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Quantum simulation of the Dirac equation

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The Dirac equation successfully merges quantum mechanics with special relativity. It provides a natural description of the electron spin, predicts the existence of antimatter 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 paradox and 'Zitterbewegung', an unexpected quivering motion of a free relativistic quantum particle. 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-ups, 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 ion 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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... However, many theoretical proposals already exist [14][15][16][17][18] to examine its effects by quantum simulation 19,20 . As one of the leading platforms for quantum information processing with long coherence time, convenient initialization and readout, as well as accurate laser or microwave control [21][22][23][24][25] , ion trap has demonstrated the quantum simulation of various phenomena such as quantum phase transitions 26,27 , many-body dynamics 26 , relativistic effects 28 and quantum field theories 29 . In this work, we report experimental realization of a prototypical SUSY QM model 14 in a trapped ion quantum simulator and demonstrate the spontaneous SUSY breaking in this model. ...
... The former can be derived by the standard procedure of first resetting the spin state to |↓〉 and then driving the phonon blue-sideband to fit the phonon number population from the spin dynamics 21 , as shown in Fig. 3a, b. As for the second term, we apply a spin-dependent force H SDF = (−Ω p /2)σ y (a + a † ) and measure the evolution of σ z (t) by observing that e iH SDF t σ z e ÀiH SDF t has a linear term in t as Ω p tσ x (a + a † ) 28 . Therefore, after preparing |ψ ± 〉, we adjust the parameters of the bichromatic Raman laser beams to create this spindependent force and fit 〈σ z (t)〉 to extract the slope at t = 0 (see Methods), from which we obtain 〈σ x (a + a † )〉, as shown in Fig. 3c. ...
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Supersymmetry (SUSY) helps solve the hierarchy problem in high-energy physics and provides a natural groundwork for unifying gravity with other fundamental interactions. While being one of the most promising frameworks for theories beyond the Standard Model, its direct experimental evidence in nature still remains to be discovered. Here we report experimental realization of a supersymmetric quantum mechanics (SUSY QM) model, a reduction of the SUSY quantum field theory for studying its fundamental properties, using a trapped ion quantum simulator. We demonstrate the energy degeneracy caused by SUSY in this model and the spontaneous SUSY breaking. By a partial quantum state tomography of the spin-phonon coupled system, we explicitly measure the supercharge of the degenerate ground states, which are superpositions of the bosonic and the fermionic states. Our work demonstrates the trapped-ion quantum simulator as an economic yet powerful platform to study versatile physics in a single well-controlled system. Quantum simulators should be able to give insight on exotic physics models such as supersymmetric extensions of Standard Model. Here, the authors demonstrate a first step in this direction, realising a prototypical SUSY model (and spontaneous SUSY breaking within it) using a trapped ion quantum simulator.
... This leaves many properties of the Weyl particles only to be analyzed theoretically, or through the idea of quantum simulation [6,7] using other well-controlled quantum systems. Indeed, quantum simulation of relativistic quantum mechanical systems has been proposed [8][9][10][11][12][13] and performed [14][15][16][17] in various physical systems like trapped ions [18][19][20]. ...
... To date, Weyl fermions have been realized in photonic crystals [21] and in condensed matter systems [22,23], but in these systems only the spectral or the transport properties can be measured [24], while direct study of the Weyl particle dynamics is still lacking. On the other hand, massive Dirac particles have been simulated in ion trap [14,15], which can reduce to the massless Weyl particles by tuning the experimental parameters. Nevertheless, to minimize the required degrees of freedom to be controlled, the experiments so far are restricted to dynamics in 1D, where interactions with external magnetic fields and evolution of spin states become trivial. ...
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Quantum simulation of 1D relativistic quantum mechanics has been achieved in well-controlled systems like trapped ions, but properties like spin dynamics and response to external magnetic fields that appear only in higher dimensions remain unexplored. Here we simulate the dynamics of a 2D Weyl particle. We show the linear dispersion relation of the free particle and the discrete Landau levels in a magnetic field, and we explicitly measure the spatial and spin dynamics from which the conservation of helicity and properties of antiparticles can be verified. Our work extends the application of an ion trap quantum simulator in particle physics with the additional spatial and spin degrees of freedom.
... This not only gives access to the average ion energy, but can also be used to measure the energy distribution, the amplitude and phase of its motion and the probability distribution in phase space, see e.g. (Wallentowitz and Vogel, 1995;Meekhof et al., 1996;Leibfried et al., 1996Leibfried et al., , 1998Leibfried et al., , 2003Lougovski et al., 2006;Santos et al., 2007;Lamata et al., 2007;Schmitz et al., 2009;Gerritsma et al., 2010;Zähringer et al., 2010;Flühmann and Home, 2020). These techniques were developed in the context of trapped ion quantum computing and rely on laser-induced qubitmotion coupling. ...
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... The former can be derived by the standard procedure of first resetting the spin state to | ↓ and then driving the phonon blue-sideband to fit the phonon number population from the spin dynamics 21 , as shown in Fig. 3a, b. As for the second term, we apply a spin-dependent force H SDF = (−Ω p /2)σ y (a + a † ) and measure the evolution of σ z (t) by observing that e iHSDFt σ z e −iHSDFt has a linear term in t as Ω p tσ x (a + a † ) 28 . Therefore, after preparing |ψ ± , we adjust the parameters of the bichromatic Raman laser beams to create this spin-dependent force and fit σ z (t) to extract the slope at t = 0 (see Methods), from which we obtain σ x (a + a † ) , as shown in Fig. 3c. ...
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Full-text available
Supersymmetry (SUSY) helps solve the hierarchy problem in high-energy physics and provides a natural groundwork for unifying gravity with other fundamental interactions. While being one of the most promising frameworks for theories beyond the Standard Model, its direct experimental evidence in nature still remains to be discovered. Here we report experimental realization of a supersymmetric quantum mechanics (SUSY QM) model, a reduction of the SUSY quantum field theory for studying its fundamental properties, using a trapped ion quantum simulator. We demonstrate the energy degeneracy caused by SUSY in this model and the spontaneous SUSY breaking. By a partial quantum state tomography of the spin-phonon coupled system, we explicitly measure the supercharge of the degenerate ground states, which are superpositions of the bosonic and the fermionic states. Our work demonstrates the trapped-ion quantum simulator as an economic yet powerful platform to study versatile physics in a single well-controlled system.
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