Quantum Simulation of the Klein Paradox with Trapped Ions
Open accessAbstract
We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external potentials we are able to simulate text book relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator. We describe extensions to solve problems that are beyond current classical computing capabilities.
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TL;DR
This paper reports a trapped-ion quantum simulation of relativistic scattering and Klein tunneling, encoding a Dirac spinor in one ion’s internal state and a collective motional mode. By engineering linear and quadratic potentials with a second ion and laser fields, the authors reconstruct wave packets frame-by-frame and observe tunneling probabilities that increase with potential gradient.
Key findings
- Klein tunneling was simulated for linear potentials with measured tunnel probabilities 0.017(7), 0.10(1), 0.32(2), and 0.45(3) for theoretical \(P_{\rm tunnel}=0, 0.03, 0.21, 0.36\), compared with numerical results in curly brackets: {0}, {0.07}, {0.22}, {0.39}.
- For quadratic potentials, the wave packet remained unconfined due to Klein tunneling: without a potential it dispersed, while with the potential its spreading was significantly restricted but could continue through Klein tunneling.
- With an initial momentum \(\langle \hat p \rangle = 0.23 \hbar/\Delta\), the wave packet oscillated in the quadratic potential and showed Klein tunneling at each turning point.
- The experiment reconstructed positive- and negative-energy components separately in the first and last frames of each sequence, enabling a frame-by-frame visualization of scattering; the initial spinor was largely positive energy with \(|\langle \psi | P_+ | \psi \rangle|^2 > 0.98\).
Methods
- Two \(^{40}\mathrm{Ca}^+\) ions were trapped in a linear Paul trap with axial frequency \(\omega_{\rm ax}=2\pi\,1.36\) MHz and radial frequency \(\omega_{\rm rad}=2\pi\,3\) MHz.
- The Dirac spinor was encoded by mapping continuous position and momentum to the quadratures of a vibrational mode and discrete components to two long-lived internal electronic states of ion 1; bichromatic laser fields simulated the free Dirac Hamiltonian, while a second ion and another bichromatic field engineered external linear or quadratic potentials.
- After simulation, a bichromatic pulse mapped position/momentum information onto ion 2’s internal state; fluorescence detection as a function of probe time and Fourier transform reconstructed the spatial probability distribution. Initial states used \(p_0=3.5\hbar/\Delta\) for linear-potential scattering and \(\langle \hat p \rangle=0\) or \(0.23\hbar/\Delta\) for quadratic-potential cases.
Limitations
- Stated in the paper: internal-state coherence time was \(\approx 3\) ms, limited by magnetic field fluctuations, while the whole simulation took up to 1.5 ms; motional coherence was limited by slow drifts changing the trap frequency by about \(\approx 25\) Hz. Both effects broadened wave packets, and systematic state-preparation errors from slowly varying settings could add structure to reconstructed data.
- Stated in the paper: for the steepest slope, states of more than 150 phonons on average were created, where the Lamb-Dicke approximation starts to fail, changing the simulated Hamiltonian and affecting reconstruction.
- Evident from the text: the demonstration simulates a single Dirac particle in one spatial dimension using two ions; the paper describes scaling to more particles and dimensions as future work.
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- Findingnone given§ Quantum simulation of the Klein paradox with trapped ions · p. 2
The generated initial spinor is comprised largely of positive energy: |<ψ|P+|ψ>|^2 > 0.98.
“The generated spinor, ψ ( x ; t = 0) ∝ e ip 0 x/ ℏ e -x 2 4∆ 2 ( 1 1 ) 1 , is comprised largely of positive energy: |〈 ψ ( x ; t = 0) | P + | ψ ( x ; t = 0) 〉| 2 > 0 . 98”
Quote in the paperSubstantial claim - Findingown data§ Quantum simulation of the Klein paradox with trapped ions · p. 3
For steeper gradients, part of the wave packet propagates into the repulsive potential via Klein tunneling.
