Entanglement in the Schwinger effect

We analyze entanglement generated by the Schwinger effect using a mode-by-mode formalism for scalar and spinor QED in constant backgrounds. Starting from thermal initial states, we derive compact, closed-form results for bipartite entanglement between particle-antiparticle partners in terms of the Bogoliubov coefficients. For bosons, thermal fluctuations enhance production but suppress quantum correlations: the logarithmic negativity is nonzero only below a (mode-dependent) critical temperature $T_c$. At fixed $T$, entanglement appears only above a critical field $E_{\text{entang}}$. For fermions, we observe a qualitatively different pattern: the fermionic logarithmic negativity is non-vanishing at finite temperature, and is monotonically suppressed by thermal noise. As a function of the electric field, it is non-monotonic, featuring a temperature-independent optimal field strength $E_*$ and decreasing on both sides of the maximum. We give quantitative estimates for analog experiments, where our entanglement criteria convert directly into concrete temperature and electric field constraints. These findings identify realistic regimes where the quantum character of Schwinger physics may be tested in the laboratory.

💡 Research Summary

The manuscript presents a comprehensive study of quantum entanglement generated by the Schwinger effect in constant electric‑field backgrounds, treating both scalar (bosonic) and spinor (fermionic) quantum electrodynamics. Using a mode‑by‑mode Bogoliubov‑transformation framework, the authors derive closed‑form expressions for the bipartite entanglement between particle–antiparticle partners for each momentum mode, starting from thermal initial states.

For bosons, the Klein‑Gordon equation in a uniform electric field is solved via parabolic‑cylinder functions, yielding Bogoliubov coefficients α_k and β_k that satisfy |α_k|²−|β_k|²=1. The mean number of created particles in mode k is |β_k|², while the logarithmic negativity (LN) – the chosen entanglement measure – depends on both β_k and the initial thermal occupation n̄_k=1/(e^{ω_k/T}−1). The authors obtain a compact analytic formula for LN that reveals a mode‑dependent critical temperature T_c(k). Above T_c the LN vanishes even though pair production continues, establishing a clear quantum‑classical crossover. At fixed temperature, a critical electric field E_entang(k) must be exceeded for entanglement to appear; this threshold lies above the usual Schwinger critical field E_crit=π m²/e and grows with temperature.

The bosonic analysis also explores the effect of preparing the vacuum in a two‑mode squeezed state. The squeezing parameter r amplifies LN by a factor involving sinh(2r), offering a practical knob to enhance quantum correlations in experimental settings.

For fermions, the Dirac equation is treated similarly, with solutions expressed in terms of Landau‑level indexed parabolic functions. The corresponding Bogoliubov coefficients α_f(k,l) and β_f(k,l) obey |α_f|²+|β_f|²=1 due to the Pauli principle. The authors employ the fermionic Gaussian formalism (Majorana covariance matrices) and the fermionic logarithmic negativity introduced in Refs.