Non-Abelian Gauge Fields in Photonic Cavities and Photonic Superfluids

We show that the TE-TM modes splitting and the structure anisotropy of a semiconductor microcavity combine into a non-Abelian gauge field for exciton-polaritons or cavity photons. The field texture can be tuned simply by rotating the sample and ranges continuously from a Rashba to a monopolar field. In the noninteracting regime, the latter leads to remarkable focusing and conical diffraction effects. In the interacting regime, the spin-orbit coupling induces a breakdown of superfluidity. The spatially homogeneous flows become unstable and dynamically evolve into spin textured states, such as stripes or domain walls.

Figure 1

Scheme of the experimental configuration necessary to produce a controllable gauge field. The angle between the crystallographic axis (the effective magnetic field $\mathbf{\Omega }$) and the horizontal is $\phi /2$ ($\phi$). A laser with a vertical inclination $\chi$ making an angle $\theta$ with the horizontal produces a propagating polariton beam with wave vector ${\mathbf{k}}_0=({\omega }_0/c)\sin \chi (\cos \theta ,\sin \theta )$, where ${\omega }_0$ is the laser frequency and $c$ is the light speed.

Figure 2

Polariton dispersion relation for $\phi =0$ ($-\mathrm{\Omega }/2$ taken as zero reference). (a) Dispersion along the $x$ direction showing the two polariton branches ${\epsilon }_{\pm }$ crossing at the magic point $k_{*}=\sqrt{\mathrm{\Omega }/2\beta }$. The green-shadowed rectangle defines the region $|{\mathbf{k}}_0-{\mathbf{k}}_{*}|\le k_{c1}$, where the transverse dispersion bends. Panels (b),(c), and (d) show the dispersion along the $y$ direction (black) and the corresponding $x$ projection of the pseudospin (blue and gray) for $k_x=\{0.4,0.97,1.0\}\mu \mathrm{m}$. In (c), a parametric process $k_y:0\to \pm {\kappa }_p$ occurs between an $X$-polarized mode and a diagonal ($D$) and an antidiagonal ($aD$) mode. In (d), we observe the linear intersection of two branches crossing each other at $k_y=0$. Panel (e) depicts the field texture of Eq. (2) in the reciprocal space and panel (f) zooms around the wave vector $k_{*}$. Parameters used are $m=4.5\times {10}^{-5}{m}_e$, $\mathrm{\Omega }=0.2\mathrm{meV}$, and $\beta =0.1\mathrm{meV}\mu {\mathrm{m}}^2$, yielding $k_{*}=1.0\mu {\mathrm{m}}^{-1}$.

Figure 3

Effects of the gauge field: (a),(b) Lensing effect with a linearly polarized polariton flow (${\mathbf{k}}_0={\mathbf{k}}_{*}$) encountering a potential barrier of size $a=10\mu \mathrm{m}$ (a) in the absence and (b) in the presence of the effective magnetic fields. The color map shows the total density $n=n_{+}+n_{-}$. (c),(d) Conical diffraction under circularly polarized pulsed excitation at ${\mathbf{k}}_0={\mathbf{k}}_{*}$: (c) $t=0\mathrm{ps}$ and (d) $t=50\mathrm{ps}$. (e),(f) Polarization domains. Gaussian linearly polarized pulsed excitation (e) at ${\mathbf{k}}_0=\mathbf{0}$ for $\mathrm{\Omega }=0$ and (f) $\mathbf{k}={\mathbf{k}}_{*}$ for $\mathrm{\Omega }\ne 0$. The color map shows the circular polarization degree ${\rho }_c=(n_{+}-n_{-})/(n_{+}+n_{-})$.

Figure 4

From (a) to (d): Transverse ($T$) and longitudinal ($L$) instabilities for a $X$-polarized condensate in the subsonic regime: real (solid) and imaginary (dashed) parts. Black (red and gray) lines depicting the lower (upper) branches ${\epsilon }_{x,y}^L$ (${\epsilon }_{x,y}^U$). $T$ instability: ${\epsilon }_x$ [panel (a)] and ${\epsilon }_y$ [panel (b)] for $k_0=0.93k_{*}>k_{c1}$. $T+L$ instability: ${\epsilon }_x$ [panel (c)] and ${\epsilon }_y$ [panel (d)] for $k_0=1.12k_{*}$. Stability diagram for $X$- [panel (e)] and $Y$-polarized [panel (f)] condensates: $v_{*}=\hslash k_{*}/m$ (black line), $v_{c1}=\hslash k_{c1}/m$ (red and light gray) and $v_{c2}=\hslash k_{c2}/m$ (blue and light gray). The points correspond to the parameters used in panels (a) and (b) and (c) and (d). We have considered ${\alpha }_{1}n=2.5\mathrm{meV}$ and ${\alpha }_2=-0.2{\alpha }_1$, $\mathrm{\Omega }=0.15\mathrm{meV}$ and $\beta =0.075\mathrm{meV}{\mu \mathrm{m}}^2$.

Figure 5

Snapshots of the diagonal polarization degree ${\rho }_D=(n_D-n_{aD})/(n_D+n_{aD})$ after the onset of instabilities. (a) $X$-polarized BEC at $v_0/c=0.6$ and (b) $Y$-polarized BEC at $v_0/c=0.45$. We have used $\mathrm{\Omega }=0.1\mathrm{meV}$ and $\beta =0.1\mathrm{meV}{\mu \mathrm{m}}^2$.