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

This paper analytically demonstrates that backreaction from analog Hawking radiation in a Bose-Einstein condensate with a step-like configuration causes the entanglement entropy of the radiation to decrease for low-energy modes, thereby providing a microscopic, unitary realization of the Page curve.

The Gist

The Big Picture: Solving the Black Hole Mystery

Imagine a black hole as a cosmic vacuum cleaner that never stops eating. In the 1970s, a physicist named Stephen Hawking discovered that these vacuum cleaners aren't actually perfect; they slowly leak energy (like steam from a kettle). This is called Hawking Radiation.

Here is the problem: If a black hole leaks energy and eventually disappears, what happens to all the information (like the history of a star or a spaceship) that fell inside?

• The Old Theory: The information vanishes forever. This breaks the fundamental rules of quantum physics (which say information can never be destroyed). This is the Information Loss Paradox.
• The Hope: Physicists believe the information must come out, but the math is incredibly hard to solve for real black holes because we don't have a complete theory of gravity yet.

The Solution: Building a Mini-Black Hole in a Lab

Since we can't easily experiment with real black holes, the authors of this paper used a clever trick: Analog Gravity.

Imagine a river flowing very fast. If the water flows faster than the speed of sound, a fish trying to swim upstream against the current can't make progress. To the fish, the water looks like a "horizon" it can't cross.

• The Setup: The authors used a Bose-Einstein Condensate (BEC). Think of this as a super-cold, super-quiet cloud of atoms acting like a single giant wave.
• The Analogy: They created a flow in this cloud where the "sound" (ripples in the atoms) moves slower than the flow of the cloud itself in one specific area. This creates a Sonic Black Hole. Sound waves (the analog of light) get trapped inside, just like light in a real black hole.

The Experiment: Watching the Entanglement

In quantum physics, when particles are created in pairs (one falling in, one escaping), they are "entangled." They are like a pair of magical dice; if you roll a 6 on one, the other must show a 1, no matter how far apart they are.

• The Problem: As the black hole evaporates, more and more of these pairs are created. If we only look at the escaping particles (the radiation), the "entanglement entropy" (a measure of how much information is hidden) keeps growing. If it keeps growing forever, the information is lost.
• The Goal: Physicists expect the entropy to go up, reach a peak, and then go back down as the black hole finishes evaporating. This curve is called the Page Curve. If the curve goes down, it means the information is coming back out.

The Secret Ingredient: Backreaction

In Hawking's original math, he assumed the black hole was a giant, unchanging stage, and the radiation was just a small actor on it. He ignored the fact that the actor might push back on the stage.

This paper asks: What if the radiation pushes back on the black hole?

• The Metaphor: Imagine a trampoline (the black hole) with a bowling ball (the radiation) rolling on it. Hawking ignored the fact that the ball actually dents the trampoline.
• The Discovery: The authors calculated that as the "sound waves" (radiation) escape, they actually change the density of the atomic cloud. Specifically, they make the cloud slightly denser.
• The Result: This change in density pushes the "horizon" (the point of no return) slightly backward. It's like the black hole is shrinking a tiny bit faster than expected because of the pressure from the escaping particles.

The Big Finding: The Curve Turns Down

The authors did the math using the microscopic rules of the atomic cloud (which are perfect and never break the laws of physics).

1. They calculated how the "backreaction" (the pushback) changes the relationship between the particles inside and outside.
2. They found that this pushback reduces the entanglement entropy.
3. The Conclusion: For a wide range of conditions, the entropy goes up, peaks, and then starts to go down.

In simple terms: The "pushback" from the escaping radiation helps the black hole spit the information back out. This confirms that the information isn't lost; it's just being returned to the universe in a way that follows the Page Curve.

Why This Matters

• It's a Proof of Concept: Real black holes are too messy to study directly. But this "sonic black hole" is a clean, controllable system where the laws of physics are known to be perfect (unitary).
• The Verdict: Since this system must preserve information, and the math shows the entropy goes down when we include backreaction, it gives strong evidence that real black holes probably do the same thing.
• The Takeaway: The universe doesn't delete data. The "pushback" of the radiation is the key mechanism that saves the information, resolving the paradox.

Summary: By building a tiny, fake black hole out of frozen atoms, the authors proved that when you account for the "recoil" of the escaping radiation, the black hole successfully returns all its secrets, solving the mystery of where the information goes.

Technical Summary

1. Problem Statement

The paper addresses the black hole information loss problem, specifically focusing on the behavior of the entanglement entropy of Hawking radiation.

• The Paradox: In standard semi-classical calculations, Hawking radiation is thermal, leading to a monotonically increasing entanglement entropy between the black hole interior and the exterior radiation. This violates unitarity if the black hole evaporates completely, as a pure initial state would evolve into a mixed state.
• The Page Curve: To resolve this, the entanglement entropy must eventually decrease (following the "Page curve") to return to zero, implying information recovery.
• The Gap: While the "island formula" and other holographic approaches suggest this behavior, a microscopic, unitary derivation within a concrete quantum system has been lacking.
• The Goal: The author aims to explicitly demonstrate, using a Bose-Einstein Condensate (BEC) analog gravity model, how backreaction (the effect of the radiation on the background geometry) causes the entanglement entropy to decrease, thereby reproducing the Page curve behavior in a unitary system.

2. Methodology

The study utilizes the Bose-Einstein Condensate (BEC) as an analog gravity system, where phonons in a flowing condensate mimic scalar fields in a black hole spacetime.

