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Cited by 302Quantum Mechanics and ApplicationsCold Atom Physics and Bose-Einstein CondensatesQuantum Computing Algorithms and Architecture

Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

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Lamata, L., León, J., Schätz, T., Solano, E.·2007·Physical Review Letters
DOI 10.1103/physrevlett.98.253005
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Abstract

We present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.

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Registry

Crossref · 10.1103/physrevlett.98.253005

Cites

24 of 24 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

5 of 20 cited sources checked · 0 supported · 0 flagged · accuracy not yet measured

  • Interpretationcitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The piecewise buildup of the 3+1 Dirac Hamiltonian allows strong reduction of experimental demands to study Dirac equations in 1+1 and 2+1 dimensions.

    “Because of our piecewise buildup of the 3 .0135 1 Dirac Hamiltonian of Eq. (2), we can strongly reduce the experimental demands to study Dirac equations in 1 .0135 1 and 2 .0135 1 dimensions [1,13].”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [1] B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, 1992). source
    • cites [13] source

      The paper’s discussion on p. 1 concerns measuring bosonic-field quadratures with a probe, and its conclusion on p. 4 concerns reducing measurements needed for that task. Searches for a piecewise Dirac Hamiltonian and for 1+1 or 2+1 dimensional Dirac equations found no passage addressing the claim. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
  • Interpretationcitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    A 1+1 axial anomaly can be produced by changing Eq. (10) into H_D^(1)= cσ_x p_x + qEx, where x ∝ (a+a†) is a motional displacement.

    “Finally, in as much as massless chiral fermions in condensed matter, we could also produce a 1 .0135 1 axial anomaly [17] by changing Eq. (10) into H .0133 1 .0134 D .0136 c.0027xpx .0135 qEx , where x / .0133 a .0135 a y .0134 is a motional displacement.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [17] H. B. Nielsen and M. Ninomiya, Phys. Lett. 130B , 389 (1983). source

      The cited paper discusses quantized ion motion and motional sidebands (p. 2), but our copy does not address a 1+1 axial anomaly or the proposed Dirac Hamiltonian. Searches for axial anomaly and for the Hamiltonian terms found no passage supporting the claim. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
  • Findingreasoning
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    For a massive particle, the 1+1 Dirac dynamics yields a Zitterbewegung oscillatory solution similar to the 3+1 case.

    “On the other hand, for a massive particle, d.0027x=dt .0136 .0137 .0027 x ; H D .0138 =i @ .0136 .0255 .0010 .0027 y , d.0027y=dt .0222 0 , and so forth. The produced set of differential equations yields a Zitterbewegung oscillatory solution similar to Eq. (7), with the evident simplification via the replacements of HD by H .0133 1 .0134 D , ~ p by px , and ~ .0011 by .0027 x .”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The 1+1-dimensional Dirac equation can be cast into a Hamiltonian involving a simultaneous JC+AJC interaction and a Stark-shift term acting on two internal levels.

    “After a unitary transformation around the x axis, transforming .0027 y into .0027 z , the 1 .0135 1 Dirac equation stemming from Eq. (2) can be cast into i @ @ j .0009 .0133 1 .0134 i =@t .0136 H .0133 1 .0134 D j .0009 .0133 1 .0134 i with”
    • Also stated in: § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion · plain
      “Our setup is now reduced to a simultaneous JC .0135 AJC interaction, .0027xpx , and a Stark-shift term .0027 z acting on two internal levels.”
    Quote in the paperSubstantial claim
  • Findingnone given
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The basic ingredients to implement the 1D Dirac dynamics with a single trapped ion are two independent electronic internal states coupled to a one-dimensional motional degree of freedom.

    “The basic ingredients to implement the 1D Dirac dynamics of Eq. (10) with a single trapped ion are two independent electronic (internal) states coupled to a onedimensional motional degree of freedom.”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 14
  • May combine two claims. 8
  • The quote only partly states this claim. 2

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

    Interpretationcitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The number of laser beams could be further reduced by electro-optical modulators to provide red and blue sidebands simultaneously.

