Vacuum polarization current in presence of intense Sauter field

TL;DR

This work tackles the problem of describing vacuum pair creation dynamics in a strong, time-dependent external field by focusing on the vacuum polarization current as a time-defined observable. Using canonical quantization of a charged scalar field in a spatially homogeneous Sauter pulse, the authors derive generalized expressions for the time-dependent particle number $\mathcal{N}(\mathbf{p},t)$ and the pair correlation function $\mathcal{C}(\mathbf{p},t)$, whose real and imaginary parts are $u(p,t)$ and $v(p,t)$, respectively. They compute the vacuum polarization current $J_{pol}(t)$ and demonstrate its robustness: while $\mathcal{N}(\mathbf{p},t)$ depends on the adiabatic basis during intermediate times, $J_{pol}(t)$ remains invariant under basis changes, making it a physically meaningful observable across all times. The results show that $u(p,t)$ and $v(p,t)$ exhibit a transient peak near the field maximum and long-lived oscillations at late times, with the polarization current tracking $u(p,t)$ and displaying undamped oscillations after the pulse, highlighting the interplay between polarization and depolarization in nonperturbative QED dynamics. These findings provide a concrete, basis-independent diagnostic tool for understanding vacuum fluctuations and real-particle formation in strong-field environments, with implications for ultrafast laser experiments and analogues in condensed matter systems.

Abstract

The quantum vacuum becomes unstable under an external field, leading to spontaneous particle-antiparticle pair creation. In canonical quantization, the time-dependent particle number, defined via Bogoliubov transformations lacks physical meaning until the external field vanishes. To address this, we explore dynamical quantities that remain well-defined at both asymptotic and intermediate times, focusing on the vacuum polarization current. Investigating this observable provides insights into the system's intermediate-time behavior. We consider pair creation in a spatially homogeneous, time-dependent, intense Sauter field. Specifically, we analyze the real and imaginary parts of the correlation function, linking them to vacuum polarization effects. The vacuum polarization current in an intense laser pulse is computed numerically, revealing that it correlates with the real part of the correlation function. Initially, the current changes sign and gradually decreases, but unlike the particle number, it does not reach a constant asymptotic value. Instead, for large times, it exhibits nearly undamped oscillations, a distinctive feature of scalar particles, oscillating strongly around zero. Additionally, we explore the uniqueness of the vacuum polarization current in the adiabatic basis, comparing different reference mode function choices. Notably, we find that the current remains independent of the basis choice.

Figures (14)

• Figure 1: Schematic picture of a tunneling process
• Figure 2: Temporal profile of the electric field \ref{['eqn33']} (left) and the corresponding vector potential with the choice $A(t=0) =0$(right) and all the units are taken in the electron mass unit.
• Figure 3: Evolution of $\mathcal{N}(p,t)$ for different field strengths $E_0 = 0.1$ (red), $0.2$ (blue) ,$0.3$(brown) and $\tau =15 [m^{-1}].$ The momentum is considered to be zero, and all the units are taken in the electron mass unit.
• Figure 4: Evolution of $\mathcal{N}(p,t)$ for different pulse duration $\tau = 11$ (magenta), $18$ (cyan) ,$25$(orange) and $E_0 =0.1E_c.$ The momentum is considered to be zero,and all the units are taken in the electron mass unit.
• Figure 5: Left panel :Evolution of $u(t)$ and Right panel :Evolution of $v(t)$ for different momentum value $p$.The field parameters are $E_0=0.1 E_c$ and $\tau =15 [m^{-1}].$