Bosonic analog of the Klein paradox

The standard Klein paradox describes how an incoming electron scatters off a supercritical electrostatic barrier that is so strong that it can generate electron-positron pairs. This fermionic system has been widely discussed in textbooks to illustrate some of the discrepancies between quantum mechanical and quantum field theoretical descriptions for the pair creation process. We compare the fermionic dynamics with that of the corresponding bosonic system. We point out that the direct counterpart of the Pauli exclusion principle (the central mechanism to resolve the fermionic Klein paradox) is stimulated emission, which leads to the resolution of the analogous bosonic paradox.

Figure 1

The final charge distribution of the bosons ${\rho }_{+}(z,t)$ for $z<0$ and antibosons ${\rho }_{-}(z,t)$ for $0<z$ at time $t=4.86\times {10}^{-3}$ a.u. as created by a supercritical potential step $V=5c^2$ in the presence of an incoming boson. The dotted line is the initial distribution of the incoming boson, ${\rho }_0(z)$. The numbers $N_{\pm }$ indicate the areas under the two densities. For comparison, the dashed line for $z<0$ shows the density created at the barrier in the absence of any incoming boson with an area of 14.74. [numerical lattice with 6000 temporal ($\Delta t=9.73\times {10}^{-7}$ a.u.) and 2048 spatial ($\Delta z=1.37\times {10}^{-3}$ a.u.) grid points, initial incoming wave packet has a spatial width $\Delta z_0=8\times {10}^{-3}$ a.u. and energy $E=2.5c^2$, the potential barrier has a width $W=6.57\times {10}^{-3}$ a.u.]

Figure 2

The total number of created boson pairs $N(t)$ as a function of time during the time interval when the incoming boson reflects at the boundary. For comparison, the dashed line is the number in the absence of any incoming boson. (All parameters as in Fig. 1.)