Entanglement in the Schwinger effect

TL;DR

This work analyzes entanglement generated by Schwinger pair production in constant backgrounds for both scalar and spinor QED using a mode-by-mode Bogoliubov framework and Gaussian state methods. It derives closed-form expressions for particle production and entanglement, showing that bosons exhibit a mode-dependent critical temperature $T_c$ and a field threshold $E_{\text{entang}}$ for entanglement, while fermions display a finite electric-field window $E_{\text{min}}<E<E_{\text{max}}$ with a temperature-independent optimal field $E_*$ and a maximum temperature $T_{\max}$ for entanglement. The results quantify how thermal noise degrades quantum correlations, how initial squeezing can enhance entanglement, and how these criteria translate into concrete constraints for analogue experiments (notably graphene) that could probe quantum aspects of Schwinger physics. The analysis also recovers the axial anomaly and provides a bridge between fundamental QED and experimentally accessible quantum signatures in solid-state platforms. Overall, the paper identifies realistic regimes where the quantum character of strong-field pair creation can be tested in the laboratory.

Abstract

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 $E_{\text{crit,entang}}$. For fermions, we observe a qualitatively different pattern: at finite $T$ entanglement exists only within a finite window $E_{\text{min}} < E < E_{\text{max}}$, with a temperature-independent optimal field strength $E_{*}$ that maximizes the logarithmic negativity. Entanglement is vanishing above $T_{\text{max}}=ω/\text{arcsinh}(1)$. 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.

Figures (8)

• Figure 1: Logarithmic Negativity vs the noise intensity parameter $n$ for the symmetric single-mode squeezed thermal state, with $\xi_1=\xi_2=\xi$, after we evolve it with the two-mode squeezing transformation. For the plot, we use the two-mode squeezing strength value $r=1$ and we study different values of the input squeezing strength $\xi$.
• Figure 2: Energy density $u_B(\omega)$ as expressed in (\ref{['eq_energy_density_bosons_no_kz']}).
• Figure 3: (a) Phase diagram-like representation of the logarithmic negativity of bosons. We clearly see that there exists a temperature $T_c/\omega$ above which there can't be any entanglement because $|\beta|$ can't be higher than $1$. (b) Same for different values of squeezing, the shaded areas correspond to regions where $L_N>0$.
• Figure 4: Critical electric fields as expressed in eqs. \ref{['eq:EcritT-particlenumber']} and \ref{['eq_E_for_entanglement_bosons']}. The shaded area corresponds to the region where the system is entangled. Above $T=T_c$, no entanglement is possible.
• Figure 5: Critical electric fields $E_{\text{crit}}$ and $E_{\text{entang}}$ as expressed in \ref{['eq:Ecrit_squeezed_bosons']} and \ref{['eq:conditionEentangxi']} in the $\vec{k}=0$ mode. The shaded area correspond to the regions where the system is entangled and are evaluated numerically. We see, as already explained in subsection \ref{['subsec:subsection-squeezing']} that the squeezing increases the entanglement and thus decreases the minimum electric field for entanglement $E_{\mathrm{entang}}$. The squeezing also stimulates the pair production, thus reducing the minimum electric field for particle creation $E_{\mathrm{crit}}$.