[Paper Review] Using Atomic Clocks to Detect Gravitational Waves

Abraham Loeb, Dan Maoz|arXiv (Cornell University)|Jan 5, 2015

Advanced Frequency and Time Standards19 citations

TL;DR

This paper proposes using an array of atomic clocks distributed along Earth's orbit to detect low-frequency gravitational waves (GWs) via their time-dilation effects, leveraging the clocks' $10^{-18}$ timing precision. The method measures differential clock-rate variations induced by GWs at mHz frequencies, offering a complementary approach to eLISA's interferometric technique with potential for high sensitivity to supermassive black hole binaries at cosmological distances.

ABSTRACT

Atomic clocks have recently reached a fractional timing precision of $<10^{-18}$. We point out that an array of atomic clocks, distributed along the Earth's orbit around the Sun, will have the sensitivity needed to detect the time dilation effect of mHz gravitational waves (GWs), such as those emitted by supermassive black hole binaries at cosmological distances. Simultaneous measurement of clock-rates at different phases of a passing GW provides an attractive alternative to the interferometric detection of temporal variations in distance between test masses separated by less than a GW wavelength, currently envisioned for the eLISA mission.

Motivation & Objective

• To explore the feasibility of detecting low-frequency gravitational waves using the extreme timing precision of modern atomic clocks.
• To address the challenge of detecting mHz gravitational waves from supermassive black hole binaries at cosmological distances.
• To propose a clock-based detection method that complements existing interferometric approaches like eLISA.
• To demonstrate that atomic clock arrays can achieve sufficient sensitivity to measure GW-induced time dilation at the $h_{00} \sim 10^{-18}$ level.
• To enable improved source localization and detection of chirping signals from inspiraling binaries through synchronized clock measurements.

Proposed method

• Deploy an array of atomic clocks along Earth's orbit at 1 AU, with separations on the order of $10^8$ km to match half-wavelengths of mHz GWs.
• Use optical lasers locked to each clock to maintain phase coherence across inter-craft baselines, enabling precise comparison of clock rates.
• Measure the beat frequency between laser signals from separated clocks, which scales as $\nu_{\text{beat}} \sim h_{00} \nu_{\text{laser}}$, to detect GW-induced timing variations.
• Leverage the fact that the relative phase difference $\Delta\phi = \mathbf{k} \cdot \Delta\mathbf{r}$ between clocks depends on the GW's propagation direction, enabling directional source localization.
• Utilize phase-locked laser links with a noise budget of $\sim 5 \times 10^{-8}$ cm/$\sqrt{\text{Hz}}$ to maintain timing stability over 1–2 AU baselines.
• Apply signal processing to separate periodic GW signals from secular trends and solar oscillations, exploiting the GW's characteristic frequency sweep (chirp) over time.

Experimental results

Research questions

• RQ1 Can atomic clocks with $10^{-18}$ fractional timing precision detect the time-dilation effect of mHz gravitational waves from supermassive black hole binaries?
• RQ2 How does the differential clock-rate variation between spatially separated clocks depend on the GW's wavelength and propagation direction?
• RQ3 Can the proposed clock-timing method achieve a signal-to-noise ratio sufficient for GW detection over integration times of $\sim 10^3$ seconds?
• RQ4 To what extent can the chirp in GW frequency due to binary inspiral be measured via clock timing to constrain source parameters like $\mathcal{M}_z$ and $D_L$?
• RQ5 How does the use of multiple uncorrelated clocks or entangled clocks improve detection sensitivity compared to a single clock pair?

Key findings

• Atomic clocks with $10^{-18}$ fractional timing precision can detect GW-induced time dilation at the level of $h_{00} \approx 9 \times 10^{-18} \left( \frac{D_L}{\text{Gpc}} \right)^{-1} \left( \frac{\mathcal{M}_z}{10^6 M_\odot} \right)^{5/3} \left( \frac{f}{\text{mHz}} \right)^{2/3}$, corresponding to detectable signals from supermassive black hole binaries at cosmological distances.
• The method achieves optimal sensitivity when clocks are separated by half a GW wavelength, $\frac{1}{2}\lambda = 1~\text{AU} \cdot (f/\text{mHz})^{-1}$, maximizing the relative timing phase difference $\Delta\phi = \pi$.
• The beat frequency between laser signals from two clocks is $\nu_{\text{beat}} \sim h_{00} \nu_{\text{laser}}$, with $\nu_{\text{laser}} \sim 10^{15}$ Hz, yielding a detectable signal with period $\lesssim 10^3$ s for mHz GWs.
• The signal-to-noise ratio improves as $\sqrt{N_{\text{cycles}}}$, meaning that integrating over $\sim 10^3$ seconds allows detection of GWs with amplitudes below the single-cycle timing precision.
• The method can detect the chirp in GW frequency due to binary inspiral, with $\Delta f / f \approx 0.3 \left( \frac{\mathcal{M}_z}{10^6 M_\odot} \right)^{5/3} \left( \frac{f}{\text{mHz}} \right)^{8/3} \left( \frac{\Delta t_{\text{obs}}}{1~\text{hour}} \right)$, providing an independent constraint on source parameters.
• The use of $N$ uncorrelated clocks improves timing precision by $1/\sqrt{N}$, and quantum entanglement could further enhance it to $1/N$, offering a scalable path to higher sensitivity.