    “The number of laser beams could be further reduced by electro-optical modulators to provide red and blue sidebands simultaneously [11,12].”
    Quote in the paperNot a substantial claim1 of 2 sources checked
    • cites [11] P. C. Haljan et al. , Phys. Rev. Lett. 94 , 153602 (2005). source

      The experimental setup description (p. 2) reports an electro-optic modulator and simultaneous red and blue sidebands in a two-beam Raman setup. But it says the electro-optic modulator bridges the hyperfine gap, while driving an acousto-optic modulator at two frequencies creates the bichromatic field. It does not state that electro-optic modulators further reduce the number of beams. (LLM, conf=0.86)

      Experimental — accuracy not yet measured
    • cites [12] C. A. Sackett et al. , Nature (London) 404 , 256 (2000). source

  • May combine two claims. The quote only partly states this claim.

    Findingcitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    In the relativistic limit, including m=0, the 1+1 Dirac Hamiltonian reduces to H_D^(1)= i ηΩ σ_x (a† - a), which produces Schrödinger cats.

    “In the relativistic limit, mc 2 .0028 pxc , which includes m .0136 0 , the 1 .0135 1 case reduces to H .0133 1 .0134 D .0136 i @ .0017 ~ .0010 .0027 x .0133 a y x .0255 ax .0134 , which produces Schro ¨dinger cats, as commented above [9,10].”
    Quote in the paperSubstantial claim2 of 2 sources checked
    • cites [9] E. Solano, R. L. de Matos Filho, and N. Zagury, Phys. Rev. Lett. 87 , 060402 (2001). source

      The discussion on pp. 2–3 describes ion interactions that generate superpositions of coherent motional states, including a single-ion odd coherent state. It does not identify the interaction as a 1+1 Dirac Hamiltonian or discuss a relativistic or massless limit; the Hamiltonian formulas in our copy are not decoded, so the stated expression cannot be verified. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
    • cites [10] E. Solano, G. S. Agarwal, and H. Walther, Phys. Rev. Lett. 90 , 027903 (2003). source

      The cavity-QED discussion (p. 2) says its one-atom interaction generates Schrödinger cat states, including at zero detuning. It does not discuss a relativistic or massless Dirac limit, and the Hamiltonian formulas in our copy are undecoded, so it cannot verify the stated Hamiltonian. (LLM, conf=0.83)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Findingreasoning
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    For a massless particle, σ_x is a constant of motion and the time evolution of the position operator is classical and does not involve Zitterbewegung.

    “For a massless particle, we can show that dx=dt .0136 .0137 x; H .0133 1 .0134 D .0138 =i @ .0136 2 .0017 .0001 ~ .0010 .0027 x and d.0027x=dt .0136 .0137 .0027 x ; H .0133 1 .0134 D .0138 =i @ .0136 0 . In consequence, .0027 x is a constant of motion and the time evolution of the position operator, x .0133 t .0134 .0136 x .0133 0 .0134 .0135 2 .0017 .0001 ~ .0010 .0027 x t , is classical and does”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims. The quote only partly states this claim.

    Findingnone given
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    Position and momentum operators are measured when choosing θ=π/2 and θ=0, respectively.

    “Then, position and momentum operators are measured when choosing .0030 .0136 .0255 .0025= 2 and .0030 .0136 0 , respectively.”
    Quote in the paperNot a substantial claim

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

    Findingreasoning
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    To apply this technique, a particular initial state of the internal states is required, so measurement of <x(t)> would have to be done in two parts.

    “To apply this technique, it is required a particular initial state of the internal states, so the measurement of h x .0133 t .0134i would have to be done in two parts, one associated with each of the suitably projected states, j.0135 .0030 i and j.0255 .0030 i .”
    Quote in the paperNot a substantial claim

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

    Findingreasoning
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The Zitterbewegung amplitude can be estimated from Eq. (7) as an equation.

