Fractional Quantum Hall States of Atoms in Optical Lattices

We describe a method to create fractional quantum Hall states of atoms confined in optical lattices. We show that the dynamics of the atoms in the lattice is analogous to the motion of a charged particle in a magnetic field if an oscillating quadrupole potential is applied together with a periodic modulation of the tunneling between lattice sites. In a suitable parameter regime the ground state in the lattice is of the fractional quantum Hall type, and we show how these states can be reached by melting a Mott-insulator state in a superlattice potential. Finally, we discuss techniques to observe these strongly correlated states.

Figure 1

(a) One period of the sequence used to create an effective magnetic field. The time evolution of the quadrupole potential is shown by the full curve, and the dashed and the dotted lines indicate the tunneling in the $x$ and the $y$ directions. For illustration the shown evolution of the tunneling is obtained from a sinusoidal variation of the lattice potential between 5 and 40 recoil energies [11]. (b),(c) Physical explanation of the procedure. (b) Tunneling in the $x$ direction is followed by a positive potential in the first and third quadrant (signs in the figure), and hence atoms will experience a lower potential by moving in the direction of the dashed arrows. (c) Same as (b) but with tunneling in the $y$ direction and opposite sign of the potential. When combined, the dashed lines in (b) and (c) make a circular cyclotron motion.

Figure 2

(a) Overlap of the ground state wave function with the Laughlin wave function (1). (b) Energy gap $\Delta E$ to the lowest excited state. In the figure we have fixed the number of particles and fluxes ($N_{\varphi }=2N$) and vary the flux per unit cell $\alpha$ by varying the size of the lattice. The shown results are for $N=2$ (●), $N=3$ (○), $N=4$ (×), and $N=5$ (□).

Figure 3

(a) Creation of the Laughlin state by melting a Mott insulator. Full lines (right axis): energy relative to the ground state of the 19 lowest excited states for different amplitudes ($V_0$) of the superlattice potential. Dashed (dotted) lines: overlap of the ground state (first excited state) with the Laughlin wave function (1) (left axis). Note that at $V_0=0$ there are two nearly degenerate states because of the combination of magnetic field and periodic boundary conditions. These two states have a 98% overlap with the two possible Laughlin states. Results are for a $6\times 6$ lattice with four atoms and eight fluxes, $\alpha \approx 0.22$. The periods of the superlattice are $p_x=p_y=3$. (b) Density distribution after expansion from the lattice (arbitrary units). For illustration we have assumed a lattice depth of ten recoil energies [11] and $\alpha \approx 0.22$ as in (a).