Synthetic magnetic fluxes on the honeycomb lattice

We devise experimental schemes that are able to mimic uniform and staggered magnetic fluxes acting on ultracold two-electron atoms, such as ytterbium atoms, propagating in a honeycomb lattice. The atoms are first trapped into two independent state-selective triangular lattices and then further exposed to a suitable configuration of resonant Raman laser beams. These beams induce hops between the two triangular lattices and make atoms move in a honeycomb lattice. Atoms traveling around each unit cell of this honeycomb lattice pick up a nonzero phase. In the uniform case, the artificial magnetic flux sustained by each cell can reach about two flux quanta, thereby realizing a cold-atom analog of the Harper model with its notorious Hofstadter’s butterfly structure. Different condensed-matter phenomena such as the relativistic integer and fractional quantum Hall effects, as observed in graphene samples, could be targeted with this scheme.

Figure 1

Laser configuration creating a honeycomb optical lattice in the ($Ox,Oy$) plane at the magic wavelength. It is made of three monochromatic linearly polarized beams with the same off-plane elevation angle $\theta$. The in-plane projections of the laser wave vectors have respective angles of ${120}^{\circ }$.

Figure 2

Hexagonal Wigner-Seitz cell of the honeycomb lattice associated with the optical potential generated by Eq. (3). The figure shows the two primitive Bravais vectors ${\mathbf{a}}_{\mathbf{i}}$, the three vectors ${\mathbf{c}}_{\mathbf{j}}$ connecting each honeycomb lattice site to its three nearest neighbors and the honeycomb lattice constant $a$.

Figure 3

Top: contour plot of the stretched magic honeycomb potential $V_m$ ($a=2/3$ in space units of ${\lambda }_{am}/2$). Bottom: plot of the antimagic standing-wave potential $V_{am}$ for a relative position $x_0=3a/8=1/4$. The antimagic potential takes its mean value half-way between the minima of the magic potential (vertical dotted lines). The superposition of these two potentials gives the state-dependent trapping potentials $V_{g,e}=V_m\pm V_{am}$. The positions of the resulting $|g\rangle$ sites and $|e\rangle$ sites are marked by the corresponding blue and red letters. Dash-dotted blue line: triangular unit cell of the state-$|g\rangle$ lattice. Dashed red line: triangular unit cell of the state-$|e\rangle$ lattice. The combination of these two regular triangular lattices produces a honeycomb lattice with alternating state-dependent sites. The unit cell of this honeycomb lattice (black solid line) is no longer regular, see Fig. 5.

Figure 4

$V_g\left(\mathbf{r}\right)$ as a function of $x$ when $y=1/\sqrt{3}$ for various $V_1/V_0$ ratios. Black thin line: magic potential for $V_1=0$. Blue thick line: $V_1=V_0$. Red dashed line: $V_1=3V_0$. Green dotted line: $V_1=5V_0$.

Figure 5

Wigner-Seitz cell of the new honeycomb structure created by the superposition of two triangular state-dependent trapping potentials. This unit cell is no longer regular and we have $a^{\prime }=0.88a$, $a^{\prime \prime }=1.07a$. The proportions on the figure have been exaggerated.

Figure 6

Top: sketch of the hexagonal plaquette alternating sites where state-$|g\rangle$ atoms are trapped (A, C, and E, open circles) and state-$|e\rangle$ atoms are trapped (B, D, and F, red-filled circles). The total phase accumulated per cell is calculated for atoms hopping clockwise (arrows). Bottom: energy diagram along the loop and the corresponding Raman transitions. All Raman transitions occur here at the angular frequency separation ${\omega }_0$ between the two hyperfine states. As a consequence, one single Raman laser beam is enough to address all sites.

Figure 7

Left: contour plot of the trapping potential $V_g\left(\mathbf{r}\right)$. Right: weak perturbing standing-wave potential which is added along $Oy$ to lift the energy degeneracy between the potential wells. The standing-wave potential is maximum for $y=0$ and its period exactly matches $\sqrt{3}a$ where $a$ is the magic honeycomb lattice constant. Every well in every other horizontal row is lifted up. Dark blue letters “G ” mark the wells which are lifted up, whereas blue letters “g” mark the unaffected ones. Accordingly, the dark red letters “E” and red letters “e” mark the respective positions of the lifted-up and unaffected wells for the trapping potential $V_e\left(\mathbf{r}\right)$. Both state-$|g\rangle$ and state-$|e\rangle$ lattices are rectangular with a two-point unit cell (blue dashed-dotted and red dashed rectangles, respectively), the corresponding Bravais vectors being $({\mathbf{a}}_1+{\mathbf{a}}_2)$ and $({\mathbf{a}}_2-{\mathbf{a}}_1)$. The resulting global honeycomb structure exhibits two different vertical alternating strips made either of the clockwise “gEGeGE” cell (dotted hexagon) or the “GegEge” one (solid hexagon).

