Recently, Markham and Vedral [Phys. Rev. A 67, 042113 (2003)] investigated the effect of beam splitting on the spin, or SU(2), coherent states for a single mode field. The spin coherent state is a binomial coherent state related to the Holstein-Primakoff realization of the su(2) Lie algebra given in terms of a set of single mode bose annihilation and creation operators. Upon beam splitting, the ordinary (or Glauber) coherent states merely split into products of ordinary coherent states with reduced amplitudes without becoming entangled, as one would expect for a classical-like field. The above authors expected the spin coherent states to go over to the ordinary coherent states in the limit of high spin, $j\ensuremath{\rightarrow}\ensuremath{\infty}$, and thus to become product states after beam splitting. But this expectation was not confirmed through numerical calculation of the entropy which, instead of going to zero, leveled off with increasing spin. In this paper we find similar behavior for SU(1,1) coherent states of the Perelomov type for large Bargman index
k, but also find that the Barut-Girardello SU(1,1) coherent states appear to rapidly become product states after beam splitting for increasing
k. We explain these results by showing that, in reality, neither the spin coherent states nor the Perelomov SU(1,1) coherent states go over to ordinary coherent states in the limits of large
j or
k, and that the Barut-Girardello coherent states merely go over to the vacuum in the large
k limit. Finally, we examine the correct limiting procedure for obtaining separable states (i.e., products of coherent states) upon beam splitting by performing contractions of the su(2) and su(1,1) Lie algebras and of their associated coherent states.
Resonant inelastic X-ray scattering (RIXS) is conventionally used to probe elementary excitations and correlations in quantum materials. Here, we propose an interferometric RIXS protocol that instead creates entanglement between spatially separated regions of a solid-state material. An incident X-ray photon is coherently split by a beam splitter into two arms illuminating opposite ends of a sample. Recombining the scattered photons at a second beam splitter and detecting an energy-selected photon then projects the material into a coherent superposition of excitations at the two ends. This protocol is the X-ray, solid-state analogue of the Duan-Lukin-Cirac-Zoller (DLCZ) heralding scheme. Using a Kramers-Heisenberg formalism with quantized photon modes, we show that orthogonal and degenerate end excitations yield maximally entangled states. We illustrate this mechanism in three models: (1) two decoupled sites yield a heralded Bell state. (2) An open tight-binding chain exhibits reflection-parity-dependent interference, providing a benchmark for the interferometer's coherence. (3) A spin-1 Haldane (AKLT) chain yields an entangled pair of fractionalized edge spins. We also discuss practical considerations and challenges for realizing the protocol in X-ray experiments.