Symbiotic versus nonsymbiotic optimization for spatial and temporal degrees of freedom in pair creation

The field-induced decay of the quantum vacuum state associated with the creation of electron-positron pairs can be caused independently by either multiphoton transitions or by tunneling processes. The first mechanism is usually induced by appropriate temporal variations of the external field while the second (Schwinger-like) process occurs if a static but spatially dependent electric field is of supercritical strength. The ultimate goal is to construct an optimal space-time profile of an electromagnetic field that can maximize the creation of particle pairs. The simultaneous optimization of parameters that characterize the spatial and temporal features of both fields suggests that the optimal two-field configuration can be remarkably similar to that predicted from two independent optimizations for the spatial and temporal fields separately.

Figure 1

(left) Surface plot of the final displacement $X(\omega ,\mathrm{\Omega })$ for $T=1$ and $V=0.01$ For better readability we have also included as a reference line the diagonal $\mathrm{\Omega }=\omega$. (right) The corresponding contour lines. The sixteen black dots are the locations of the first 16 $i$ points, discussed in Secs. 3b and 3c.

Figure 2

The connection pathway between the $i$ point $i_{2,2}=\{\omega ,\mathrm{\Omega }\}=\left\{9.425,7.725\right\}$ and the true optimum for the $\left\{A=1,B=1\right\}$ space-time forced system given by {9.560, 8.816}. The pathway was linearly parametrized as a function of ε as {${\omega }_{\mathrm{opt}}^{\left\{1,\varepsilon \right\}}$, ${\mathrm{\Omega }}_{\mathrm{opt}}^{\left\{\varepsilon ,1\right\}}$} with $0\le \varepsilon \le 1$.

Figure 3

The disappearance of optima for the system with decreasing initial velocity V. (a) The Hessian index $h$ as an indicator of the transition from a maximum ($h<1$) to a saddle point ($h=1$) as a function of V. The three insets show the corresponding contour plots of X(ω,Ω) for three different values of the velocity. For reference, the white dot is the $i$ point, $i_{2,3}=\left\{9.425,14.07\right\}$, whose coordinates are independent of the velocity. (b) The orbits of the center of the ellipse and the saddle point in the {ω,Ω} plane with decreasing velocity.

Figure 4

The final number N(T) of created electron-positron pairs after the interaction of the vacuum state as a function of the frequency ω of a time-dependent electric field given by the vector potential $A\left(t\right)=F_{0}c/\omega \exp \left[-{\left(t-T/2\right)}^4/\left(0.02T^4\right)\right]\cos \left(\omega t\right)$, shown in the inset for $\omega =c^2$. For reference the three dashed vertical lines point to ${\omega }_{\mathrm{opt}}^{\left\{1,0\right\}}=0.844$, 1.26, and 2.51. ($F_0=0.3c^3$, $T=0.006$ a.u., $L=1.5$ a.u., the electric field was chosen to be nonzero for a spatial region of length L/4, $N_x=4096$, $N_t=6000$).

Figure 5

The final number N(T) of created electron-positron pairs after the interaction of the vacuum state as a function of the spatial extension W of the electric field $E\left(t\right)=-dV/dx$, where $V\left(x\right)=V_0\left[1-\tanh \left(x/W\right)\right]/2$ and $V_0=2.5c^2{\left(Wc/0.3\right)}^{1/2}$, as shown in the inset. The analytical predictions are based on the steady-state decay rate given by Eq. (4.3). ($T=0.006$ a.u.; V($x$) was turned on smoothly in time over 0.0002 a.u.).

Figure 6

The final number N(T) of created electron-positron pairs after the interaction of the vacuum state as a function of amplitude $V_b$ of the bump. Here V($x$) corresponds to a smooth two-step potential as described in the text and shown in the inset. The open circles are the approximate and rate-based predictions according to the analytical expression Eq. (4.5) ($V_0=2.5c^2$, $T=0.006$ a.u.; the potential was turned on smoothly over a time 0.0001 a.u.).

Figure 7

The locations of the true optima (open circles) and the corresponding $i$ points (open squares) for N(T) in the (ω,W) plane. In the inset we show the corresponding contour plots around the true optima. The $i$ points are located at $i_1=\{\omega /c^2,Wc\}=\left\{0.844,0.375\right\}$, $i_2=\left\{1.26,0.375\right\}$, $i_3=\left\{2.52,0.375\right\}$, and the corresponding true optima are at {0.880, 0.372}, {1.25, 0.392}, and {2.48, 0.423}. ($F_0=0.3c^3$, $V_0=2.5c^2$, $T=0.006$ a.u.; the potential V($x$) was turned on smoothly over a time 0.0001 a.u.).