Demonstrating the thermal noise limit in a backaction-free bulk acoustic wave resonator chain for gravitational wave detection

Author(s)

Albani, G., Borghesi, M., Canonica, L., Carobene, R., De Guio, F., Faverzani, M., Ferreiro Iachellini, N., Ferri, E., Gerosa, R., Ghezzi, A., Giachero, A., Labranca, D., Mariani, L., Maifredi, R., Nucciotti, A., Rozza, D., de Fatis, T. Tabarelli

Abstract

We report the full sub-Kelvin characterization of the BAUSCIA high-frequency gravitational-wave antenna, based on cryogenic Bulk Acoustic Wave resonators coupled to SQUID amplifiers for signal readout. The end-to-end performance analysis of the experimental chain, from the sensor to the DAQ, demonstrates that resonator thermal noise dominates the noise budget at temperatures well below 100 mK for all investigated modes. The measurements also verify proper thermalization of the resonator modes and rule out significant SQUID backaction on the strain sensor, confirming that the system operates close to its fundamental limits with comparable resonator thermal noise and additive readout noise. From this characterization, we derive projected strain sensitivities in the range $1$--$10 \times 10^{-21}\,\mathrm{Hz}^{-1/2}$ for several resonant frequencies between 5 and 20~MHz, with corresponding fundamental thermal limits in the range $0.7$--$1.4 \times 10^{-21}\,\mathrm{Hz}^{-1/2}$. Equipped with three resonators and synchronized to an absolute global time reference for multi-site coincidences, the antenna is ready to enter a pilot science run.

