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Quantum simulation of the Dirac equation

Abstract

The Dirac equation1 successfully merges quantum mechanics with special relativity. It provides a natural description of the electron spin, predicts the existence of antimatter2 and is able to reproduce accurately the spectrum of the hydrogen atom. The realm of the Dirac equation—relativistic quantum mechanics—is considered to be the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects, such as Klein’s paradox3 and ‘Zitterbewegung’, an unexpected quivering motion of a free relativistic quantum particle4. These and other predicted phenomena are key fundamental examples for understanding relativistic quantum effects, but are difficult to observe in real particles. In recent years, there has been increased interest in simulations of relativistic quantum effects using different physical set-ups5,6,7,8,9,10,11, in which parameter tunability allows access to different physical regimes. Here we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion7 set to behave as a free relativistic quantum particle. We measure the particle position as a function of time and study Zitterbewegung for different initial superpositions of positive- and negative-energy spinor states, as well as the crossover from relativistic to non-relativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate textbook examples of relativistic quantum physics.

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Figure 1: Expectation values, 〈 (t )〉, for particles with different masses.
Figure 2: Zitterbewegung for a state with non-zero average momentum.
Figure 3: Time evolution of a negative-energy eigenstate with λC = 1.2 Δ.

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Acknowledgements

We gratefully acknowledge support from the Austrian Science Fund, the European Commission (Marie-Curie programme) and the Institut für Quanteninformation GmbH. E.S. acknowledges support of Universidad del País Vasco - Euskal Herriko Unibertsitatea grant GIU07/40 and European Union project EuroSQIP. This material is based upon work supported in part by Intelligence Advanced Research Projects Activity. We thank H. Häffner and M. Baranov for comments on the manuscript.

Author Contributions G.K., F.Z. and R.G. performed the experiment and analysed the data; E.S. provided the theoretical analysis; C.F.R. and R.G. designed the experiment and carried out numerical simulations; R.B., G.K., F.Z., R.G. and C.F.R. contributed to the experimental set-up; and all authors co-wrote the paper.

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Correspondence to C. F. Roos.

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Gerritsma, R., Kirchmair, G., Zähringer, F. et al. Quantum simulation of the Dirac equation. Nature 463, 68–71 (2010). https://doi.org/10.1038/nature08688

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