Klein tunneling and Dirac potentials in trapped ions

We propose the quantum simulation of the Dirac equation with potentials, allowing the study of relativistic scattering and Klein tunneling. This quantum relativistic effect permits a positive-energy Dirac particle to propagate through a repulsive potential via the population transfer to negative-energy components. We show how to engineer scalar, pseudoscalar, and other potentials in the $1+1$ Dirac equation by manipulating two trapped ions. The Dirac spinor is represented by the internal states of one ion, while its position and momentum are described by those of a collective motional mode. The second ion is used to build the desired potentials with high spatial resolution.

Figure 1

Setup for simulating a Dirac particle in an external potential. A string of two trapped ions is manipulated with addressed laser beams, coupling to each ion’s internal state and the collective motion. The Hamiltonian for a free Dirac particle and the potentials $A$, $V$, and $\mathrm{Ṽ}$ can be implemented directly by the laser light impinging on ion 1. The electrostatic potential $\varphi$ is implemented by red and blue sidebands applied to the auxiliary ion to create an interaction $\propto {\sigma }_2^x\mathrm{x̂}$, with the auxiliary ion prepared in an eigenstate of ${\sigma }_2^x$.

Figure 2

Klein tunneling in a repulsive potential, ${\varphi }_{\mathrm{el}}(x)={\upsilon }_{\mathrm{el}}x.$ (a) In position space, the particle may bounce back (I) or enter the forbidden region by reducing its kinetic energy (II). (b) In momentum space, I and II correspond to the particle having positive energy or turning into an antiparticle.

Figure 3

Scattering events for three effective masses, $m=0$, 0.5, and 1 (left to right). We plot the total probability density as a function of space $x$ and time $t.$ We have used natural units $c=\hslash =1,$ and a fixed potential slope $c{\upsilon }_{\mathrm{el}}=1.$