Artificial gauge field for photons in coupled cavity arrays

We propose and characterize solid-state photonic structures where light experiences an artificial gauge field. A nontrivial phase for photons tunneling between adjacent sites of a coupled cavity array can be obtained by inserting optically active materials in the structure or by inducing a suitable coupling of the propagation and polarization degrees of freedom. We also discuss the feasibility of observing strong gauge field effects in the optical spectra of realistic systems, including the Hofstadter butterfly spectrum.

Figure 1

Scheme of the coupled DBR microcavity configuration to generate the artificial magnetic field. (a) Two-dimensional square lattice of cavities: In the sketch, four cavities containing an oblique mirror each are connected by phase elements embedded between DBR mirrors. Within the Landau gauge, the artificial magnetic field arises from the $x$ dependence of the tunneling phase in the $y$ direction. (b) Simplest two-cavity setup. (c) For the two-cavity setup of panel (b), transmission spectrum (blue solid line) for ${\sigma }_{+}$ incident light; relative phase $\phi$ of the field in the two cavity layers (red dashed line). The outer (inner) DBRs contain 19 (10) periods of alternate layers of refractive index $n_1=3.6$ and $n_2=2.9$ and optical thickness ${\lambda }_{\mathrm{Br}}/4$, where ${\lambda }_{\mathrm{Br}}=2\pi c/{\omega }_{\mathrm{Br}}$, ${\omega }_{\mathrm{Br}}$ being the Bragg frequency. The cavity layers are ${\lambda }_{\mathrm{Br}}/2$ thick and have $n_{\mathrm{cav}}=2.9$. They also contain a circularly birefringent medium of thickness $d_{\sigma }/{\lambda }_{\mathrm{Br}}=0.21/\pi$ with $n_{{\sigma }_{\pm }}=1.2,2$. The angle $\theta$ between the optical axes of the half-wave slabs is arbitrarily chosen to be $3\pi /22$. The same effect is observed in the presence of an optically active slab embedded in between the inner DBRs.

Figure 2

Laterally patterned planar microcavity scheme to generate the artificial magnetic field. (a) Spatial dependence of the scalar potential $V_{sc}\left(x\right)$ (solid line) and the amplitude of the $xy$ component of the vector field (dashed lines); different dashed lines correspond to the components of amplitude $V_s(x-x_j)$ centered at $x_j$ having different angles ${\zeta }_j$ with ${\zeta }_{\max }=2\pi /3$. Effective photon mass $m$ was taken to be $5\times {10}^{-5}$ times the electron mass. The $z$ component of the vector field is $V_z\left(x\right)=-1.21$ meV. (b) Spatial dependence of the ground-state expectation value of different spin components $\langle S_i\rangle =\frac{1}{2}\langle {\widehat{\sigma }}_i\rangle$. Inset shows the corresponding loop on the Poincaré sphere.

Figure 3

(a) Average total number $n_T$ of transmitted photons as a function of $\Delta {\omega }_p={\omega }_p-{\omega }_{\circ }$. The system considered is a $3\times 3$ lattice with flux quanta per plaquette $\alpha =\frac{1}{3}$ and $\gamma /J=0.05$. Different (green solid, blue dashed, red dot-dashed) curves correspond to a driving of amplitude $\overline{F}/J=0.005$ localized on different (2,3,5) sites according to the enumeration given in the inset. Solid black line shows the sum over all nine sites. (b) Real-space profile $|{\psi }^{\left(l\right)}{\left(\mathbf{r}\right)|}^2$ of eigenmodes $|E_l\rangle$, where Gaussian real-space basis functions were assumed at each site for illustrative purposes.

Figure 4

(a) Color plot of the total transmission $n_T$ as a function of flux quanta per plaquette $\alpha$ and pump frequency $\Delta {\omega }_p/J$ for a $10\times 10$ lattice with $\overline{F}/J=0.005$ and $\gamma /J=0.05$. (b) Cut of the color plot along the constant $\Delta {\omega }_p$ line indicated in (a) as a black line. (c) Photon occupation number $\langle {\widehat{b}}_i^{†}{\widehat{b}}_i\rangle$ pattern for a pump resonant with the ground state and localized on a single site at the lattice center $(21\times 21$ lattice, magnetic flux quanta per plaquette $\alpha =\frac{1}{3}$, $\gamma /J=0.01$).

Figure 5

Schematic arrangement of mirrors forming the cavity. Central mirror is located at 45 deg with respect to $x$ and $y$ axes, its center being at a distance $L$ from the external mirrors.

Figure 6

Solid and dashed double-headed arrows symbolize two different modes that are weakly coupled by the central mirror. (a) $|r_{\circ }|\ll 1$ and (b) $t_{\circ }\ll 1$.

Figure 7

Total transmission $|E_t{|}^2=|E_{t_1}{|}^2+|E_{t_2}{|}^2+{\left|E_{t_3}\right|}^2$ as a function of $\omega /{\omega }_{\mathrm{r}}$ with $t=0.1$ and $L=\pi c/4n{\omega }_{\mathrm{r}}$. The red dashed line shows the case for $|r_{\circ }|=0$ and the blue solid line is for $|r_{\circ }|=0.05$.

Figure 8

Magnitude and phase of the complex valued electric fields inside the cavity as a function of $\omega /{\omega }_{\mathrm{r}}$ for $t=0.1$, $|r_{\circ }|=0.05$, and $L=\pi c/4n{\omega }_{\mathrm{r}}$. Phases are defined in the range $[-\pi ,\pi )$. The $y$ axis can also be used to read off $\left|E\right|$. Vertical dashed lines correspond to peak positions in the total transmission of Fig. 7. (a) Arguments of $E_1$, $E_2$, $E_5$, and $E_6$ are shown. Only $|E_1|$ is depicted as other magnitudes behave very similarly. (b) Arguments of $E_3$, $E_4$, $E_7$, and $E_8$ along with $|E_3|$ are shown.