Quantum entanglement of Hawking-Partner modes in expanding cavities

TL;DR

This work uses Gaussian-state methods to study Hawking-like entanglement generated in a 1D cavity with a moving boundary that mimics horizon formation. Entanglement is quantified via logarithmic negativity for a given mode against the rest, and via an HSU-based reconstruction of the Hawking partner for pure states, revealing that partners do not generally commute and thus cannot describe the full multimode state in a simple pairwise way. The expanding cavity acts as a multimode squeeze device, with Hawking–partner pairs behaving predominantly as two-mode squeezed states and purification concentrated in low-energy (infrared) modes; the entanglement structure remains robust to initial squeezing and moderate thermal noise, and the HSU approach offers substantial computational advantages. The results have implications for analogue gravity experiments and deepen understanding of multimode entanglement during horizon-like dynamics.

Abstract

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.

Figures (18)

• Figure 1: Hawking modes and their partners: these plots correspond to the trajectories given by Eq. \ref{['eq:Traj-1plt']}, with $\epsilon=0.375$ and $\kappa=33.3$. We show the (modulus squared of the) Bogoliubov coefficients of a Hawking mode (upper panel) and the ones of its partner (lower panel) for a fixed $in$ frequency $I=20$ and $N\to\infty$ via Richardson extrapolation of the simulations with $N=256,\,512,\,1024$, and as functions of the $out$ frequencies in the interval $[1,100]$.
• Figure 2: Hawking modes and their partners: these plots correspond to the trajectories given by Eq. \ref{['eq:Traj-1plt']}, with $\epsilon=0.375$ and $\kappa=33.3$. The upper panel shows the Bogoliubov coefficients in Eq. \ref{['eq:bh-mode']} of all Hawking modes and the lower panel those in Eq. \ref{['eq:bp-mode']}, for the corresponding partner modes as functions of the $out$ frequencies in the interval $[1,100]$ and in the limit $N\to\infty$ (via Richardson extrapolation of $N=256,512,1024$). Besides, we include in both cases the fitting expression in Eq. \ref{['eq:betas-fit2']}.
• Figure 3: Logarithmic negativity: this plot corresponds to the trajectories given by Eq. \ref{['eq:Traj-1plt']}, with $\epsilon=0.375$ and $\kappa=33.3$. It shows the logarithmic negativity between a given Hawking mode $I$ and its partner in the limit $N\to\infty$ via Richardson extrapolation out of simulations with total numbers of modes given by $N=256,\,512,\,1024$.
• Figure 4: $1\times(N-1)$ Logarithmic negativity: These plots correspond to the trajectories given by Eq. \ref{['eq:Traj-1plt']}, with $\epsilon=0.375$ and $\kappa=33.3$. It shows the limit $N\to\infty$ via Richardson extrapolation, out of simulations with $N=256,\,512,\,1024$, for five initial one-mode squeezed states with squeezing intensities $r=10^{-4}$, $r=10^{-3}$, $r=10^{-2}$, $r=10^{-1}$, and $r=1$.
• Figure 5: $1\times(N-1)$ Logarithmic negativity: These plots correspond to the trajectories given by Eq. \ref{['eq:Traj-1plt']}, with $\epsilon=0.375$ and $\kappa=33.3$. It shows the Logarithmic Negativity in the asymptotic future in the limit $N\to\infty$ via Richardson extrapolation (out of simulations with $N=256,\,512,\,1024$ total modes) for five initial two-mode squeezed states with squeezing intensities: $r=10^{-4}$, $r=10^{-3}$, $r=10^{-2}$, $r=10^{-1}$, and $r=1$.