For commensurate high-Tc SQUID arrays (Ns rows of Np parallel Josephson junctions, all loops of equal area), numerical simulations by Nieves and Müller (Supercond. Sci. Technol. 37, 105003 (2024)) found that the low-frequency voltage noise spectral density of 1D parallel arrays violates the naive Johnson-noise scaling Sv(0)∼Ns/NpS_v(0)\sim N_s/N_p, following instead Sv(0)∼Ns/Np0.3S_v(0)\sim N_s/N_p^{0.3}, while 2D arrays approximately recover the expected law; the rms flux noise likewise deviates from the expected (NsNp)−1/2(N_sN_p)^{-1/2} behavior. These benchmarks were established only for idealized arrays up to (20,20)(20,20) with identical junctions at a single temperature (77 K). The open problem is to determine the true noise-scaling law of such arrays: whether the observed scalings persist, evolve, or saturate in substantially larger arrays; how realistic fabrication disorder (statistical spreads in critical current and resistance) modifies the noise scaling and the transfer-function/flux-noise trade-off; what microscopic mechanism — such as the finite Josephson-junction interaction radius invoked by Kornev et al. — produces the anomalous 1D exponent and how it manifests in 2D; and to establish the scalings' generality across operating regimes.

Background

A superconducting quantum interference device (SQUID) is a superconducting loop interrupted by Josephson junctions — weak links whose electrical behavior is characterized by a critical current IcI_c and a normal-state resistance RR. Biased with a dc current, a SQUID converts applied magnetic flux into a periodic voltage response and forms the basis of the most sensitive magnetometers. Its noise performance is quantified by the voltage noise spectral density Sv(f)S_v(f) (with Sv(0)S_v(0) the low-frequency white-noise floor), the transfer function vˉϕ=∂vˉ/∂ϕa\bar v_\phi=\partial\bar v/\partial\phi_a (the slope of the mean voltage versus applied flux at the operating point), and the rms flux noise Sϕ1/2=Sv1/2/vˉϕS_\phi^{1/2}=S_v^{1/2}/\bar v_\phi. The noise and optimization of the single dc SQUID have been understood since the classical analysis of Tesche and Clarke. To increase voltage response and reduce noise, many SQUIDs can be wired into commensurate arrays — arrays in which every loop has the same area, so the voltage-flux response retains the single flux-quantum Φ0\Phi_0 periodicity — with NsN_s rows of junctions connected in series and NpN_p junctions in parallel in each row. Such arrays, typically made of the high-transition-temperature (TcT_c) cuprate YBCO operating at 77 K with overdamped junctions, are used for magnetometry and low-noise amplification; this contrasts with superconducting quantum interference filters (SQIFs), which deliberately use incommensurate loop areas to obtain an absolute-field response. If the Johnson (thermal) noise voltages of the NsNpN_sN_p junctions were statistically independent, one would expect Sv(0)∝Ns/NpS_v(0)\propto N_s/N_p and Sϕ1/2∝(NsNp)−1/2S_\phi^{1/2}\propto(N_sN_p)^{-1/2}, so that enlarging the array would monotonically improve sensitivity.

