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Modern Physics Letters A

On the κ-deformed Dirac oscillator in (2 + 1)-dimensions

YASSINE CHARGUI

Published Mar 21, 2019• 9 citations
Publisher

Abstract

We show that the (2 + 1)-dimensional [Formula: see text]-Dirac oscillator is equivalent, to the first-order of the deformation parameter, to a couple of simultaneous Jaynes–Cummings (JC) and Anti-Jaynes–Cummings (AJC) models with opposite chiralities, where the former is entirely due to the deformation. The energy eigenvalues and the corresponding eigenfunctions of system are obtained in a simple manner from the solutions of the non-deformed Dirac oscillator by mapping the complete system to a single AJC model through a similarity transformation. The pseudo-supersymmetry and other interesting features of the system are also discussed.

DOI
10.1142/s0217732319500895

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We show that the (2 + 1)-dimensional [Formula: see text]-Dirac oscillator is equivalent, to the first-order of the deformation parameter, to a couple of simultaneous Jaynes–Cummings (JC) and Anti-Jaynes–Cummings (AJC) models with opposite chiralities, where the former is entirely due to the deformation. The energy eigenvalues and the corresponding eigenfunctions of system are obtained in a simple manner from the solutions of the non-deformed Dirac oscillator by mapping the complete system to a single AJC model through a similarity transformation. The pseudo-supersymmetry and other interesting features of the system are also discussed.
The theory of non-Hermitian systems and the theory of quantum deformations have attracted a great deal of attention in the past decades. In general, non-Hermitian Hamiltonians are constructed by an ad hoc manner. Here, we study the (2+12+12+1) Dirac oscillator and show that in the context of the $\ensuremath{\kappa}$-deformed Poincar\'e-Hopf algebra its Hamiltonian is non-Hermitian but has real eigenvalues. The non-Hermiticity stems from the $\ensuremath{\kappa}$-deformed algebra. From the mapping in Bermudez et al., Phys. Rev. A 76, 041801(R) (2007), we propose the $\ensuremath{\kappa}$-Jaynes-Cummings and $\ensuremath{\kappa}$-anti-Jaynes-Cummings models, which describe an interaction between a two-level system with a quantized mode of an optical cavity in the $\ensuremath{\kappa}$-deformed context. We find that the $\ensuremath{\kappa}$ deformation modifies the Zitterbewegung frequencies and the collapses and revivals of quantum oscillations. In particular, the total angular momentum in the zzz direction is not conserved anymore, as a direct consequence of the deformation.
The charged Dirac oscillator on a noncommutative plane coupling to a uniform perpendicular magnetic held is studied in this paper. We map the noncommutative plane to a commutative one by means of Bopp shift and study this problem on the commutative plane. We find that this model can be mapped onto a quantum optics model which contains Anti—Jaynes—Cummings (AJC) or Jaynes—Cummings (JC) interactions when a dimensionless parameter ζ (which is the function of the intensity of the magnetic held) takes values in different regimes. Furthermore, this model behaves as experiencing a chirality quantum phase transition when the dimensionless parameter ζ approaches the critical point. Several evidences of the chirality quantum phase transition are presented. We also study the non-relativistic limit of this model and find that a similar chirality quantum phase transition takes place in its non-relativistic limit.
The charged Dirac oscillator on a noncommutative plane coupling to a uniform perpendicular magnetic field is studied in this paper. We map the noncommutative plane to a commutative one by means of Bopp shift and study this problem on the commutative plane. We find that this model can be mapped onto a quantum optics model which contains Anti–Jaynes–Cummings(AJC) or Jaynes–Cummings(JC) interactions when a dimensionless parameterζ(which is the function of the intensity of the magnetic field) takes values in different regimes. Furthermore, this model behaves as experiencing a chirality quantum phase transition when the dimensionless parameter ζ approaches the critical point. Several evidences of the chirality quantum phase transition are presented. We also study the non-relativistic limit of this model and find that a similar chirality quantum phase transition takes place in its non-relativistic limit.
We study generalized Dirac oscillators with complex interactions in (1+1) dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are  pseudo Hermitian with respect to certain metric operators  . Exact solutions of the generalized Dirac Oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with Anti Jaynes Cummings type model and by spin flip it can also be identified with Jaynes Cummings type model.
A map from the charged Dirac oscillator on a noncommutative plane interacting with a homogeneous magnetic field onto a quantum optics model is proposed. We find that both anti-Jaynes-Cummings (AJC) and Jaynes-Cummings (JC) interactions are included in this mapped quantum optics model. A competition between AJC and JC interactions will happen when the intensity of the magnetic field changes. Contrary to previous studies, the competition is symmetrical since the phonons that take part in the competition are only associated with the excitation of the Dirac oscillator. Furthermore, this model will experience a quantum phase transition when the intensity of the magnetic field crosses the critical point. However, the critical point is different from the one in the model of the charged nonrelativistic harmonic oscillator interacting with a homogenous magnetic field on a noncommutative plane. It also deviates from its commutative counterpart because of spatial noncommutativity. Some characteristics of the quantum phase transition are studied. Then we investigate the nonrelativistic limit of this model. Our results show that its nonrelativistic counterpart inherits similar properties of the chirality quantum phase transition from the relativistic one.