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Cited by 15Quantum Mechanics and ApplicationsOrbital Angular Momentum in OpticsQuantum and Classical Electrodynamics

Optical simulation of the free Dirac equation

Open access
Silva, Thais L., Taillebois, E. R. F., Gomes, R. M., Walborn, S. P., Avelar, Ardiley T.·2019·Physical Review A
DOI 10.1103/physreva.99.022332
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Abstract

We present a classical optics simulation of the one-dimensional Dirac equation for a free particle. Positive- and negative-energy components are represented by orthogonal polarizations of a free propagating beam, while the spatial profile represents the spatial wave function of the particle. Using a completely tunable time parameter, we observed the oscillation of the average value of the Dirac position operator---known as Zitterbewegung. We are also able to measure the so-called mean-position operator which is a single-particle observable and presents no oscillations. Our work opens the way for optical simulation of interesting phenomenon of relativistic systems, as well as condensed-matter physics, without any requirement for specially engineered medium.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper reports a classical-optics simulation of the 1+1D free Dirac equation, encoding spinor components in orthogonal polarizations of a propagating light beam and the spatial wavefunction in its transverse profile. Using a tunable time parameter, the authors observed Zitterbewegung in the Dirac position operator and measured the Foldy-Wouthuysen mean-position operator, which shows no such oscillation.

Key findings

  • The experiment observed the oscillatory Zitterbewegung behavior of the average Dirac position operator \(\langle \hat{x}_D(t)\rangle\) for Compton lengths \(\lambda_C/\Delta = 1, 3, 5, 7, 10, 100\), with experimental points agreeing with theoretical/numerical predictions.
  • For \(\lambda_C/\Delta = 5\), the mean-position operator \(\langle \hat{x}_{FW}(t)\rangle\) showed no Zitterbewegung and fit a linear time dependence with \(R^2 = 99.6\%\); positive- and negative-energy projections gave independent linear mean trajectories.
  • Fitting \(\langle \hat{x}_D(t)\rangle\) with \(vt + A\sin(\omega t + \delta)\) showed the Zitterbewegung amplitude increasing linearly with \(\lambda_C\), frequency proportional to \(1/\lambda_C\), and mean velocity falling quadratically with \(\lambda_C\) for large masses, approaching the speed of light for \(\lambda_C = 100\Delta\).
  • The small deviation from perfect linear behavior in \(\langle \hat{x}_{FW}(t)\rangle\) was explained by a numerical simulation assuming 95% modulation efficiency in each SLM.

Methods

  • Classical optical setup: a 632.8 nm He-Ne laser with a Gaussian initial spinor (\(a=-b=1\), \(\Delta = 48.6\,\mu\text{m}\), \(\lambda R/\pi = (2.2\Delta)^2\)), two Holoeye reflective SLMs each divided into halves, quarter- and half-wave plates at \(45^\circ\), cylindrical lenses with 150 mm focal distance, and a CCD camera.
  • Simulation protocol: the vertical transverse coordinate represented the particle position \(x\), horizontal/vertical polarizations represented the two spinor components, SLMs and wave plates implemented the Foldy-Wouthuysen transformation and inverse transformation, and momentum-dependent phases implemented the diagonal free evolution \(\exp[\pm i t \varepsilon(p)/\hbar]\).
  • Analysis: \(\langle \hat{x}_D\rangle(t) = \sum_{i=1,2}\int dx\, x |\psi_{Di}(x,t)|^2\) was computed from CCD intensity profiles, and the data were fitted with \(vt + A\sin(\omega t + \delta)\) for each Compton wavelength and compared with theory/numerical solutions.

Limitations

  • The paper states that there is no exact Foldy-Wouthuysen transformation for the non-free Dirac equation, so the technique is limited if one tries to implement the actual FWT for a potential problem; Appendix B treats only a particular class of potentials.
  • The paper states that a general 3+1D simulation would require a third beam coordinate not available in the scheme, so only a particular class of initial states is considered; the 2+1D extension also does not present spin effects.
  • The paper states that cylindrical lenses can introduce non-separability, making the initial state not entirely pure, though the authors conclude these effects are negligible; it also notes that SLMs do not modulate all incident light, with 95% efficiency used to explain small deviations. Evident from the text: the work is a classical-optics simulation rather than a quantum simulation.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physreva.99.022332

Cites

25 of 25 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 13 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § IV. RESULTS · p. 4

    The experimental ZB data agree with theoretical predictions.

