Light-Matter Interactions in Synthetic Magnetic Fields: Landau-Photon Polaritons

We study light-matter interactions in two-dimensional photonic systems in the presence of a spatially homogeneous synthetic magnetic field for light. Specifically, we consider one or more two-level emitters located in the bulk region of the lattice, where for increasing magnetic field the photonic modes change from extended plane waves to circulating Landau levels. This change has a drastic effect on the resulting emitter-field dynamics, which becomes intrinsically non-Markovian and chiral, leading to the formation of strongly coupled Landau-photon polaritons. The peculiar dynamical and spectral properties of these quasiparticles can be probed with state-of-the-art photonic lattices in the optical and the microwave domain and may find various applications for the quantum simulation of strongly interacting topological models.

Figure 1

(a) Sketch of a system of two-level emitters coupled to a photonic lattice with a synthetic magnetic field $B$. The magnetic field is implemented by adjusting the hopping phases ${\varphi }_{ij}$ between neighboring lattice sites such that around each plaquette ${\sum }_{\square }{\varphi }_{ij}=2\pi \alpha$. (b) The projected density of states, $\rho ({\overrightarrow{r}}_e,\omega )$, is plotted on a logarithmic scale (arbitrary units) as a function of $\alpha$ and for a lattice of $M=20\times 20$ sites. For this plot, ${\overrightarrow{r}}_e/l_0=(10,10)$ and each resonance is represented by a broadened $\delta$ function with a finite width of $\gamma /J\approx {10}^{-3}$.

Figure 2

(a) Evolution of the excited-state population, $p_e(t)$, of an emitter located at ${\overrightarrow{r}}_e/l_0=(25,25)$ in a lattice of $50\times 50$ sites. The parameters are $\alpha =0$ and ${\delta }_e/J=1.35$ (blue line), $\alpha =0.08$ and ${\delta }_e/J=1.35$ (orange line), and $\alpha =0.08$ and ${\delta }_e/J=1.76$ (green dashed line). (b) Plot of the lowest eigenfrequencies ${\omega }_{\lambda }$ of the two photonic lattices as used for the simulation shown in blue and orange in (a). The dashed black lines indicate the corresponding emitter’s frequencies. (c) Photon density, $|\phi ({\overrightarrow{r}}_i,t_{\pi }){|}^2$, combined with the profile of the photon current, $⟨{\overrightarrow{j}}_p⟩({\overrightarrow{r}}_i,t_{\pi })$, at time $t_{\pi }=\pi /(2\mathrm{\Omega })$, for $\alpha =0.08$ and ${\omega }_e={\omega }_{\ell =0,1,2}$. For all plots $g/J=0.14$ and for each lattice site in the bulk (on the edge) a photon decay rate of ${\gamma }_p/J=4\times {10}^{-4}$ (${\gamma }_{\mathrm{edge}}/J\sim {10}^{-1}$) has been introduced [34].

Figure 3

(a) The spectrum of $H_{\mathrm{LPP}}^{(\ell )}$ in the single- and two- excitation sector for $N=3$ equidistant emitters with varying spacing $d$ and for $\ell =1$ and ${\omega }_e={\omega }_1$. (b) Zoom of the lower polaritonic band of the emitter’s excitation spectrum, ${\mathcal{S}}_e^n(\omega )$, for a $N=4\times 4$ square lattice of emitters with open boundaries, where ${\overrightarrow{r}}_e^n$ is the location of one of the four inner emitters (the full spectrum is reflection symmetric around ${\omega }_e$). For this plot, $\alpha =0.08$, ${\gamma }_e/\mathrm{\Omega }=0.02$, and $\ell =3$. The color scale is normalized to the maximum value. (c) Plot of the two-photon correlation function $C({\overrightarrow{r}}_i,{\overrightarrow{r}}_e^1)$ for the different two-photon eigenstates are indicated in (a). The green crosses represent the emitters position, and the red circle marks the reference emitter’s position ${\overrightarrow{r}}_e^1$.

Figure 4

(a) Evolution of the excited state populations $p_e^n(t)$ of $N=3$ emitters arranged in a triangle of length $d/l_0=4$. For this plot $\ell =0$, and $\alpha =1/(16\sqrt{3})\approx 0.036$, such that the enclosed effective flux is ${\theta }_{△}\simeq \pi /2$ and the dipole-dipole interactions become fully chiral (see Supplemental Material [34] for details). In the upper panel the initial state contains two excitations in emitter 1 and 2. In the lower panel the initial state contains just one excitation in emitter 1. (b) Single-excitation spectrum of $H_{\mathrm{eff}}$ for a square lattice of $20\times 20$ emitters and normalized to the nearest-neighbor coupling strength $\tilde{J}=|{\tilde{J}}_{12}|$. The two spectra are obtained for the spacings $d/l_0=2$ (${\alpha }_{\mathrm{eff}}=0.32$) and $d/l_0=5.39$ (${\alpha }_{\mathrm{eff}}=2.32$) and in both cases $\alpha =0.08$ and $\ell =0$ has been assumed.