arxiv: 2604.14824 · v3 · submitted 2026-04-16 · ✦ hep-th · gr-qc

Recognition: unknown

Decrease of the entanglement entropy of the Hawking radiation induced by backreaction in the Bose-Einstein condensate

Tsunehide Kuroki

Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:03 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords Bose-Einstein condensateanalog Hawking radiationentanglement entropybackreactionPage curveBogoliubov coefficientsstep-like configuration

0 comments

The pith

Backreaction from analog Hawking radiation decreases its entanglement entropy in a Bose-Einstein condensate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to demonstrate that backreaction effects in a Bose-Einstein condensate reduce the entanglement entropy of analog Hawking radiation, helping realize the Page curve in a unitary system. Using the microscopic Hamiltonian for a step-like configuration, the authors derive an explicit backreaction term. They combine this with known Bogoliubov coefficients to compute the entropy analytically and show a decrease for low-energy modes across a wide parameter range. A sympathetic reader would care because this provides a concrete, microscopic example of how backreaction can address information loss in analog black hole models.

Core claim

By deriving an explicit form of backreaction from the microscopic theory of the Bose-Einstein condensate in a step-like configuration and combining it with known Bogoliubov coefficients, the entanglement entropy of the analog Hawking radiation is computed and shown to decrease as expected due to the backreaction for sufficiently low energy modes over a wide range of the characterizing parameter.

What carries the argument

The backreaction term derived from the microscopic BEC Hamiltonian in the step-like configuration, combined with Bogoliubov coefficients to modify and compute the entanglement entropy of the Hawking radiation.

If this is right

The entanglement entropy of the Hawking radiation decreases due to the inclusion of backreaction.
This decrease occurs specifically for sufficiently low energy modes.
The decrease holds over a wide range of the parameter describing the step-like configuration.
The microscopic unitary theory of the BEC allows explicit reproduction of Page-curve-like behavior in the analog system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach may allow similar analytic entropy calculations in other analog gravity setups that possess microscopic Hamiltonians.
Experimental measurements of entanglement in BEC systems could directly test the predicted entropy decrease.
The result suggests that backreaction effects can be tracked mode-by-mode to connect analog models with black hole unitarity.

Load-bearing premise

The backreaction derived from the microscopic theory can be combined directly with the Bogoliubov coefficients without higher-order corrections or breakdown of the approximations for the low-energy modes and parameter ranges considered.

What would settle it

A computation or BEC experiment that includes the backreaction and finds the entanglement entropy does not decrease (or increases) for the low-energy modes in the step-like configuration.

Figures

Figures reproduced from arXiv: 2604.14824 by Tsunehide Kuroki.

**Figure 1.** Figure 1: In x > 0 region for ω > 0, ω − vk and q ǫ (0) k (ǫ (0) k + 2λρ0) are depicted as functions of the momentum k. The two intersections are the solution to (3.15). One of them is an outgoing mode with a positive group velocity, and the other is an ingoing one with a negative group velocity. 2. x < 0 In the left region, 0 < cs < −vl and the straight line and the curve again intersect only at two points for arbi… view at source ↗

**Figure 2.** Figure 2: In x < 0 region for ω > 0, we again have two modes, The right-moving (k > 0) mode is ingoing one with a positive group velocity, while the left-moving (k < 0) mode is outgoing with a negative group velocity. However, in this case, we also have two intersections for negative ω satisfying −ωc < ω < 0 as shown in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: In x < 0 region for −ωc < ω < 0, we still have two intersections. They are the rightmoving modes (k > 0). One of them is an ingoing mode with a positive group velocity, and the other is an outgoing one with a negative group velocity. They are associated with antiparticle excitations. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: In x < 0 region for ω < 0, momenta associated with antiparticle excitations are also identified by considering the − sign in (3.14) and ω → −ω. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: δ as a function of D in (2.25). As shown in [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗

read the original abstract

We analytically study the effect of backreaction from analog Hawking radiation on its entanglement entropy in the Bose-Einstein condensate (BEC). The backreaction is expected to play an essential role in the decrease of the entanglement entropy and in realizing the Page curve. Since the BEC theory has microscopic Hamiltonian and thus exhibits unitarity, it is desirable to reproduce the Page curve explicitly by using the Hamiltonian. In order to analyze this in a concrete example, we study the BEC with a step-like configuration that has been extensively studied in the literature. By using the microscopic theory, we derive an explicit form of backreaction from analog Hawking radiation. Combining it with the known results of the Bogoliubov coefficients, we analytically compute the entanglement entropy of the Hawking radiation, and show that it decreases as expected due to the backreaction for sufficiently low energy modes over a wide range of the parameter characterizing the step-like configuration.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives an explicit backreaction term from the microscopic BEC Hamiltonian in a step-like flow and combines it with existing Bogoliubov coefficients to show an analytic drop in entanglement entropy for low-energy Hawking modes.

