Vacuum polarization current in presence of intense Sauter field

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.

💡 Research Summary

This paper investigates the quantum vacuum instability leading to spontaneous particle-antiparticle pair creation (the Sauter-Schwinger effect) under a strong, time-dependent external electric field. The central critique of existing approaches is that the commonly used “time-dependent number of created particles,” defined via Bogoliubov transformations in canonical quantization, lacks a clear physical interpretation at intermediate times while the external field is still present. It only becomes meaningful asymptotically, after the field is switched off. To overcome this conceptual hurdle, the authors propose studying dynamical quantities that remain well-defined throughout the entire evolution, focusing specifically on the “vacuum polarization current.”

The theoretical framework is built upon scalar Quantum Electrodynamics (QED) in (1+1) dimensions, coupled to a spatially homogeneous, time-dependent electric field in the temporal gauge. The canonical quantization procedure is revisited, highlighting the ambiguity in defining particle states (and thus the vacuum) when time-translation symmetry is broken by the external field. This ambiguity is encoded in the choice of mode functions χ(p,t) used to expand the quantum field. The authors parameterize these functions using a WKB-like ansatz, which naturally incorporates different orders of the adiabatic approximation (e.g., zeroth-order vs. first-order), corresponding to different natural choices for a time-dependent vacuum.

The key dynamical observables analyzed are the number of created particles N(p,t) and, more importantly, the correlation function C(p,t) = ⟨0| â†(p,t) b†(-p,t)|0⟩. The real part u(p,t) of this correlation function is linked to vacuum polarization effects. The total vacuum polarization current J_pol(t) is derived and is shown to be connected to the momentum-integrated behavior of u(p,t).

For a concrete analysis, the Sauter-type electric field profile, E(t) = E₀ sech²(t/τ), is employed. This field allows for analytical insights and precise numerical computation. The main numerical findings are striking:

1. As expected, the particle number N(p,t) grows during the pulse and settles to a constant asymptotic value after the field vanishes.
2. In contrast, the vacuum polarization current J_pol(t) exhibits profoundly different dynamics. After an initial transient phase where it changes sign and decreases, it does not approach a constant value for large times (t ≫ τ). Instead, it displays persistent, nearly undamped oscillations around zero. This oscillatory behavior is identified as a distinctive feature of scalar (bosonic) particles in this context.
3. The study further examines the potential dependence of these results on the choice of the adiabatic basis (i.e., the choice of reference mode functions χ(p,t)). A significant result is that the computed vacuum polarization current J_pol(t) is found to be independent of this choice. Whether one uses a basis corresponding to a higher-order (e.g., second-order) adiabatic vacuum or a lower-order one, the resulting current is identical. This establishes J_pol(t) as a robust, basis-independent physical observable.

In conclusion, this work successfully identifies and characterizes the vacuum polarization current as a valuable observable for probing the real-time dynamics of pair production in strong fields. Its unique oscillatory asymptotic behavior provides a clear signature distinct from the particle number, and its invariance under changes of the adiabatic vacuum confirms its physical robustness. This approach offers a promising pathway to understand intermediate-time quantum vacuum phenomena beyond the limitations of the asymptotic particle number concept.