Role of Postselection in Generating Vortex States of Particles of Arbitrary Mass, Spin, and Energy

There exist many methods to obtain twisted photons with an orbital angular momentum up to the soft X-ray region. However, despite the big interest in generating massive particles with the phase vortices, the available diffraction techniques are only applicable to the moderately relativistic beams of electron microscopes, to cold neutrons, or to non-relativistic atoms. To bring matter waves physics into the high energy domain, one needs to develop alternative generation methods, applicable for ultrarelativistic energies and for composite particles. Here, we show that the vortex states of in principle arbitrary particles, including $X$- and $\gamma$-rays, muons, hadrons, nuclei, and ions, can be generated during photon emission in helical undulators, via Cherenkov radiation, in collisions of charged particles with intense laser beams, in such scattering or annihilation processes as $e\mu \to e\mu, ep \to ep, e^-e^+ \to p\bar{p}$, and so forth. We argue that the key element in obtaining them is the postselection protocol due to entanglement between a pair of final particles and it is largely not the process itself. The quantum state of a final particle -- be it a photon, a proton, or a nucleus -- becomes twisted if the azimuthal angle of the other particle momentum is measured with a large error or is not measured at all, which is often the case for small scattering angles. As a result, the requirements to the incoming beam transverse coherence can be greatly relaxed, which enables the generation of highly energetic vortex beams at accelerators and synchrotron radiation facilities, thus making them a new tool for hadronic and spin studies. This technique can also facilitate the development of sources of hard X-ray and $\gamma$-range twisted photons for nuclear and particle physics.