Entanglement Scaling Demonstrates Area-To-Volume Law Transition in Sauter-Schwinger Effect

Scientists have long sought to understand how entanglement behaves in extreme conditions, and a new study sheds light on this very question in the context of the Sauter-Schwinger effect , a process predicting particle creation from a strong electric field. S. Mahesh Chandran from Seoul National University and Karthik Rajeev from the University of Edinburgh, along with their colleagues, present the first detailed numerical investigation of entanglement entropy during this nonperturbative quantum electrodynamic (QED) phenomenon. Their research demonstrates a crucial transition in entanglement scaling, moving from area-law behaviour in weak fields to a volume-law scaling in strong-field regimes, indicating a fundamental change in the correlations arising from spontaneous pair production. This finding is significant because it reveals how entanglement structure is fundamentally altered by intense fields and offers valuable insights for exploring related phenomena in high-energy physics.

Experiments show that weak-field regimes are dominated by area-law states, consistent with established quantum field theory, but as the field strength increases, a remarkable change occurs. This detailed analysis provides crucial insights into the underlying correlation structure of the vacuum as it decays under extreme electromagnetic conditions. This research establishes a clear connection between nonperturbative pair creation in strong fields and the spatial entanglement of quantum states, a relationship previously poorly understood.

Scientists prove that the behaviour of the low-energy pair-creation spectrum directly influences this entanglement scaling, offering a new lens through which to interpret the dynamics of vacuum decay. The work opens exciting possibilities for understanding quantum phenomena in extreme environments, such as those found near neutron stars or potentially achievable with next-generation high-intensity lasers. Furthermore, the findings have implications for emerging experimental platforms, including condensed-matter analogues designed to model strong-field QED effects. Next-generation ultra-high-intensity lasers are expected to approach the critical Schwinger field strength, creating unprecedented opportunities to study these phenomena, and this theoretical work provides essential guidance for interpreting experimental results.,.

Entanglement dynamics via cylindrical modes in strong QED

The research team engineered a nonperturbative approach to examine how entanglement entropy scales with spatial bipartitioning, employing cylindrical modes defined by positive- and negative-energy functions: φ±κkzm(x) = √2πeikzzeimφJm(κρ)v±κkz(t). These modes were constructed using the Klein-Gordon inner product, ensuring orthonormality through careful normalization of the vk functions, achieving relations like (φ+k, φ+k′) = ±(2π)3δm,m′δ(kz −k′z)δ(κ′ −κ)κ. Experiments employed a cylindrical basis to analyse the entanglement, meticulously verifying its completeness by demonstrating that the sum over all modes yields the identity operator: Σk −i φ+k(t, x)∂tφ+∗k(t, x′) −φ−k(t, x)∂tφ−∗k(t, x′) = δ(φ −φ′)δ(z −z′)δ(ρ −ρ′)ρ. The study pioneered a discretization technique, imposing periodic boundary conditions along the z-axis (kzL = 2πl) and Dirichlet conditions at the radial boundary (Jm(κR) = 0), resulting in discrete modes φ±nlm(x) normalized to a volume V = πR2L.

This discrete mode sum prescription, 1/V Σm,l,n, was crucial for computations of entanglement entropy and related observables. The team observed an intermediate power-law scaling that smoothly interpolates between area and volume laws, linking this behaviour to the low-energy pair-creation spectrum. This innovative approach enables a deeper understanding of quantum phenomena in curved spacetime and offers potential applications to experimental analogues, paving the way for future investigations incorporating backreaction effects and exploring more complex entanglement measures like logarithmic negativity.

Entanglement entropy transitions with strong-field QED

Experiments measured the entanglement entropy to quantify correlations arising from vacuum decay in the presence of an intense electric field, providing insights into the quantum nature of this phenomenon. The team developed a framework to investigate geometric entanglement generated by nonperturbative pair production, achieving a detailed understanding of how entanglement evolves dynamically. Measurements confirm that the entanglement entropy is calculated using the symplectic eigenvalues of the reduced covariance matrix, obtained from the original covariance matrix by considering a subsystem of modes. Specifically, the symplectic eigenvalues {γk} are derived from the eigenvalues of the matrix iΩinσ, where Ωin represents the symplectic matrix associated with the subsystem.

Scientists recorded a clear connection between the observed entanglement scaling and the low-energy pair-creation spectrum, providing a physical interpretation of the results. The study meticulously details the use of Gaussian states to describe the dynamics of the scalar-QED vacuum under a time-dependent electric field, employing quadrature field operators {φi, πi} that satisfy commutation relations [φi, πj] = iδij. The covariance matrix Σ, defined as Σij = 1/2 Tr {Qi − Qi, Qj − Qj}ρ, fully characterizes the Gaussian state, with Q representing a 2N-dimensional vector of quadrature operators. Tests prove that the entanglement entropy, calculated as S = −Tr [ρin log(ρin)], is directly linked to the symplectic spectrum obtained through Williamson’s theorem, where σ = MσMT = diag(γ1, ., γn) O O diag(γ1, ., γn).

Furthermore, the research establishes a basis for extending the framework to infinite bosonic modes, crucial for applications in quantum field theory. The Sauter-Schwinger effect was investigated using a gauge potential A(t) = −E0τ [1 + tanh(t/τ)] dz, resulting in an electromagnetic field F(t) = E0 cosh2(t/τ) dz ∧ dt. The dimensionless parameters governing the system, the classical nonlinearity parameter ξ = |eE0|τ/μ, the adiabaticity parameter η = 1/(μτ), and the quantum nonlinearity parameter χ = |eE0|/μ2, were carefully defined and utilized in the analysis. This breakthrough delivers a novel approach to probing quantum correlations in strong-field QED, opening avenues for future investigations of related phenomena and providing a deeper understanding of vacuum pair production.