Determining $G$ with Laser Spectroscopy to 38 ppb

This paper proposes a precision measurement using a 3 mW tunable diode laser and a 1 T permanent-magnet assembly in a Mach-Zehnder interferometer to detect axion-photon conversion, aiming to determine the gravitational constant $G$ with a 38 ppb accuracy—a 600-fold improvement over current standards—by measuring the axion Compton frequency.

The Gist

The Big Idea: Weighing Gravity with a Laser

Imagine you want to measure the weight of a feather, but your scale is so sensitive that a single breath of wind throws off the reading. That is the current problem with measuring $G$ (the Universal Gravitational Constant). It tells us how strongly gravity pulls on things. For decades, scientists have tried to measure it using heavy metal weights and twisting wires, but the results are messy and imprecise.

This paper proposes a radical new way to measure gravity. Instead of using heavy weights, the author, Noah Bray-Ali, suggests using a laser and a hypothetical particle called an axion. The goal is to measure gravity with 600 times more precision than we can today, using a "table-top" experiment that fits on a desk.

The Cast of Characters

1. The Laser: Think of this as a very precise flashlight. It shines a beam of infrared light (a color we can't see) through a vacuum.
2. The Magnet: A giant, super-strong magnet (1 Tesla, about 20,000 times stronger than a fridge magnet) sits in the path of the laser.
3. The Axion: This is the star of the show. Axions are ghostly, invisible particles that physicists think might make up "dark matter." They are so light and elusive that we've never caught one directly.
4. The "Magic" Connection: The paper claims there is a secret, mathematical link between the "mass" (or frequency) of the axion and the strength of gravity ($G$). If we can find the exact "note" the axion hums at, we can instantly calculate the exact strength of gravity.

How the Experiment Works (The Analogy)

Imagine you are trying to tune a radio to a specific, very quiet station. You turn the dial slowly, listening for a tiny crackle that says, "You're close!"

1. The Setup:
The experiment uses a Mach-Zehnder interferometer. Imagine a road that splits into two lanes.

• Lane A (The Control): The laser beam goes straight through.
• Lane B (The Sensing Arm): The laser beam goes through the giant magnet.

The two beams are then recombined. Usually, they are set up to cancel each other out perfectly, creating a "dark port" (a place where no light should be detected).

2. The Interaction:
As the laser light passes through the magnet in Lane B, the paper suggests something magical happens. The light photons (particles of light) briefly turn into axions and then turn back into light.

• The Analogy: Imagine a runner (the photon) running through a tunnel (the magnet). For a split second, the runner turns into a ghost (the axion) and then turns back into a runner. Because they spent a moment as a ghost, they come out slightly out of step with the runner in Lane A.

3. The Detection:
Because the runners are out of step, they don't cancel each other out perfectly anymore. A tiny, flickering amount of light appears in the "dark port."

• The laser frequency is wiggled (modulated) very fast.
• When the laser hits the exact "frequency" of the axion (about 122 Terahertz), that flickering light gets slightly brighter.
• This is the "crackle" on the radio.

Why This Changes Everything

Currently, our best measurement of gravity has a margin of error. It's like saying a building is 100 meters tall, plus or minus 22 centimeters.

This new method aims to measure it like this: 100 meters, plus or minus 0.003 centimeters.

The "600-Fold" Improvement:
The paper claims this experiment could improve our knowledge of gravity by a factor of 600.

• Current Method: Like trying to measure the thickness of a hair using a ruler.
• New Method: Like using a laser micrometer to measure the thickness of a hair.

The "Secret Sauce" (The Math)

The author connects the dots using a complex formula (Equation 1 in the paper). It's a bit like a recipe:

• If you know the Planck Constant (the size of the quantum world), the Speed of Light, and the Mass of a Proton (all of which we know perfectly), and you measure the Axion's Frequency...
• ...you can solve for $G$ (Gravity).

The paper argues that because we know the other ingredients so well, the only thing stopping us from knowing gravity perfectly is finding the exact "frequency" of the axion. This experiment is designed to find that frequency in just a couple of hours.

The Bottom Line

This is a proposal for a "Year One" experiment. It doesn't require a massive particle collider or a mountain of money. It requires a:

• Tunable laser (about $3,000).
• A custom-made permanent magnet.
• A very sensitive detector.

If successful, this "table-top" experiment could rewrite the textbooks on gravity, turning one of the least precise numbers in physics into one of the most precise, all by listening to the faint "hum" of a ghost particle.

Technical Summary

1. Problem Statement

The primary challenge addressed is the persistent lack of precision in measuring the universal gravitational constant ($G$).