“For a shallow potential gradient, the particle is almost completely reflected (b) and for steeper gradients (c,d) part of the wave packet propagates into the repulsive potential via Klein tunneling (color online).”
Quote in the paperSubstantial claim - Findingown data§ Quantum simulation of the Klein paradox with trapped ions · p. 3
For a shallow potential gradient, the particle is almost completely reflected.
“For a shallow potential gradient, the particle is almost completely reflected (b) and for steeper gradients (c,d) part of the wave packet propagates into the repulsive potential via Klein tunneling (color online).”
Quote in the paperSubstantial claim - Findingown data§ Quantum simulation of the Klein paradox with trapped ions · p. 1
Quantum simulations of relativistic scattering dynamics using trapped ions were performed.
“We report on quantum simulations of relativistic scattering dynamics using trapped ions.”
- Also stated in: § Quantum simulation of the Klein paradox with trapped ions · p. 1 · plain
“In this paper we perform quantum simulations of the scattering dynamics of relativistic quantum particles.”
Quote in the paperSubstantial claim - Also stated in: § Quantum simulation of the Klein paradox with trapped ions · p. 1 · plain
- Study design and conductnone given§ Quantum simulation of the Klein paradox with trapped ions · p. 3
The motional coherence is limited by slow drifts, which can change the trap frequency by about ≈ 25 Hz.
“The motional coherence is limited by slow drifts, which can change the trap frequency by about ≈ 25 Hz.”
Quote in the paperNot a substantial claim
- Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 2
For ultra-relativistic particles, the probability for tunneling is given by P_tunnel = e^{-2πΓ}, with Γ = m^2 c^3 / 2 ħ g.
“For ultra-relativistic particles, the probability for tunneling is given by P tunnel = e -2 π Γ , with Γ = m 2 c 3 / 2 ℏ g .”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [16] F. Sauter, Z. Phys. 69 , 742 (1931). source
Not checked in the free run.
- cites [24] J. Casanova, J. J. Garcıa-Ripoll, R. Gerritsma, C. F. Roos, and E. Solano, Phys. Rev. A 82 , 020101(R) (2010). source
Not checked in the free run.
- cites [25] C. Wittig, J. Phys. Chem. B 109 , 8428 (2005). source
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- Interpretationcitation§ Quantum simulation of the Klein paradox with trapped ions · p. 3
The authors are able to create approximately quadratic electric potentials which, due to Klein tunneling, are nonconfining in the relativistic limit.
“We are also able to create approximately quadratic electric potentials which, due to Klein tunneling, are nonconfining in the relativistic limit [18].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18] R. Giachetti and E. Sorace, Phys. Rev. Lett. 101 , 190401 (2008). source
Not checked in the free run.
- Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 2
For linear potentials eφ(x)=gx the situation is conceptually equivalent to Landau Zener tunneling.
“For linear potentials eφ ( x ) = gx the situation is conceptually equivalent to Landau Zener tunneling [16, 24, 25], as shown in Fig 1(a).”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [16] F. Sauter, Z. Phys. 69 , 742 (1931). source
Not checked in the free run.
- cites [24] J. Casanova, J. J. Garcıa-Ripoll, R. Gerritsma, C. F. Roos, and E. Solano, Phys. Rev. A 82 , 020101(R) (2010). source
Not checked in the free run.
- cites [25] C. Wittig, J. Phys. Chem. B 109 , 8428 (2005). source
Not checked in the free run.
- Study design and conductown data§ Quantum simulation of the Klein paradox with trapped ions · p. 3
For the steepest slope, states of more than 150 phonons on average were created, for which the Lamb-Dicke approximation starts to fail.
“For the steepest slope, states of more than 150 phonons on average were created, for which the Lamb-Dicke approximation starts to fail.”
Quote in the paperNot a substantial claim
- May combine two claims. 11
- Classified as probably not a substantial claim. 11
- The quote only partly states this claim. 1
May combine two claims.
