Quantum entanglement of Hawking-Partner modes in expanding cavities

This article investigates quantum entanglement generated within a one-dimensional cavity where one boundary undergoes prescribed acceleration, a setup designed to mimic aspects of Hawking radiation. We quantify quantum correlations using logarithmic negativity for bipartitions where subsystem $A$ is a given mode and subsystem $B$ is the rest of the system. For initial pure states, we also consider a given mode and reconstruct its partner using the Hotta-Schützhold-Unruh formula, obtaining identical results. Interestingly, this last method offers notable computational efficiency. However, partner modes do not commute, due to the nontrivial multimode entanglement structure. Hence, a pairwise description will not be suitable for describing the full system. Besides, our findings reveal that the expanding cavity effectively acts as a squeezing device, with Hawking-partner pairs largely behaving as two-mode squeezed states. We checked that, in our setting, purification of Hawking modes is predominantly a low-energy process, with high-energetic particles contributing negligibly to the partner modes. Indeed, in both small and large acceleration regimes of the boundaries, quantum entanglement decreases toward the ultraviolet modes, indicating that higher-energy particles are more challenging to entangle and hence less probable to contribute in the purification process. Besides the initial vacuum state, we also consider one-mode squeezed and two-mode squeezed states, in order to confirm if quantum entanglement can be stimulated. Moreover, we analyze its robustness against initial thermal noise. Our analysis is based on numerical simulations and does not assume any approximation beyond the validity of our numerical algorithms. We conclude with a discussion about the possible implementation and observation of our results in the laboratory.

💡 Research Summary

The paper investigates quantum entanglement generated in a one‑dimensional cavity whose right boundary follows a prescribed accelerated trajectory, an analogue of Hawking radiation. Starting from the classical massless Klein‑Gordon field with Dirichlet conditions at moving walls, the authors derive mode‑mixing equations that contain time‑dependent coupling matrices Rₙₘ and Sₙₘ. These equations are solved numerically using an embedded Prince‑Dormand‑Runge‑Kutta (8,9) scheme for up to 1024 modes, with Richardson extrapolation to approximate the N → ∞ limit. Quantization proceeds by selecting a positive‑frequency subspace, constructing orthonormal mode bases, and relating in‑ and out‑states via Bogoliubov coefficients α and β; particle creation is encoded in β.

Gaussian state formalism is employed because the entire dynamics can be captured by the first two moments (mean vector μ and covariance matrix σ). Logarithmic negativity is used to quantify entanglement between a chosen mode (subsystem A) and the remainder of the field (subsystem B). In addition, the Hotta‑Schützhold‑Unruh partner formula is applied to reconstruct the partner mode of a given Hawking‑like mode, yielding identical entanglement values but with far lower computational cost. A key finding is that partner modes do not commute, reflecting a genuinely multipartite entanglement structure that cannot be reduced to independent two‑mode pairs.

Numerical results are presented for both small and large boundary accelerations. In all cases the expanding cavity acts as an effective squeezing device: Hawking‑partner pairs resemble two‑mode squeezed states, and the purification of a Hawking mode is dominated by low‑energy excitations. As mode frequency increases toward the ultraviolet, logarithmic negativity drops sharply, indicating that high‑energy particles contribute negligibly to the purification process.

The authors also explore non‑vacuum initial conditions: single‑mode squeezed states, two‑mode squeezed states, and thermal states. Increasing the squeezing parameter r enhances entanglement, confirming that the process can be stimulated. Thermal noise reduces entanglement but does not destroy it unless the mean occupation number becomes large.

Finally, the feasibility of laboratory implementation is discussed. Superconducting circuits with SQUID‑terminated waveguides, as well as Bose‑Einstein condensate or optical fiber analogues, are suggested as platforms where the required boundary motion (effective acceleration) can be engineered via tunable inductance or refractive index modulation. The study concludes that expanding cavities provide a controllable test‑bed for Hawking‑like particle creation and for probing how multipartite quantum correlations encode information that would otherwise be lost in black‑hole evaporation.