• Microscopic Framework:

  • The system is defined by a microscopic Hamiltonian for a BEC in $d$-spatial dimensions, ensuring manifest unitarity.
  • The background is described by the Gross-Pitaevskii (GP) equation, and fluctuations (analog Hawking radiation) are described by the Bogoliubov-de Gennes (BdG) equation.
  • Backreaction is incorporated by expanding the field $\phi$ in powers of a formal parameter $\tilde{\hbar}$, treating the backreacted background $\tilde{\phi}_0$ as distinct from the radiation fluctuation $\delta\tilde{\phi}$.

• Setup:

  • A step-like configuration is used, where the flow velocity $v(x)$ changes abruptly (or smoothly in the limit) across a horizon at $x=0$.
  • The flow is subsonic ($|v| < c_s$) for $x>0$ and supersonic ($|v| > c_s$) for $x<0$, creating an analog black hole horizon.
  • The analysis focuses on low-energy modes ($\omega \ll c_s/L$, where $L$ is the scale of the step).

• Analytical Approach:

  1. Mode Identification: The author identifies ingoing and outgoing modes in the asymptotic regions ($x \to \pm \infty$) and distinguishes between particle and antiparticle modes (associated with negative frequency solutions in the supersonic region).
  2. Bogoliubov Transformation: Using results from previous literature (specifically [7]), the author derives the linear transformation (Bogoliubov coefficients $\alpha, \beta, R, B, \dots$) connecting the "in" vacuum to the "out" modes.
  3. Entanglement Entropy Calculation: The entanglement entropy ($S_{EE}$) is calculated by tracing out the interior modes. It is expressed in terms of the Bogoliubov coefficients, specifically the squeezing parameter $r$ and the mixing angle $\theta$.
  4. Backreaction Analysis:
     • The author derives the explicit form of the backreaction on the background density $\rho_0$ and phase $\theta_0$ by solving the modified GP equation including the two-point function of the fluctuations $\langle \delta\phi^\dagger \delta\phi \rangle$.
     • This backreaction induces a shift in the sound speed $c_s$ and the horizon location.
     • The change in the background parameters is fed back into the Bogoliubov coefficients to calculate the shift in entanglement entropy.

3. Key Contributions

• Microscopic Derivation of Backreaction: The paper provides an explicit derivation of the backreaction term directly from the microscopic BEC Hamiltonian, showing that the backreaction manifests as an increase in the condensate density $\rho_0$.
• Explicit Entanglement Entropy Formula: The author derives a closed-form expression for the entanglement entropy of the analog Hawking radiation in terms of the Bogoliubov coefficients, utilizing a specific basis transformation to simplify the reduced density matrix.
• Quantitative Demonstration of Entropy Decrease: By combining the derived backreaction with the known Bogoliubov coefficients for a step-like potential, the paper analytically proves that the entanglement entropy decreases for a wide range of parameters.

4. Key Results

• Density Increase and Horizon Shrinkage: The backreaction calculation reveals that the radiation causes the condensate density to increase ($\delta\rho_0 > 0$). Since the sound speed $c_s \propto \sqrt{\rho_0}$, the sound speed increases. Consequently, the sonic horizon (where $|v| = c_s$) recedes (moves inward), effectively shrinking the black hole.
• Modification of Bogoliubov Coefficients: The change in the background parameters ($c_s$ and flow velocities) modifies the Bogoliubov coefficient $\beta$, which controls the rate of Hawking pair creation.
• Entropy Reduction:
  • The entanglement entropy $S_{EE}$ is a monotonically increasing function of $|\beta|^2$.
  • The analysis shows that for low-energy modes and a wide range of the step-parameter $D$ (characterizing the velocity jump), the backreaction induces a negative correction to $|\beta|^2$.
  • Specifically, the quantity $\delta$ (representing the relative change in $|\beta|^2$) is found to be negative for most values of $D$ (except near $D \approx 1$).
  • Conclusion: As a result, the entanglement entropy decreases due to the backreaction, consistent with the expected Page curve behavior required to resolve the information paradox.
• Limitations: The decrease is not observed near the limit $D \to 1$ (where the incoming mode does not sufficiently convert to the outgoing mode), suggesting that higher-order corrections in the low-energy expansion are needed in that specific regime.

5. Significance

• Unitary Resolution in Analog Systems: This work provides a concrete, unitary example where the information loss paradox is resolved via backreaction. It demonstrates that in a system with a known microscopic Hamiltonian, the entanglement entropy naturally decreases without invoking non-local effects or exotic physics, provided the backreaction is accounted for.
• Validation of the Page Curve: It offers strong evidence that the "Page curve" is a generic feature of unitary quantum systems mimicking black holes, supporting the idea that backreaction is the mechanism driving the entropy decrease.
• Distinction between Analog and Gravity: The paper highlights that while the mechanism (backreaction reducing entropy) is robust, the specific details (like the separability of Hilbert spaces in BEC vs. the non-separability in gravity) differ. This helps isolate universal features of the information paradox from model-dependent artifacts.
• Future Directions: The results suggest that calculating higher-order terms in the low-energy expansion could resolve the anomalies near $D=1$ and further refine the understanding of how microscopic quantum gravity (or its analogs) preserves unitarity.

In summary, Kuroki's paper successfully bridges the gap between microscopic quantum mechanics and black hole thermodynamics by showing that backreaction in a BEC analog system leads to a decrease in entanglement entropy, offering a tangible mechanism for the recovery of information.