Unitary equivalence between ordinary intelligent states and generalized intelligent states

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Abstract

Ordinary intelligent states (OISs) hold equality in the Heisenberg uncertainty relation involving two noncommuting observables {A,B}, whereas generalized intelligent states (GISs) do so in the more generalized uncertainty relation, the Schrödinger-Robertson inequality. In general, OISs form a subset of GISs. However, if there exists a unitary evolution U that transforms the operators {A,B} to a new pair of operators in a rotation form, it is shown that an arbitrary GIS can be generated by applying the rotation operator U to a certain OIS. In this sense, the set of OISs is unitarily equivalent to the set of GISs. It is the case, for example, with the su(2) and the su(1,1) algebras which have been extensively studied, particularly in quantum optics. When these algebras are represented by two bosonic operators (nondegenerate case), or by a single bosonic operator (degenerate case), the rotation, or pseudorotation, operator U corresponds to phase shift, beam splitting, or parametric amplification, depending on two observables {A,B}.

Original languageEnglish
Article number053834
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Volume76
Issue number5
DOIs
Publication statusPublished - 29 Nov 2007

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equivalence
operators
algebra
quantum optics
set theory
phase shift

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Physics and Astronomy(all)

Cite this

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title = "Unitary equivalence between ordinary intelligent states and generalized intelligent states",
abstract = "Ordinary intelligent states (OISs) hold equality in the Heisenberg uncertainty relation involving two noncommuting observables {A,B}, whereas generalized intelligent states (GISs) do so in the more generalized uncertainty relation, the Schr{\"o}dinger-Robertson inequality. In general, OISs form a subset of GISs. However, if there exists a unitary evolution U that transforms the operators {A,B} to a new pair of operators in a rotation form, it is shown that an arbitrary GIS can be generated by applying the rotation operator U to a certain OIS. In this sense, the set of OISs is unitarily equivalent to the set of GISs. It is the case, for example, with the su(2) and the su(1,1) algebras which have been extensively studied, particularly in quantum optics. When these algebras are represented by two bosonic operators (nondegenerate case), or by a single bosonic operator (degenerate case), the rotation, or pseudorotation, operator U corresponds to phase shift, beam splitting, or parametric amplification, depending on two observables {A,B}.",
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