Prefactor in the dynamically assisted Sauter-Schwinger effect

The probability of creating an electron-positron pair out of the quantum vacuum by a strong electric field can be enhanced tremendously via an additional weaker time-dependent field. This dynamically assisted Sauter-Schwinger effect has already been studied in several works. It has been found that the enhancement mechanism depends on the shape of the weaker field. For example, a Sauter pulse $1/\cosh (\omega t{)}^2$ and a Gaussian profile $\exp (-{\omega }^2{t}^2)$ exhibit significant, qualitative differences. However, so far most of the analytical studies were focused on the exponent entering the pair-creation probability. Here, we study the subleading prefactor in front of the exponential using the worldline instanton method. We find that the main features of the dynamically assisted Sauter-Schwinger effect, including the dependence on the shape of the weaker field, are basically unaffected by the prefactor. To test the validity of the instanton approximation, we compare the number of produced pairs to a numerical integration of the full Riccati equation.

Figure 1

Predicted number density of produced pairs by a single Sauter pulse with Keldysh parameter $\gamma$. The points depict the worldline instanton approximation (15), and the lines show the exact analytical result. The field strength increases from bottom to top.

Figure 2

Imaginary part of the effective action (11) in the assisted Sauter Schwinger effect for different pulse shapes $h(\chi )$, Keldysh parameters $\gamma$, and relative field strengths $ϵ$. The color scale spans from ${10}^{-40}$ (blue area, bottom left corners) to ${10}^{-16}$ (yellow area, top right corners). Note the dependence of the threshold ${\gamma }^{\text{crit}}$ on $ϵ$ for the cosine and Gauss profiles, while ${\gamma }^{\text{crit}}\approx \text{const}$ for the Sauter and Lorentz profiles.

Figure 3

Density of produced pairs from a strong, slow and another weak, fast Sauter pulse. The strong field strength is $E/E_{\mathrm{S}}=0.033$, corresponding to an intensity of $\approx 5\times {10}^{26}\mathrm{W}/{\mathrm{cm}}^2$. The blue surface (lying above for large $\gamma$ and small $ϵ$) is the instanton result (11), and the orange surface shows the numerical integration of the full Riccati equation.

Figure 4

Number density of produced pairs for a Lorentzian pulse. The dots represent the numerical results for different values of $ϵ$, and the lines represent the approximations (26) and (27) for $\gamma <1$ and (29) for $\gamma >1$. $ϵ$ increases from bottom to top.

Figure 5

Comparison of the analytic solution (line) and numerical results (points) of the Riccati equation for a single Sauter pulse with $E=E_{\mathrm{S}}/100$ and $k_3=0$. The top plot shows the relative deviation from the analytic result.