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arXiv:0704.2315 · Physical Review A

Exact mapping of the 2+1 Dirac oscillator onto the Jaynes-Cummings model: Ion-trap experimental proposal

A. Bermúdez, Miguel Angel Martin-Delgado, E. Solano

Published Oct 2, 2007• 232 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.76.041801

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The Dirac oscillator is a relativistic quantum system, characterized by its linearity in both position and momentum. Moreover, considering (1+1)(1{+}1)(1+1) and (2+1)(2{+}1)(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 study the relations between massless Dirac fermions in an electromagnetic field and atoms in quantum optics. After getting the solutions of the energy spectrum, we show that it is possible to reproduce the 2D Dirac Hamiltonian, with all its quantum relativistic effects, in a controllable system as a single trapped ion through the Jaynes--Cummings and anti-Jaynes--Cummings models. Also we show that under certain conditions the evolution of the Dirac Hamiltonian provides us with Rashba spin-orbit and linear Dresselhaus couplings. Considering the multimode multiphoton Jaynes-Cummings model interacting with N modes of electromagnetic field prepared in general pure quantum states, we analyze the Rabi oscillation. Evaluating time evolution of the Dirac position operator, we determine the Zitterbewegung frequency and the corresponding oscillating term as function of the electromagnetic field.
We study the relations between massless Dirac fermions in an electromagnetic field and atoms in quantum optics. After getting the solutions for the energy spectrum, we show that it is possible to reproduce the 2D Dirac Hamiltonian, with all its quantum relativistic effects, in a controllable system such as a single trapped ion, through the Jaynes–Cummings and anti-Jaynes–Cummings models. Also we show that under certain conditions, the evolution of the Dirac Hamiltonian provides us with Rashba spin–orbit and linear Dresselhaus couplings. Considering the multimode multiphoton Jaynes–Cummings model interacting with N modes of an electromagnetic field prepared in general pure quantum states, we analyze the Rabi oscillation. Evaluating the time evolution of the Dirac position operator, we determine the Zitterbewegung frequency and the corresponding oscillating term as a function of the electromagnetic field.
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.