Degeneracies of discrete and continuum states with the Dirac sea in the pair-creation process

By solving the quantum field theoretical Dirac and Klein-Gordon equations numerically we analyze the different roles of discrete and continuum states for the pair-creation process from the vacuum. If this process is solely induced by discrete states that have become degenerate with the negative energy continuum, the complex scaling method can predict the time scales associated with the bosonic as well as fermionic growth of the particle yield. We show that for bosons the long-time exponential growth behavior is not affected by additional continuum-continuum degeneracies, while for fermions they characterize the asymptotic (linear) growth. We illustrate the impact of these degeneracies on the temporally resolved energy and spatial distributions of the created particles.

Figure 1

The effect of the transformation $x\to x\exp \left(i\theta \right)$ on the spectrum of the relativistic Hamiltonian. The bound states as well as the threshold where the continuum begins are invariant. The continuous portion of the spectrum “rotates” about the threshold by −θ and the complex resonance eigenvalues may be exposed. Such eigenvalues are originally “hidden” on a higher Riemann sheet for θ = 0, but they will become exposed if the cuts are appropriately moved.

Figure 2

The behavior of complex energy as a function of the complex scaling angle θ. The left (right) panel is for fermions (bosons). The scale angle θ changes from 0.13 to 0.27 rad for fermions and from 0.20 to 0.30 rad for bosons. The θ interval of any two adjacent dots in the figures is the same.

Figure 3

Sketch of the bound state resonance responsible for the pair creation. The gray region is the continuum and the dotted or solid lines in the potential well $qV\left(x\right)$ represent bound states. The bound states in the potential well that have dived into the negative continuum (solid lines) are associated with the pair creation.

Figure 4

The energy spectrum of the Dirac-Hamiltonian as a function of potential height $V_0$ for a fermionic system. The parameters are $W$ = 0.3/$c$ and $D$ = 3.2/$c$. It is apparent that there is no discrete energy in the Dirac sea unless $V_0$ exceeds 2.35$c^2$.

Figure 5

The number of created electrons as a function of the interaction time with (A) $V_0$ = 2.15$c^2$, (B) $V_0$ = 2.66$c^2$, and (C) $V_0$ = 3.6$c^2$. The other parameters are $W$ = 0.3/$c$ and $D$ = 3.2/$c$.

Figure 6

The positron emission spectrum as a function of the kinetic energy $E_k$. [Same parameters as in Fig. 5 for curve (C) and $t$ = 5.7 × 10${}^{-3}$.]

Figure 7

Snapshots of the spatial density ρ($x$,$t$) for the created electrons. The left (right) graph is for one (two) resonance states in Fig. 5. The parameters are the same as in Fig. 5.

Figure 8

The energy density ρ($E$) of the (Hermitian) Hamiltonian $H^F$ as a function of energy $E$. In this case, two bound states are embedded in the Dirac sea. The density is peaked at $E_1$ = −2.16$c^2$ (with a peak height of 116 97) and $E_2$ = −1.33$c^2$ (peak height 5246). ($W$ = 0.3/$c$ and $D$ = 3.2/$c$, $V_0$ = 3.6$c^2$.)

Figure 9

The final number of created particles as a function of the interaction time for fermions (left) and bosons (right) with different $V_0$. For fermions the parameters are as in Fig. 5 and for bosons we have (a) $V_0$ = 2.15$c^2$ (no resonances); (b) $V_0$ = 2.5$c^2$ (one resonance); (c) $V_0$ = 3.0$c^2$ (two resonances); (d) $V_0$ = 3.7$c^2$ (three resonances). ($W$ = 0.3/$c$ and $D$ = 3.2/$c$.)

Figure 10

The energy spectrum of the Hamiltonian as a function of potential height $V_1$ for the bosonic system. ($W$ = 0.3/$c$ and $D$ = 3.2/$c$ and $V_2$ = 0.95$c^2$.)

Figure 11

The final number of created bosons as a function of the interaction time for (a) $V_1$ = 2.04$c^2$, (b) $V_1$ = 2.34$c^2$, (c) $V_1$ = 2.84$c^2$, and (d) $V_1$ = 3.54$c^2$. The other parameters are $W$ = 0.3/$c$, $D$ = 3.2/$c$, and $V_2$ = 0.95$c^2$.

Figure 12

The antiboson emission spectrum as a function of the kinetic energy $E_k$ at various moments in time $t_i$ = $i$ × 1.5 × 10${}^{-4}$, $i$ = 1,2,…,6. [Same parameters as in Fig. 11 for curve (d), $V_1$ = 3.54$c^2$, $W$ = 0.3/$c$, $D$ = 3.2/$c$, and $V_2$ = 0.95$c^2$.]

Figure 13

The area under the antiboson emission spectrum of Fig. 12 between $E_k$ = 0 and $E_k$ = 0.59$c^2$ as a function of time. [Same parameters as in Fig. 11 for curve (d).]

Figure 14

Snapshots of the spatial density for the created bosons. The left (right) panel is for the long-time limit $t$ = 1.5 × 10${}^{-3}$ (1.65 × 10${}^{-3}$). The parameters are the same as curves (c) and (d) in Fig. 11.