Figures

\textbf{Left:} Electrical equivalent representation of the setup used for squid calibration: $\bar{V}^2 / \Delta f = 4 k_B T R$ is the Johnson-Nyquist noise of the input resistor. \textbf{Right:} Measured voltage \gls{psd} of the resistor's noise alongside best-fit curves using Eq.~\eqref{eq:RL_psd}. The corner frequency $f_0$ directly constrains the SQUID input inductance $L_{\text{in}}$.
Caption \textbf{Left:} Electrical equivalent representation of the setup used for squid calibration: $\bar{V}^2 / \Delta f = 4 k_B T R$ is the Johnson-Nyquist noise of the input resistor. \textbf{Right:} Measured voltage \gls{psd} of the resistor's noise alongside best-fit curves using Eq.~\eqref{eq:RL_psd}. The corner frequency $f_0$ directly constrains the SQUID input inductance $L_{\text{in}}$.
\textbf{Left:} Electrical equivalent circuit of a single BAW acoustic mode coupled to the DC SQUID input circuit. The voltage generator $4 k_B T R_{\lambda}$ models the thermal noise of the motional resistance $R_{\lambda}$. \textbf{Centre:} Measured voltage noise PSD of the 3B mode ($f_{\lambda} = 5.689\text{ MHz}$) at four distinct cryostat temperatures, along with Lorentzian best fits (dashed lines). \textbf{Right:} Extrapolation of the Lorentzian peak power spectral density ($A^2$) versus bath temperature $T_{\text{bath}}$, highlighting the threshold temperature $T_{\text{min}} \sim 18\text{ mK}$ where the thermal noise equals the additive readout noise for the 3B mode.
Caption \textbf{Left:} Electrical equivalent circuit of a single BAW acoustic mode coupled to the DC SQUID input circuit. The voltage generator $4 k_B T R_{\lambda}$ models the thermal noise of the motional resistance $R_{\lambda}$. \textbf{Centre:} Measured voltage noise PSD of the 3B mode ($f_{\lambda} = 5.689\text{ MHz}$) at four distinct cryostat temperatures, along with Lorentzian best fits (dashed lines). \textbf{Right:} Extrapolation of the Lorentzian peak power spectral density ($A^2$) versus bath temperature $T_{\text{bath}}$, highlighting the threshold temperature $T_{\text{min}} \sim 18\text{ mK}$ where the thermal noise equals the additive readout noise for the 3B mode.
\textbf{Left:} Mode effective temperature $T_{\text{eff}}$ as a function of the cryostat bath temperature $T_{\text{bath}}$ for the $n=3$ harmonic family. The dashed diagonal represents thermal equilibrium ($T_{\text{eff}} = T_{\text{bath}}$); linear fits confirm unity slopes within uncertainties, demonstrating complete mode thermalization and negligible backaction. (Data points for 3B and 3C are horizontally offset by $+15\text{ mK}$ and $+30\text{ mK}$ for visual clarity.) \textbf{Right:} Input-referred flux noise spectrum $\sqrt{S_{\Phi\Phi}}$ around the 3B mode resonance, comparing experimental data (black points) against the Lorentzian fit (red line), total imprecision noise floor (gray line), and the theoretical SQUID backaction contribution calculated via Tesche-Clarke theory (blue dashed line).
Caption \textbf{Left:} Mode effective temperature $T_{\text{eff}}$ as a function of the cryostat bath temperature $T_{\text{bath}}$ for the $n=3$ harmonic family. The dashed diagonal represents thermal equilibrium ($T_{\text{eff}} = T_{\text{bath}}$); linear fits confirm unity slopes within uncertainties, demonstrating complete mode thermalization and negligible backaction. (Data points for 3B and 3C are horizontally offset by $+15\text{ mK}$ and $+30\text{ mK}$ for visual clarity.) \textbf{Right:} Input-referred flux noise spectrum $\sqrt{S_{\Phi\Phi}}$ around the 3B mode resonance, comparing experimental data (black points) against the Lorentzian fit (red line), total imprecision noise floor (gray line), and the theoretical SQUID backaction contribution calculated via Tesche-Clarke theory (blue dashed line).
\textbf{Left:} Projected single-tone strain sensitivity at $T_{\text{ref}} = 20$ mK of the prototype BAUSCIA channel measured in this work. Filled markers indicate the estimates based on the total measured noise, whereas open markers indicate the thermal limit set by the resonator thermal noise. \textbf{Right:} Projected single-tone strain sensitivities of the BAUSCIA array equipped with three prototype BAW resonators at a reference temperature $T_{\text{ref}} = 20\text{ mK}$. Grey shaded areas indicate the different monitored BAW modes of the three resonators, with each mode labelled by its overtone number and mode family.
Caption \textbf{Left:} Projected single-tone strain sensitivity at $T_{\text{ref}} = 20$ mK of the prototype BAUSCIA channel measured in this work. Filled markers indicate the estimates based on the total measured noise, whereas open markers indicate the thermal limit set by the resonator thermal noise. \textbf{Right:} Projected single-tone strain sensitivities of the BAUSCIA array equipped with three prototype BAW resonators at a reference temperature $T_{\text{ref}} = 20\text{ mK}$. Grey shaded areas indicate the different monitored BAW modes of the three resonators, with each mode labelled by its overtone number and mode family.
References
  • [1] B. P. Abbott et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett., vol. 116, p. 061102, Feb 2016. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.116.061102
  • [2] A. G. Abac et al., “GWTC-4.0: Updating the gravitational-wave transient catalog with observations from the first part of the fourth LIGO–Virgo–KAGRA observing run,” The Astrophysical Journal Letters, vol. 1004, no. 2, p. L22, jun 2026. [Online]. Available: https://doi.org/10.3847/2041-8213/ae2c74
  • [3] M. Maggiore et al., “Science case for the Einstein Telescope,” J. Cosmol. Astropart. Phys., vol. 2020, no. 03, p. 050, 2020.