    “The agreement between the experimental ZB data and the theoretical predictions confirm that our optical setup is well suited for the study of 1+1D relativistic dynamical systems, the theoretical extension to larger dimensions being discussed in the Appendix.”
    • Also stated in: § IV. RESULTS · p. 4 · plain
      “Our experimental results are in agreement with the expected linear dependence of amplitude and inverse dependence of frequency on λ C for small λ C [23], as can be seen in Fig.4-(b).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [23]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • The authors state this as an assumption: The x and y intensity distributions remain separable throughout all the apparatus.
  • Findingown data
    § IV. RESULTS · p. 4

    The ZB visibility increases for smaller values of λ_C.

    “This is consistent with the fact that the ZB visibility in Fig.3 increases for smaller values of λ C .”
    Quote in the paperNot a substantial claim
  • Findingreasoning
    § V. DISCUSSION AND CONCLUSIONS · p. 6

    The experiment decomposes the unitary evolution into operations realizable with off-the-shelf optical components using the FWT.

    “Our experiment is based on the diagonalization of the Dirac Hamiltonian using the FWT, which allowed for the decomposition of the unitary evolution into operations that are realizable with off-the-shelf optical components.”
    Quote in the paperSubstantial claim
  • Findingown data
    § IV. RESULTS · p. 3

    The non-separability effects are negligible for the present experiment.

    “Since our experimental results agree well with theory, we conclude that these effects are negligible for the present experiment.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The x and y intensity distributions remain separable throughout all the apparatus.
  • Findingown data
    § Optical simulation of the free Dirac Equation · p. 1

    The average value of the Dirac position operator exhibits oscillatory behavior (Zitterbewegung) in the experiment.

    “Using a completely tunable time parameter, we observed the oscillation of the average value of the Dirac position operator-known as Zitterbewegung (ZB).”
    • Also stated in: § IV. RESULTS · p. 3 · plain
      “In Fig.2-(a) we present the mean position 〈 ˆ x D 〉 ( t ) as a function of t , the ZB being evidenced by the oscillatory behavior.”
    • Also stated in: § V. DISCUSSION AND CONCLUSIONS · p. 6 · plain
      “Adjusting the tunable time parameter we observed the oscillatory ZB phenomenon for Dirac's position operator.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The x and y intensity distributions remain separable throughout all the apparatus.
    • Rests on an assumption: The horizontal spatial degrees of freedom on the transverse plane play no relevant role in the experiment.
  • May combine two claims. 15
  • Classified as probably not a substantial claim. 3

  • May combine two claims.

    Findingown data
    § Optical simulation of the free Dirac Equation · p. 1

    The mean-position operator is a single-particle observable and presents no oscillations.

    “We are also able to measure the so called mean-position operator which is a single-particle observable and presents no oscillations.”
    • Also stated in: § I. INTRODUCTION · p. 1 · plain
      “This observable is often called mean-position operator and does not exhibit the oscillatory behavior characteristic of the ZB, a result that generates doubts concerning its actual existence.”
    • Also stated in: § IV. RESULTS · p. 5 · plain
      “The ZB is not present for these mean trajectories and a linear behavior in time is observed, as it was expected.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The x and y intensity distributions remain separable throughout all the apparatus.

  • May combine two claims.

    Findingreasoning
    § IV. RESULTS · p. 5

    The experimental data for x_FW,+ and x_FW,- show two independent mean trajectories with no ZB and linear behavior.

    “As is expected from the independence of the two problems in the SP description, we have two independent mean trajectories corresponding to the free evolution of the particle and the corresponding anti-particle. The ZB is not present for these mean trajectories and a linear behavior in time is observed, as it was expected.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The x and y intensity distributions remain separable throughout all the apparatus.

  • May combine two claims.

    Findingown data
    § IV. RESULTS · p. 4

    The mean velocity falls quadratically with λ_C for large masses (small λ_C) and approaches the speed of light for very small masses (λ_C = 100∆).

