Instantaneous rest-frame transformation method for temporally induced pair creation

We introduce a computational method to determine the rate of the electron-positron pair creation triggered by a time-dependent subcritical external field. It is based on constructing a Lorentz transformation to an instantaneous rest frame, for which the pair-creation rate can be determined by standard techniques. We will discuss the accuracy and efficiency of this method by comparing its predictions with exact time-dependent quantum field theoretical solutions to the Dirac and Klein-Gordon equations for various space-time dependent external fields.

Figure 1

The number of particle pairs $N$($t$) as a function of time $t$ that were created by a moving potential barrier $V$($x$,$t$) = $U$($x$ – $v$t) moving with three velocities $v$. The top (bottom) is for fermions (bosons). $V_0$ = 1.9c${}^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 1.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 8000 grid points.

Figure 2

The pair-creation rate Γ as a function of velocity of the potential barrier $V$($x$,$t$) = $U$($x$ – $v$t). When the barrier's speed exceeds the critical velocity (0.312 25$c$), the creation rate increases with velocity. The exact rates are obtained from the slopes of the numerically obtained $N$($t$), while the IRFT method is based on Hund's rule. $V_0$ = 1.9$c^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 1.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 8000 grid points.

Figure 3

The creation rate Γ of electron-positron pairs as a function of time $t$ that were created by a moving potential barrier $V$($x$,$t$) = $U$($x$ – $v$t) whose velocity $v$ increased slowly from $v$ = 0 to $v$ = 0.8$c$ during a time interval of duration $T_v$, as shown by the top curve. For comparison, the prediction due to the IRFT method is shown by the dashed line. $T_v$ = 0.0024 a.u., $V_0$ = 1.9$c^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 2.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 16 000 grid points.

Figure 4

The creation rate Γ of electron-positron pairs as a function of time $t$ that were created by a moving potential barrier $V$($x$,$t$) = $U$($x$ – $v$t) whose velocity $v$ was reversed at time $T_v$, as shown by the top curve. For comparison, the prediction due to the IRFT method is shown by the dashed line. (a) reversal at time $T_v$ = 0.0009 a.u.; (b) reversal at time $T_v$ = 0.0019 a.u. $V_0$ = 1.9c${}^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 2.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 16 000 grid points.

Figure 5

Temporal snapshots of spatial density of the electrons (solid) and positrons (dotted line) at four moments in time. The vertical dashed line is the location of the force field. (a) $t$ = 2.0 × 10${}^{-4}$ a.u.; (b) $t$ = 8.9 × 10${}^{-4}$ a.u., the time when the velocity of the potential 0.8$c$ was reversed; (c) $t$ = 1.02 × 10${}^{-3}$ a.u., after the reversal; (d) $t$ = 2.51 × 10${}^{-3}$ a.u., after the force field has caught up with the priorly created electrons. $V_0$ = 1.9$c^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 2.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 16 000 grid points.

Figure 6

The creation rate Γ of particle pairs as a function of time $t$ that were created by a moving potential barrier $V$($x$,$t$) = $U$($x$ – $v$t) whose velocity alternates instantly at times $T=0.0009n$ between ±0.08$c$ (bottom), ±0.4$c$ (middle), and ±0.64$c$ (top). For comparison, the prediction due to the IRFT method is shown by the dashed line. $V_0$ = 1.9$c^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 1.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 8000 grid points.

Figure 7

The creation rate Γ of particle pairs as a function of time $t$ that were created by a moving potential barrier $V$($x$,$t$) = $U$($x$ – $v$t) whose velocity alternates instantly at times $T=0.0009n$ between 0.5$c$ and −0.5$c$. The widths of the potentials are $w$ = 0.55/$c$ (top), 0.75/$c$ (middle), and 1/$c$ (bottom). For comparison, the prediction due to the IRFT method is shown by the dashed line. $V_0$ = 1.9$c^2$, the numerical box of length $L$ = 1.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 8000 grid points.

Figure 8

The creation rate Γ of particle pairs as a function of time $t$ that were created by a potential barrier $V$($x$,$t$) = $U$($x$ – $v$t) whose velocity oscillates periodically with frequency ω = 0.836$c^2$ between ±0.35$c$ (bottom), ±0.55$c$ (middle), and ±0.8$c$ (top). For comparison, the prediction due to the IRFT method is shown by the dashed line. $V_0$ = 1.9c${}^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 1.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 8000 grid points.

Figure 9

The creation rate Γ of particle pairs as a function of time $t$ that were created by a potential barrier $V$($x$,$t$) = $U$($x$ – $v$t) whose velocity oscillates periodically with frequency ω = 0.836$c^2$ with a constant background velocity. For comparison, the prediction due to the IRFT method is shown by the dashed line. $V_0$ = 1.9$c^2$, $w$ = 0.5/$c$, the numerical box of length $L$ = 1.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 8000 grid points.

Figure 10

The number of particle pairs $N$ as a function of time $t$ that were created by a potential well $V$($x$,$t$) = $U$($x$ – $v$t) moving with three velocities. The top (bottom) is for fermions (bosons). The inset shows the long-time behavior of the boson on a logarithmic scale. $V_0$ = 1.6$c^2$, $w$ = 0.3/$c$, $D$ = 6.4/$c$, the numerical box of length $L$ = 1.0 a.u. was discretized into $N_x$ = 1024 grid points, and the time was sampled at $N_t$ = 8000 grid points.

Figure 11

The final number of particle pairs $N$ as a function of the velocity $v$ of the moving potential well $V$($x$,$t$) = $U$($x$ – $v$t). The top (bottom) is for fermions (bosons) at final time $t$ = 0.008 a.u. ($t$ = 0.0065 a.u.). The parameters are the same as in Fig. 10.

Figure 12

The energy spectrum as a function of the velocity $v$ for the fermionic (top) and bosonic (bottom) systems. $V_0$ = 1.6$c^2$, $w$ = 0.3/$c$, $D$ = 6.4/$c$, and the numerical box of length $L$ = 0.5 a.u. was discretized into $N_x$ = 512 grid points.