left(x\right)\widehat{\psi }\left(0\right)\rangle$ for $V/E_{\text{r}}=2$. The decay is exponential in the insulator (left inset, $\gamma =7)$ and algebraic in the superfluid (right inset, $\gamma =1.25)$. Left: Correlation length in the insulator. Right: Luttinger parameter in the superfluid extracted from the decay of the correlation function (solid black) and from the formula $K=\pi \sqrt{n_{\text{s}}\kappa }$ (dashed red). The QMC results are the points, connected by straight lines to guide the eye. Note that the inflection of the correlation length curve close to the transition is due to finite size effects.

Figure 3

Phase diagram in the $g\text{−}V$ plane at commensurability $\left(na=1\right)$. The big black and small green points (both with error bars) are the Monte Carlo and experimental results, respectively. The solid blue line is the BH prediction using exact Wannier functions and the critical value ${(J/U)}_{\text{c}}=0.297\pm 0.01$ (the shaded area corresponds to this error bar). The red cross indicates the critical interaction strength $g_{\text{c}}$ such that $K\left(g_{\text{c}}\right)=2$ in the Lieb-Liniger model. Along the whole transition line the effective Luttinger parameter remains $K=2$. The dashed red line is the result of the bare sine-Gordon theory [30].

Figure 4

Critical momentum $p_{\text{c}}$ versus scattering length $a_{3\text{D}}$ for four lattice depths: From right to left, $V/E_{\text{r}}=1$ (red), 2 (orange), 2.8 (green), 4 (blue). A piecewise fit (solid lines) determines the critical values for the superfluid–Mott insulator transition (circles): respectively, $a_{\text{c}}/a_0=392\pm 12,214\pm 6,160\pm 30,122\pm 8$. The error bars derive from the procedure to obtain $p_{\text{c}}$ [38]. Inset: time evolution of the momentum distribution peak $p$ for $a_{3\text{D}}=109a_0$ and $V/E_{\text{r}}=2$. The solid line is the theoretical damped oscillation fitting the data for $p<p_{\text{c}}$ before the dynamical instability sets in. The error bars comprise the imaging resolution and the statistical uncertainties.