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arXiv:1103.5297 · Journal of Statistical Mechanics Theory and Experiment

Massless Dirac fermions in an electromagnetic field

Ahmed Jellal, Abderrahim El Mouhafid, Mohammed Daoud A

Published Jan 30, 2012• 9 citations
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Abstract

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.

DOI
10.1088/1742-5468/2012/01/p01021

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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.
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.
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 dynamics of the 2+12+12+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 wave packets in a relativistic Dirac oscillator is compared to that of the Jaynes-Cummings model. The strong spin-orbit coupling of the Dirac oscillator produces the entanglement of the spin with the orbital motion similar to what is observed in the model of quantum optics. The collapses and revivals of the spin which result extend to a relativistic theory our previous findings on nonrelativistic oscillator where they were known under the name of `spin-orbit pendulum'. There are important relativistic effects (lack of periodicity, zitterbewegung, negative energy states). Many of them disappear after a Foldy-Wouthuysen transformation.
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.
The Dirac‐Moshinsky oscillator is an elegant example of an exactly solvable quantum relativistic model that under certain circumstances can be mapped onto the Jaynes‐Cummings model in quantum optics. In this work we show, how to do this in detail. Then we extend it by considering its coupling with an external (isospin) field and find the conditions that maintain solvability. We use this extended system to explore entanglement in relativistic systems and then identify its quantum optical analog: two different atoms interacting with an electromagnetic mode. We show different aspects of entanglement which gain relevance in this last system, which can be used to emulate the former.