Findingown data§ Quantum simulation of the Klein paradox with trapped ions · p. 3The measured tunnel probabilities for the four slope gradients (Ω̃2/2π = 0, 22, 50, 76 kHz) are 0.017(7), 0.10(1), 0.32(2), and 0.45(3), respectively.
“The measured tunnel probabilities are given by 0.017(7) { 0 } , 0.10(1) { 0 . 07 } , 0.32(2) { 0 . 22 } and 0.45(3) { 0.39 } , here we put results obtained by numerical calculations in curly brackets.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Quantum simulation of the Klein paradox with trapped ions · p. 1By precisely engineering a range of external potentials, the authors were able to simulate textbook relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator.
“By precisely engineering a range of external potentials we are able to simulate text book relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ Quantum simulation of the Klein paradox with trapped ions · p. 3Without a potential, the particle moves to the right with constant velocity.
“Without a potential, the particle moves to the right with constant velocity (a).”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim. May combine two claims.
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1For V > E + mc2 the particle can propagate undamped in the potential barrier, by turning into its anti-particle; this effect is known as the Klein paradox.
“However, for V > E + mc 2 the particle can propagate undamped in the potential barrier, by turning into its anti-particle. This effect is known as the Klein paradox.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [15] O. Klein, Z. Phys. 53 , 157 (1929). source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1For a slightly larger barrier, E + mc2 > V > E - mc2, the particle should completely reflect.
“For a slightly larger barrier, E + mc 2 > V > E -mc 2 , the particle should completely reflect.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [15] O. Klein, Z. Phys. 53 , 157 (1929). source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1Recently, a quantum simulation of the dynamics of a free relativistic particle has been performed in our group.
“Recently, a quantum simulation of the dynamics of a free relativistic particle has been performed in our group [14].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [14] R. Gerritsma, G. Kirchmair, F. Z¨ ahringer, E. Solano, R. Blatt, and C. F. Roos, Nature 463 , 68 (2010). source
Not checked in the free run.
May combine two claims.
Study design and conductcitation§ Quantum simulation of the Klein paradox with trapped ions · p. 3This changes the simulated Hamiltonian somewhat, while also affecting the reconstruction.
“This changes the simulated Hamiltonian somewhat, while also affecting the reconstruction [28].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [28] F. Z¨ ahringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, Phys. Rev. Lett. 104 , 100503 (2010).
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1For barrier heights smaller than the kinetic energy of the electron, V < E - mc2, the particle is predicted to partially transmit.
“For barrier heights smaller than the kinetic energy of the electron, V < E -mc 2 , the particle is predicted to partially transmit.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [15] O. Klein, Z. Phys. 53 , 157 (1929). source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Interpretationnone given§ Quantum simulation of the Klein paradox with trapped ions · p. 3Scaling up the quantum simulation is conceptually straightforward: additional Dirac particles can be simulated by adding more ions and motional modes, and each extra spatial dimension requires an additional motional mode.
“Scaling up our quantum simulation is conceptually straightforward: additional Dirac particles can be simulated by adding more ions and motional modes. Each extra spatial dimension requires an additional motional mode.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ Quantum simulation of the Klein paradox with trapped ions · p. 3Performing classical simulations at the same resolution achieved by the quantum simulations requires describing a 2 × 200 dimensional Hilbert space; adding just one Dirac particle increases the Hilbert space to ≈ 16 × 10^4 dimensions, which would require 10^3 GB of memory just to store the Hamiltonian operator with 64-bit precision.
“Performing classical simulations at the same resolution achieved by the quantum simulations requires describing a 2 × 200 dimensional Hilbert space (one qubit and 200 harmonic oscillator states). Adding just one Dirac particle increases the Hilbert space to ≈ 16 × 10 4 dimensions, which would require 10 3 GB of memory just to store the Hamiltonian operator with 64-bit precision.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1Simulating quantum mechanics using conventional computers rapidly becomes intractable as the physical systems to which it is applied grow larger.
