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arXiv:2106.16244 · Physical Review A

Effects of quantum deformation on the Jaynes-Cummings and anti-Jaynes-Cummings models

Gustavo M. Uhdre, Danilo Cius, Fabiano M. Andrade

Published Jan 6, 2022• 10 citations

Abstract

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 ($2+1$) 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 $z$ direction is not conserved anymore, as a direct consequence of the deformation.

DOI: 10.1103/physreva.105.013703

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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+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

z

direction is not conserved anymore, as a direct consequence of the deformation.

It is shown that various variants of the deformed Jaynes-Cummings model (JCM) correspond to the JCM with an intensity-dependent coupling characterized by two additional phenomenological parameters (p,q). The standard JCM is obtained for p=q=1. The quantum collapse and revival effects and the squeezing properties of a particular variant of the (p,q)-deformed Jaynes-Cummings model are studied numerically. The model is based on the q-deformed oscillator algebra ${\mathit{AA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$-${\mathit{qA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$A=1 that interpolates between Fermi-Dirac and Bose-Einstein statistics. If the cavity field is prepared initially in a q-deformed coherent state, the quantum collapse and revival effects are observed only for q\ensuremath{\approxeq}1. For q>1, the atomic inversion 〈${\mathrm{\ensuremath{\sigma}}}_{3}$(t)〉 exhibits chaoticlike behavior, which is a feature observed also in other q-deformed JCM's. Strong squeezing is observed only for small positive and small negative q values. If q\ensuremath{\approxeq}1, the squeezing is very weak. In the limit q=1\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon} with \ensuremath{\varepsilon}\ensuremath{\ll}1, the algebra can be interpreted as describing a small violation of Bose-Einstein statistics in the JCM.

The Dirac oscillator is a relativistic quantum system, characterized by its linearity in both position and momentum. Moreover, considering

(1{+}1)

and

(2{+}1)

dimensions, the system can be mapped onto the Jaynes-Cummings and anti-Jaynes-Cummings models, as illustrated in an exact manner by Bermudez \textit{et al.} [\href{ https://doi.org/10.1103/PhysRevA.76.041801}{Phys. Rev. A 76, 041801(R) (2007)}]. Using the optical counterparts of the Dirac oscillator, we analyze an extension of the model that incorporates a time-dependent frequency. We focus on the consequences of these time modulations on the angular momentum observables and spin-orbit entanglement. Noticeable changes in the \emph{Zitterbewegung} are found. We show that a specific choice of time dependence yields aperiodic evolution of the observables, whereas an alternative choice allows analytical solutions.

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.

We study the dynamics of the

2+1

Dirac oscillator exactly and find spin oscillations due to a Zitterbewegung of purely relativistic origin. We find an exact mapping of this quantum-relativistic system onto a Jaynes-Cummings model, describing the interaction of a two-level atom with a quantized single-mode field. This equivalence allows us to map a series of quantum optical phenomena onto the relativistic oscillator and vice versa. We make a realistic experimental proposal, in reach with current technology, for studying the equivalence of both models using a single trapped ion.

The dynamics of wavepackets in a relativistic Dirac oscillator (DO) is compared with that of the Jaynes-Cummings model. The strong spin-orbit coupling of the DO produces the entanglement of the spin with the orbital motion similar to that observed in the model of quantum optics. The resulting collapses and revivals of the spin extend to a relativistic theory our previous findings on a nonrelativistic oscillator where they were known as spin-orbit pendulum . The Foldy-Wouthuysen transformation can be performed exactly for the DO. It produces the well known smoothing effect over the Compton wavelength. Thus, after this transformation, zitterbewegung disappears just as the components of the WP associated to negative energy states.