Optical lattice quantum Hall effect

We explore the behavior of interacting bosonic atoms in an optical lattice subject to a large artificial magnetic field. We extend earlier investigations of this system where the number of magnetic flux quanta per unit cell $\alpha$ is close to a simple rational number [R. N. Palmer and D. Jaksch, Phys. Rev. Lett. 96, 180407 (2006)]. Interesting topological states such as the Laughlin and Read-Rezayi states can occur even if the atoms experience a weak trapping potential in one direction. An explicit numerical calculation near $\alpha =1/2$ shows that the system exhibits a striped vortex lattice phase of one species, which is analogous to the behavior of a two-species system for small $\alpha$. We also investigate methods to probe the encountered states. These include spatial correlation functions and the measurement of noise correlations in time-of-flight expanded atomic clouds. Characteristic differences arise which allow for an identification of the respective quantum Hall states. We furthermore discuss that a counterintuitive flow of the Hall current occurs for certain values of $\alpha$.

Figure 1

(Color online) Numerical approximation to the ground state of the ${\alpha }_c=1/2$ vortex lattice phase, calculated using imaginary time propagation of a Gutzwiller ansatz. (a) Density of the atom distribution, black stands for no density, white means high superfluid density, red (hardly visible) means high Mott insulator density. (b) Population of the two “layers,” black and white depict the two components, color encodes the phase of the superposition of those two components. The parameters are $\alpha =0.48$, $U=0.1J$, and $\tilde{\nu }=21.2$.

Figure 2

Numerically calculated time-of-flight expansions in symmetric gauge of (a) and (c) a Laughlin and Read-Rezayi state $\left(\nu =1/2,1\right)$, and (b) and (d) a 221 and NASS state $\left(\tilde{\nu }=2/3,4/3\right)$. The initial states are circular with diameter 100 lattice sites for the outer ring and 50 for the inner one. The artificial magnetic field is chosen such that (a) $\alpha =0.005$, (b) $\alpha =0.505$, (c) $\alpha =0.01$, and (d) $\alpha =0.51$. The resulting larger fields $\Omega$ or $\tilde{\Omega }$ in (c) and (d), respectively, lead to a wider expansion of the cloud. The shown expansions do not include the $\tilde{w}$ slow decay, as the speed of this is implementation dependent. A darker color corresponds to a higher density of particles, and figures (a) and (b) are to the same brightness scale, as are (c) and (d). Shown is the first Brillouin zone, and $\mathbf{K}=m_0\mathbf{X}/\hslash t$.

Figure 3

Two-point functions for continuum fractional quantum Hall states: $\nu =1/2$ Laughlin state (solid), $\tilde{\nu }=2/3$ 221 state with $g_{11}=g_{22}$ (dashed) and $g_{12}=g_{21}$ (dotted), and $\nu =1$ Read-Rezayi state (dash-dotted).

Figure 4

Numerically calculated shot noise correlations for an infinite system in symmetric gauge. The states are (a) Laughlin, (b) $\nu =1$ Read-Rezayi, (c) $\nu =3/2$ Read-Rezayi, (d) 221, (e) $\tilde{\nu }=4/3$ NASS, and (f) $\tilde{\nu }=3/2$ Read-Rezayi. The artificial magnetic field was chosen to give $\alpha =0.01$ in (a)–(c), and $\alpha =0.51$ in (d)–(f). Gray corresponds to no correlation, whereas black corresponds to 100% anticorrelation. We cannot rule out that the four dips at $\Delta k=\left(\pm \pi /4d,\pm \pi /4d\right)$ are artifacts arising from our approximations.

Figure 5

Dimensionless Hall current $I_d=I/\left(2a{m}^2\sqrt{2u\tilde{\Omega }/{\hslash }^3}/\omega \right)$ against dimensionless number of atoms per unit length $N_d=N/\left(2Lm\sqrt{2u{\tilde{\Omega }}^3/{\hslash }^3}/\omega \right)$, where $\mathit{L}$ is the length of the system, which is contained in a 1D harmonic trap $V\left(x,y\right)=m{\omega }^2{y}^2/2+yma$ at $\alpha \approx 1/2$. The linear term $yma$ is turned on after the atoms have come to equilibrium in the trap, and the curves are for no disorder (dotted straight line), maximal disorder (constant density of states, black curve), and Lorentzian disorder of width $\left(1/5\right)um\tilde{\Omega }/2\pi \hslash$ (gray curve). The corresponding plot for small $\alpha$ is qualitatively similar [9].