The Bose-Hubbard polaron from weak to strong coupling

Abstract

We investigate the zero-temperature properties of a mobile impurity immersed in a bath of bosonic particles confined to a square lattice. We analyze the regimes of attractive and repulsive coupling between the impurity and the bath particles for different strengths of boson-boson interactions in the bath, using exact large-scale quantum Monte-Carlo simulations in the grand canonical ensemble. For weak coupling, the polaron mass ratio is found to decrease around the Mott insulator (MI) to superfluid (SF) transition of the bath, as predicted by recent theory, confirming the possible use of the impurity as a probe for the transition. For strong coupling in the MI regime, instead, the impurity is found to modify the bath density by binding to an extra bath particle or a hole, depending on the sign of the polaron-bath interactions. While the binding prevent the aforementioned use of the polaron mass ratio as an MI-SF transition probe, we show that it can be used instead as a probe of the binding itself. Our exact numerical results provide a benchmark for comparing lattice Bose polaron theories and are relevant for experiments with cold atoms trapped in optical lattices, where the presence of a confining harmonic potential can be modeled by a slowly varying local chemical potential.

Figures (4)

• Figure 1: (a) Schematic representation of the different states of the system in the MI regime. For $\mu-U < U_{IB} < \mu$, the impurity weakly perturbs the bath and remains mobile. For $U_{IB} < \mu - U$ ($U_{IB} > \mu$), the impurity binds to an extra bath particle (hole), forming a mobile bound pair that propagates via correlated (anti-correlated) hops, respectively (colored arrows). (b) QMC snapshots of a single impurity (red) in a bath of 33 bosons (blue) in a harmonic trap with $\Omega/t \approx 1.777$ at $2dt/U = 0.05$, for $U_{IB}/U = -1$ (left), 0.2 (center), and 0.5 (right). Marker areas are proportional to the local occupations $\langle n_I\rangle$ and $\langle n_B\rangle$. $\Delta E$ is the energy cost of moving a bath particle from the edge to the trap center.(c) Same parameters as in (b), but marker areas represent the deviation from the impurity-free bath, $\langle n_B\rangle - \langle n_{B,\text{pure}}\rangle$; green (purple) indicates enhanced (reduced) bath density.
• Figure 2: Top: Polaron energy shift $E_0$ as a function of impurity-bath interaction strength $U_{IB}/U$ across different bath phases: (a) Mott Insulator (MI) $(2dt/U = 0.05)$, (b) near the MI-SF transition (T) $(2dt/U = 0.225)$, and (c) superfluid (SF) $(2dt/U = 0.35)$. Solid points represent QMC estimates of $E_0$, while dashed lines show theoretical shifts. $E_{+1} = 2U_{IB} + U - \mu$. Insets: Impurity-bath density-density correlation $C_{IB}(r=0)$, scaled by $L^2$, showing local bath occupation changes. Bottom: effective mass ratio $M/M^*$ as a function of $U_{IB}/U$, illustrating how impurity-bath interactions modify the impurity's effective mass in different bath phases: (d) Mott Insulator (MI), (e) near the MI-SF transition (T), and (f) superfluid (SF). All plots are for an $8\times8$ lattice. Vertical dashed lines indicate $U_{IB}/U$ values at which energy level crossings occur. Error bars that are not visible are within the marker size.
• Figure 3: Polaron mass ratio $M/M^*$ as a function of $2dt/U$ for (a) attractive and (b) repulsive branches. Different curves correspond to varying impurity-bath interaction strengths $U_{IB}/U$. Insets: zooms on black dotted boxes, showing results consistent with Colussi_2023. Dashed grey vertical line: MI-SF transition point of the pure bath (no impurity present). Dashed black line: guide for the eye of the transition between the polaron and dimer regimes for strong repulsive interaction. All plots are for a $8\times8$ lattice.
• Figure 4: (a) Polaron energy $E_p/U$, (b) bath-induced shift $E_0$ normalised by the mean field shift $U_{IB}\langle n_b \rangle$, (c) mass ratio $M/M^*$ vs $\mu/U$ for $U_{IB}/U=0.1$ (solid blue line, circular markers) and $-0.1$ (dashed red line, square markers). Dashed vertical lines represent the MI-SF transition points of the bath based on estimated bath density and superfluid fraction. Dotted vertical lines are guides for the eyes for the position of the sharp features. Here, $2dt/U=0.05$. All plots are for an $8\times8$ lattice. The $\mu/U$ scale is a pseudo-log10 centered at 0.5 and scaled by a factor 0.3.