ArticlePDF Available

Quantum simulation of the Dirac equation

Authors:

Abstract and Figures

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.
Content may be subject to copyright.
A preview of the PDF is not available
... 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. ...
Article
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. 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. ...
Preprint
Full-text available
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. ...
Preprint
Full-text available
Experimental setups that study laser-cooled ions immersed in baths of ultracold atoms merge the two exciting and well-established fields of quantum gases and trapped ions. These experiments benefit both from the exquisite read-out and control of the few-body ion systems as well as the many-body aspects, tunable interactions, and ultracold temperatures of the atoms. However, combining the two leads to challenges both in the experimental design and the physics that can be studied. Nevertheless, these systems have provided insights into ion-atom collisions, buffer gas cooling of ions and quantum effects in the ion-atom interaction. This makes them promising candidates for ultracold quantum chemistry studies, creation of cold molecular ions for spectroscopy and precision measurements, and as test beds for quantum simulation of charged impurity physics. In this review we aim to provide an experimental account of recent progress and introduce the experimental setup and techniques that enabled the observation of quantum effects.
... 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. ...
Preprint
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.
Article
The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’s equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac’s equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrödinger equation, and the other is the NKE Schrödinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author’s opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.
Article
Non-Hermitian (NH) quantum theory has been attracting increased research interest due to its featured properties, novel phenomena, and links to open and dissipative systems. Typical NH systems include PT-symmetric systems, pseudo-Hermitian systems, and their anti-symmetric counterparts. In this work, we generalize the pseudo-Hermitian systems to their complex counterparts, which we call pseudo-Hermitian-φ-symmetric systems. This complex extension adds an extra degree of freedom to the original symmetry. On the one hand, it enlarges the non-Hermitian class relevant to pseudo-Hermiticity. On the other hand, the conventional pseudo-Hermitian systems can be understood better as a subgroup of this wider class. The well-defined inner product and pseudo-inner product are still valid. Since quantum simulation provides a strong method to investigate NH systems, we mainly investigate how to simulate this novel system in a Hermitian system using the linear combination of unitaries in the scheme of duality quantum computing. We illustrate in detail how to simulate a general P-pseudo-Hermitian-φ-symmetric two-level system. Duality quantum algorithms have been recently successfully applied to similar types of simulations, so we look forward to the implementation of available quantum devices.
Preprint
Full-text available
Electrical energy is considered as a fundamental parameter for inclusion in Fermi gas theory, in addition to thermal energy. It is argued that electrical energy can move some electrons to above the Fermi Level, providing free charges to carry the electrical current, even at absolute zero temperature. The Drude model, Ohm's law, quantum resistance, and the electrical resistivity due to electron-electron scattering appear naturally as a consequence of the theoretical description, which is based on the quantization of the angular momentum and the Fermi-Dirac distribution, considering total energy as ${\epsilon}$ = k$_B$$T + {\Phi}_0$$I$. The electrical and magnetic forces acting on an electron are related to the ratio between the Fermi velocity and the speed of light and show that the electron motion is due to helical paths. Considering the center of mass description for the Bohr atom, it was possible to show that the magnetic force is related to the electrical force as $F_M$ = (${\alpha}$/${\pi}$) $F_E$, which demonstrates that the electrons move in helical paths along the orbit. The helical motion naturally provides for quantization of the magnetic flux, the spin of the electron, and the first correction term of the anomalous magnetic moment. Applying the model to describe the electron-electron scattering allows prediction of the behavior of the electrical resistivity of many metals at low temperatures, which is in excellent agreement with empirical observations.
Chapter
Experimental setups that study laser-cooled ions immersed in baths of ultracold atoms merge the two exciting and well-established fields of quantum gases and trapped ions. These experiments benefit both from the exquisite read-out and control of the few-body ion systems as well as the many-body aspects, tunable interactions, and ultracold temperatures of the atoms. However, combining the two leads to challenges both in the experimental design and the physics that can be studied. Nevertheless, these systems have provided insights into ion-atom collisions, buffer gas cooling of ions and quantum effects in the ion-atom interaction. This makes them promising candidates for ultracold quantum chemistry studies, creation of cold molecular ions for spectroscopy and precision measurements, and as test beds for quantum simulation of charged impurity physics. In this review we aim to provide an experimental account of recent progress and introduce the experimental setup and techniques that enabled the observation of quantum effects.
