Quantum phases in tunable state-dependent hexagonal optical lattices

We study the ground-state properties of ultracold bosonic atoms in a state-dependent graphenelike honeycomb optical lattice, where the degeneracy between the two triangular sublattices A and B can be lifted. We discuss the various geometries accessible with this lattice setup and present a scheme to control the energy offset with external magnetic fields. The competition of the on-site interaction with the offset energy leads to Mott phases characterized by population imbalances between the sublattices. For the definition of an optimal Hubbard model, we demonstrate a scheme that allows for the efficient computation of Wannier functions. Using a cluster mean-field method, we compute the phase diagrams and provide a universal representation for arbitrary energy offsets. We find good agreement with the experimental data for the superfluid to Mott insulator transition.

Figure 1

Setup and possible geometries of versatile spin-dependent hexagonal lattices. (a) Setup of the three running laser beams in the $xy$ plane. The quantization axis of the system is defined by a homogeneous magnetic field B. Its orientation can be quantified by the Euler angles $\alpha$ and $\gamma$. The linear polarization of the three beams encloses an angle $\theta$ with the lattice plane. (b) For $\theta =\pi /2$ (polarization perpendicular to lattice plane) the resulting potential is a triangular lattice with $\pi$-polarized light. The light intensity is strongly modulated. (c) Purely state-dependent polarization lattice in the case of $\theta =arccos(1/3)/2$, depicted here for atoms trapped at ${\sigma }^{-}$ sites and $\alpha =0$. Note that compared with (b) a much weaker confinement arises. (d) With $\theta =0$ the potential forms a spin-dependent honeycomb lattice. The alternating circular polarization lifts the degeneracy of the sublattice sites A and B $\left(\alpha =0\right)$. The lattice spacing (533 and 320 nm) are exemplary given for a laser wavelength of ${\lambda }_L=830$ nm.

Figure 2

Stirring of lattice wells in the polarization lattice by rotating the quantization field. In the case of a purely state-dependent potential it is possible to stir ${\sigma }^{+}$ and ${\sigma }^{-}$ lattice sites around each other by rotating the quantization axis in the lattice plane (i.e., for $\alpha =\pi /2$).

Figure 3

Band structure of the tunable honeycomb lattice. (a) The rotation of the magnetic field axis with respect to the lattice plane alters the projection of the light field onto the atomic spin states. Thus, the energy offset $\epsilon$ between the sublattice sites A and B can be continuously adjusted by the Euler angle $\alpha$ in the honeycomb lattice system. This behavior can be described by an effective magnetic quantum number $m$. (b) and (c) Density of the Bloch functions $|{\varphi }_{\mathbf{k}}^{\left(n\right)}\rangle$ for the four lowest bands plotted for the momentum $\mathbf{k}=\Gamma$ or K (see Fig. 4). The results are shown for a lattice depth of $V_0=3E_{\mathrm{R}}$ and an effective magnetic quantum number (b) $m=0$ and (c) $m=1$, where the deep blue color corresponds to nodes in the wave function with zero density. Indicated in white are symmetry axes and the sign of the wave function, where the circular arrows indicate a complex value with a rotating complex phase.

Figure 4

(a) Band structure $E_{\mathbf{k}}^{\left(n\right)}$ showing the four lowest bands at a lattice depth $V_0=3E_{\mathrm{R}}$ for $m=0,0.1$, 1 in unit of the recoil energy $E_{\mathrm{R}}$. (b) Close up of Dirac cones for $m=0,0.02,...,0.1$.

Figure 5

(a)–(h) Illustration of processes within and beyond the Hubbard model: (a) the tunneling matrix element $J$, (b) the site-offset ${\epsilon }_B$, (c) and (d) the next-nearest-neighbor tunneling $J_{\mathrm{A}\mathrm{A}/\mathrm{B}\mathrm{B}}$, (e) and (f) the on-site interactions $U_{\mathrm{A}}$ and $U_{\mathrm{B}}$, (g) the density-induced tunneling ${\Delta J}_{\mathrm{A}/\mathrm{B}}$, and (h) next-neighbor interaction $U_{\mathrm{A}\mathrm{B}}$. (i) and (j) Parameters dependent on the variational parameter $s$ for (i) $m=0$ and (j) $m=0.02$ at a lattice depth of $V_0=12E_{\mathrm{R}}$. Shown are the absolute values and negative signs are indicated by dashed lines. The blue circles depict the mean ${({\Delta J}_{\mathrm{A}}^2+{\Delta J}_{\mathrm{B}}^2+U_{\mathrm{A}\mathrm{B}}^2)}^{1/2}$, which is used as the localization criterion.

