[Paper Review] Schwinger Pair Production in Pulsed Electric Fields

Sang Pyo Kim, H. W. Lee+1 more|arXiv (Cornell University)|Jul 22, 2012

Laser-Plasma Interactions and Diagnostics40 references3 citations

TL;DR

This paper presents a numerical study of Schwinger pair production in pulsed electric fields using the evolution operator formalism in scalar quantum electrodynamics (QED). By solving the time-dependent Schrödinger equation and expressing the Hamiltonian in terms of Minkowski vacuum creation and annihilation operators, the authors compute the exact quantum state and pair production rate, revealing momentum spectrum structures and time-dependent dynamics that depend on field polarity and configuration, including oscillatory behavior post-interaction and distinct pair dominance patterns in di-polar fields.

ABSTRACT

We numerically investigate the temporal behavior and the structure of longitudinal momentum spectrum and the field polarity effect on pair production in pulsed electric fields in scalar quantum electrodynamics (QED). Using the evolution operator expressed in terms of the particle and antiparticle operators, we find the exact quantum states under the influence of electric pulses and measure the number of pairs of the Minkowski particle and antiparticle. The number of pairs, depending on the configuration of electric pulses, exhibits rich structures in the longitudinal momentum spectrum and undergoes diverse dynamical behaviors at the onset of the interaction but always either converges to a momentum-dependent constant or oscillates around a momentum-dependent time average after the completion of fields.

Motivation & Objective

• To investigate the temporal dynamics and momentum spectrum structure of Schwinger pair production in pulsed electric fields using exact quantum state evolution.
• To examine the role of field polarity—mono-polar vs. di-polar—on pair production, particularly whether the field configuration affects the final state and momentum distribution.
• To resolve the ambiguity on whether produced pairs are Minkowski or adiabatic states in non-adiabatic, time-varying fields.
• To analyze the long-term behavior of pair production, including oscillations after field cessation and convergence to time-averaged values.
• To explore the conditions under which the system returns to the Minkowski vacuum, especially in symmetric di-polar configurations.

Proposed method

• Formulate the Hamiltonian for scalar QED in momentum space as an infinite set of time-dependent oscillators with frequencies modulated by the pulsed electric field.
• Express the evolution operator using $SU(1,1)$ algebraic structure in terms of Minkowski vacuum creation and annihilation operators: $\prod_k e^{\xi_k(t)\hat{a}^\dagger_k\hat{b}^\dagger_{-k}} e^{i\gamma_k(t)(\hat{a}^\dagger_k\hat{a}_k + \hat{b}_{-k}\hat{b}^\dagger_{-k})} e^{\eta_k(t)\hat{a}_k\hat{b}_{-k}}$.
• Solve the time-dependent Schrödinger equation directly to obtain the exact quantum state evolved from the Minkowski vacuum under the influence of the pulsed field.
• Compute the number of pairs via the number operator expectation value in the evolved state, tracking time and momentum dependence.
• Classify electric fields into mono-polar (e.g., Sauter, Gaussian, oscillating Gaussian) and di-polar (e.g., two Sauter pulses, solitonic gauge potential) configurations to compare pair production dynamics.
• Use the gauge potential (integral of electric field) to distinguish between fields with non-zero asymptotic potential (mono-polar) and those vanishing at infinity (di-polar), linking to vacuum return and symmetry.

Experimental results

Research questions

• RQ1 How does the longitudinal momentum spectrum of produced pairs depend on the shape and polarity of pulsed electric fields in scalar QED?
• RQ2 Does the field polarity (mono-polar vs. di-polar) influence the dominance of positive or negative momentum pairs in the final state?
• RQ3 Why does the oscillating Gaussian electric field produce a characteristic bunching of pairs around symmetric positive and negative momenta with separation equal to the field’s angular frequency?
• RQ4 To what extent do the number of pairs oscillate after the field is switched off, and does this behavior depend on the field’s asymptotic gauge potential?
• RQ5 Under what conditions does the system return to the Minkowski vacuum after pair production, and how is this related to the field’s temporal symmetry?

Key findings

• In mono-polar fields (e.g., Sauter and Gaussian), the number of pairs increases during the field pulse and then oscillates around a momentum-dependent time average, consistent with asymptotic solutions.
• The longitudinal momentum spectrum in mono-polar fields exhibits substructures for small momenta, particularly in the Sauter and Gaussian fields, with a notable bunching effect in the oscillating Gaussian field centered at momenta symmetric about zero, separated by the field’s angular frequency.
• In di-polar fields with vanishing asymptotic gauge potential (e.g., two Sauter pulses, solitonic potential), the number of pairs increases and then decreases to a constant, showing simpler momentum spectrum structures.
• For the di-polar Sauter field, the order of pulse polarity determines momentum dominance: a positive-first pulse leads to positive-momentum dominance, while a negative-first pulse leads to negative-momentum dominance.
• The solitonic gauge potential produces symmetric pair production in time and fully returns the system to the Minkowski vacuum without residual pairs, indicating a complete vacuum-to-vacuum transition.
• The evolution operator formalism successfully captures the exact quantum state and pair number, providing a robust framework for analyzing non-adiabatic pair production beyond approximations like the kinetic or Wigner methods.