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Journal of Physics Conference Series

Pisot-Fibonacciq-coherent states

Jean‐Pierre Gazeau, M. A. del Olmo

Published Mar 1, 2011• 0 citations
Publisher

Abstract

A family of q -coherent states is constructed allowing us to obtain a new quantized version of the harmonic oscillator. These q -states are normalized and form an overcomplete set resolving the unity with respect to the appropriate Jackson measure if 0 < q < 1. We only consider here those values of q such that q −1 is a Pisot number. In this case the q -deformed integers ([ n ] q ) form Fibonacci-like sequences of integers. We study the main physical characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, probability distributions and related statistical features and semi-classical phase space trajectories whose periodicity is related with the fact that q is an algebraic number.

DOI
10.1088/1742-6596/284/1/012027

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A family of q -coherent states is constructed allowing us to obtain a new quantized version of the harmonic oscillator. These q -states are normalized and form an overcomplete set resolving the unity with respect to the appropriate Jackson measure if 0 < q < 1. We only consider here those values of q such that q −1 is a Pisot number. In this case the q -deformed integers ([ n ] q ) form Fibonacci-like sequences of integers. We study the main physical characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, probability distributions and related statistical features and semi-classical phase space trajectories whose periodicity is related with the fact that q is an algebraic number.
We construct a new family of q -deformed coherent states | z ⟩ q , where 0 < q < 1. These states are normalizable on the whole complex plane and continuous in their label z . They allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated Stieltjes power-moment problem and expressed in terms of one of the two Jacksons's q -exponentials. They also permit exact evaluation of matrix elements of physically-relevant operators. We use this to show that the photon number statistics for the states is sub-Poissonian and that they exhibit quadrature squeezing as well as an enhanced signal-to-quantum noise ratio over the conventional coherent state value. Finally, we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics.
It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock–Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock–Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.
Maths-type q-deformed coherent states with q>1q > 1q>1 allow a resolution of unity in the form of an ordinary integral. They are sub-Poissonian and squeezed. They may be associated with a harmonic oscillator with minimal uncertainties in both position and momentum and are intelligent coherent states for the corresponding deformed Heisenberg algebra.
The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.