Fractional quantum Hall states of Rydberg polaritons

We propose a scheme for realizing fractional quantum Hall states of light. In our scheme, photons of two polarizations are coupled to different atomic Rydberg states to form two flavors of Rydberg polaritons that behave as an effective spin. An array of optical cavity modes overlapping with the atomic cloud enables the realization of an effective spin-$1/2$ lattice. We show that the dipolar interaction between such polaritons, inherited from the Rydberg states, can be exploited to create a flat, topological band for a single spin-flip excitation. At half filling, this gives rise to a photonic (or polaritonic) fractional Chern insulator—a lattice-based, fractional quantum Hall state of light.

Figure 1

(a) Quasi-two-dimensional cloud of atoms (red disk) overlaps with an array of cavity modes (with cavity axis $\widehat{z}$) at a plane tilted relative to $\widehat{z}$. The overlaps (red balls) allow one to define a square-lattice array of Rydberg polaritons. Each polariton can be in state $|\Uparrow \rangle$ or $|\Downarrow \rangle$. The resulting spin model has a fractional quantum Hall ground state. (b) To achieve a topological flat-band structure, single-atom dressed states $|\uparrow \rangle$ and $|\downarrow \rangle$ are constructed as linear combinations of several Rydberg levels with spatially dependent coefficients $s,v$, and $w$. A weak dc electric field along $\widehat{z}$ is assumed. (c) The $|\Uparrow \rangle$ and $|\Downarrow \rangle$ polaritons are created by coupling $|\uparrow \rangle$ and $|\downarrow \rangle$ states to ${\mathcal{E}}_{\uparrow }$ (${\sigma }^{-}$-polarized) and ${\mathcal{E}}_{\downarrow }$ (${\sigma }^{+}$-polarized) photonic modes, respectively. The flip-flop ($|\uparrow \downarrow \rangle \to |\downarrow \uparrow \rangle$) dipolar interaction yields the fractional quantum Hall polariton Hamiltonian.

Figure 2

(a) Checkerboard lattice used to create the fractional Chern insulator. The parameters $s,v$, and $w$ used to define $|\uparrow \rangle$ are different on the two sublattices ($a$ and $b$). (b) Topological flat band for Rydberg polaritons (Chern number $C=-1$) featuring a flatness $\approx 10$.

Figure 3

(a) Fractional Chern insulator of Rydberg polaritons. For a $6\times 4$ lattice with six particles with periodic boundary conditions, the eigenstates in momentum sector $n_2+4n_1$, where $(k_x,k_y)=(n_1/3,n_2/2-n_1/3)\pi ,n_1=0,1,2$, and $n_2=0,1,2,3$. The two degenerate ground states (red) at $(k_x,k_y)=(0,0)$ and $(0,\pi )$ are separated from the other states by a gap. (b) The momentum-resolved eigenstates for the same parameters as in (a), except now with only five particles instead of six. We observe a clear gap above the 36 low-energy quasihole states (red), in agreement with the quasihole counting formula $Q$ given in the text [1, 53].

Figure 4

Levels of ${}^{87}\mathrm{Rb}$ for generating the $\Uparrow$ (a) and $\Downarrow$ (b) polaritons. In particular, $|\uparrow \rangle =s|3\rangle +v|4\rangle +w(|5\rangle +|6\rangle )/\sqrt{2}$, while $|\downarrow \rangle =(|2\rangle -|1\rangle )/\sqrt{2}$. All the labeled states except for $|e_{\uparrow }\rangle$ have nuclear spin quantum number $m_I=3/2$. Green and magenta are microwave fields coupling Rydberg states to Rydberg states. Blue, red, and orange are optical fields coupling $5P$ to $5S$ and to Rydberg states. $|g\rangle =|F=2,m_F=2\rangle =|m_J=1/2,m_I=3/2\rangle$; $|e_{\downarrow }\rangle =|F=3,m_F=3\rangle =|m_J=3/2,m_I=3/2\rangle$. The state $|s\rangle$ in (b) is used virtually. The positions of Rydberg levels in (b) relative to those in (a) are drawn to minimize the crowding of the figure: in reality, $69D_{3/2}$ lies between $70P_{3/2}$ and $71S_{1/2}$.