Optical-lattice implementation scheme of a bosonic topological model with fermionic atoms

We present a scheme to implement a Fermi-Hubbard-like model in ultracold atoms in optical lattices and analyze the topological features of its ground state. In particular, we show that the ground state for appropriate parameters has a large overlap with a lattice version of the bosonic Laughlin state at filling factor one-half. The scheme utilizes laser assisted and normal tunneling in a checkerboard optical lattice. The requirements on temperature, interactions, and hopping strengths are similar to those needed to observe the Néel antiferromagnetic ordering in the standard Fermi-Hubbard model in the Mott insulating regime.

Figure 1

Amplitudes ${\tilde{t}}_{mn}$ of the hopping terms (2) in the Fermi-Hubbard-like Hamiltonian (1). The (green) bullets are the lattice sites, and each line combining two sites represents that a fermion can hop between the sites with the amplitude given in the table on the right, where $t$ and $t^{\prime }$ are real numbers. The rule for choosing the phases is that the product of the three $\tilde{t}$ factors appearing in the hopping terms that move a fermion in the counterclockwise direction around a triangle consisting of one horizontal line, one vertical line, and one diagonal line (i.e., any triangle that is a lattice translation of any of the four triangles marked in the figure) should always be $-i{t}^2{t}^{\prime }$. Unless otherwise specified, we number the lattice sites row-wise starting from the lower left corner as shown.

Figure 2

Overlap per site $|\langle {\psi }_0|\psi \rangle {|}^{1/N}$ between the lowest energy eigenstate ${\psi }_0$ of the Hamiltonian in (5) (with $U>0$) and the FQH-like state $\psi$ in (8) as a function of $t/U$ and $t^{\prime }/t$ for (a) a $4\times 3$ lattice, (b) a $4\times 4$ lattice, and (c) a $4\times 5$ lattice. The Schrieffer-Wolff transformation leading from (1) to (5) is valid when $\left|t\right|/U\ll 1$ and $|t^{\prime }|/U\ll 1$, and we note that part of the area displaying an almost perfect overlap is within this region.

Figure 3

Flatness $F$ [see (14)] of the lowest energy band of $H_{\mathrm{kin},\sigma }$ [see (2)] as a function of the ratio of the two hopping strengths $t^{\prime }$ and $t$.

Figure 4

Checkerboard optical-lattice potential. Fermions in the red spin-up or in the red spin-down state see the dark gray (red) potential in (a) and are hence trapped on the lattice sites that belong to the dark gray (red) sublattice shown in (b). Fermions in the blue spin-up or in the blue spin-down state see the light gray (blue) potential in (a) and are hence trapped on the light gray (blue) lattice sites in (b). A hop between light gray (blue) and dark gray (red) lattice sites can be accomplished via a Raman transition that changes the internal state of the fermion as illustrated schematically in part (a) of the figure.

Figure 5

Schematic illustration of the laser beams needed for the implementation. The laser beams originating from the eight dark gray boxes produce the checkerboard optical lattice in the $xy$ plane, and the laser beams originating from the six light gray boxes implement the hopping terms and take care of trapping in the $z$ direction.

Figure 6

The atoms are assumed to have a ${}^2{S}_{1/2}$ ground-state manifold and a ${}^2{P}_{1/2}$ excited-state manifold, which gives the fine-structure states shown in the figure. The drawing also displays the transitions driven by off-resonant left and right circularly polarized light and linearly $z$-polarized light, respectively.

Figure 7

Hyperfine structure of the ground-state manifold and the encoding of the red and blue spin-up and -down states. The states are labeled $|L,F,m_F\rangle$ as on the left-hand side of (41). The vertical position of each state is the energy of the state in the checkerboard optical lattice when ${\beta }^2=2$, ${\alpha }^2=1.2$, and $V_0>0$. The states that see a potential with minima at the blue sites for these parameters are shown in light gray (blue) and are assumed to be trapped on a blue site. The states that see a potential with minima at the red sites for these parameters are shown in dark gray (red) and are assumed to be trapped on a red site. Possible differences in zero point energy on the different lattice sites and the trapping in the $z$ direction are not taken into account in this drawing. Such effects will, however, not spoil the symmetry ensuring that the two $\hslash \delta$'s are the same. The hyperfine splitting $\hslash {\Delta }_{\mathrm{HF}}$ is not to scale.

Figure 8

Minimum and maximum values of $V_{3/2,m_F}/V_0$ for $m_F=-3/2,-1/2,1/2,3/2$, and ${\beta }^2=2$ as a function of ${\alpha }^2$ in the region where (46) is fulfilled.

Figure 9

Implementation of the Fermi-Hubbard-like model (1) with laser assisted tunneling on the checkerboard optical lattice. The kinetic-energy part is implemented with the two $z$-polarized standing-wave laser fields along, respectively, the $x$ and $y$ axis with frequency ${\omega }_r$ and a standing-wave laser field (not shown) along the $z$ direction with frequency ${\omega }_b$ containing left and right circularly polarized components. Note that it is possible to use the same set of lasers to implement both $H_{\mathrm{kin},\uparrow }$ and $H_{\mathrm{kin},\downarrow }$, and that $H_{\mathrm{int}}$ comes from the interaction between two fermions with opposite spin sitting on the same site.

Figure 10

Raman transitions used for laser assisted tunneling between nearest-neighbor sites (the frequencies of the lasers are not to scale).