Spin-Geometric Nonreciprocal Quantum Routing from Pauli–Lindblad Dynamics

This side note shows how a spin's orientation alone can direct microwaves one way or the other—acting as a programmable quantum router or circulator without magnets or external phases. Two crossed spin couplings create a natural quadrature, and a reciprocal link closes the loop to turn that geometry into real directionality. Flip the spin, and the signal swaps direction, giving ±33.8 dB isolation with only 2.8 dB loss forward. Make the couplings collinear, and the effect vanishes, restoring full reciprocity. All passive, no gain.