read the letter

The main new element here is the explicit backreaction derived from the microscopic Hamiltonian for the step-like BEC configuration, then inserted into the entanglement entropy formula. This produces a concrete analytic result that the entropy decreases for sufficiently low-energy modes across a wide parameter range, which aligns with expectations for unitary evolution toward Page-curve behavior in analog Hawking radiation. That is a useful step in a model that has a clear microscopic origin and unitarity built in from the start. The calculation stays analytic and avoids heavy numerics, which keeps it transparent for readers who already know the prior Bogoliubov work on this setup. The stress-test concern about whether the backreaction can be added without shifting the mode functions or the Bogoliubov transformation itself is the central point to check. The abstract claims the combination works for low-energy modes, but any letter to the authors should ask for an explicit check that the correction to the condensate profile or dispersion stays small enough that the unmodified coefficients remain accurate to the order needed for the entropy decrease. If that justification is only order-of-magnitude or limited to a narrow window, the result is more conditional than the abstract suggests. The paper is aimed at people working on analog gravity and black-hole information in condensed-matter systems. It is narrow enough that most readers outside that niche will not need it, but inside the subfield it supplies a calculable example that can be compared to other approaches. I would send it to peer review because the analytic claim is sharp enough to be worth referee time, even if the consistency argument around the backreaction insertion needs tightening.

Referee Report

1 major / 1 minor

Summary. The manuscript analytically derives an explicit backreaction term from the microscopic Hamiltonian of a Bose-Einstein condensate with a step-like flow configuration. It combines this term with known analytic Bogoliubov coefficients for the unperturbed case to compute the entanglement entropy of the analog Hawking radiation, claiming an explicit decrease for sufficiently low-energy modes over a wide range of the step-like parameter.

Significance. If the central combination holds without uncontrolled corrections, the result supplies a concrete, Hamiltonian-based demonstration that backreaction reduces Hawking entanglement entropy in a unitary analog system, advancing explicit realizations of Page-curve behavior in analog gravity. The microscopic starting point and analytic control are strengths that distinguish it from purely effective-field treatments.

major comments (1)

[The section combining the backreaction term with the Bogoliubov coefficients] The central claim rests on inserting the derived backreaction directly into the entanglement-entropy formula while retaining the pre-existing Bogoliubov coefficients. The manuscript must demonstrate quantitatively that the backreaction remains a small perturbation that does not appreciably shift the mode functions, effective horizon location, or the Bogoliubov transformation itself for the low-energy modes and parameter range considered; otherwise the analytic combination undercounts or overcounts the unitary evolution of the reduced density matrix.

minor comments (1)

[Abstract and introduction] Clarify the precise definition of the step-like parameter and its range of validity in the abstract and introduction to aid readers unfamiliar with the prior literature on this configuration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We are pleased that the referee recognizes the potential significance of our Hamiltonian-based approach to analog Hawking radiation and the Page curve. We respond to the major comment below.

read point-by-point responses

Referee: The central claim rests on inserting the derived backreaction directly into the entanglement-entropy formula while retaining the pre-existing Bogoliubov coefficients. The manuscript must demonstrate quantitatively that the backreaction remains a small perturbation that does not appreciably shift the mode functions, effective horizon location, or the Bogoliubov transformation itself for the low-energy modes and parameter range considered; otherwise the analytic combination undercounts or overcounts the unitary evolution of the reduced density matrix.

