Abstract: Interest continues to grow in photonic and phononic analogues of topological electronic phases. These systems are typically noninteracting and have the same band structure and edge-state structure as their fermionic counterparts. In this talk, I’ll discuss recent theory work in my group on a class of bosonic systems where this correspondence fails. They involve using parametric ​“two-photon” driving and have Hamiltonians that superficially resemble those of topological superconductors. Among the surprising effects that emerge are the presence of  topologically protected instabilities that can be harnessed for nonreciprocal quantum amplification and effective non-Hermitian dynamics in a bosonic analogue of the Kitaev-Majorana chain. I’ll discuss how these ideas could be realized in a variety of different experimental platforms, including superconducting quantum circuits and optomechanics.