    “As is expected, the mean velocity falls quadratically with λ C for large masses (small λ C ), while it is close to the speed of light for very small masses ( λ C = 100∆).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data
    § III. SIMULATION PROTOCOL AND EXPERIMENT · p. 3

    The initial state parameters were determined to be ∆=48.6 μm and λR/π = (2.2∆)^2.

    “Using a beam profiler, we determined the initial state parameters to be ∆=48 , 6 µm and λR/π = (2 . 2∆) 2 .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § IV. RESULTS · p. 4

    The amplitude and frequency of ZB agree with expected dependence on λ_C for small λ_C.

    “Our experimental results are in agreement with the expected linear dependence of amplitude and inverse dependence of frequency on λ C for small λ C [23], as can be seen in Fig.4-(b).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [23]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Findingnone given
    § Optical simulation of the free Dirac Equation · p. 1

    In the simulation, positive and negative energy components are represented by orthogonal polarizations of a free propagating beam, and the spatial profile represents the spatial wave function.

    “Positive and negative energy components are represented by orthogonal polarizations of a free propagating beam, while the spatial profile represents the spatial wave function of the particle.”
    • Also stated in: § V. DISCUSSION AND CONCLUSIONS · p. 6 · plain
      “where the beam's spatial profile plays the role of the particle's wavefunction, and its orthogonal polarization components are associated to spinor components.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The reference frame is fixed.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    The state produced in the experiment had zero average momentum.

    “The state produced in this experiment had zero avarage momentum, but simply shifting the momentum in all SLMs phases by the same ∆ p can be interpreted as if the state has non-vanishing average momentum.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    Shifting the momentum in all SLM phases by the same ∆p can be interpreted as the state having non-vanishing average momentum.

    “simply shifting the momentum in all SLMs phases by the same ∆ p can be interpreted as if the state has non-vanishing average momentum.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § III. SIMULATION PROTOCOL AND EXPERIMENT · p. 3

    The time coordinate is a programmable parameter, allowing measurements for as many time values as desired.

    “Since the time coordinate also comes up as a programmable parameter, we could in principle take measurements for as many time values as we wish inside a time interval.”
    • Also stated in: § III. SIMULATION PROTOCOL AND EXPERIMENT · p. 3 · plain
      “It is worth noting that the evolved state is accessible for any time value.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § I. INTRODUCTION · p. 1

    The present approach performs evolution in the diagonalized Foldy-Wouthuysen representation and permits switching between FWR and Dirac representation, unlike previous direct simulations.

    “Previous works perform a direct simulation of the 1+1D Dirac Hamiltonian, while the present approach performs the evolution in the diagonalized Foldy-Wouthuysen representation (FWR) and permits one to switch back and forth between this and Dirac's representation, allowing us to investigate the behavior of both the Dirac position and the mean-position operators.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    The method can be extended to 2+1D and some 3+1D initial states.

    “Albeit here we present the theory and results for the 1+1D case, the extension for 2+1D and for some initial states in 3+1D is straightforward as shown in the Appendix.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § IV. RESULTS · p. 4

    The optical setup permits investigation in the Foldy-Wouthuysen representation and from a single-particle perspective.

    “Beside serving as a proof-of-concept, the proposed setup permits to investigate the system in the FWR, an interesting possibility since it allows to describe the dynamics of the system according to the single-particle perspective, i.e. assigning physical sense only to the projections of single-particle operators over the subspaces of definite sign of energy.”
    • Also stated in: § V. DISCUSSION AND CONCLUSIONS · p. 6 · plain
      “Using our experimental FWT, we were also able to address this phenomenon from a single-particle perspective, where the position description is given by the positive and negative energy projections of the single-particle mean-position operator.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The reference frame is fixed.

  • May combine two claims.

    Interpretationreasoning
    § 2. Simulation of 3+1 dimensional Dirac Equation for a particular class of initial states · p. 7

    The FWR only mixes components two by two, allowing simulation with two beams without joint transformation.

    “we notice that the FWR only mixes the components two by two what makes possible to simulate it using two beams without any joint transformation between them.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § Optical simulation of the free Dirac Equation · p. 1

    The work opens the way for optical simulation of relativistic systems and condensed-matter physics without specially engineered medium.

