Simulating frustrated antiferromagnets with quadratically driven QED cavities

We propose a class of quantum simulators for antiferromagnetic spin systems based on coupled photonic cavities in the presence of two-photon driving and dissipation. By modeling the coupling between the different cavities through a hopping term with negative amplitude, we solve numerically the quantum master equation governing the dynamics of the open system and determine its nonequilibrium steady state. Under suitable conditions, the steady state can be described in terms of the degenerate ground states of an antiferromagnetic Ising model. When the geometry of the cavity array is incommensurate with the antiferromagnetic coupling, the steady state presents properties which bear full analogy with those typical of the spin-liquid phases arising in frustrated magnets.

Figure 1

A sketch of the array of cavities in the presence of a two-photon driving with amplitude $G$, one-photon losses with rate $\gamma$, two-photon losses with rate $\eta$, and hopping amplitude $J<0$. In the limit of a strong driving field, the photons in each cavity form a coherent state with phase $\alpha$ or $-\alpha$. The coupling with $J<0$ tends to favor the emergence of states with opposite phases between neighboring cavities, but this condition cannot be satisfied in the frustrated system made up of three mutually coupled cavities. This system bears strong analogies to three antiferromagnetically interacting Ising spins.

Figure 2

The first-order correlation function $g_{1,2}^{\left(1\right)}$ as a function of the amplitude of the two-photon driving for the systems with $N=2$ and $N=3$ cavities. Inset: behavior of $g_{1,2}^{\left(1\right)}$ vs $G$ in the regime of large $G/\gamma$ for the case of $N=3$ cavities, plotted on a log-log scale: the results show a power-law dependence $|1/3+g_{1,2}^{\left(1\right)}|\sim G^{\nu }$, with $\nu =-1.02\pm 0.02$. The other Hamiltonian parameters are $U/\gamma =10, \mathrm{\Delta }/\gamma =J/\gamma =-10$.

Figure 3

The von Neumann entropy $S$ as a function of the amplitude of the two-photon driving for the systems with $N=2$ and $N=3$ cavities. Inset: behavior of $S$ vs $G$ in the regime of large $G/\gamma$ for the case of $N=3$ cavities, plotted on a log-log scale: the results show a power-law dependence $S-ln\left(6\right)\sim G^{\mu }$, with $\mu =-2.28\pm 0.08$. The other Hamiltonian parameters are $U/\gamma =10, \mathrm{\Delta }/\gamma =J/\gamma =-10$.

Figure 4

(a) The von Neumann entropy $S$ and (b) the first-order coherence correlation function $g_{1,2}^{\left(1\right)}$ as a function of the amplitude of the two-photon driving rescaled with the one-photon loss rate $G/\gamma$ for the system with $N=3$ cavities and for different values of the Hamiltonian parameters. (c) and (d) The same data as a function of the two-photon driving rescaled with the nonlinearity $G/U$.

Figure 5

The entanglement negativity $\mathcal{N}$ as a function of the amplitude of the two-photon driving for the systems with $N=2$ and $N=3$ cavities. The other Hamiltonian parameters are $U/\gamma =10, \mathrm{\Delta }/\gamma =J/\gamma =-10$.

Figure 6

Field amplitude $|\langle {\widehat{a}}_1\rangle |$ (rescaled by $|{\alpha }_0|$) as a function of the amplitude $F$ of the applied one-photon driving for the systems with $N=2$ and $N=3$ cavities. The other parameters of the Hamiltonian are $U/\gamma =10, \mathrm{\Delta }/\gamma =J/\gamma =-10, G/\gamma =60$.