Authors: We agree that a quantitative demonstration of the perturbative nature of the backreaction is necessary to justify retaining the unperturbed Bogoliubov coefficients. In the revised manuscript, we will include an additional analysis estimating the magnitude of the backreaction-induced corrections to the mode functions and the effective horizon position. For the low-energy modes and the parameter range of the step-like flow where the decrease in entanglement entropy is observed, these corrections are shown to be small, of order O( (Hawking flux / background energy density) ), which is much less than unity. This supports the validity of our analytic combination without uncontrolled corrections to the unitary evolution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper first derives an explicit backreaction term directly from the microscopic BEC Hamiltonian for the step-like flow. This term is then inserted into the entanglement entropy expression that already employs the standard analytic Bogoliubov coefficients obtained from prior literature on the non-backreacted case. The observed decrease for low-energy modes is an explicit computational result of that insertion rather than a definitional identity, a fitted parameter renamed as a prediction, or a reduction to any self-citation chain. No equations are shown to be equivalent by construction, and the cited Bogoliubov coefficients are external, falsifiable results independent of the present backreaction derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the analog horizon model, the perturbative treatment of backreaction, and the applicability of Bogoliubov transformations to the step-like configuration.

axioms (2)

domain assumption The BEC system is described by a microscopic Hamiltonian that is unitary.
Invoked to justify reproducing the Page curve explicitly.
domain assumption The step-like density profile creates a valid analog black hole horizon whose radiation can be treated with Bogoliubov methods.
Stated as extensively studied in the literature.

pith-pipeline@v0.9.0 · 5451 in / 1291 out tokens · 33864 ms · 2026-05-10T11:03:51.891345+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 21 canonical work pages

[1]

Breakdown of Predictability in Gravitat ional Collapse,

S. W. Hawking, “Breakdown of Predictability in Gravitat ional Collapse,” Phys. Rev. D 14 (1976), 2460-2473

1976
[2]

Almheiri, T

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian an d A. Tajdini, “The entropy of Hawking radiation,” Rev. Mod. Phys. 93, no.3, 035002 (2021) [arXiv:2006.06872 [hep-th]]

work page arXiv 2021
[3]

Seeing Page Curves and Islands with Blinders On,

H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randal l and M. Riojas, “Seeing Page Curves and Islands with Blinders On,” [arXiv:2602.06543 [h ep-th]]

work page arXiv
[4]

Geng, SciPost Phys.19, no.6, 146 (2025) [arXiv:2312.13336 [hep-th]]

H. Geng, “Graviton mass and entanglement islands in low s pacetime dimensions,” SciPost Phys. 19, no.6, 146 (2025) [arXiv:2312.13336 [hep-th]]

work page arXiv 2025
[5]

Geng, (2025), arXiv:2502.08703 [hep-th]

H. Geng, “The mechanism behind the information encoding for islands,” JHEP 03, 037 (2026) [arXiv:2502.08703 [hep-th]]

work page arXiv 2026
[6]

Experimental black hole evaporation,

W. G. Unruh, “Experimental black hole evaporation,” Phy s. Rev. Lett. 46 (1981), 1351-1353

1981
[7]

Macher and R

J. Macher and R. Parentani, “Black/White hole radiation from dispersive theories,” Phys. Rev. D 79 (2009), 124008 [arXiv:0903.2224 [hep-th]]

work page arXiv 2009
[8]

Bogoliubov Theor y of acoustic Hawking radiation in Bose-Einstein Condensates,

A. Recati, N. Pavloﬀ and I. Carusotto, “Bogoliubov Theor y of acoustic Hawking radiation in Bose-Einstein Condensates,” Phys. Rev. A 80 (2009), 043603 [arXiv:0907.4305 [cond- mat.quant-gas]]. 25

work page arXiv 2009
[9]

Average entropy of a subsystem,

D. N. Page, “Average entropy of a subsystem,” Phys. Rev. L ett. 71 (1993), 1291-1294 [arXiv:gr- qc/9305007 [gr-qc]]; D. N. Page, “Information in black hole radiation,” Phys. Rev . Lett. 71 (1993), 3743-3746 [arXiv:hep-th/9306083 [hep-th]]

work page arXiv 1993
[10]

Quan tum back-reaction in dilute Bose-Einstein condensates,

R. Schutzhold, M. Uhlmann, Y. Xu and U. R. Fischer, “Quan tum back-reaction in dilute Bose-Einstein condensates,” Phys. Rev. D 72, 105005 (2005) [arXiv:cond-mat/0503581 [cond- mat.other]]

work page arXiv 2005
[11]

Dynamical aspects of analogue gravity: The Backreaction of quantum ﬂuctua- tions in dilute Bose-Einstein condensates,

U. R. Fischer, “Dynamical aspects of analogue gravity: The Backreaction of quantum ﬂuctua- tions in dilute Bose-Einstein condensates,” Lect. Notes Ph ys. 718, 93-113 (2007) [arXiv:cond- mat/0512537 [cond-mat.other]]

work page arXiv 2007
[12]

