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Quantum Optical Inspired Models for Unitary Black Hole Evaporation

Paul M. Alsing

TL;DR

This work addresses the black hole information paradox by constructing a tractable, Gaussian, optically inspired model that can reproduce Page-curve behavior within unitary evolution. The authors model the black hole as a single-mode squeezed state interacting with external vacuum modes through a sequence of beam splitters, followed by post-interaction squeezing to preserve thermality, and track the dynamics with a symplectic formalism. By exploring constant and occupancy-dependent reflectivity profiles, as well as a full beam-splitter-plus-squeezing (BSSQ) implementation, they demonstrate how information transfer from the BH to Hawking radiation emerges around the Page time while maintaining near-thermal HR at early times and allowing the BH to completely evaporate. The work further quantifies BH-HR correlations via purity and log-negativity and analyzes temporal auto-correlations, highlighting how the model captures key features of information flow and provides a foundation for future, energy-consistent extensions and alternative evaporation scenarios.

Abstract

In this work, we describe optically inspired models for unitary black hole (BH) evaporation. The goal of these models are (i) to be operationally simple, (ii) approximately preserve the thermal nature of the emitted Hawking Radiation (HR), and (iii) attempt to reproduce the Page Curve that purports that information flows forth from the BH when it has evaporated to approximately half its initial mass. We concentrate on modeling the BH as a single mode squeezed state successively interacting, by means of beam splitters and squeezers, with vacuum modes near the horizon, giving rise to entangled pairs representing the external Hawking radiation and its partner particle inside the horizon. Since all states and operations are Gaussian throughout, we use a symplectic formalism to track the evolution of the composite system through the evolving means and variances of their quadrature operators. This allows us to easily compute correlations and entanglement between the BH and the HR, as well as calculate correlations between the BH at early and late times.

Quantum Optical Inspired Models for Unitary Black Hole Evaporation

TL;DR

This work addresses the black hole information paradox by constructing a tractable, Gaussian, optically inspired model that can reproduce Page-curve behavior within unitary evolution. The authors model the black hole as a single-mode squeezed state interacting with external vacuum modes through a sequence of beam splitters, followed by post-interaction squeezing to preserve thermality, and track the dynamics with a symplectic formalism. By exploring constant and occupancy-dependent reflectivity profiles, as well as a full beam-splitter-plus-squeezing (BSSQ) implementation, they demonstrate how information transfer from the BH to Hawking radiation emerges around the Page time while maintaining near-thermal HR at early times and allowing the BH to completely evaporate. The work further quantifies BH-HR correlations via purity and log-negativity and analyzes temporal auto-correlations, highlighting how the model captures key features of information flow and provides a foundation for future, energy-consistent extensions and alternative evaporation scenarios.

Abstract

In this work, we describe optically inspired models for unitary black hole (BH) evaporation. The goal of these models are (i) to be operationally simple, (ii) approximately preserve the thermal nature of the emitted Hawking Radiation (HR), and (iii) attempt to reproduce the Page Curve that purports that information flows forth from the BH when it has evaporated to approximately half its initial mass. We concentrate on modeling the BH as a single mode squeezed state successively interacting, by means of beam splitters and squeezers, with vacuum modes near the horizon, giving rise to entangled pairs representing the external Hawking radiation and its partner particle inside the horizon. Since all states and operations are Gaussian throughout, we use a symplectic formalism to track the evolution of the composite system through the evolving means and variances of their quadrature operators. This allows us to easily compute correlations and entanglement between the BH and the HR, as well as calculate correlations between the BH at early and late times.
Paper Structure (17 sections, 28 equations, 21 figures)

This paper contains 17 sections, 28 equations, 21 figures.

Table of Contents

  1. Introduction
  2. On the Page Information in the BH radiation
  3. Review of Page's results
  4. An Optically Inspired Model for Unitary BH Evaporation: The Model
  5. The role of BS alone
  6. $R_k\equiv R=1-\epsilon$ for all $k$
  7. $R_k = (N_{b_k}-R_{\textrm{\scriptsize offset}})/N_{b_0}$
  8. Case $r=2.5:$
  9. Case $r=7.5:$
  10. The full model: BSSQ: BS plus Squeezing with $R_k = (N_{b_k}-R_{\textrm{\scriptsize offset}})/N_{b_0}$
  11. Occupation numbers and variances
  12. BH-HR correlations: Purity and Log Negativity
  13. Quantum Correlation Plots
  14. Temporal AutoCorrelation Plots
  15. Conclusion and Future Directions
  16. ...and 2 more sections

Figures (21)

  • Figure 1: Page entropy of BH radiation (solid) $S_{m,n}$ and information (dashed) $I=\ln m - S_{m,n}$ vs $\ln m$ for $m n = 291,600$ after Page:1993b). The interpretation is that the information in the BH leaks out into the outgoing radiation at roughly half the evaporation (Page) time of the BH.
  • Figure 2: BH Evaporation Circuit
  • Figure 3: Summary of black hole evaporation beam splitter and single mode squeezing for each interaction event labeled by integer $k$.
  • Figure 4: Summary of black hole evaporation procedure for each interaction event labeled by integer $k$.
  • Figure 5: Occupation numbers ($r=2.5$): (left) $N_{b_k}$ (BH) and (right) $N_{a_k}$ (HR) for $R_k = 0.999$ for all $k$.
  • ...and 16 more figures