Beam splitting and entanglement: Generalized coherent states, group contraction, and the classical limit

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.