The infor mation loss problem: an analogue gravity perspective,

S. Liberati, G. Tricella and A. Trombettoni, “The infor mation loss problem: an analogue gravity perspective,” Entropy 21, no.10, 940 (2019) [arXiv:1908.01036 [gr-qc]]

work page arXiv 2019
[13]

Backreaction in an analogue black hole experiment,

S. Patrick, H. Goodhew, C. Gooding and S. Weinfurtner, “ Backreaction in an analogue black hole experiment,” Phys. Rev. Lett. 126, no.4, 041105 (2021) [arXiv:1905.03045 [gr-qc]]

work page arXiv 2021
[14]

Back-rea ction in canonical analogue black holes,

S. Liberati, G. Tricella and A. Trombettoni, “Back-rea ction in canonical analogue black holes,” Appl. Sciences 10 (2020) no.24, 8868 [arXiv:2010.09966 [gr-qc]]

work page arXiv 2020
[15]

Low frequency analogue Hawking radiation: The Bogoliubov- de Gennes model,

A. Coutant and S. Weinfurtner, “Low frequency analogue Hawking radiation: The Bogoliubov- de Gennes model,” Phys. Rev. D 97 (2018) no.2, 025006 [arXiv:1707.09664 [gr-qc]]

work page arXiv 2018
[16]

Tripartite Entanglement of Hawk ing Radiation in Dispersive Model,

Y. Nambu and Y. Osawa, “Tripartite Entanglement of Hawk ing Radiation in Dispersive Model,” Phys. Rev. D 103 (2021), 125007 [arXiv:2101.11764 [gr-qc]]

work page arXiv 2021
[17]

Finite-Temper ature Models of Bose-Einstein Conden- sation

Proukakis, Nick P., and Brian Jackson. “Finite-Temper ature Models of Bose-Einstein Conden- sation.” Journal of Physics B: Atomic, Molecular and Optica l Physics, vol. 41, no. 20, Oct. 2008, p. 203002

2008
[18]

Quantum nonlinear eﬀects in the number-conserving analog gravity of Bose-Einstein condensates,

K. Pal and U. R. Fischer, “Quantum nonlinear eﬀects in the number-conserving analog gravity of Bose-Einstein condensates,” Phys. Rev. D 110, no.11, 116022 (2024) [arXiv:2410.13596 [gr- qc]]

work page arXiv 2024
[19]

Spacetime analogue of Bose-Einstein condensates: Bogoliubov-de Gennes formu lation,

Y. Kurita, M. Kobayashi, T. Morinari, M. Tsubota and H. I shihara, “Spacetime analogue of Bose-Einstein condensates: Bogoliubov-de Gennes formu lation,” Phys. Rev. A 79 (2009), 043616 [arXiv:0810.3088 [cond-mat.other]]

work page arXiv 2009
[20]

Macher and R

J. Macher and R. Parentani, “Black hole radiation in Bos e-Einstein condensates,” Phys. Rev. A 80 (2009), 043601 [arXiv:0905.3634 [cond-mat.quant-gas]]

work page arXiv 2009
[21]

Step-like discontinuities i n Bose-Einstein condensates and Hawking radiation: the hydrodynamic limit,

A. Fabbri and C. Mayoral, “Step-like discontinuities i n Bose-Einstein condensates and Hawking radiation: the hydrodynamic limit,” Phys. Rev. D 83 (2011), 124016 [arXiv:1004.4876 [gr-qc]]

work page arXiv 2011
[22]

Step-like discon tinuities in Bose-Einstein condensates and Hawking radiation: dispersion eﬀects,

C. Mayoral, A. Fabbri and M. Rinaldi, “Step-like discon tinuities in Bose-Einstein condensates and Hawking radiation: dispersion eﬀects,” Phys. Rev. D 83 (2011), 124047[arXiv:1008.2125 [gr-qc]]. 26

work page arXiv 2011
[23]

Evanescent Mod es and Step-like Acoustic Black Holes,

J. B. Curtis, G. Refael and V. Galitski, “Evanescent Mod es and Step-like Acoustic Black Holes,” Annals Phys. 407 (2019), 148-165 [arXiv:1801.01607 [cond-mat.quant-gas] ]

work page arXiv 2019
[24]

Particle creation and entanglem ent in a dispersive model with step velocity proﬁle,

Y. Osawa and Y. Nambu, “Particle creation and entanglem ent in a dispersive model with step velocity proﬁle,” Phys. Rev. D 107 (2023) no.10, 105005 [arXiv:2204.08684 [gr-qc]]. 27

work page arXiv 2023