Microwave resonator for measuring time-reversal symmetry breaking at cryogenic temperatures

TL;DR

This work presents a microwave method to measure time-reversal symmetry breaking via the polar Kerr effect in unconventional superconductors, using a deformable TE111 cavity whose two degenerate modes act as polarization states. By interrogating circularly polarized microwaves, TRSB manifests as a nonreciprocal difference between forward and reverse transmission, quantified through the TRSB parameter ω_tau and translated to Kerr angles via a cavity-perturbation framework. The system achieves cryogenic operation, a base temperature of 20 mK, and high sensitivity (δθ_K ≈ 810 nrad) by leveraging in-situ quadrupolar tuning and careful microwave design that suppresses spurious perturbations. Demonstrations with ferrite and CrGeTe3 validate the method, showing clear nonreciprocity and TRSB signatures under controlled magnetic fields and temperatures, offering a microwave alternative to optical Sagnac interferometry for probing TRSB in superconductors.

Abstract

We present a microwave-frequency method for measuring polar Kerr effect and spontaneous time-reversal symmetry breaking (TRSB) in unconventional superconductors. While this experiment is motivated by work performed in the near infrared using zero-loop-area Sagnac interferometers, the microwave implementation is quite different, and is based on the doubly degenerate modes of a TE$_{111}$ cavity resonator, which act as polarization states analogous to those of light. The resonator system has $in$-$situ$ actuators that allow quadrupolar distortions of the resonator shape to be controllably tuned, as these compete with the much smaller perturbations that arise from TRSB. The most reliable way to the detect the TRSB signal is by interrogating the two-mode resonator system with circularly polarized microwaves, in which case the presence of TRSB shows up unambiguously as a difference between the forward and reverse transmission response of the resonator - i.e., as a breaking of reciprocity. We illustrate and characterize a coupler system that generates and detects circularly polarized microwaves, and then show how these are integrated with the TE$_{111}$ resonator, resulting in a dilution refrigerator implementation with a base temperature of 20 mK. We show test data on yttrium-iron-garnet (YIG) ferrite and the van der Waals ferromagnet CrGeTe$_3$ as an illustration of how the system operates, then present data showing system performance under realistic conditions at millikelvin temperatures.

Figures (13)

• Figure 1: The vertical (left column) and horizontal (right column) polarizations of the doubly degenerate TE$_{111}$ mode of a cylindrical cavity. The electric field is purely transverse, and has maximum magnitude in the mid plane of the resonator, as shown in the top row of plots. The magnetic field forms loops that extend in the $z$ direction, but is purely transverse at the end walls, as shown in the bottom row. The sample, indicated by the square, is located in the center of an end wall, at a maximum of the transverse magnetic field. When the modes are superimposed to form circular polarizations, the field at the sample rotates continuously, clockwise or counterclockwise.
• Figure 2: Schematic representation of the four independent types of perturbation to the degenerate TE$_{111}$ modes of a cylindrical cavity resonator, along with the corresponding matrix perturbation in the $z$ basis. (a) common-mode perturbations, which affect both TE$_{111}$ polarizations equally; (b) $xy$-type quadrupolar distortions; (c) time-reversal-symmetry-breaking perturbations, which have opposite effect on left- and right-circular polarizations; and (d) ($x^2 - y^2$)-type quadrupolar distortions.
• Figure 3: Perturbations to (a) an undistorted cylindrical resonator are controllably introduced via a set of eight drive rods and contact pads to generate (b) an $x^2 - y^2$ distortion and (c) an $xy$ distortion of the resonator walls. (d) A photograph of the BeCu resonator walls showing a controlled and reversible pattern of distortion.
• Figure 4: Schematic of the TRSB resonator system. From left to right, microwaves are: launched from an input coaxial cable; linearly polarized; converted to circularly polarized microwaves by a $\tfrac{1}{4}$-wave tube; coupled into and out of the TE$_{111}$ modes of a cavity resonator; passed through a second $\tfrac{1}{4}$-wave tube; and then a second linear polarizer, with that linear polarization coupled into the output coaxial cable. The combined effect of the pair of linear polarizers and $\tfrac{1}{4}$-wave tubes is to interrogate the resonator with right-circularly polarized microwaves in the forward direction, and left-circularly polarized microwaves in the reverse direction. Any perturbation to the resonator that breaks time-reversal symmetry is revealed as a breaking of reciprocity.
• Figure 5: Cutaway schematic (left) and finite element simulation (right) of a pair of quarter-wave tubes separated by a section of cylindrical waveguide, illustrating the conversion from linear to circular to 90-degree-rotated linear polarizations. Arrows denote a snapshot of the local electric field polarization along the guide.