Bilayer fractional quantum Hall states with dipoles

Using the example of dysprosium atoms in an optical lattice, we show how dipolar interactions between magnetic dipoles can be used to obtain fractional quantum Hall states. In our approach, dysprosium atoms are trapped one atom per site in a deep optical lattice with negligible tunneling. Microwave and spatially dependent optical dressing fields are used to define an effective spin-$\frac{1}{2}$ or spin-1 degree of freedom in each atom. Thinking of spin-$\frac{1}{2}$ particles as hard-core bosons, dipole-dipole interactions give rise to boson hopping, topological flat bands with Chern number 1, and the $\nu =\frac{1}{2}$ Laughlin state. Thinking of spin-1 particles as two-component hard-core bosons, dipole-dipole interactions again give rise to boson hopping, topological flat bands with Chern number 2, and the bilayer Halperin (2,2,1) state. By adjusting the optical fields, we find a phase diagram, in which the (2,2,1) state competes with superfluidity. Generalizations to solid-state magnetic dipoles are discussed.

Figure 1

(a) Dysprosium atoms are loaded one atom per site into a square lattice in the $X-Y$ plane. (b) The quantization axis of the atoms $\widehat{z}$ (red) is determined by the polarization of an applied optical field discussed below (no static magnetic or electric fields are applied). To obtain the $x-y-z$ coordinate system, one rotates the $X-Y-Z$ coordinate system by ${\mathrm{\Phi }}_0$ around the $\widehat{Z}$ axis and then by ${\mathrm{\Theta }}_0$ around the $\widehat{y}$ axis. (c) The relevant ${}^{161}\mathrm{Dy}$ level structure (the vertical axis shows the energy) used for obtaining the flat band with Chern number $C=1$ and the corresponding $\nu =\frac{1}{2}$ Laughlin state. The lower hyperfine levels labeled by $F$ are in the $J=8$ ground electronic state. Since the nuclear spin is $I=\frac{5}{2},F$ runs from $\frac{11}{2}$ to $\frac{21}{2}$. To avoid overcrowding the figure with unused energy levels, we do not show $F=\frac{11}{2}$, as well as most of the Zeeman levels associated with other $F$ levels. Far-detuned light (not shown) defines the quantization axis by providing an ac Stark shift $\propto M^2$ with an $F$-dependent coefficient. The 421-nm light (magenta), which drives a transition to the $F^{\prime }=\frac{17}{2}$ hyperfine level of the $4f^{10}\left({}^5{I}_8\right)6s6p\left({}^1{P}_1^o\right){(8,1)}_9^o$ state (only 2 out of 18 Zeeman states are used and hence shown), and the microwaves (black) define the dark dressed states $\left|\downarrow \right.〉$ and $\left|\uparrow \right.〉$. The dressing optical field is different on the two sublattices [green and yellow in (a)] making the state $\left|\uparrow \right.〉$ sublattice dependent. Levels $\left|1\right.〉$ and $\left|2\right.〉$ comprising state $\left|\downarrow \right.〉$ are color coded red, while levels $\left|3\right.〉,\left|4\right.〉,\left|5\right.〉$, and $\left|6\right.〉$ comprising state $\left|\uparrow \right.〉$ are color coded blue.

Figure 2

(a) Flat topological band with Chern number $C=1$. The flatness of the band (band gap divided by bandwidth) is $\approx 8$. (b) The momentum-resolved eigenvalues for the $\nu =\frac{1}{2}$ fractional Chern insulator with six hard-core bosons on a $6\times 4$ torus. The momentum sector $n_2+4n_1$ corresponds to $(k_x,k_y)=(n_1/3,n_2/2-n_1/3)\pi ,n_1=0,1,2$, and $n_2=0,1,2,3$. The spectrum features a gap separating two degenerate ground states at $(k_x,k_y)=(0,0)$ and $(0,\pi )$ (blue) from the other states by a gap.

Figure 3

(a) Magnetic field required to tune the hyperfine coupled NV states to their desired resonances. (b) Optical dressing $M$ scheme which enables sufficient control to realize topological flat bands.

Figure 4

Control of solid-state spins. (a) Two-dimensional array of implanted NV centers near diamond 110 surface, with NV lattice spacing $a=20$ nm. A uniform applied electric field and a dielectric nanostructure patterned on the surface modulate the displacement field at each NV center. (b) Schematic optical dressing scheme. The left panel corresponds to the bare level structure and the right panel corresponds to the shifted level structure in the presence of gating. Spin states $\left|\mp 1\right.〉$ in the electronic ground state are coupled to the $\left|A_2\right.〉$ excited state with ${\sigma }^{\pm }$ polarized light. We have left out the hyperfine structure for simplicity. Modulated displacement field brings the two frequencies (red and blue) of optical dressing in and out of resonance. (c) Patterned dielectric on the surface of diamond (with an indium tin oxide (ITO) back gate) which can yield a local dc field on the order of ${10}^3\mathrm{V}$/cm as shown in (d).

Figure 5

(a) Flat topological band with Chern number $C=2$. The flatness of the band (band gap divided by bandwidth) is $\approx 11$. (b) The momentum-resolved eigenvalues for the $\nu =\frac{1}{3}$ fractional Chern insulator with four two-component hard-core bosons on a $3\times 4$ torus. The momentum sector corresponds to $(k_x,k_y)=(2n_1/3,n_2/2)\pi ,n_1=0,1,2$, and $n_2=0,1,2,3$. The spectrum features a gap separating the three degenerate ground states at $(k_x,k_y)=(0,0)$, ($2\pi /3$,0), and ($4\pi /3$,0) (blue) from the other states by a gap.

Figure 6

$C=2$ phase diagram as a function of $({\mathrm{\Theta }}_0,{\mathrm{\Phi }}_0)$, which specify the direction of the quantization axis. Red circle indicates the point in the (2,2,1) phase corresponding to the spectrum in Fig. 5.

Figure 7

The level structure of ${}^{161}\mathrm{Dy}$ relevant for implementing the $C=2$ flat band and the resulting Halperin (2,2,1) state.