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Physical Review A

Symplectic transformations in then-mode coherent-state representation using integration within an ordered product of operators

Fan Hong‐Yi, Xu Zhihua

Published Oct 1, 1994• 33 citations
Publisher

Abstract

We derive unitary-operator images corresponding to classical phase-space symplectic transformations in the n-mode canonical coherent-state representation. The evaluation can be greatly facilitated by the technique of integration within an ordered product of operators, showing clearly the connection between symplectic transformations and the corresponding normally ordered unitary operators. Applications of these operators results in the construction of generalized coherent states.

DOI
10.1103/physreva.50.2921

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We derive unitary-operator images corresponding to classical phase-space symplectic transformations in the n-mode canonical coherent-state representation. The evaluation can be greatly facilitated by the technique of integration within an ordered product of operators, showing clearly the connection between symplectic transformations and the corresponding normally ordered unitary operators. Applications of these operators results in the construction of generalized coherent states.
Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product" technique, and the mapping is manifestly apparent in the derivation. New generalized coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.
The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.
The normally ordered Hilbert space image of the general (complex) linear similarity transformation in phase space is obtained in a coherent state representation. Although preserving the commutator of a and a + , these quantum mechanical images of classical transformations are generally not unitary. Remarkably, although the kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, squeezed state analogues produced using the similarity transformation still satisfy an overcompleteness relation. The results are extended to two-mode Fock space and simple examples of the utility of the similarity transformation are presented. The evaluation of the normally ordered operators is greatly facilitated by the use of integration within ordered products.