  • [4] M. Evans et al., “A horizon study for Cosmic Explorer: Science, observatories, and community impact,” 2021. [Online]. Available: https://arxiv.org/abs/2109.09882
  • [5] M. Colpi et al., “LISA definition study report,” 2024. [Online]. Available: https://arxiv.org/abs/2402.07571
  • [6] G. Agazie et al., “The NANOGrav 15 yr data set: Evidence for a gravitational-wave background,” Astrophys. J. Lett., vol. 951, p. L8, 2023.
  • [7] J. Antoniadis et al., “The second data release from the european pulsar timing array: Iii. search for gravitational wave signals,” Astron. Astrophys., vol. 678, p. A50, 2023.
  • [8] D. J. Reardon et al., “Search for an isotropic gravitational-wave background with the Parkes pulsar timing array,” Astrophys. J. Lett., vol. 951, p. L6, 2023.
  • [9] H. Xu et al., “Searching for the nano-Hertz stochastic gravitational wave background with the Chinese pulsar timing array data release i,” Res. Astron. Astrophys., vol. 23, p. 075024, 2023.
  • [10] N. Aggarwal, O. D. Aguiar, A. Bauswein et al., “Challenges and opportunities of gravitational-wave searches at MHz to GHz frequencies,” Living Reviews in Relativity, vol. 24, no. 1, p. 4, 2021. [Online]. Available: https://doi.org/10.1007/s41114-021-00032-5
  • [11] N. Aggarwal, D. Blas, V. Domcke, S. A. R. Ellis, G. Franciolini et al., “Challenges and opportunities of gravitational wave searches above 10 kHz,” Living Rev. Relativ., 2025.
  • [12] G. Franciolini, A. Maharana, and F. Muia, “Hunt for light primordial black hole dark matter with ultrahigh-frequency gravitational waves,” Phys. Rev. D, vol. 106, p. 103520, Nov 2022. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.106.103520
  • [13] A. Arvanitaki and S. Dubovsky, “Exploring the string axiverse with precision black hole physics,” Phys. Rev. D, vol. 83, p. 044026, Feb 2011. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.83.044026
  • [14] G. D’Amico, A. A. Geraci, N. Kaloper, and A. Westphal, “Very-high-frequency gravitational waves from multimonodromy inflation,” Phys. Rev. D, vol. 114, p. 023540, Jul 2026. [Online]. Available: https://link.aps.org/doi/10.1103/q4pv-zj9c
  • [15] D. Blas, J. Casalderrey-Solana, D. Mateos, and M. Sanchez-Garitaonandia, “Megahertz gravitational waves from neutron star mergers,” Phys. Rev. Lett., vol. 136, p. 101401, Mar 2026. [Online]. Available: https://link.aps.org/doi/10.1103/6yz9-94ql
  • [16] K. Bleau, J. Kopp, J. Lee, and J. van de Vis, “High-frequency gravitational waves from phase transitions in nascent neutron stars,” 2026. [Online]. Available: https://arxiv.org/abs/2603.18153
  • [17] M. R. Haque, F. Iocco, and L. Visinelli, “Primordial black hole interpretation of the sub-solar merger event S251112cm,” 2026. [Online]. Available: https://arxiv.org/abs/2603.25795
  • [18] J. Manley, D. J. Wilson, R. Stump, D. Grin, and S. Singh, “Searching for scalar dark matter with compact mechanical resonators,” Phys. Rev. Lett., vol. 124, p. 151301, Apr 2020. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.124.151301
  • [19] A. Arvanitaki, A. Madden, and K. Van Tilburg, “Piezoaxionic effect,” Phys. Rev. D, vol. 109, p. 072009, Apr 2024. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.109.072009
  • [20] T. Trickle, “Piezoelectric bulk acoustic resonators for dark photon detection,” 2025. [Online]. Available: https://arxiv.org/abs/2501.05504
  • [21] G. Albani, M. Borghesi, L. Canonica et al., “An Array of Bulk Acoustic Wave Sensors as a High-Frequency Antenna for Gravitational Waves,” Galaxies, vol. 13, no. 4, p. 94, 2025. [Online]. Available: https://www.mdpi.com/2075-4434/13/4/94
  • [22] W. M. Campbell, M. Goryachev, and M. E. Tobar, “The multi-mode acoustic gravitational wave experiment: MAGE,” Scientific Reports, vol. 13, p. 10638, 2023. [Online]. Available: https://doi.org/10.1038/s41598-023-35670-y
  • [23] M. Goryachev, E. N. Ivanov, F. van Kann, S. Galliou, and M. E. Tobar, “Observation of the fundamental Nyquist noise limit in an ultra-high Q-factor cryogenic bulk acoustic wave cavity,” Applied Physics Letters, vol. 105, no. 15, Oct 2014. [Online]. Available: http://dx.doi.org/10.1063/1.4898813
  • [24] K. S. Van Dyke, “The piezo-electric resonator and its equivalent network,” Proceedings of the Institute of Radio Engineers, vol. 16, no. 6, pp. 742–764, 1928.
  • [25] H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev., vol. 83, pp. 34–40, Jul 1951. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.83.34
  • [26] M. Goryachev and M. E. Tobar, “Gravitational wave detection with high frequency phonon trapping acoustic cavities,” Phys. Rev. D, vol. 90, p. 102005, Nov 2014. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.90.102005
  • [27] ——, “Erratum: Gravitational wave detection with high frequency phonon trapping acoustic cavities [Phys. Rev. D 90, 102005 (2014)],” Phys. Rev. D, vol. 108, p. 129901, Dec 2023. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.108.129901
  • [28] C. D. Tesche and J. Clarke, “DC SQUID: Noise and optimization,” Journal of Low Temperature Physics, vol. 29, no. 3–4, pp. 301–331, Nov. 1977.
  • [29] ——, “DC SQUID: Current noise,” Journal of Low Temperature Physics, vol. 37, no. 5–6, pp. 397–403, Nov. 1979.
  • [30] V. Ankel, C. Bartram, J. Begin et al., “Noise limits for DC SQUID readout of high-Q resonators below 300 MHz,” Journal of Applied Physics, vol. 138, no. 9, p. 094505, 09 2025. [Online]. Available: https://doi.org/10.1063/5.0280831
  • [31] D. Drung, C. Hinnrichs, and H.-J. Barthelmess, “Low-noise ultra-high-speed dc SQUID readout electronics,” Superconductor Science and Technology, vol. 19, no. 5, p. S235, feb 2006. [Online]. Available: https://doi.org/10.1088/0953-2048/19/5/S15