“Simulating quantum mechanics using conventional computers rapidly becomes intractable as the physical systems to which it is applied grow larger.”
Quote in the paperSubstantial claim0 of 3 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Study design and conductreasoning§ Quantum simulation of the Klein paradox with trapped ions · p. 3Both effects cause broadening of the wave packets, whereas systematic errors in state preparation due to slowly varying experimental settings can cause additional structure in the reconstructed data.
“Both effects cause broadening of the wave packets, whereas systematic errors in state preparation due to slowly varying experimental settings can cause additional structure in the reconstructed data.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1A proposed solution is to use highly controlled laboratory quantum systems themselves to perform such simulations.
“A proposed solution is to use highly controlled laboratory quantum systems themselves to perform such simulations [1-3].”
Quote in the paperSubstantial claim0 of 3 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1In its original form, the Klein paradox considers a relativistic electron described by the Dirac equation, with total energy E and rest mass energy mc2, hitting a step-shaped potential barrier of height V.
“In its original form [15] the Klein paradox considers a relativistic electron described by the Dirac equation, with total energy E and rest mass energy mc 2 , hitting a stepshaped potential barrier of height V .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [15] O. Klein, Z. Phys. 53 , 157 (1929). source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ Quantum simulation of the Klein paradox with trapped ions · p. 1A particularly promising approach uses trapped ions, with which several quantum simulations have recently been performed.
“A particularly promising approach uses trapped ions [9-11] with which several quantum simulations have recently been performed.”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [9] D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. BenKish, J. Britton, W. M. Itano, B. Jelenkovi´ c, C. Langer, T. Rosenband, et al., Phys. Rev. Lett. 89 , 247901 (2002). source
Not checked in the free run.
- cites [10] D. Porras and J. I. Cirac, Phys. Rev. Lett. 92 , 207901 (2004). source
Not checked in the free run.
- cites [11] M. Johanning, A. F. Var´ on, and C. Wunderlich, J. Phys. B 42 , 154009 (2009). source
Not checked in the free run.
May combine two claims.
Study design and conductnone given§ Quantum simulation of the Klein paradox with trapped ions · p. 3The internal state coherence time for the ions is ≈ 3 ms, limited by magnetic field fluctuations, while the whole simulation takes up to 1.5 ms.
“The internal state coherence time for the ions is ≈ 3 ms, limited by magnetic field fluctuations, while the whole simulation takes up to 1.5 ms.”
Quote in the paperNot a substantial claimMay combine two claims. The quote only partly states this claim.
Interpretationown data§ Quantum simulation of the Klein paradox with trapped ions · p. 3For a positive energy particle with initial momentum <p>=0.23 ħ/∆ under a quadratic potential, the scattering dynamics correspond to that of a quantum relativistic mass on a spring given a small initial kick and show Klein tunneling at each turning point.
“The scattering dynamics in this case correspond to that of a quantum relativistic mass on a spring that is given a small initial kick and show Klein tunneling at each turning point.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationown data§ Quantum simulation of the Klein paradox with trapped ions · p. 3In a quadratic potential, the wavepacket is still unconfined, due to Klein tunneling, but it spreads more slowly than in the free case.
“The dynamics are clearly different in this case: the wavepacket is still unconfined, due to Klein tunneling, but it spreads more slowly.”
Quote in the paperSubstantial claim
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?
Earlier studies raised a debate about the existence of Dirac surface states, i.e. relativistic surface Tamm states which disappear in the nonrelativistic limit.
“In earlier studies, the Dirac-Kronig-Penney model also attracted some attention and caused a lively debate about the existence of so-called Dirac surface states, i.e. relativistic surface Tamm states which disappear in the nonrelativistic limit [66,70,77,78,79,80,81,82].”
