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Journal of Modern Optics

Quasi coherent state of the Dirac oscillator

Piu Ghosh, Pinaki Roy

Published Jan 2, 2021• 4 citations
Publisher

Abstract

We construct quasi coherent state (QCS) of the (1+1) dimensional Dirac oscillator as eigenstates of matrix annihilation operators. It has been shown that the QCS exhibits non-classical features like quadrature squeezing. It has also been found that for a general complex coherence parameter the quadratures are correlated and the QCS satisfies Schrödinger–Robertson uncertainty relation.

DOI
10.1080/09500340.2021.1876261

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We construct quasi coherent state (QCS) of the (1+1) dimensional Dirac oscillator as eigenstates of matrix annihilation operators. It has been shown that the QCS exhibits non-classical features like quadrature squeezing. It has also been found that for a general complex coherence parameter the quadratures are correlated and the QCS satisfies Schrödinger–Robertson uncertainty relation.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
The coherent states for the Kratzer-Fues potential, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed using the algebraic approach. The method is extended to include the rotating Kratzer-Fues oscillator. Construction of such states is important in the theory of molecular interactions with coherent electromagnetic fields, for example, in calculation of the dynamic alternation of the refractive index due to the interactions of the laser pulse with the coherent vibration-rotational states of the diatomic molecule.
Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1
The main characteristics of the quantum oscillator coherent states including the two-particle Calogero interaction are investigated. We show that these Calogero coherent states are the eigenstates of the second-order differential annihilation operator which is deduced via R-deformed Heisenberg algebra or Wigner-Heisenberg algebraic technique and correspond exactly to the pure uncharged-bosonic states. They possess the important properties of non-orthogonality and completeness. The minimum uncertainty relation for the Calogero interaction coherent states is investigated. New sets of even and odd Wigner oscillator coherent states are pointed out.