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AIP conference proceedings

Quon theories in quantum optics

A. I. Solomon

Published Jan 1, 2000• 0 citations
Publisher

Abstract

We start from the viewpoint of quantum canonical transformations, analogues of the usual classical canonical transformations. We show that in the quantum case, such transformations lead naturally to the definition of some important quantum optics states, including the vacuum, coherent states, squeezed vacuum and squeezed states. We then go on to consider quons, operators satisfying deformed versions of the canonical commutation relations. Extending the idea of a canonical transformation to deformed quantum commutators gives quon analogues of these states, quon states. We finally show that this approach provides us with defining relations for some quantum groups.

DOI
10.1063/1.1337723

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We characterize point transformations in quantum mechanics from the mathematical viewpoint. To conclude that the canonical variables given by each point transformation in quantum mechanics correctly describe the extended point transformation, we show that they are all selfadjoint operators in L2(Rn)L^2(\mathbb{R}^n)L2(Rn) and that the continuous spectrum of each coincides with R\mathbb{R}R. They are also shown to satisfy the canonical commutation relations.
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Abstract These lecture notes provide a comprehensive exploration of coherent and squeezed states, which are central to quantum optics and continuous-variable quantum information. We begin with the traditional formalism, where displacement and squeezing operators are defined via the creation and annihilation operators. Coherent states are presented as quantum analogues of classical fields with minimal uncertainty, while squeezed states are introduced as nonclassical resources that reduce quantum noise below the vacuum limit, enabling next-generation quantum technologies. A central feature of these notes is a parallel development from a group-theoretical standpoint. Here, these operators are defined abstractly by their algebraic properties, revealing their structure as bosonic realizations of the Weyl–Heisenberg and SU(1,1)SU(1,1)SU(1,1) S U ( 1 , 1 ) structures, respectively. This advanced perspective not only provides a more rigorous mathematical foundation but also leads to new insights, including a proof of the uniqueness of the single-mode squeezing operator and a no-go theorem for certain multi-mode squeezing configurations. Designed for both pedagogical clarity and advanced study, these notes bridge the foundational operator algebra of quantum optics with the modern group-theoretical structures essential for research in continuous-variable quantum systems.