    “Similarly, we can estimate from Eq. (7) the Zitterbewegung amplitude associated with h ~ r .0133 t .0134i as”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The required internal states can be composed of two ground-state hyperfine levels of an earth alkaline atomic ion, e.g., 25Mg+ with |F=3,mf=-3> and |F=2,mf=-2> as |a> and |b>.

    “The required states could be composed by two ground-state hyperfine levels of an earth alkaline atomic ion, e.g., of 25 Mg .0135 by j F .0136 3; mf .0136 .0255 3 i and j F .0136 2; mf .0136 .0255 2 i , j a i and j b i , respectively, as depicted in Fig. 1.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The 1+1-dimensional case could be reached by current experiments, keeping most striking results available.

    “We focus now on the 1 .0135 1 -dimensional case, which could be reached by current experiments, keeping most striking results available.”
    Quote in the paperNot a substantial claim

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

    Interpretationcitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    All necessary Raman beams could be derived from a single laser source, considering available laser intensities.

    “Considering the available laser intensities [21], all necessary Raman beams could be derived from a single laser source.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [21] A. Friedenauer et al. , Appl. Phys. B 84 , 371 (2006). source

  • May combine two claims.

    Premisecitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The Zitterbewegung frequency and amplitude for a real electron are ω_ZB ~ 10^21 Hz and R_ZB ~ 10^-3 Å, which are out of experimental reach.

    “The predicted values for a real electron, ! ZB .0024 10 21 Hz and R ZB .0024 10 .0255 3 .0023 A , are out of experimental reach, the effect has never been observed, and its existence is even questioned by quantum field theory considerations.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [1] B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, 1992). source

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

    Premisecitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The Dirac equation in 3+1 dimensions can be expressed in a supersymmetric representation.

    “This is easily seen if we express the Dirac equation”
    • Also stated in: § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion · plain
      “in its ''supersymmetric'' representation [1]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [1] B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, 1992). source

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

    Premisecitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The Zitterbewegung is a helicoidal motion of a free Dirac particle, a natural consequence of noncommutativity of velocity operator components.

    “The Zitterbewegung is a known quantum-relativistic effect consisting of a helicoidal motion of a free Dirac particle, a natural consequence of the noncommutativity of its velocity operator components, c.0011 i , with i .0136 x , y , z .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [1] B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, 1992). source

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

    Premisecitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The one-particle solutions of the Dirac equation predict Zitterbewegung and Klein's paradox.

    “However, the one-particle solutions of the Dirac equation in relativistic quantum mechanics predict some astonishing effects, like the Zitterbewegung and the Klein's paradox.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [1] B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, 1992). source

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationnone given
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The 1+1 case has minimal experimental needs while keeping most striking predictions.

    “We have studied the 1 .0135 1 case, where experimental needs are minimal while keeping most striking predictions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The relativistic energy related to H+V may be recast into (pc)^2 = (E+V - V + mc^2)(E+V - V - mc^2), which is positive when either both factors are positive or negative.

    “This stems from the fact that the relativistic energy related to H .0133 1 .0134 V may be recast into .0133 pc .0134 2 .0136 .0133 E .0133 1 .0134 V .0255 V .0135 mc 2 .0134.0133 E .0133 1 .0134 V .0255 V .0255 mc 2 .0134 , which is positive when either both factors are positive or negative.”
    Quote in the paperSubstantial claim

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

    Premisecitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    A recent proposal for realizing fast measurements of motional quadratures relies on measuring population of an ionic internal level at short probe-motion interaction times with high precision.

    “A recent proposal for realizing fast measurements of motional quadratures [14] relies on the possibility of measuring the population of an ionic internal level, Pa , at short probemotion interaction times .0028 with high precision [15].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [14] P. Lougovski, H. Walther, and E. Solano, Eur. Phys. J. D 38 , 423 (2006). source
    • cites [15] D. M. Meekhof et al. , Phys. Rev. Lett. 76 , 1796 (1996). source

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    The required potential V can be produced through a fast and homogeneous Stark shift in both internal levels.

