Synthetic Dimensions with Magnetic Fields and Local Interactions in Photonic Lattices

We discuss how one can realize a photonic device that combines synthetic dimensions and synthetic magnetic fields with spatially local interactions. Using an array of ring cavities, the angular coordinate around each cavity spans the synthetic dimension. The synthetic magnetic field arises as the intercavity photon hopping is associated with a change of angular momentum. Photon-photon interactions are local in the periodic angular coordinate around each cavity. Experimentally observable consequences of the synthetic magnetic field and of the local interactions are pointed out.

Figure 1

Schematic illustration of hoppings (a) in the $x\text{−}\theta$ plane and (b) in the $x\text{−}w$ plane when $c=1$.

Figure 2

Energy levels as a function of $D/J$ for (a) $c=1$ and (b) $c=2$. The solid lines show the numerically obtained first (a) five and (b) six energy eigenvalues, and the dotted lines are the analytical approximations for $D/J\ll 1$.

Figure 3

Cyclotron motion of a wave packet for $N_x=31$, $D=0.1J$, $c=1$, $UN=20J$, and $E_x=0.1J$, where $N$ is the total number of photons. In both (a) and (b), the horizontal axis is $x$ and the vertical axis is $\theta$. The initial condition of the wave packet is plotted in (a). In (b), the center-of-mass motion of the wave packet from $t=0$ to $4\pi /{\omega }_c$ is plotted. The solid line is from the numerical simulation of the dynamics of the wave packet, and the dashed line is the analytical prediction from Eq. (11).

Figure 4

Free expansion of a Gaussian wave packet for $c=1$ and $D=0.1J$ after the expansion time of $\pi /{\omega }_c$. (a) The initial wave packet. (b) The expansion in the absence of the interaction. (c) The expansion in the presence of the contact interaction with $UN=20J$. (d) The expansion in the presence of the infinite-range interaction with $U_{\text{inf}}N=5J$. The horizontal axes are $x$, and the vertical axes are $\theta$.