Quantized Microwave Faraday Rotation

The interaction of light with matter in the presence of a static magnetic field gives rise to a wide variety of phenomena, including the Faraday effect whereby the polarization of light is rotated by magnetically induced circular birefringence. We report quantitative microwave Faraday rotation measurements conducted with a high-mobility two-dimensional electron gas in a GaAs/AlGaAs semiconductor heterostructure. Cyclotron motion of charge carriers in a two-dimensional electron gas leads to a large Faraday effect in the microwave frequency range. The magnitude of Faraday rotation is suppressed at higher frequencies by the kinetic inductance that arises from electron inertia. As with the Hall effect, a continuous classical Faraday effect is observed as well as a quantized Faraday effect. The high electron mobility 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. In the quantum regime, the Faraday rotation $\theta_F$ is naturally quantized in units of the fine structure constant $\alpha\simeq \frac{1}{137}$, giving a geometric prescription for determining $\alpha$. Electromagnetic confinement leads to Faraday rotation that is quantized in units of an effective $\alpha^*$ whose value is on the order of its free space value. Engineering of the electromagnetic field distribution and impedance, including resonant cavity effects, could hence be used to purposefully enhance or suppress Faraday rotation.