Density-dependent hopping for ultracold atoms immersed in a Bose-Einstein-condensate vortex lattice

Both mixtures of atomic Bose-Einstein condensates and systems with atoms trapped in optical lattices have been intensely explored theoretically, mainly due to the exceptional developments on the experimental side. We investigate the properties of ultracold atomic impurities (bosons) immersed in a vortex lattice of a second Bose-condensed species. In contrast to the static optical-lattice configuration, the vortex lattice presents intrinsic dynamics given by its Tkachenko modes. These excitations induce additional correlations between the impurities, which consist of a long-range attractive potential and a density-dependent hopping, described here in the framework of an extended Bose-Hubbard model. We compute the quantum phase diagram of the impurity species through a Gutzwiller ansatz and through the mean-field approach, and separately identify the effects of the two additional terms, i.e., the shift and the deformation of the Mott-insulator lobes. The long-range attraction, in particular, induces the existence of a triple point in the phase diagram, in agreement with previous quantum Monte Carlo calculations [Chaviguri et al., Phys. Rev. A 95, 053639 (2017)].

Figure 1

Phase diagram of the EBH model in Eq. (11), featuring continuous MI/SF transitions (solid lines) and discontinuous MI/MI transitions (dashed lines). We show the separate effects of the long-range attraction $q$ [panel (a), with $p=0]$ and of the density-dependent hopping $p$ [panel (b), with $q=0]$, where the triple point and the MI/SF boundaries are computed through Eqs. (24) and (33), respectively. (c) Density $\langle {\widehat{n}}_i\rangle$ for $q=p=0.1$, computed through the Gutzwiller method (color codes, with $n_{\max }=6$), compared with the analytical phase boundaries.

Figure 2

From top to bottom: Solid blue: on-site energy $U$. Solid gray: “Gaussian” hopping $J$. Dashed black: induced hopping $J_E$. Dashed blue: long-range potential $V_E$. We set $a_B/l_z^B\approx 0.05$ with ${\mathrm{\Gamma }}_L\approx 1$ for ${}^{87}\mathrm{Rb}$ atoms and assume a unitary filling of the lattice sites, $n_i=n_j=1$.

Figure 3

From top to bottom: Solid brown: “dynamic” hopping $J_P$. Solid red: long-range potential $V$. Dashed brown: “static” induced hopping $J_E$. Dashed red: long-range “static” potential $V_E$. We use the mixture Na-Rb, where Rb atoms are considered as impurities. In addition, we assume a unitary filling of the lattice sites ($n_i=n_j=1$) and ${\mathrm{\Gamma }}_L\approx 1$.

Figure 4

Phase diagram of the EBH model at $q=0.1$, with $p=0.05$ (a) and $p=0.5$ (b). The condensate density is computed through the Gutzwiller approach (color code, obtained with $n_{\max }=6$). Also shown are the standard MF theory [solid red line, Eq. (33)] and the single-site MF approach [dotted white line, Eq. (E9)].