Experimental test of an extension of the Rosenzweig–Porter model to mixed integrable-chaotic systems experiencing time-reversal invariance violation

We report on the theoretical and experimental investigations of the transition of a typical quantum system with mixed regular-integrable classical dynamics to one with violated time-reversal ($\mathcal{T}$) invariance. The measurements are performed with a flat superconducting microwave resonator with circular shape in which chaoticity is induced by using either long antennas or inserting two circular disks into the cavity, and by magnetizing a ferrite disk placed at its center, which leads to violation of $\mathcal{T}$ invariance. We propose an extension of the Rosenzweig–Porter (RP) model, which interpolates between mixed regular-chaotic instead of integrable dynamics and fully chaotic dynamics with violated $\mathcal{T}$-invariance, and derive a Wigner-surmise like analytical expression for the corresponding nearest-neighbor spacing distribution. We propose a procedure involving this result and those for the RP model to determine the size of $\mathcal{T}$-invariance violation and chaoticity and validate it employing the experimental eigenfrequency spectra.