Noncompeting Channel Approach to Pair Creation in Supercritical Fields

The Dirac and Klein-Gordon equations are solved on a space-time grid to study the strong-field induced pair creation process for bosons and fermions from the vacuum. If the external field is sufficiently strong to induce bound states that are embedded in the negative energy continuum, a complex scaling technique of the Hamiltonian can predict the longtime behavior of the dynamics. In the case of multiple bound states this technique predicts the occurrence of a new collective time scale. The longtime behavior of the pair creation is not determined by a single (most important) channel, but collectively by the sum of all individual widths of the embedded states.

Figure 1

The energy spectrum as a function of the strength of the external potential. For $V_0>2.42c^2$. the first discrete state has “dived” into the negative energy continuum. For $V_0=3.6c^2$, there are two discrete states (close to energies $E_2\approx -1.3c^2$ and $E_1\approx -2.1c^2$) embedded in the negative energy continuum. [$V(x)$ is given in the text, with $W=0.3/c$, $D=3.2/c$.]

Figure 2

Energy density $\rho (E)$ of the total (Hermitian) Hamiltonian $H_D$ as a function of energy. Two bound states are embedded in the Dirac Sea. The density is peaked at $E_1=-2.16c^2$ (with a peak height of $11697$) and $E_2=-1.33c^2$ (peak height 5246). [$W=0.3/c$ and $D=3.2/c$ and $V_0=3.6c^2$.]

Figure 3

The final number of created particles as a function of the interaction time for fermions with ($A$) $V_0=3.6c^2$ and ($B$) $V_0=2.66c^2$. The inset presents $n(t)\equiv N(t\to \infty )-N(t)$ for longer times with a logarithmic $y$ axis. The slopes of the matched straight lines are $-0.08602c^2$ and $-0.17247c^2$. [The other parameters are $W=0.3/c$ and $D=3.2/c$.]

Figure 4

The positron emission spectrum as a function of the kinetic energy $E_k$. [Same parameters as in Fig. 1 and $T=5.7\times {10}^{-3}\mathrm{a}.\mathrm{u}.$]