    “We suggest to produce potential V with the required characteristics through a fast and homogeneous Stark shift in both internal levels.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

    In 1929, Klein noticed the anomalous behavior of Dirac particles in regions where a high potential energy V exists.

    “In 1929, Klein noticed [16] the anomalous behavior of Dirac particles in regions where a high potential energy V exists: H .0133 1 .0134 V .0136 H .0133 1 .0134 D .0135 V 1 2 .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [16] O. Klein, Z. Phys. 53 , 157 (1929). source
18 of 18

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Non-relativistic electrons in graphene monolayer obey the massless relativistic dispersion known as the Dirac cone, leading to realization of analogies to Zitterbewegung and Klein tunneling in low-energy condensed-matter physics and atom optics.

    “Recently, the non-relativistic electrons in graphene monolayer were found to obey the massless relativistic dispersion known as the Dirac cone, leading to the realization of analogies to Zitterbewegung and Klein tunneling in low-energy condensed-matter physics and atom optics 4 -19 .”

    From Observation of interaction-induced phenomena of relativistic quantum mechanics (2021) · cites this paper as [8]

    Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

    Experimental — accuracy not yet measured

  • Zitterbewegung is a signature of the interference of particle and antiparticle states.

    “that is a signature of the interference of particle and antiparticle states”

    From Photonic cellular automaton simulation of relativistic quantum fields: Observation of <i>Zitterbewegung</i> (2024) · cites this paper as [79]

    Experimental — accuracy not yet measured

  • Zitterbewegung is the oscillation of the expected value of the position operator X.

    “namely the oscillation of the expected value of the position operator X [32-34, 77-91].”

    From Photonic cellular automaton simulation of relativistic quantum fields: Observation of <i>Zitterbewegung</i> (2024) · cites this paper as [79]

  • Zitterbewegung (ZB) is the rapid trembling motion of a free Dirac electron caused by the interference between positive and negative energy state components.

    “Zitterbewegung (ZB) refers to the rapid trembling motion of a free Dirac electron caused by the interference between positive and negative energy state components [1, 2].”

    From Photonic analog of Zitterbewegung in binary waveguide arrays (2010) · cites this paper as [2]

  • There has not been any progress in the investigation on the zitterbewegung of photons since some preliminary studies have been presented.

    “Contrary to these, there has not achieved any progress in the investigation on the ZB of photons since some preliminary studies have been presented [30].”

    From Photon wave function and<i>Zitterbewegung</i> (2009) · cites this paper as [30]

    The passages under “Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion” (page not provided) discuss Zitterbewegung in a trapped-ion simulation of Dirac particles. Searches for photonic Zitterbewegung and preliminary studies found no passage addressing whether research on photons had progressed. (LLM, conf=0.76)

    Experimental — accuracy not yet measured

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Related

Authors
Lamata, L., León, J., Schätz, T., & Solano, E.
Affiliations
  • Consejo Superior de Investigaciones Científicas (ES)
  • Max Planck Institute of Quantum Optics (DE)
  • Pontificia Universidad Católica del Perú (PE)
ORCID
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
2007-06-22
Volume
98
Issue
25
Pages
253005-253005
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.98.253005
OpenAlex
W1968873885
Cited by
300
References
24
Citations by year
2026: 2 · 2025: 13 · 2024: 7 · 2023: 4 · 2022: 16 · 2021: 17 · 2020: 16 · 2019: 19 · 2018: 17 · 2017: 20 · 2016: 18 · 2015: 21 · 2014: 21 · 2013: 14 · 2012: 21
FWCI
9.38
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • DIGITAL.CSIC (Spanish National Research Council (CSIC)) (submittedVersion, other-oa)
  • Max Planck Institute for Plasma Physics (submittedVersion)
  • MPG.PuRe (Max Planck Society) (submittedVersion)
  • arXiv (Cornell University)
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 › Cold Atom Physics and Bose-Einstein Condensates
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
Dirac equation, trapped ions, quantum simulation, spin-1/2 particles, Zitterbewegung, Klein paradox, relativistic effects

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