Inverse stability problem in beam dynamics

Beam stability is conventionally described by means of the stability diagram, which is a threshold line in a complex plane of a particular intensity parameter, the coherent tune shift; the diagram separates stable and unstable states. The incoherent tune spread of the beam particles normally causes the transverse stability diagram to be an asymmetric bell-shaped curve above the real axis; for a given optical nonlinearity, the diagram is determined by the beam distribution function only. Recently, such a diagram was measured by means of an antidamper, see Antipov et al., Phys. Rev. Lett. 126, 164801 (2021). Such measurements open the door to a new method of beam diagnostics, finding the beam phase space density as a solution of an inverse stability problem. The main idea, methods of its implementation, possible obstacles and optimizations are discussed.