Figure 8

Top: clockwise “gEGeGE” plaquette. Open and grey-filled circles mark the unaffected and lifted-up $|g\rangle$ sites; red-filled and dark-red-filled circles mark the unaffected and lifted-up $|e\rangle$ sites. Bottom: energy diagram for the Raman transitions around the plaquette. Because the wells are lifted differently, three different angular frequencies, $\omega$, ${\omega }_{+}$ and ${\omega }_{-}$, are now needed to activate the hops around this plaquette.

Figure 9

Top: clockwise “GegEge” plaquette. Open and grey-filled circles mark the unaffected and lifted-up $|g\rangle$ sites; red-filled and dark-red-filled circles mark the unaffected and lifted-up $|e\rangle$ sites. Bottom: energy diagram for the Raman transitions around the plaquette. Because the wells are lifted differently, three different angular frequencies, ${\omega }_0$, ${\omega }_{+}$, and ${\omega }_{-}$, are now needed to activate the hops around this plaquette.

Figure 10

Staggered magnetic field case. Contour plot of the magic honeycomb lattice $V_m^{\prime }\left(\mathbf{r}\right)$ and the antimagic standing-wave potential $V_{am}^{\prime }\left(\mathbf{r}\right)$ with spatial period $3\tilde{a}$, $\tilde{a}$ being the magic honeycomb lattice constant. The blue letters “g” mark the positions of the state-$|g\rangle$ sites for the trapping potential $V_g^{\prime }=V_m^{\prime }+V_{am}^{\prime }$; red letters “e” mark the positions of the state-$|e\rangle$ sites for the trapping potential $V_e^{\prime }=V_m^{\prime }-V_{am}^{\prime }$. Both state-$|g\rangle$ and state-$|e\rangle$ lattices are rectangular but with a two-point unit cell (blue dashed-dotted and red dashed rectangles, respectively), the corresponding Bravais vectors being $({\mathbf{a}}_1+{\mathbf{a}}_2)$ and $({\mathbf{a}}_2-{\mathbf{a}}_1)$. The Raman laser beams only connect horizontal nearest-neighbor state-$|g\rangle$ and state-$|e\rangle$ sites. The resulting honeycomb structure exhibits two different alternating vertical strips made either of the clockwise cell “eeggge” (dotted hexagon) or the “ggeeg” one (solid hexagon).

Figure 11

Staggered magnetic field case. Trapping potential $V_g^{\prime }\left(\mathbf{r}\right)$ as a function of $x$ when $y=\sqrt{3}\tilde{a}/2$ for various $V_1^{\prime }/V_0^{\prime }$ ratios. Black thin line: magic honeycomb potential ($V_1^{\prime }=0$). Blue thick line: $V_1^{\prime }=V_0^{\prime }$. Red dashed line: $V_1^{\prime }=3V_0^{\prime }$. Green dotted line: $V_1^{\prime }=5V_0^{\prime }$.

Figure 12

Staggered magnetic field case. Top: clockwise “eeggge” plaquette. The red-filled circles mark the state-$|e\rangle$ sites; open circles mark the state-$|g\rangle$ sites. Bottom: energy diagram for the Raman transitions around the plaquette. A single Raman laser beam at frequency ${\omega }_0$ is required here to activate hopping along the horizontal “e-g” links. Hopping along the vertical direction is due to direct tunneling through a potential barrier.

Figure 13

The cell used to derive the Harper Hamiltonian for the simplest uniform magnetic flux configuration. Red circles: state-$|e\rangle$ sites. Open circle: state-$|g\rangle$ site. The arrows on the links indicate the hop directions considered to write down the hop operator $T_n$, see Eq. (28). The Raman wave vectors implied in the hops are indicated along each link. The hop operator $T_n^{†}$ operates on the same cell but with reversed hop directions and thus complex conjugate hopping amplitudes. The Harper Hamiltonian is obtained by superposing the hop operators obtained by repeated translations along the Bravais lattice $\mathcal{B}$ spanned by vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, see Eq. (27).

Figure 14

The cell used to derive the Harper Hamiltonian in the general case. Red circles: state-$|e\rangle$ sites. Dark red circles: lifted-up state-$|e\rangle$ sites. Open circles: state-$|g\rangle$ sites. Grey circles: lifted-up state-$|g\rangle$ sites. The arrows on the links indicate the hop directions considered to write down the hop operator $T_n$, see Eq. (28). The Raman wave vectors implied in the hops are indicated along each link. The hop operator $T_n^{†}$ operates on the same cell but with reversed hop directions and thus complex conjugate hopping amplitudes. The Harper Hamiltonian is obtained by superposing the hop operators obtained by repeated translations along the rectangular Bravais lattice $\mathcal{B}$ spanned by vectors ${\mathbf{a}}_1+{\mathbf{a}}_2$ and ${\mathbf{a}}_2-{\mathbf{a}}_1$, see Eq. (27).