Article
Full-text available
The so-called Klein paradox-unimpeded penetration of relativistic particles through high and wide potential barriers-is one of the most exotic and counterintuitive consequences of quantum electrodynamics. The phenomenon is discussed in many contexts in particle, nuclear and astro-physics but direct tests of the Klein paradox using elementary particles have so far proved impossible. Here we show that the effect can be tested in a conceptually simple condensed-matter experiment using electrostatic barriers in single- and bi-layer graphene. Owing to the chiral nature of their quasiparticles, quantum tunnelling in these materials becomes highly anisotropic, qualitatively different from the case of normal, non-relativistic electrons. Massless Dirac fermions in graphene allow a close realization of Klein's gedanken experiment, whereas massive chiral fermions in bilayer graphene offer an interesting complementary system that elucidates the basic physics involved.
Article
Full-text available
The control of internal and motional quantum degrees of freedom of laser cooled trapped ions has been subject to intense theoretical and experimental research for about three decades. In the realm of quantum information science the ability to deterministically prepare and measure quantum states of trapped ions is unprecedented. This expertise may be employed to investigate physical models conceived to describe systems that are not directly accessible for experimental investigations. Here, we give an overview of current theoretical proposals and experiments for such quantum simulations with trapped ions. This includes various spin models (e.g., the quantum transverse Ising model, or a neural network), the Bose-Hubbard Hamiltonian, the Frenkel-Kontorova model, and quantum fields and relativistic effects.
Article
Full-text available
To gain deeper insight into the dynamics of complex quantum systems we need a quantum leap in computer simulations. We cannot translate quantum behaviour arising from superposition states or entanglement efficiently into the classical language of conventional computers. The solution to this problem, proposed in 1982 (ref.1), is simulating the quantum behaviour of interest in a different quantum system where the interactions can be controlled and the outcome detected sufficiently well. Here we study the building blocks for simulating quantum spin Hamiltonians with trapped ions. We experimentally simulate the adiabatic evolution of the smallest non-trivial spin system from paramagnetic into ferromagnetic order with a quantum magnetization for two spins of 98. We prove that the transition is not driven by thermal fluctuations but is of quantum-mechanical origin (analogous to quantum fluctuations in quantum phase transitions). We observe a final superposition state of the two degenerate spin configurations for the ferromagnetic order (|+|), corresponding to deterministic entanglement achieved with 88 fidelity. This method should allow for scaling to a higher number of coupled spins, enabling implementation of simulations that are intractable on conventional computers.
Article
Full-text available
We propose an optical lattice scheme which would permit the experimental observation of Zitterbewegung (ZB) with ultracold, neutral atoms. A four-level tripod variant of the setup for stimulated Raman adiabatic passage (STIRAP) has previously been proposed for generating non-Abelian gauge fields. Dirac-like Hamiltonians, which exhibit ZB, are simple examples of such non-Abelian gauge fields; we show how a variety of them can arise, and how ZB can be observed, in a tripod system. We predict that the ZB should occur at experimentally accessible frequencies and amplitudes.
Book
This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics.
Article
Es wird die Reflexion von Elektronen an einem Potentialsprung nach der neuen Diracschen Dynamik untersucht. Bei sehr groen Werten des Potentialsprungs dringen der Theorie zufolge Elektronen gegen die auf sie wirkende elektrische Kraft durch die Sprungflche und kommen auf der anderen Seite mit einer negativen kinetischen Energie an. Dies drfte als ein besonders schroffes Beispiel der von Dirac hervorgehobenen Schwierigkeit der relativistischen Dynamik zu betrachten sein.
Article
It is shown, for the first time, that the Zitterbewegung of photons can appear near the Dirac point in a two-dimensional photonic crystal. The superiority of such a phenomenon for photons is that it can be found in different scaling structures with wide frequency regions. It can be observed by measuring the time dependence of the transmission coefficient through photonic crystal slabs. Thus, it is particularly suited for experimentally observing this effect. We have observed such a phenomenon by exact numerical simulations, confirming a long-standing theoretical prediction.