• Current Status: The current best determination of $G$ (CODATA 2022) has a relative uncertainty of approximately 22 parts per million (ppm).
• Limitations: Existing methods rely on macroscopic mechanical systems (e.g., tungsten source masses and silica-fiber torsional oscillators), which are prone to "artifactual" errors and environmental noise.
• Goal: To achieve a 600-fold improvement in precision, reducing the relative uncertainty of $G$ from 22 ppm to 38 parts per billion (ppb) within the first year of the proposed program.

2. Methodology

The proposal outlines a novel metrological approach that links the gravitational constant to the axion Compton frequency ($\nu_A$) via fundamental constants, rather than measuring gravity directly through mechanical forces.

Theoretical Framework

The core of the proposal is a derived relationship between $G$ and the axion Compton frequency ($\nu_A$), based on Quantum Chromodynamics (QCD) and axion electrodynamics:
$$G = \frac{\bar{h}c^5 x_k^3 y_T^{10}}{(h\nu_A)^5 (k\check{T})^7}$$
Where:

• $h, c$: Planck constant and speed of light (exact values).
• $x_k, y_T$: Dimensionless constants related to black-body radiation and QCD vacuum topology.
• $\check{T}$: The Big Bang temperature, implicitly determined by $\nu_A$ and nucleon masses ($m_p, m_n$).

By measuring $\nu_A$ with high precision, $G$ can be calculated with high precision, bypassing the need for massive source weights.

Experimental Setup

The experiment utilizes a table-top, free-space Mach–Zehnder interferometer to detect the conversion of photons into axions (and vice versa) in a strong magnetic field.

• Light Source: A tunable external-cavity diode laser operating at 2458 nm (near-infrared) with 3 mW power and a 1 MHz linewidth.
• Magnetic Field: A custom-built permanent magnet assembly providing a 1 Tesla (T) dipole field over a 40 cm length, with a 6 mm pole gap.
• Beam Geometry: The laser beam is expanded to a waist of 3 mm as it passes through the magnetic field.
• Detection Principle:
  1. Axion Conversion: According to axion electrodynamics ($L_{A\gamma} = g_{A\gamma\gamma}\phi_A \mathbf{E}_{NIR} \cdot \mathbf{B}_{DC}$), photons in the laser beam convert into axions while traversing the magnetic field.
  2. Modulation: The laser frequency is modulated at 1 kHz across a 65 MHz window centered on the predicted axion frequency ($\nu_A \approx 121.94$ THz).
  3. Signal: When the laser frequency matches $\nu_A$, the photon-axion conversion rate peaks, causing a minute modulation in the optical power at the interferometer's "dark port."
  4. Lock-in Amplification: A phase-locked interferometer (using Electro-Optical Modulators at 4 MHz for phase locking and 1 kHz for frequency modulation) detects this power modulation ($\Delta P_{NIR}$) using a photodiode.

3. Key Contributions

• Novel Metrology: This is the first proposal to use axion-photon interactions specifically to determine the value of a fundamental coupling constant ($G$), rather than just searching for axion dark matter.
• Theoretical Link: It establishes a rigorous mathematical link between the axion Compton frequency and $G$, utilizing the QCD vacuum topological susceptibility and nucleon masses.
• Feasible Table-Top Design: Unlike current $G$ measurements requiring massive, isolated setups, this experiment uses a compact, optical bench-top configuration with commercially available components (laser, permanent magnets).
• Precision Target: It targets a measurement of $\nu_A$ to within the laser linewidth (1 MHz), which translates directly to a 38 ppb determination of $G$.

4. Predicted Results

Based on the parameters provided in the proposal:

• Predicted Axion Frequency: $\nu_A \approx 121,944,463 \pm 65$ MHz (derived from current $G$ values).
• Signal Strength: The expected modulation in laser power at the dark port is approximately 12.0 fW (femtowatts).
• Integration Time: A signal-to-noise ratio (SNR) of 5 is achievable in approximately 2 hours of integration time, assuming a detector noise-equivalent power (NEP) of $200 \text{ fW}/\sqrt{\text{Hz}}$.
• Final Precision:
  • Relative uncertainty in $\nu_A$: $\approx 8 \times 10^{-8}$ (8 ppb).
  • Resulting relative uncertainty in $G$: 38 ppb.

5. Significance

• Order of Magnitude Improvement: Achieving 38 ppb would represent a ~600-fold reduction in uncertainty compared to the current CODATA 2022 value.
• Paradigm Shift: It moves the measurement of $G$ from the domain of classical mechanics (rigid bodies, torsion balances) to quantum metrology (atom-photon interactions), potentially eliminating systematic errors associated with macroscopic mass distributions.
• Validation of Fundamental Physics: Successfully measuring $\nu_A$ would not only refine $G$ but also provide a direct experimental test of the relationship between the QCD vacuum, the Big Bang temperature, and the axion mass, bridging particle physics and cosmology.
• Feasibility: The proposal demonstrates that a massive leap in precision can be achieved using modest power lasers and permanent magnets, suggesting a new, scalable path for future fundamental constant measurements.