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Nuclear Physics B - Proceedings Supplements

Multimode two-photon coherent states and unitary representation of the symplectic group

Horace P. Yuen

Published Mar 1, 1989• 12 citations
Publisher
DOI
10.1016/0920-5632(89)90462-3

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Ingrid C. Daubechies, Theoretische Natuurkunde

cause the substitution has taken us out of it) is, up to some constant factor, a unitary operator. The family of these oper­ ators gives us our projective representation. We work with intrinsic and coordinate-free notations differing from the no­ tations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the de­ quantization of these operators, we proceed then to compute the classical functions corresponding to the symplectic transformations. This calculation of classical functions for symplectic transformations has been done for one-param­ eter subgroups of the symplectic group.8.9 One then only

•Jan 1, 2012•0 citations
It is shown that the SU(1,1)-like and SU(2)-like two-photon coherent states can be combined to form a O(3,2)-like two-photon states. Since the O(3,2) group has many subgroups, there are also many new interesting new coherent and squeezed two-photon states. Among them is the two-photon sheared state whose symmetry property is like that for the two-dimensional Euclidean group. There are now two-phonon coherent states which may exhibit symmetries not yet observed for photons, including sheared states. Let us note that both SU(1,1) and S(3,2) are isomorphic to the symplectic groups Sp(2) and Sp(4) respectively, and that symplectic transformations consist of rotations and squeeze transformations.
We present an explicit construction of coherent states for an arbitrary irreducible representation of the unitary symplectic group USp (4). Three different families of coherent states are obtained, corresponding to the subgroups U (1) × U (1), U (2) and SU (2) × SU (2). The symplectic structure on the manifold of coherent states is obtained, and canonical coordinates are used to express the classical limit of quantum observables. One of the families is seen to provide a trivial classical limit.
cause the substitution has taken us out of it) is, up to some constant factor, a unitary operator. The family of these oper­ ators gives us our projective representation. We work with intrinsic and coordinate-free notations differing from the no­ tations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the de­ quantization of these operators, we proceed then to compute the classical functions corresponding to the symplectic transformations. This calculation of classical functions for symplectic transformations has been done for one-param­ eter subgroups of the symplectic group.8.9 One then only