Realization of pure gyration in an on-chip superconducting microwave device

Synthetic materials that emulate tight-binding Hamiltonians have enabled a wide range of advances in topological and non-Hermitian physics. A crucial requirement in such systems is the engineering of non-reciprocal couplings and synthetic magnetic fields. More broadly, the development of these capabilities in a manner compatible with quantum-coherent degrees of freedom remains an outstanding challenge, particularly for superconducting circuits, which are highly sensitive to magnetic fields. Here we demonstrate that pure gyration -- a non-reciprocal coupling with exactly matched magnitude but non-reciprocal $π$ phase contrast -- can be realized between degenerate states using only spatio-temporal modulation. Our experiments are performed using microwave superconducting resonators that are modulated using dc-SQUID arrays. We first show the existence of continuous exceptional surfaces in modulation parameter space where coupling with arbitrarily-large magnitude contrast can be achieved, with robust volumes of $π$ phase contrast contained within. We then demonstrate that intersection of these volumes necessarily gives rise to new continuous surfaces in parameter space where pure gyration is achieved. With this we experimentally demonstrate $>58$ dB isolation and the first on-chip gyrator with only superconducting circuit elements. Our method is fully agnostic to physical implementation (classical or quantum) or frequency range and paves the way to large-scale non-reciprocal metamaterials.