Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

We present a method of simulating the Dirac equation in $3+1$ dimensions for a free spin-$1/2$ particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified $1+1$ case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein’s paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.

Figure 1

Energy levels of a ${}^{25}\mathrm{Mg}^{+}$ ion confined in a trap [22, 23]. Shown are the relevant types of two-photon Raman transitions between states $|a,n=1⟩$ and $|b,n=0,1,2⟩$: red sideband (dotted line), carrier (solid line) and blue sideband (dashed line); see also the resonant transition for state sensitive detection. We subsumed excited levels of the $P_{1/2}$ and $P_{3/2}$ states.