Figure 1

(Color online). (a) The atomic levels and the interactions between atoms and laser fields. (b) Schematic representation of the experimental setup with the two laser beams incident on the cloud of atoms. (c) Schematic of the square optical lattice and the designed phase factor (denoted by arrows). (d) The scheme of overlapping the two state-selective optical lattices.Reuse & Permissions

Figure 2

(Color online). Energy dispersion for cold fermionic atoms in a square optical lattice. (a) shows the energy dispersion and (b) represents the profiles of the energy dispersion with $k_z=\pi /2l$ [blue (deep gray) line] and $k_x=\pi /2l$ [green (light gray) star], for the ideal case $t_a=t_b=t_1=t$, $l_x=l_z=l$, $\gamma =\pi$. (c) shows the energy dispersion and (d) represents the profiles of the energy dispersion with $k_z=\pi /2l$ [blue (deep gray) line] and $k_x=\pi /2l$ [green (light gray) line], for the anisotropic case $t_a=t_b=2t_1/3=2t/3$, $l_x=l_z=l$, $\gamma =\pi$. (e) shows the energy dispersion and (f) represents the profiles of the energy dispersion with $k_z=2\pi /5l$ [blue (deep gray) line] and $k_x=\pi /2l$ [green (light gray) line], for the anisotropic case $t_a=t_b=t_1=t$, $l_z=5l_x/4=5l/4$, $\gamma =\pi$. (g) shows the energy dispersion and (h) represents the profiles of the energy dispersion with $k_z=2\pi /5l$ [blue (deep gray) line] and $k_x=\pi /2l$ [green (light gray) line], for the anisotropic case with $t_a=t_b=t_1=t$, $l_x=l_z=l$, $\gamma =6\pi /5$.Reuse & Permissions

Figure 3

(Color online). The Bragg spectroscopies for the ideal case, (a) $t_a=t_b=t_1=t$, $l_x=l_z=l$, $\gamma =\pi$, and the anisotropic cases, (b) $t_a=t_b=2t_1/3=2t/3$, $l_x=l_z=l$, $\gamma =\pi$, (c) $t_a=t_b=t_1=t$, $l_z=5l_x/4=5l/4$, $\gamma =\pi$, (d) $t_a=t_b=t_1=t$, $l_x=l_z=l$, $\gamma =6\pi /5$ and $q=\pi /10l$. Here, we represent the Bragg spectroscopies with the blue (deep gray) lines for the momentum difference $\mathbf{q}$ in the $x$ direction and with the green (light gray) stars or the green (light gray) lines for $\mathbf{q}$ in the $z$ direction. The frequency difference $\omega$ is expressed in units of $q{v}_0$ and the dynamic structure factor $S\left(\mathbf{q},\omega \right)$ is expressed in units of $q/8{\pi }^{2}n{v}_0$ with $n$ being the number density of atoms in the system.Reuse & Permissions