Figure 6

(a) The Wannier functions $|w_{\mathrm{A}}\rangle$ and $|w_{\mathrm{B}}\rangle$ for sublattices A and B at $V_0=3E_{\mathrm{R}}$ for the symmetric case $m=0$. The plotted area contains $5\times 5$ unit cells. The white contour lines in the upper plot for $|w_{\mathrm{A}}\rangle$ indicate the zero crossings of the Wannier function. The lower plots show $|w_{\mathrm{B}}\rangle$ on a linear and its absolute value on a logarithmic scale. Since both sublattices are identical for $m=0,|w_{\mathrm{A}}\rangle$ and $|w_{\mathrm{B}}\rangle$ are identical under reflection. (b) The Wannier functions for the strongly nonsymmetric case $m=1$.

Figure 7

Hubbard and extended-Hubbard parameters for effective magnetic quantum numbers (a) $m=0$,(b) $m=0.02$, and (c) $m=1$. Within the Hubbard model (15), only the on-site interactions $U_{\mathrm{A}}$ (red) and $U_{\mathrm{B}}$ (dark red), the tunneling matrix element $J$ (blue), and the site-offset ${\epsilon }_B$ (brown) contribute. Additional processes are the density-induced tunneling ${\Delta J}_{\mathrm{A}/\mathrm{B}}$ (black/gray), the next-nearest-neighbor tunneling $J_{\mathrm{A}\mathrm{A}/\mathrm{B}\mathrm{B}}$, and the next-neighbor interaction $U_{\mathrm{A}\mathrm{B}}$. Shown are the absolute values and negative signs are indicated by dashed lines.

Figure 8

(a) The amplitude on- and off-site processes (see Figs. 7 and 5) as a function of the effective magnetic quantum number $m$ for $V_0=5E_{\mathrm{R}},10E_{\mathrm{R}}$, and $15E_{\mathrm{R}}$. (b) Nonlogarithmic plot of the site-offset $\epsilon$ showing its linear dependency. (c) Hubbard model with tunneling $J$, on-site interaction $U_{\mathrm{A}}=U_{\mathrm{B}}$, and tunable site-offset $\epsilon$. (d) The change between ferromagnetic and antiferromagnetic coupling of the next-nearest-neighbor tunneling on a linear scale.

Figure 9

(a)–(d) Superfluid to Mott insulator transition for $m=0,0.02,0.1$, and 1, where $\mu$ is the chemical potential. The Mott lobes are shown as a function of the lattice depth $V_0$ (axis at the bottom) and the ratio $J/U$ (top). Mott phases with integer and half-integer filling $\rho$ exist for nonequivalent sublattices $\mathrm{A}$ and $\mathrm{B}$ with the occupation numbers $(n_{\mathrm{A}},n_{\mathrm{B}})$ (gray). The results for the Hubbard model (15) are shown as solid lines and for the extended model (16) as dashed lines. The phase boundary within the Hubbard model depends only on the ratio $J/U$ assuming $U=U_{\mathrm{A}}\approx U_{\mathrm{B}}$. (e) Lattice structure of the honeycomb lattice and (f) mapping on a square lattice with reduced bonds. The depicted cluster of sites is used for the cluster Gutzwiller calculation, where the double arrow indicates periodic boundary conditions.

Figure 10

Universal phase diagram for arbitrary values of $m$ in the $\epsilon /U\text{−}J/U$ plane for (a) half-integer and (b) integer filling factors up to $\rho =4$. The intersections of the Mott lobes with the function $\epsilon /U$ for different values of $m$ (dashed lines) corresponds to the SF-MI transition point at the tip of the Mott lobe in Fig. 9. (c) Phase diagram combining half-integer and integer filling factors for site-offsets $0<\epsilon <U$.

Figure 11

Comparison with the experimental data from Ref. [3], which depicts the visibility after time-of-flight (circles) for $m=0$ (blue) and $m=1$ (green). The vertical lines are the theoretical predictions for the Mott insulator transition for filling factor $\rho$ in both cases. The solid lines show the result for the Hubbard and the dashed lines for the extended model. The experimental data are in good agreement with transitions at a filling factor of $\rho =1$.