Photon counting readout for detection and inference of gravitational waves from neutron star merger remnants

This paper proposes and demonstrates through simulations that a photon-counting readout scheme, which quantizes signals and noise into discrete photon distributions, can effectively detect rare, low signal-to-noise ratio gravitational waves from neutron star merger remnants and significantly improve constraints on neutron star radii compared to traditional homodyne techniques.

The Gist

The Big Picture: Listening to the "Aftermath" of a Cosmic Crash

Imagine two neutron stars (super-dense city-sized balls of matter) crashing into each other. This collision sends out ripples in space-time called gravitational waves.

Scientists are very interested in what happens after the crash. The wreckage (the "remnant") vibrates and emits a specific type of gravitational wave. If we can listen to this "aftermath song," we can learn the secrets of how matter behaves under extreme pressure—essentially figuring out the recipe for the densest stuff in the universe.

The Problem:
These "aftermath songs" are very quiet. They are like trying to hear a whisper in a hurricane. Current and future detectors are great, but the "noise" of the universe (and the detectors themselves) is often louder than the signal. Most of the time, the signal is so faint that standard listening methods just hear static.

The Old Way: The "Microphone" (Homodyne Readout)

Currently, gravitational wave detectors work like a very sensitive microphone. They measure the continuous "volume" of the light bouncing inside the machine.

• How it works: It measures the average flow of light waves.
• The Flaw: Because the signal is so weak, it gets drowned out by "quantum static" (random jitters in the light particles called photons). It's like trying to hear a whisper while someone is shaking a bag of marbles next to your ear. The shaking (noise) is so loud that you can't tell if the whisper is there.

The New Idea: The "Click Counter" (Photon Counting)

The authors propose a different way to listen. Instead of measuring the continuous volume of light, they suggest counting the individual clicks of light particles (photons) arriving at the detector.

• The Analogy: Imagine you are in a dark room.
  • The Microphone (Old Way): You try to measure the average brightness of the room. If there is a tiny bit of light (the signal) mixed with a lot of random flickering (noise), you can't tell the difference.
  • The Click Counter (New Way): You put on night-vision goggles that only see individual sparks. You wait. If you see a spark at a very specific time and place, you know it's a signal. Even if the room is mostly dark, a single spark is a clear "Yes, something happened!"

Why This Works for "Whispers"

The paper argues that for these specific, very faint signals (which happen at high pitches, above 1,000 Hz), counting the sparks is actually better than measuring the volume.

1. The "One Spark" Rule: In the old method, if the signal is too weak, it just looks like part of the background noise. In the new method, if even one single photon (spark) arrives that matches the pattern of the signal, the detector can say, "I found it!"
2. The Odds: The authors ran computer simulations and found that for a signal that is 100 times too quiet to be heard by the old method, there is still about a 1 in 100 chance that a single spark will appear. If you watch enough crashes, you will eventually catch those lucky sparks.

The Result: Building a Better Picture

The researchers didn't just look at one crash; they simulated watching 10,000 crashes.

• The Old Method: Even after watching 10,000 crashes, the "microphone" method was still very fuzzy. It couldn't pin down the size of the neutron star wreckage very well.
• The New Method: By "stacking" all those single sparks from the 10,000 crashes, the new method was able to measure the size of the neutron star twice as accurately as the old method.

The Catch (The "Classical Noise" Problem)

This new "Click Counter" method has one strict rule: It only works if the room isn't too noisy with other things.

• Quantum Noise: The random jitters of light (which the new method handles well).
• Classical Noise: Real-world vibrations, heat, and electronic hums.

If the detector is shaking too much (high classical noise), the "click counter" will get confused by false sparks. The paper shows that if we can build detectors that are super-stable (low classical noise), this new method is a game-changer. If the noise is too high, the old microphone is still better.

Summary

The paper suggests that for listening to the faint, high-pitched "aftermath" of neutron star crashes, we should stop trying to measure the "volume" of the light and start counting the individual light particles.

It's like switching from trying to hear a whisper in a storm by measuring the wind speed, to simply waiting for a single, distinct leaf to blow past your ear. If you wait long enough and have a quiet enough room, you can hear the whisper that everyone else missed. This allows scientists to learn more about the universe's densest matter than ever before.

Technical Summary

1. Problem Statement

The Challenge of Post-Merger Detection:
Binary Neutron Star (BNS) mergers emit gravitational waves (GWs) during the inspiral, merger, and post-merger phases. While the inspiral is well-detected, the post-merger signal (emitted at frequencies $>1$ kHz) carries crucial information about the dense matter equation of state (EoS). However, these signals are expected to be extremely weak.

• Current Limitations: Next-generation detectors (e.g., Cosmic Explorer, Einstein Telescope) are projected to observe hundreds of thousands of BNS mergers annually, but only a tiny fraction (approx. 1 per year) will have a post-merger Signal-to-Noise Ratio (SNR) $>5$.
• The "Sub-threshold" Problem: For the vast majority of events, the post-merger SNR will be $<1$. Under standard readout schemes (homodyne detection), these sub-threshold signals yield uninformative posteriors on source properties because the noise dominates the signal.
• Noise Dominance: Above 1 kHz, detector sensitivity is limited by quantum shot noise (photon counting noise), not classical noise (seismic, thermal). Standard homodyne readout treats this noise as an additive Gaussian background, which obscures weak signals.

2. Methodology

The authors propose and analyze an alternative readout scheme: Photon Counting via Temporal-Mode Basis.

