Probing time-reversal symmetry breaking at microwave frequencies

Motivated by experiments carried out in the near infrared using zero-loop-area Sagnac interferometers, we explore electromagnetic signatures of time-reversal symmetry breaking (TRSB) at microwave frequencies, using as a prototypical example a semiclassical conductor in a magnetic field. TRSB is generically accompanied by a skew-symmetric term in the electrodynamic response tensors (permittivity, conductivity, surface impedance), imparting a nonreciprocal phase shift to left- and right-circularly polarized electromagnetic waves reflected from the surface of such a material. We show that TRSB manifests as a difference in the surface reactance experienced by circularly polarized waves, and can be detected using a doubly degenerate resonator mode, such as the TE$_{111}$ mode of a cylindrical cavity. In addition to the frequency splitting induced by TRSB we show that, when interrogated by circularly polarized microwaves, the forward and reverse transmission responses of such a resonator break reciprocity, providing a crucial signature that distinguishes true Faraday effects (i.e., circular birefringence) from non-TRSB effects such as linear birefringence. In the limit that the sample is larger than the spot size (i.e., larger than the diameter of the microwave cavity) we show that the TRSB resonator has sensitivity to polar Kerr angle comparable to that of the zero-loop-area Sagnac, and should provide complementary insights into unconventional superconductors such as UPt$_3$ and Sr$_2$RuO$_4$ that have been observed to spontaneously break time-reversal symmetry.