Coherent States for Kronecker Products of Non Compact Groups: Formulation and Applications

We introduce and study the properties of a class of coherent states for the group SU(1,1) X SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We restrict ourselves to the discrete series representations of SU(1,1). These are the generalization of the ‘Barut Girardello’ coherent states to the Kronecker Product of two non-compact groups.The resolution of the identity and the analytic phase space representation of these states is presented. This phase space representation is based on the basis of products of ‘pair coherent states’ rather than the standard number state canonical basis. We discuss the utility of the resulting ‘bi-pair coherent states’ in the context of four-mode interactions in quantum optics. 1 FORMULATION 1.1 Coupling of Pair coherent states in the fock state basis For two mode systems the traditional SU(1,1) coherent states which have been extensively studied in the context of squeezing have been the Caves-Schumaker states [1], defined by the relation |ζ >= exp(ζab − ζab)|0, 0 >, (1) In addition to these states many authours [2] [3] have considered the SU(1,1) coherent states of two mode systems or the ‘pair coherent states’ which were simultaneous eigenstates of ab and aa− bb ab|ζ, q > = ζ |ζ, q >, Q1|ζ, q > = q1|ζ, q > . (2) These can be mapped onto the SU(1,1) group by means of the two Boson realisation: K 1 = a b , K 1 = ab , K z 1 = 1 2 (aa+ bb+ 1) , (3) which form an SU(1,1) algebra with the commutation relations