Zitterbewegung of relativistic electrons in a magnetic field and its simulation by trapped ions

One-electron $3+1$ and $2+1$ Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion, Zitterbewegung (ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron ZB: it introduces intraband frequency components into the motion, influences all the frequencies, and makes the motion stationary (not decaying in time) in case of the $2+1$ Dirac equation. Finally, simulations of the $2+1$ Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers, we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.

Figure 1

Calculated motion of wave packet with the second nonzero component during first 200 $t_c\simeq 0.25$ attoseconds of motion for various wave packet parameters. The magnetic field corresponds to $L={\lambda }_c$. Packet parameters are $d_x=1.5{\lambda }_c$, $d_z=1.8{\lambda }_c$, $k_{0x}=0.998{\lambda }_c^{-1}$, $k_{0z}=0$. Time scale is in $t_c=\hslash /(m{c}^2)=1.29\times {10}^{-21}\mathrm{s}$ units, while position is in ${\lambda }_c=\hslash /(mc)=3.86\times {10}^{-13}\mathrm{m}$ units.

Figure 2

Calculated motion of wave packet with the second nonzero component and nonzero velocity in the $z$ direction. Packet parameters are$d_x=2.0{\lambda }_c$, $d_y=1.8{\lambda }_c$, $d_z=1.5{\lambda }_c$, $k_{0x}=0.673{\lambda }_c^{-1}$.

Figure 3

Trajectories of wave packets with the second nonzero component for $3+1$ Dirac equation in various magnetic fields. Packet parameters are $d_x=0.632(B_b/B{)}^{0.5}{\lambda }_c$, $d_y=0.569(B_b/B{)}^{0.5}{\lambda }_c$, $d_z=0.474(B_b/B{)}^{0.5}{\lambda }_c$, $k_{0z}=0$, $k_{0x}=0.999(B/B_b{)}^{0.5}{\lambda }_c^{-1}$, where $B_b=2\times {10}^{10}\mathrm{T}$. The products $L{k}_{0x}$, $d_x{k}_{0x}$, $d_y{k}_{0x}$, and $d_z{k}_{0x}$ are the same for all figures. In all cases the packet motion is transient, but for lower magnetic fields the decay of oscillations is slow.

Figure 4

Calculated ZB components of electron motion in a magnetic field in two very different time scales. Packet parameters are $d_x=20000{\lambda }_c$, $d_y=18000{\lambda }_c$, $d_z=15000{\lambda }_c$, $k_{0x}=0.5L^{-1}=8.72\times {10}^7{\mathrm{m}}^{-1}$, $k_{0z}=0$. The ZB oscillations decay as $t^{-1/2}$, but they have very small amplitudes. Note the collapse-and-revival character of ZB oscillations.

Figure 5

Calculated motion of two-component wave packet simulated by trapped ${}^{40}\mathrm{Ca}^{+}$ ions for three values of effective rest energies $\hslash \Omega$. Trap parameters are $\eta =0.06$, $\tilde{\Omega }=2\pi \times 68\mathrm{kHz}$, $\Delta \simeq 96\AA$; packet parameters are $k_{0x}={\Delta }^{-1}$, $d_y=\Delta \sqrt{2}$, $d_x=0.9d_y$. Simulations correspond to $\kappa =\hslash {\omega }_c/2m{c}^2=16.65$ (a), 0.26 (b), 0.116 (c), respectively. Positions are given in $L=\sqrt{2}\Delta$ units. Oscillations do not decay in time.

Figure 6

Trajectories of electron wave packet in a constant magnetic field for various simulated rest energies $\hslash \Omega$, as calculated for $2+1$ Dirac equation. Trap and packet parameters are the same as in Fig. 5. Positions are given in magnetic radius $L$. In the nonrelativistic limit (a), the ZB is practically absent. As the rest energy decreases, the motion becomes more relativistic and the ZB (interband) frequency components become stronger. The ratio $\kappa$ defined in Eq. (110) is (a) 0.0018, (b) 0.029, (c) 0.116, (d) 1.05.

Figure 7

Simulated ZB motion of one-component packets in the regime $\kappa \gg 1$. Simulated gap frequency is $\Omega =2\pi \times 1000\mathrm{Hz}$; other trap and packet parameters are as in Fig. 5. Upper part: packet with the first nonzero component; lower part: packet with the second nonzero component. Note largely different magnitudes of the $x$ oscillations in the two cases.

Figure 8

Collapse-and-revival cycles of packet motion for simulations using $3+1$ DE (a) and $2+1$ DE (b). Packet parameters are $d_x=d_y=d_z=L$, $k_{0x}={\Delta }^{-1}$. Trap parameters are as in Fig. 5, simulated gap frequency is $\Omega =2\pi \times 12000\mathrm{Hz}$. Note transient character of motion for the $3+1$ DE and persistent oscillations for the $2+1$ DE. In both cases, the collapse-and-revival cycles appear.

Figure 9

Calculated time evolution of dynamic averages: (a) $⟨{\widehat{A}}_1(t){⟩}^{2,2}$, (b) $⟨{\widehat{A}}_2(t){⟩}^{2,2}$, (c) $⟨{\widehat{A}}_1^{+}(t){⟩}^{2,2}$, (d) $⟨{\widehat{A}}_2^{+}(t){⟩}^{2,2}$, as given in Eqs. (e2, e3, e4, e5), for $2+1$ DE. Trap parameters as in Fig. 5, simulated gap frequency $\Omega =2\pi \times 4000\mathrm{Hz}$. Packet parameters are $d_x=0.63{\lambda }_c$, $d_y=0.57{\lambda }_c$, $k_{0x}=0.999{\lambda }_c^{-1}$. Motion is plotted for $0<t<8\mathrm{ms}$.