Barut - Girardello coherent states for u ( p , q ) and sp ( N , R ) and their macroscopic superpositions

The Barut-Girardello coherent states (BG CS) representation is extended to the noncompact algebras u(p, q) and sp(N,R) in (reducible) quadratic boson realizations. The sp(N,R) BG CS take the form of multimode ordinary Schrödinger cat states. Macroscopic superpositions of 2n−1 sp(N,R) CS (2n canonical CS (CCS), n = 1, 2, . . .) are pointed out which are overcomplete in the N mode Hilbert space and the relation between the CCS and the u(p, q) BG type CS representations is established. The sets of u(p, q) and sp(N,R) BG CS and their discrete superpositions contain many states studied in quantum optics (even and odd N mode CS, pair CS) and provide an approach to quadrature squeezing, alternative to that of intelligent states. New subsets of weakly and strongly nonclassical states are pointed out and their statistical properties (first and second order squeezing, photon number distributions) are discussed. For specific values of the angle parameters and small amplitude of the CCS components these states approaches multimode Fock states with 1, 2 or 3 bosons/photons. It is shown that eigenstates of a non-Hermitian operator A2 (generalized cat states) can exhibit squeezing of the quadratures of A.