Quantitative Measurements of Quantized Microwave Faraday Rotation.

We report {\it quantitative} microwave Faraday rotation measurements conducted with a high-mobility two-dimensional electron gas (2DEG) in a GaAs/AlGaAs semiconductor heterostructure.In a magnetic field, the Lorentz force acting on the charge carriers leads to a large classical Faraday effect that can be quantitatively understood by a Drude model accounting for the kinetic response in the microwave frequency range. The high electron mobility of the 2DEG enables a large single-pass Faraday rotation of $\theta_F^{max} \simeq 45^\circ$ $(\simeq0.8$~rad) to be achieved at a modest magnetic field of $B \simeq 100$~mT, leading to a colossal Verdet constant of $V\sim 10^{8}$~rad~T$^{-1}$~m$^{-1}$. As with the Hall effect, a continuous classical Faraday effect is observed as well as a quantized Faraday effect. In the quantum regime, the Faraday rotation $\theta_F$ is naturally quantized in units of the fine structure constant $\alpha\simeq \frac{1}{137}$. Electromagnetic confinement leads to Faraday rotation that is quantized in units of an effective $\alpha^*$ whose value is $\alpha^*/\alpha = 2.80(4)$. This enhancement is shown to be in good agreement with expectations from a Drude model that includes the kinetic inductance of the 2DEG in the waveguide.