From Classical simulation of relativistic quantum mechanics in periodic optical structures (2011) · cites this paper as [77]
The paper discusses the Klein paradox and relativistic scattering from electrostatic potentials (pp. 1–3). Searches for Dirac-Kronig-Penney models, surface Tamm states, and their nonrelativistic limit found no passage addressing the claimed debate. (LLM, conf=0.76)
Experimental — accuracy not yet measuredA quantum simulation of the Dirac equation in trapped ions is a useful playground to analyze different regimes and to verify in a table-top experiment predicted phenomena such as Zitterbewegung or the Klein paradox.
“A quantum simulation of the Dirac equation is a useful playground to analyze different regimes and to verify in a table-top experiment predicted phenomena such as Zitterbewegung [9,10] or the Klein paradox [12,13].”
From Relativistic quantum mechanics with trapped ions (2011) · cites this paper as [13]
The experimental simulation of the Klein paradox relied on the manipulation of two trapped ions.
“The experimental simulation of the Klein paradox relied on the manipulation of two trapped ions [13].”
From Relativistic quantum mechanics with trapped ions (2011) · cites this paper as [13]
The implementation of the proposed quantum simulations requires the validity of the Lamb-Dicke approximation, which for Ca+ ions in a trap with secular frequency of ∼1 MHz and manipulated with 729 nm laser light is well justified (η ∼ 0.05) [13].
“The implementation of the proposed quantum simulations requires the validity of the Lamb-Dicke approximation, which for Ca + ions in a trap with secular frequency of ∼1 MHz and manipulated with 729 nm laser light is well justified (the Lamb-Dicke parameter η ∼ 0.05 1) [13].”
From Relativistic quantum mechanics with trapped ions (2011) · cites this paper as [13]
In recent experiments [13], it has been shown that the motional state of the ions can be coherently manipulated while reaching states that have >100 phonons on average, and the coherence can be sustained for more than 10 ms.
“In recent experiments [13], it has been shown that the motional state of the ions can be coherently manipulated while reaching states that have >100 phonons on average, and the coherence can be sustained for more than 10 ms.”
From Relativistic quantum mechanics with trapped ions (2011) · cites this paper as [13]
The spinor degrees of freedom can be encoded in long-lived electronic states of Ca+ ions, as was done before in quantum simulations of relativistic quantum mechanics [10,13].
“The spinor degrees of freedom can be encoded in long-lived electronic states of this ion, as was done before in quantum simulations of relativistic quantum mechanics [10,13].”
From Relativistic quantum mechanics with trapped ions (2011) · cites this paper as [13]
The degree of quantum control is even better in ultracold-ion systems, but these are so far limited to few ions.
“The degree of quantum control is even better in ultracold-ion systems, but these are so far limited to few ions [21-23, 26, 27].”
From Can one trust quantum simulators? (2012) · cites this paper as [27]
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References 27 · Cited by 204
- Authors
- Gerritsma, R., Lanyon, B. P., Kirchmair, G., Zähringer, F., Hempel, C., Casanova, J., García-Ripoll, J. J., Solano, E., Blatt, R., & Roos, C. F.
- Affiliations
- Austrian Academy of Sciences (AT)
- University of the Basque Country (ES)
- Ikerbasque (ES)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2011-02-11
- Volume
- 106
- Issue
- 6
- Pages
- 060503-060503
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2072216248
- Cited by
- 204
- References
- 27
- Citations by year
- 2026: 2 · 2025: 4 · 2024: 7 · 2023: 4 · 2022: 15 · 2021: 5 · 2020: 11 · 2019: 12 · 2018: 13 · 2017: 17 · 2016: 25 · 2015: 15 · 2014: 18 · 2013: 9 · 2012: 28
- FWCI
- 12.82
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- DIGITAL.CSIC (Spanish National Research Council (CSIC)) (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
- Klein paradox, Klein tunneling, trapped ions, quantum simulation, Dirac equation, relativistic scattering, analog quantum simulation, scattering potential
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