Fractional quantum Hall effect in optical lattices

We analyze a recently proposed method to create fractional quantum Hall (FQH) states of atoms confined in optical lattices [A. Sørensen et al., Phys. Rev. Lett. 94, 086803 (2005)]. Extending the previous work, we investigate conditions under which the FQH effect can be achieved for bosons on a lattice with an effective magnetic field and finite on-site interaction. Furthermore, we characterize the ground state in such systems by calculating Chern numbers which can provide direct signatures of topological order and explore regimes where the characterization in terms of wave-function overlap fails. We also discuss various issues which are relevant for the practical realization of such FQH states with ultracold atoms in an optical lattice, including the presence of a long-range dipole interaction which can improve the energy gap and stabilize the ground state. We also investigate a detection technique based on Bragg spectroscopy to probe these systems in an experimental realization.

Figure 1

(Color online) (a) The overlap of the ground state with the Laughlin wave function. For small $\alpha$ the Laughlin wave function is a good description of the ground state for positive interaction strengths. The inset shows the same result of small $U$. (b) The energy gap for $N∕N_{\varphi }=1∕2$ as a function of interaction $U∕J$ from attractive to repulsive. For a fixed $\alpha$, the behavior does not depend on the number of atoms. The inset defines the particle numbers, lattice sizes, and symbols for both parts (a) and (b).

Figure 2

(Color online) (a) The energy gap as a function of $\alpha U$ and $\alpha$ for a fixed number of atoms $(N=4)$. The gap is calculated for the parameters marked with dots, and the surface is an extrapolation between the points. (b) Linear scaling of the energy gap with $\alpha U$ for $U⪡J$, $\alpha \lesssim 0.2$. The results are shown for $N=2$ (◻), $N=3$ (∗), $N=4$ (+), and $N=5$ (▿). The gap disappears for noninteraction system, and increases with increasing interaction strength $(\propto \alpha U)$ and eventually saturates to the value in the hard-core limit.

Figure 3

(Color online) Ground-state overlap with the Laughlin wave function. (a) and (b) are for three and four atoms on a lattice, respectively. $\alpha$ is varied by changing the size of the lattice (the size in the two orthogonal directions differ at most by unity). The Laughlin state ceases to be a good description of the system as the lattice nature of the system becomes more apparent, $\alpha \gtrsim 0.25$. The overlap is only calculated at the positions shown with dots, and the color coding is an extrapolation between the points.

Figure 4

(a) Twist angles of the toroidal boundary condition. In addition to the flux going through the surface there may also be a flux inside the torus or going through the hole in the middle. When encircling these fluxes the wave function acquires an extra phase represented by the boundary conditions in Eq. (11). (b) Redefining the vector potential around the singularities: ${\mathcal{A}}_j$ is not well defined everywhere on the torus of the boundary condition. Therefore, another vector field $\mathcal{A}^{\prime }{}_j$ with different definition should be introduced around each singularity $({\theta }_1^n,{\theta }_2^n)$ of ${\mathcal{A}}_j$. ${\mathcal{A}}_j$ and ${\mathcal{A}}_j^{\prime }$ are related to each other by a gauge transformation $\chi$, and the Chern number depends only on the loop integrals of $\chi$ around those singularities regions; cf. Eq. (16).

Figure 5

(Color online) Chern number associated with low-lying energy states in the presence of impurities. Due to the impurity potential (a), the twofold-degenerate ground state splits and the wave function overlap with the Laughlin state drops to 52% and 65% for the first and second energy states, respectively. The results are for three atoms on a $6\times 6$ lattice $(\alpha =0.17)$ in the hard-core limit. (b) $\Omega ({\theta }_1,{\theta }_2)$ for the first level has no vorticity. However, for the second level, as shown in (c), $\Omega ({\theta }_1,{\theta }_2)$ has vorticity equal to 1 associated with regions where either ${\Lambda }_{\varphi }$ or ${\Lambda }_{\varphi }^{\prime }$ vanishes.

Figure 6

(Color online) (a) shows the argument of $\Omega ({\theta }_1,{\theta }_2)$ as arrows for fixed $\Phi$ and ${\Phi }^{\prime }$. (b) and (c) Surface plots of $\det {\Lambda }_{\Phi }$ and $\det {\Lambda }_{\Phi }^{\prime }$ (blue is lower than red). ${\theta }_1$ and ${\theta }_2$ change from zero to $2\pi$. These plots have been produced for three atoms with $N_{\varphi }=6$ $(\alpha =0.24)$ in the hard-core limit on a $5\times 5$ lattice. The total vorticity corresponding to each of the reference wave functions ($\Phi$ or ${\Phi }^{\prime }$) indicates a Chern number equal to 1.

Figure 7

(Color online) $\Omega ({\theta }_1,{\theta }_2)$ for fixed $\Phi$ and ${\Phi }^{\prime }$. ${\theta }_1$ and ${\theta }_2$ change from zero to $2\pi$. This plot has been produced for four atoms with $N\varphi =8$ in the hard-core limit on a $5\times 5$ lattice $(\alpha =0.32)$. Although there are more vortices here compared to Fig. 6, the total vorticity corresponding to each of the trial functions ($\Phi$ or ${\Phi }^{\prime }$) indicates a Chern number equal to 1.

Figure 8

(Color online) Low-lying energy levels as a function of twist angles. For high $\alpha$ the degeneracy of the ground state is a function of twist angles. The shown results are for five atoms on a $5\times 5$ lattice; i.e., $\alpha =0.4$ (a) shows first three energy manifolds as a function of the toroidal boundary condition angles and (b) shows a cross section of (a) at ${\theta }_2=\pi$ for seven lowest-energy levels. The first and second energy levels get close to each other at ${\theta }_1={\theta }_2=\pi$.

Figure 9

(Color online) (a) The overlap of the ground state with the Laughlin wave function (dashed lines) and four low-lying energies of the system (solid lines) versus the dipole-dipole interaction for four atoms on a $6\times 6$ lattice. (b) Gap enhancement for a fixed repulsive dipole-dipole interaction strength $U_{\mathit{\text{dipole}}}=5J$ versus $\alpha$. The results are shown for $N=2$ (◻), $N=3$ (∗), and $N=4$ (+).

Figure 10

(Color online) The overlap of the first four low-lying energy states with the Laughlin wave function for the case of $\nu =1∕4$ on a torus. The dashed (dotted) line shows the overlap for the system without (with) a dipole interaction $(U_{\mathit{\text{dipole}}}=5J)$. The Laughlin state is only a good description for a more dilute lattice $\alpha \lesssim 0.1$ compared to $\nu =1∕2$. The dipole interaction stabilizes the system and the overlap is more significant for higher values of $\alpha \lesssim 0.2$.

Figure 11

(Color online) (a) Structure factor and (b) energy spectrum for a $11\times 11$ lattice with three atoms on a torus. Points show the momentums allowed by the boundary conditions. The dotted line in (b) shows for comparison the low-energy spectrum of a free particle which equals $J{\left(qa\right)}^2$.

Figure 12

Proposal for realizing a rotating optical lattice. Four AOMs (black boxes) change the direction of the lattice beams, which are subsequently focused in the middle of the setup by four lenses (gray). Simultaneously varying the four diffraction angles in the AOMs will generate a rotating optical lattice.