Quantum Simulation of the Two-Dimensional Weyl Equation in a Magnetic Field

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.

Figure 1

Schematic of simulating a 2D Weyl fermion using ion trap. We apply a pair of Raman laser beams on a trapped ion, with different frequency components resonant to the red and the blue sidebands of both the $x$ and $y$ oscillation modes, to couple the internal states of the ion to its spatial oscillation modes. By further tuning the phase of these frequency components in the Raman laser, we can choose to couple to ${\widehat{\sigma }}_x$ or ${\widehat{\sigma }}_y$ of the internal states, and to different quadratures $\widehat{x}$ or $\widehat{p}$ of each mode, which finally gives us the Hamiltonian in Eq. (2).

Figure 2

Linear dispersion relation of a free Weyl particle and its spectrum in a magnetic field. (a) Spin dynamics $⟨{\widehat{\sigma }}_y(t)⟩$ from an initial state $|+z⟩|{\alpha }_x=ip/\sqrt{2}⟩|{\alpha }_y=0⟩$ with $B=0$ in Eq. (2). Each data point consists of 500 experimental shots and the error bars are estimated from the standard deviation of 10 repetitions. The solid curves are theoretical predictions under the same parameters. (b) From the slope of the early-time spin dynamics (inset), we can extract the average energy $E(p)$ for a free Weyl particle with momentum $p$. A linear dispersion relation is observed as the black fitting line. The error bars are standard deviations of 5 repetitions. (c) Spin dynamics $⟨{\widehat{\sigma }}_z(t)⟩$ in a magnetic field with $eB=1$ from an initial state $|+z⟩|{\alpha }_x=i⟩|{\alpha }_y=0⟩$. Because of the larger number of data points and the longer evolution time, here we only repeat each point by 200 times and the error bars are 1 standard deviation of the average value. The red curve is the theoretical prediction with the motional decoherence included. (d) The energy spectrum of a Weyl fermion in a magnetic field can be obtained by a Fourier transform of the spin dynamics. The first three peaks at $2E_0=0$, $2E_1=2\pi \times 8.3$ and $2E_2=2\pi \times 11.7\mathrm{kHz}$ agree well with the theoretical $E_n\propto \sqrt{n}$ scaling, while higher peaks are more difficult to distinguish due to the frequency resolution limited by the total evolution time of $600\mu \mathrm{s}$. Inset is the ideal result for an evolution time of 5 ms without decoherence.

Figure 3

Spin and spatial dynamics and conservation of helicity. (a),(b) Spin dynamics in a magnetic field $eB=1$ from the initial state $|+x⟩|{\alpha }_x=i⟩|{\alpha }_y=0⟩$. Each point is measured by 500 times and the error bar is estimated as 1 standard deviation of the average. Solid curves are theoretical results under the same parameters. (c),(d) Evolution of the kinetic momentum $\widehat{\mathit{\pi }}=\widehat{\mathit{p}}-e\widehat{\mathit{A}}$ under the same condition. Each quadrature is measured by applying an additional spin-dependent force and fitting the early-time evolution (see Supplemental Material [27]), with error bars estimated from one standard deviation of the fitting. The error bar of $⟨{\widehat{\pi }}_y⟩=⟨{\widehat{p}}_y⟩-⟨\widehat{x}⟩$ is further computed from those of $⟨{\widehat{p}}_y⟩$ and $⟨\widehat{x}⟩$ as shown in the inset. (e) We plot the ratio between the $x$ and $y$ components of the spin (red) and the kinetic momentum (blue). The two values are close to each other and follow the same tendency when the ratios by themselves change orders of magnitudes, although small difference exists. Solid curves are the ideal theoretical results. (f) We further compute the azimuthal angles of the spin and the kinetic momentum in the $x\text{−}y$ plane. The data points distribute around the diagonal, which indicates that the spin and the kinetic momentum roughly align with each other during the time evolution in a magnetic field, namely, the conservation of helicity.

Figure 4

Trajectories of 2D Weyl particles with opposite helicities. Using the method of Fig. 3, we can plot the trajectory in a magnetic field $eB=1$ from the initial state $|+x⟩|{\alpha }_x=i⟩|{\alpha }_y=0⟩$ (red) and $|-x⟩|{\alpha }_x=i⟩|{\alpha }_y=0⟩$ (blue). These correspond to a Weyl particle with positive helicity and an antiparticle with negative helicity, respectively. Theoretically, the particles circle in real space with 2D Zitterbewegung as shown by the solid curves. Here our measurement precision cannot resolve the Zitterbewegung, but it can clearly be seen that the particle and the antiparticle have opposite charges as they rotate in opposite directions in a magnetic field.