Fig. 1: Topological nonreciprocal hysteretic winding.
From: Topological hysteretic winding for temporal anti-lasing
a Schematic of experimental setup. A ferrite-based nonreciprocal circulator is periodically modulated by an external magnetic field using a low-frequency function generator (sine wave with period Tm, modulation amplitude Vm, and DC bias V0). The magnetization hysteresis in the circulator ferrite translates to a hysteresis winding loop of the microwave scattering parameters, that can be measured with a vector network analyser. The third port of the circulator is matched. b Illustration of topological hysteretic winding. The scattering matrix topology is characterized by its determinant \(\det (S)\), whose evolution is plotted with phase (color) and magnitude (height) in the 2D parameter space of frequency detuning Δf0 and Zeeman splitting ΔZeeman. The singularity \(\det (S)=0\) corresponds to nonreciprocal coherent perfect absorption (CPA, marked by a star). Depending on the modulation speed and amplitude, one obtains either a non-hysteretic path, or hysteretic loops. Such loops can wind about the CPA point or not, with the winding number being defined from the total phase accumulation of \(\det (S)\) over a modulation period. c Experimentally measured trivial (V0 = 0 V) and topological (V0 = 3.5 V) hysteretic winding for microwave photon scattering at 5 GHz with Tm = 10 ms and Vm = 6 V. The complex value of \(\det (S)\) is mapped onto polar coordinates. d Experimental data showing the accumulation of the topological winding charge Q over 2 s at different modulation periods of Tm = 10, 2, 1 ms.