A. Conceptual Shift

• Homodyne Readout (Standard): Measures the continuous amplitude of the electromagnetic field (fringe light). The output is a continuous time-series where quantum noise manifests as additive Gaussian noise.
• Photon Counting Readout (Proposed): Measures the occupation number (discrete photon counts) in specific temporal modes (frequency-time basis functions).
  • The detector is configured to filter the output light into a set of orthogonal temporal modes ($d_k(f)$).
  • Instead of measuring amplitude, the system counts discrete photons in each mode.
  • This changes the statistical nature of the noise: in the regime where classical noise is negligible, the only way to detect a photon is if the signal (or classical noise) excites the vacuum state.

B. Statistical Framework

• Likelihood Function: The authors derive a new likelihood function (Eq. 31) based on the probability of observing $N_k$ photons in a specific mode $k$.
  • The expected number of photons $\bar{N}_k$ is the sum of signal photons ($\bar{N}_{sig}$) and classical noise photons ($\bar{N}_{cl}$).
  • The distribution is a convolution of a Poisson distribution (signal) and a geometric-like distribution (classical noise).
• Noise Scaling:
  • In homodyne, the uncertainty scales linearly with the total noise power spectral density ($S_{total} = S_{classical} + S_{quantum}$).
  • In photon counting, the Fisher Information for stochastic or marginalized parameters scales as $\sqrt{S_{classical} \cdot S_{quantum}}$. If $S_{classical} \ll S_{quantum}$ (the case above 1 kHz), photon counting offers a fundamental advantage in scaling.

C. Implementation Details

• Basis Design: They construct a set of 400 orthonormal temporal-mode filters (damped sinusoids) to match the expected post-merger spectral peaks (1.5–4 kHz).
• Simulation: They simulate $10^4$ BNS post-merger events using the APR4 EoS model, assuming a Cosmic Explorer (CE) detector configuration with 1.5 MW arm power and 10 dB squeezing (8 dB effective above 1 kHz).
• Inference: They perform Bayesian hierarchical inference to constrain the radius of a 1.6 $M_\odot$ neutron star ($R_{1.6}$), a proxy for the EoS, by combining data from many sub-threshold events.

3. Key Contributions

1. Theoretical Framework: Established the formal statistical likelihood for photon counting readout in GW detectors, replacing the standard Gaussian likelihood with a discrete photon-counting distribution.
2. Single-Event Inference: Demonstrated that photon counting can extract meaningful information from signals with SNR $\ll 1$ (e.g., SNR $\approx 0.2$).
   • While homodyne yields no information for SNR < 1, photon counting can detect a signal if even a single photon is generated by the signal (which happens with $\sim 1$ in 100 probability for SNR 0.2).
3. Hierarchical Population Analysis: Showed that stacking these "serendipitous" single-photon detections across a population significantly tightens constraints on the neutron star EoS.
4. Noise Regime Analysis: Clarified that the advantage of photon counting is strictly dependent on the hierarchy $S_{classical} \ll S_{quantum}$. If classical noise dominates, the benefit vanishes.

4. Key Results

• Single Event Performance:
  • For an SNR of 1, homodyne readout provides no constraint on signal parameters.
  • For the same SNR, photon counting has an $\sim 18\%$ chance of generating at least one photon. If a photon is detected, the peak frequency and time offset can be constrained.
  • For SNR 0.2, photon counting detects $\sim 1$ in 100 events, whereas homodyne detects 0.
• Population Constraints (EoS):
  • Using $10^4$ simulated events (equivalent to 0.01 to 1.5 years of observation):
    • Homodyne (10 dB squeezing): Yields a broad posterior for $R_{1.6}$ with a 68% credible interval width of 1.25 km.
    • Photon Counting (Design classical noise): Yields a significantly tighter posterior with a 68% credible interval width of 0.51 km.
  • Improvement: Photon counting provides a ~2.5x reduction in uncertainty compared to the standard homodyne design sensitivity.
• Classical Noise Sensitivity:
  • The advantage of photon counting is highly sensitive to classical noise levels. If classical noise is reduced by an order of magnitude, the photon counting constraint tightens further to 0.25 km.
  • Conversely, if classical noise is high, the "background" of noise photons overwhelms the signal photons, negating the advantage.

5. Significance and Implications

• Unlocking Sub-Threshold Science: This work suggests that the "rare" high-SNR post-merger detections are not the only path to EoS constraints. By utilizing the "tail" of the distribution (thousands of sub-threshold events), photon counting can extract scientific value currently considered lost to noise.
• Hardware Requirements: The study highlights that future detectors must prioritize reducing classical noise (leakage, thermal, etc.) to maximize the benefit of photon counting. It also notes that the technology for temporal-mode filtering (quantum memory/metrology) is not yet mature but is a necessary direction for development.
• Strategic Detector Design: The results argue for a hybrid approach: use standard homodyne for low frequencies ($<1$ kHz, where classical noise dominates) and switch to photon counting for high frequencies ($>1$ kHz, where quantum noise dominates).
• Statistical Insight: The paper provides a crucial insight into hierarchical inference: when nuisance parameters (like phase) must be marginalized over for every event, the information scaling degrades. Photon counting, being optimal for incoherent/stochastic estimation, mitigates this degradation better than homodyne.

In conclusion, the paper presents a compelling case that photon counting is a viable and superior alternative to homodyne readout for high-frequency GW astronomy, potentially transforming the ability to constrain the nuclear equation of state using the vast population of sub-threshold neutron star merger remnants.