Parabose Oscillator Research Articles

Para-Bose oscillator algebras of odd orders: x-representations and Wigner functions for coherent and cat states and their photon-added and photon-subtracted counterparts

Para-Bose oscillator algebras of odd orders: x-representations and Wigner functions for coherent and cat states and their photon-added and photon-subtracted counterparts

Generalized para-Bose states

In this paper, we construct integrals of motion in a para-Bose formulation for a general time-dependent quadratic Hamiltonian, which, in its turn, commutes with the reflection operator. In this context, we obtain generalizations for the squeezed vacuum states (SVS) and coherent states (CS) in terms of the Wigner parameter. Furthermore, we show that there is a completeness relation for the generalized SVS owing to the Wigner parameter. In the study of the probability transition, we found that the displacement parameter acts as a transition parameter by allowing access to odd states, while the Wigner parameter controls the dispersion of the distribution. We show that the Wigner parameter is quantized by imposing that the vacuum state has even parity. We apply the general results to the case of the time-independent para-Bose oscillator and find that the mean values of the coordinate and momentum have an oscillatory behavior similarly to the simple harmonic oscillator, while the standard deviation presents corrections in terms of the squeeze, displacement, and Wigner parameters.

Quantum algebra from generalized q-Hermite polynomials

In this paper, we discuss new results related to the generalized discrete q-Hermite II polynomials h˜n,α(x;q), introduced by Mezlini et al. in 2014. Our aim is to give a continuous orthogonality relation, a q-integral representation and an evaluation at unity of the Poisson kernel, for these polynomials. Furthermore, we introduce q-Schrödinger operators and we give the raising and lowering operator algebra corresponding to these polynomials. Our results generate a new explicit realization of the quantum algebra suq(1,1), using the generators associated with a q-deformed generalized para-Bose oscillator.

Nonlinear coherent states of the para-Bose oscillator and their non-classical features

The construction of nonlinear coherent states via a unitary displacement operator is possible only for a few quantum-mechanical systems. In this paper, we define two non-unitary and a unitary displacement operators with the help of corresponding f -deformed bosonic annihilation and creation operators. While the action of the non-unitary displacement type operators on the vacuum state of field results in two new families of nonlinear coherent states (NLCSs), the unitary displacement operator reproduces Wigner-Heisenberg coherent states of the Gilmore-Perelomov type. We prove that the introduced NLCSs satisfy the resolution of the identities through positive definite measures. We also examine the non-classical properties of the obtained NLCSs by evaluating Klyshko’s criterion, Mandel’s parameter, quadrature squeezing and Wigner quasi-probability distribution function, in detail. Finally, we propose a simple scheme for the physical generation of the introduced NLCSs of the Gilmore-Perelomov type.

Simulating para-Fermi oscillators

Quantum mechanics allows for a consistent formulation of particles that are neither bosons nor fermions. These para-particles are rather indiscernible in nature. Recently, we showed that strong coupling between a qubit and two field modes is required to simulate even order para-Bose oscillators. Here, we show that finite-dimensional representations of even order para-Fermi oscillators are feasible of quantum simulation under weak coupling. This opens the door to their potential implementation in different contemporaneous quantum electrodynamics platforms. We emphasize the intrinsic value of para-particles for the quantum state engineering of bichromatic field modes. In particular, we demonstrate that binomial two field mode states result from the evolution of para-Fermi vacuum states in the quantum simulation of these oscillators.

Nonclassical and semiclassical para-Bose states

Motivated by the proposal to simulate para-Bose oscillators in a trapped-ion setup [C. Huerta Alderete and B. M. Rodr\'iguez-Lara, Phys. Rev. A 95, 013820 (2017)], we introduce an overcomplete, nonorthogonal basis for para-Bose Hilbert spaces. The states spanning these bases can be experimentally realized in the trapped-ion simulation via time evolution. The para-Bose states show both nonclassical and semiclassical statistics on their Fock state distribution, asymmetric field quadrature variances, and do not minimize the uncertainty relation for the field quadratures. These properties are analytically controlled by the para-Bose order and the evolution time; both parameters might be feasible for fine tuning in the trapped-ion quantum simulation.

Quantum simulation of driven para-Bose oscillators

Quantum mechanics allows paraparticles with mixed Bose-Fermi statistics that have not been experimentally confirmed. We propose a trapped-ion scheme whose effective dynamics are equivalent to a driven para-Bose oscillator of even order. Our mapping suggest highly entangled vibrational and internal ion states as the laboratory equivalent of quantum simulated parabosons. Furthermore, we show the generation and reconstruction of coherent oscillations and para-Bose analogs of Gilmore-Perelomov coherent states from population inversion measurements in the laboratory frame. Our proposal, apart from demonstrating an analog quantum simulator of para-Bose oscillators, provides a quantum state engineering tool that foreshadows the potential use of paraparticle dynamics in the design of quantum information systems.

Meixner Polynomials and ID Para-Bose Oscillator

In this paper, the matrix elements of the para-squeeze operator in the eigenstate basis of the para-Bose oscillator are computed explicitly. These matrix elements are seen to involve the Meixner polynomials. The underlying framework allows for the systematic derivation of the recurrence relations, difference equation, lowering and rising operators and generating functions which these polynomials satisfy.