    “Our work opens the way for optical simulation of interesting phenomenon of relativistic systems, as well as condensed-matter physics, without any requirement for specially engineered medium.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    A single-particle position observable exists for Dirac's theory and is obtained using the Foldy-Wouthuysen transformation.

    “For Dirac's theory, a SP position observable exists and is obtained using the so-called Foldy-Wouthuysen transformation (FWT) [16] - a momentum dependent unitary transformation that diagonalizes Dirac's Hamiltonian and is at the kernel of important algorithms used to obtain quantum relativistic corrections [17, 18].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [16] L. L. Foldy, S. A. Wouthuysen, Phys. Rev. 78(1), 29 (1950).

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The Dirac equation predicts Zitterbewegung and Klein's paradox, which impede the single-particle interpretation.

    “In this case, the Dirac equation predicts intriguing phenomena, for instance the ZB [3] and Klein's paradox [4], which impede the single-particle (SP) interpretation of the Dirac equation.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [3] E. Schrödinger, Sitzungsber. Preuss. Akad. Wiss. Phys.Math. Kl. 24, 418 (1930).

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [4] Klein, O. Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Z. Phys. 53, 157 (1929). source

  • May combine two claims.

    Premisecitation
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    Continuous variable quantum simulators can implement Gaussian Hamiltonians but non-Gaussian operations are difficult.

    “Typically in continuous variable quantum simulators one can implement Gaussian Hamiltonians, but non-Gaussian operations (third order and above) are quite difficult [24].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [24] D.S. Tasca, R.M. Gomes, F. Toscano, P.H. Souto Ribeiro, and S.P. Walborn, Phys. Rev. A 83, 052325 (2011).

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

17 of 17

  • The authors state this as an assumption.

    The reference frame is fixed.

    “Although a spinor is a mathematical object which transforms very specifically under a reference frame change, it is not a concern for this simulation since the reference frame is assumed to be fixed.”

  • The authors state this as an assumption.

    The x and y intensity distributions remain separable throughout all the apparatus.

    “This procedure assumes that x and y intensity distributions remain separable troughout all the apparatus.”

  • The paper relies on this without support.

    The horizontal spatial degrees of freedom on the transverse plane play no relevant role in the experiment.

    “The horizontal spatial degrees of freedom on the transverse plane play no relevant role in the experiment.”
3 of 3

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Mean-position decomposition repeats the positive-energy projection

    Minor

    The stated decomposition contains two positive-energy terms and no negative-energy term, whereas the paper describes the mean position as including both projections.

  • Two different methods are given for obtaining the one-dimensional distribution

    Minor

    Taking the intensity at a fixed y coordinate is a slice; marginalizing over y integrates across y. The results text and Fig. 2 caption give different procedures for the plotted distribution.

  • Appendix time-evolution arguments conflict with the defined phase operator

    Minor

    The appendix passes iεt/ħ as the argument to P, but P is defined to apply exp(if). That argument therefore produces a real exponential rather than the dynamical phase described in the main text.

3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. E. Schrödinger, 1930, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., 24, 418

Cited by

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Related

Authors
Silva, T. L., Taillebois, E. R. F., Gomes, R. M., Walborn, S. P., & Avelar, A. T.
Affiliations
  • Universidade Federal de Goiás (BR)
  • Universidade Federal do Rio de Janeiro (BR)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2019-02-25
Volume
99
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 2469-9926
  • 2469-9934
DOI
10.1103/physreva.99.022332
OpenAlex
W2918023906
Semantic Scholar
e102dbfb7f16f6e88d0400de5e2008227c5d934f
Cited by
14
References
23
Citations by year
2025: 4 · 2024: 2 · 2023: 1 · 2022: 2 · 2021: 4 · 2020: 1
FWCI
0.77
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
  • Conselho Nacional de Desenvolvimento Científico e Tecnológico (#459339/2014-1, 140832/2016-5, 3070027/2013-9)
  • Fundação de Amparo à Pesquisa do Estado de Goiás (#201710267000503)
  • Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (010.002997/2014, 010.002911/2014, 202.790/2017)
  • Instituto Nacional de Ciência e Tecnologia de Informação Quântica
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Orbital Angular Momentum in Optics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
Keywords
Dirac equation, optical simulation, classical optics, polarization, positive and negative energy states, Zitterbewegung, relativistic quantum mechanics

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