Pfaffian states in coupled atom-cavity systems

Coupled atom-cavity arrays, such as those described by the Jaynes-Cummings-Hubbard model, have the potential to emulate a wide range of condensed-matter phenomena. In particular, the strongly correlated states of the fractional quantum Hall effect can be realized. At some filling fractions, the fraction quantum Hall effect has been shown to possess ground states with non-Abelian excitations. The most well studied of these states is the Pfaffian state of Moore and Read G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991), which is the ground state of a Hall liquid with a three-body interaction. We show how an effective three-body interaction can be generated within the cavity QED framework, and that a Pfaffian-like ground state of these systems exists.

Figure 1

(a) Three-level atom configuration for inducing a three-body interaction in the JCH lattice. (b) Energy per excitation in the three-level atom for $\ell =1–5$ (blue, orange, green, red, and purple, respectively). Excitations in the $\ell =1$ and $\ell =2$ excitation subspace have the same energy cost. For higher excitation numbers, the energy per particle increases. (c) Energy per excitation as a function of ${\beta }_2$ [same colors as (b)] for ${\mathrm{\Delta }}_{}/{\beta }_1=-5$.

Figure 2

Energy gap for the first five excited states in the three-level JCH model on a $4\times 4$ lattice with four particles and four flux quanta ($\nu =1$). There is a three-dimensional quasidegeneracy in the ground state, indicative of a Pfaffian-like state.

Figure 3

Band gap and Pfaffian overlap as a function of ${\beta }_2/{\beta }_1$ (at ${\mathrm{\Delta }}_{}/{\beta }_1=-5$) and ${\mathrm{\Delta }}_{}/{\beta }_1$ (with ${\beta }_2/{\beta }_1=\sqrt{2}$) for four particles on $4\times 4$ (blue long-dashed line), $4\times 5$ (orange dashed line), $5\times 5$ (green dot-dashed line), $6\times 5$ (purple solid line), and $6\times 6$ (brown dotted line) lattice.