Bose Description of the Deformed Parabose Oscillator with Z p -Graded Reflection Symmetry

In this paper Bose description of the deformed parabose oscillator with Z p -graded reflection symmetry is discussed. New lowering and raising operators are obtained from p Bose operators and their adjoint ones. These new step operators are shown to be satisfy the Heisenberg relation of Wigner algebra type.

The algebra of dual −1 Hahn polynomials and the Clebsch-Gordan problem of sl−1(2)

The algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}$\end{document}H of the dual −1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl−1(2). The dual −1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q → −1 limit of the dual q-Hahn polynomials. The Hopf algebra sl−1(2) has four generators including an involution, it is also a q → −1 limit of the quantum algebra slq(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}$\end{document}H, a two-parameter generalization of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {u}(2)$\end{document}u(2) with an involution as additional generator, is first derived from the recurrence relation of the −1 Hahn polynomials. It is then shown that \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}$\end{document}H can be realized in terms of the generators of two added sl−1(2) algebras, so that the Clebsch-Gordan coefficients of sl−1(2) are dual −1 Hahn polynomials. An irreducible representation of \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}$\end{document}H involving five-diagonal matrices and connected to the difference equation of the dual −1 Hahn polynomials is constructed.

Finite oscillator models: the Hahn oscillator

A new model for the finite one-dimensional harmonic oscillator is proposed based upon the algebra . This algebra is a deformation of the Lie algebra extended by a parity operator, with the deformation parameter α. A class of irreducible unitary representations of is constructed. In the finite oscillator model, the (discrete) spectrum of the position operator is determined, and the position wavefunctions are shown to be dual Hahn polynomials. Plots of these discrete wavefunctions display interesting properties, similar to those of the parabose oscillator. We show indeed that in the limit, when the dimension of the representations goes to infinity, the discrete wavefunctions tend to the continuous wavefunctions of the parabose oscillator.

A Bose description of the 1-D para-Bose and para-Fermi oscillators

In the Fock space of two Bose operators all the irreducible representations of both the 1-D para-Bose and para-Fermi oscillators are constructed. Bose states of the form |p + k–1, kn > (|p – k, kn), n = 1,2, are shown to stand for states of k-parabosons (k-parafermions) of order p. For n = 1 or n = 2 the various subspaces may be visualized in the plane as either straight-lines or parabolae, respectively.

The Wigner distribution function for the one-dimensional parabose oscillator

In the beginning of the 1950s, Wigner introduced a fundamental deformation from the canonical quantum mechanical harmonic oscillator, which is nowadays sometimes called a Wigner quantum oscillator or a parabose oscillator. Also, in quantum mechanics the so-called Wigner distribution is considered to be the closest quantum analogue of the classical probability distribution over the phase space. In this paper, we consider which definition for such a distribution function could be used in the case of non-canonical quantum mechanics. We then explicitly compute two different expressions for this distribution function for the case of the parabose oscillator. Both expressions turn out to be multiple sums involving (generalized) Laguerre polynomials. Plots then show that the Wigner distribution function for the ground state of the parabose oscillator is similar in behaviour to the Wigner distribution function of the first excited state of the canonical quantum oscillator.

New Deformation of Para-Bose Statistics

We propose commutation relations for a single mode ĝ-deformed para-Bose oscillator. In this new deformation of para-Bose statistics the distribution function has the same form as in the para-Bose statistics. Furthermore, we show analogies between the coherent states of ĝ-deformed and q-deformed para-Bose statistics.

Parabose Coherent States Based on Green's Ansatz**The project supported by the Natural Science Foundation of Jiangxi Province of China

The coherent states of parabose oscillator of order p based on the well-known Green's ansatz have been constructed. Furthermore, it is shown that the completeness relation for these coherent states may be expressed in the form of matrix.

SQUEEZED STATES FOR PARABOSONS

Squeezed states for parabosons are constructed since we know that a parabose oscillator can be parametrized in terms of bosonic operators. We also discuss some of their important properties.

The q-deformed oscillator representations and their coherent states of the su(1, 1) algebra

We present various oscillator representations of the q-deformed su(1, 1) algebra such as the Holstein-Primakoff, the Dyson, the Fock-Bargmann, the anyonic, and the parabose oscillator representations and discuss their coherent states with the resolution of unity.

Coherent states for parabosons

Taking into consideration that the Fock space of a parabose oscillator may be described by bilinear commutation relations involving creation and annihilation operators, we construct the coherent states for odd and even order parabosons.

Generalized deformed para-Bose oscillator and its coherent states

Generalized deformed commutation relations for a single-mode para-Bose oscillator are constructed. The connection of generalized deformed para-Bose oscillators with para-Bose oscillators is determined. From these, the energy spectrum of generalized deformed para-Bose oscillators and the coherent states of the annihilation operators are discussed.

GENERALIZED DEFORMED PARA-BOSE OSCILLATOR AND NONLINEAR ALGEBRAS

Generalized deformed commutation relations for a single mode para-Bose oscillator and for a system of two para-Bose oscillators are constructed. It turns out that generalized deformed para-Bose oscillators are not, in general, exactly independent. Furthermore, we also discuss about the Fock space corresponding to generalized deformed para-Bose oscillators. Finally, we show how SU(2) and SU(1, 1) generators can be constructed in terms of generalized deformed para-Bose creation and annihilation operators. The algebras SU(2) and SU(1, 1) of generalized deformed oscillators14,18 are the special cases of generalized deformed para-Bose oscillators algebras but, interestingly, they have the same form.