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Massless Dirac fermions in an electromagnetic field

Jellal, A., El Mouhafid, A., Mohammed Daoud A

Published Jan 1, 2012• 9 citations
OpenAlex

Abstract

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.

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Massless Dirac fermions in an electromagnetic field

Ahmed Jellal, Abderrahim El Mouhafid, Mohammed Daoud A

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.

arXiv:1103.5297•Journal of Statistical Mechanics Theory and Experiment•Jan 30, 2012•9 citations

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

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.

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Optical perspective on the time-dependent Dirac oscillator

Thiago T. Tsutsui, Alison A. Silva, Antonio S. M. de Castro +1 more

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.

quant-pharXiv:2512.18904•Frontiers in Physics•Dec 23, 2025•0 citations

The Dirac Oscillator: A relativistic version of the Jaynes Cummings model

P. Rozmej, Robert Arvieu

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.

•Jan 1, 1999•64 citations

The Dirac oscillator. A relativistic version of the Jaynes-Cummings model

P Rozmej, Robert Arvieu

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.

arXiv:quant-ph/9903073•Journal of Physics A Mathematical and General•Jan 1, 1999•100 citations

Graphical Proposal for a Quantum Simulation of the Two-Dimensional Dirac Oscillator: Optical Trapped-Ion and Microwave Implementations

Abdelmalek Boumali

Abstract The two-dimensional Dirac oscillator is one of the most transparent exactly solvable models in relativistic quantum mechanics. Its Hamiltonian is linear in both momentum and posi- tion and therefore has no direct elementary-particle realization in ordinary harmonic traps. This extended proposal develops two complementary laboratory routes. The first is an optical trapped-ion implementation in which the electronic two-level system encodes the Dirac bispinor and two transverse motional modes encode the two spatial degrees of freedom. Bichromatic red- and blue-sideband laser fields synthesize the Jaynes-Cummings and anti-Jaynes-Cummings cou- plings required by the oscillator. The second route is a microwave-based proposal. Two meanings of “microwave implementation” are distinguished: a microwave-driven trapped-ion implementa- tion using magnetic-gradient-induced spin-motion coupling, and a microwave-resonator/tight- binding analogue based on coupled dielectric resonators. The former remains a genuine single- ion quantum simulator and can, in principle, implement the same effective Hamiltonian with improved phase stability and reduced optical spontaneous-emission errors. The latter is experi- mentally simpler for spectral measurements and can serve as a second proposal, but it is a wave analogue rather than a single-particle trapped-ion quantum simulation. We do not claim the first mapping of the 2 + 1 Dirac oscillator onto trapped-ion dynamics, which is already established in the literature; rather, the contribution is a consolidated, experimentally oriented roadmap that compares the optical trapped-ion, microwave-driven trapped-ion, and microwave-resonator routes, with explicit feasibility conditions, an observable strategy, a corrected graphical map- ping, an explicit separation of the ideal Hamiltonian from its leading correction terms, a worked order-of-magnitude estimate for a 171 Yb+ ion, and an explicit quantitative cross-walk to the trapped-ion and microwave experiments on which the scheme builds. We identify the advan- tages and limitations of each route and propose a staged experimental program for measuring Zitterbewegung, the relativistic square-root spectrum, and chirality-dependent dynamics.

•Jun 23, 2026•0 citations
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.
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 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.
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.
Abstract The two-dimensional Dirac oscillator is one of the most transparent exactly solvable models in relativistic quantum mechanics. Its Hamiltonian is linear in both momentum and posi- tion and therefore has no direct elementary-particle realization in ordinary harmonic traps. This extended proposal develops two complementary laboratory routes. The first is an optical trapped-ion implementation in which the electronic two-level system encodes the Dirac bispinor and two transverse motional modes encode the two spatial degrees of freedom. Bichromatic red- and blue-sideband laser fields synthesize the Jaynes-Cummings and anti-Jaynes-Cummings cou- plings required by the oscillator. The second route is a microwave-based proposal. Two meanings of “microwave implementation” are distinguished: a microwave-driven trapped-ion implementa- tion using magnetic-gradient-induced spin-motion coupling, and a microwave-resonator/tight- binding analogue based on coupled dielectric resonators. The former remains a genuine single- ion quantum simulator and can, in principle, implement the same effective Hamiltonian with improved phase stability and reduced optical spontaneous-emission errors. The latter is experi- mentally simpler for spectral measurements and can serve as a second proposal, but it is a wave analogue rather than a single-particle trapped-ion quantum simulation. We do not claim the first mapping of the 2 + 1 Dirac oscillator onto trapped-ion dynamics, which is already established in the literature; rather, the contribution is a consolidated, experimentally oriented roadmap that compares the optical trapped-ion, microwave-driven trapped-ion, and microwave-resonator routes, with explicit feasibility conditions, an observable strategy, a corrected graphical map- ping, an explicit separation of the ideal Hamiltonian from its leading correction terms, a worked order-of-magnitude estimate for a 171 Yb+ ion, and an explicit quantitative cross-walk to the trapped-ion and microwave experiments on which the scheme builds. We identify the advan- tages and limitations of each route and propose a staged experimental program for measuring Zitterbewegung, the relativistic square-root spectrum, and chirality-dependent dynamics.