Fractional Quantum Hall Physics in Jaynes-Cummings-Hubbard Lattices

Jaynes-Cummings-Hubbard arrays provide unique opportunities for quantum emulation as they exhibit convenient state preparation and measurement, as well as in situ tuning of parameters. We show how to realize strongly correlated states of light in Jaynes-Cummings-Hubbard arrays under the introduction of an effective magnetic field. The effective field is realized by dynamic tuning of the cavity resonances. We demonstrate the existence of Laughlin-like fractional quantum Hall states by computing topological invariants, phase transitions between topologically distinct states, and Laughlin wave function overlap.

Figure 1

(a) Schematic of a square JCH lattice with a constant effective magnetic field. Photons moving around a plaquette acquire a phase $\Delta \varphi$. (b) A single-mode photonic cavity with frequency $\omega$ coupled to a two-level atom with strength $\beta$. (c) Scheme for breaking TRS in photonic cavities: a potential $V=[V^{\mathrm{DC}}+V^{\mathrm{AC}}\sin ({\omega }^{\mathrm{rf}}t+\Delta \varphi y)]x$ ($x$ and $y$ in units of the lattice spacing) is applied to the cavities (indicated by thick vertical green arrows) by dynamically tuning $\omega$. The phase offset, $\Delta \varphi$, along $y$ results in the synthetic magnetic field seen in (a).

Figure 2

Single particle spectrum of the JCH lattice with (a) $\Delta =0$ and (b) $\Delta =3$. In each case $\kappa =1$ and $\beta =1$. The spectra comprise two transformed Hofstadter butterflies. Color indicates the projection into the photonic modes.

Figure 3

(a) Energy gaps, (b) Laughlin wave function overlap, and (c) Chern numbers in the JCH FQHE as a function of the detuning $\Delta$. For $\Delta \ll 0$, the effective on-site energy is very high. In the opposite case, where $\Delta \gg 0$, the effective on-site energy is much lower than the energy from the pressure and the gap is due to multiple site occupancy. Dashed lines indicate the transition from a fractional $1/2$ state to a noninteracting particle state as determined by the Chern number. Colors correspond to configurations in Table : i, black; ii, dark blue; iii, red; iv, green; $v$, light blue.