Demixing effects in mixtures of two bosonic species

Motivated by recent experiments on two-component systems, we investigate the ground-state phase diagram of a mixture of two bosonic species by means of path-integral quantum Monte Carlo simulations by a two-worm algorithm. The mixture is trapped in a square lattice at different filling conditions. Various quantum phases are stabilized depending on the interplay between intra- and interspecies interactions and on the filling factors. We show that the ground-state phase diagram at half-filling features a demixed superfluid phase and a demixed Mott-insulator phase when the interspecies interaction becomes greater than the intraspecies repulsion, and a double-superfluid phase or a supercounterflow otherwise. We show that demixing, characterized by spatial separation of the two species, can be detected experimentally through the effects of anisotropy revealed by time-of-flight images. We also study how demixing effects depend on the filling factor of the two components. Finally, we find that the supercounterflow phase is preserved in the presence of unbalanced populations.

Figure 1

Ground-state phase diagram of twin bosonic species trapped in a uniform optical lattice at $\nu =\frac{1}{2}$. Four phases are stabilized: double superfluid (2SF), supercounterflow (SCF), demixed Mott insulator (dMI), and demixed superfluid (dSF). Symbols mark the phase boundaries as calculated from Monte Carlo simulations (see text). Whenever not visible, error bars lie within the symbol size. The inset shows the scaled total superfluidity ${\rho }_{\mathrm{tot}}$ as a function of the interspecies interaction for system sizes $L=8$, 16, 24, and 36 and $U/t=20$. The intersection between curves corresponding to different system sizes gives the transition point for the 2SF-SCF transition.

Figure 2

Demixing parameter $\mathrm{\Delta }$ as a function of $U_{ab}/t$ for $U/t=5$, 10, and 15 (circles, squares, and triangles, respectively) across the 2SF-dSF transition. A jump of 4 orders of magnitude is clearly visible when $U_{ab}/t\sim U/t$, signaling the onset of the demixed phase dSF at the expense of the 2SF phase.

Figure 3

Quantum-statistical average of the particle numbers $\langle n_{ai}\rangle$ and $\langle n_{bi}\rangle$. The $x$ and $y$ axes denote the $x$ and $y$ coordinates on the lattice. The color code is displayed in the the bar on the right-hand side. (a) 2SF phase at $U/t=10$ and $U_{ab}/t=7$ with density uniformly distributed in the lattice. (b) dSF phase at $U/t=15$ and $U_{ab}/t=20$ with two components occupying spatially separated regions of the lattice. (c) dMI phase at $U/t=20$ and $U_{ab}/t=22$ with two components occupying spatially separated regions of the lattice. Compenetration between the two regions is noticeable.

Figure 4

Momentum distributions $n_k$ in the first Brillouin zone for the 2SF (a), dSF (b), SCF (c), and dMI (d)] phases. The insets show the corresponding quantum-statistical average of the density of the two components within the lattice. The dSF phase shows an evident anisotropy along the $x$ and $y$ directions. This is expected due to the anisotropy of the spatial separation.

Figure 5

$\mathrm{\Delta }$ parameter as a function of total filling $n$ for the dSF (circles, $U/t=10,U_{ab}/t=15$, and $L=24$) and 2SF (triangles, $U/t=10,U_{ab}/t=7$, and $L=24$) phases. Insets represent the average densities of the two components for $n=0.3$ (left) and $n=1.3$ (right).

Figure 6

Supercounterfluidity stiffness as a function of $N_b/N_a$ ($U/t=20,U_{ab}/t=17$, and $L=24$). The dashed line is a cubic polynomial least-square fit, while the dotted lines represent 1-$\sigma$ deviation contours from the fit.