Nieves and Müller (Supercond. Sci. Technol. 37, 105003 (2024)) simulated commensurate arrays using the resistively shunted junction (RSJ) model — the standard overdamped junction equation including a Johnson noise source — coupled through the full inductance matrix of the SQUID loops, with thermal noise strength Γ=2πkBT/(Φ0Ic)\Gamma=2\pi k_BT/(\Phi_0I_c) and screening parameter βL=2IcLs/Φ0\beta_L=2I_cL_s/\Phi_0 (comparing loop self-inductance LsL_s to the flux quantum), at T=77T=77 K with Ic=20 μAI_c=20\,\mu\mathrm{A}, R=10 ΩR=10\,\Omega, and identical junction parameters. They found that 1D parallel arrays (Ns=1N_s=1) violate the naive law: Sv(0)S_v(0) scales as Ns/Np0.3N_s/N_p^{0.3} rather than Ns/NpN_s/N_p, while 2D arrays follow Ns/NpN_s/N_p fairly well and approach the Johnson-noise bound 4ΓNs/Np4\Gamma N_s/N_p for large NsN_s. The rms flux noise deviates from (NsNp)−1/2(N_sN_p)^{-1/2} for both 1D and 2D arrays when the transfer function is optimized, and within the array sizes examined ((1,2)(1,2), (1,20)(1,20) and (20,20)(20,20)) no single choice of bias current and applied flux simultaneously maximizes vˉϕ\bar v_\phi and minimizes Sϕ1/2S_\phi^{1/2}: the flux-noise minimum approximately coincides with the transfer-function maximum for the 1D arrays examined but not for the (20,20)(20,20) array. Kornev et al. had earlier attributed such 1D deviations to a finite Josephson-junction interaction radius — an effective range over which junctions coupled in parallel lock their dynamics — but did not examine 2D arrays; a related coupling radius has since been characterized for the array transfer function by Galí Labarias et al. (IEEE Trans. Appl. Supercond. 32, 1600205 (2022)), who showed that the maximum transfer function plateaus beyond a junction-count set by the normalized loop impedance. The noise simulations of Nieves and Müller were capped at (Ns,Np)=(20,20)(N_s,N_p)=(20,20) by the computation time required for accurate noise spectra, assumed perfectly identical junctions, and used a single operating temperature, leaving the size range, disorder dependence, temperature generality, and microscopic mechanism of the observed scaling laws unestablished.

Problem Statement

Determine the low-frequency noise scaling of commensurate high-TcT_c SQUID arrays as a function of geometry (Ns,Np)(N_s,N_p) beyond the ideal-homogeneous 77 K benchmark of Nieves and Müller (2024), whose simulations were capped at (20,20)(20,20) arrays with identical junctions. The problem has one overarching objective — an established noise-scaling law for realistic high-TcT_c SQUID arrays usable as a magnetometer design rule — with the following open aspects:

  1. Size extension: establish whether the anomalous 1D law Sv(0)∼Ns/Np0.3S_v(0)\sim N_s/N_p^{0.3} and the near-ideal 2D law Sv(0)∼Ns/NpS_v(0)\sim N_s/N_p (with Sϕ1/2S_\phi^{1/2} versus its expected (NsNp)−1/2(N_sN_p)^{-1/2} behavior) persist, evolve, or saturate for substantially larger arrays (NpN_p and NsN_s well beyond 20), by simulation, theory, or both.
  2. Disorder robustness: quantify how realistic fabrication disorder — statistical spreads in critical current IcI_c and normal resistance RR — modifies these scaling laws and the trade-off between maximizing the transfer function vˉϕ\bar v_\phi and minimizing the flux noise Sϕ1/2S_\phi^{1/2} at the array operating point.
  3. Microscopic origin: identify and quantify the mechanism behind the anomalous 1D exponent — in particular whether the finite Josephson-junction interaction radius invoked by Kornev et al. explains it, and how that interaction (e.g., through the coupling radius set by the normalized loop impedance) manifests in 2D arrays so as to approximately restore the expected scaling.
  4. Generality: establish the range of operating temperatures, IcI_c regimes, βL\beta_L values, and inductance-matrix geometries over which the benchmarks hold.

A solution resolves the task when it establishes the noise-scaling law for the size-extension and disorder-robustness aspects (1 and 2); the microscopic-origin and generality aspects (3 and 4) are complementary extensions that are not required for completion.

References

  1. O. A. Nieves and K.-H. Müller, 'Noise scaling in SQUID arrays', Superconductor Science and Technology 37, 105003 (2024). DOI: 10.1088/1361-6668/ad70dd, https://doi.org/10.1088/1361-6668/ad70dd; arXiv:2409.19837, https://arxiv.org/abs/2409.19837
  2. O. A. Nieves, M. A. Galí Labarias, A. C. Keser, K.-H. Müller and E. E. Mitchell, 'Effect of Josephson junction parameter spread on the performance of SQUID arrays', Superconductor Science and Technology 39, 015012 (2026). DOI: 10.1088/1361-6668/ae3b19, https://doi.org/10.1088/1361-6668/ae3b19; arXiv:2508.07685, https://arxiv.org/abs/2508.07685
  3. M. A. Galí Labarias, K.-H. Müller and E. E. Mitchell, 'The Role of Coupling Radius in 1-D Parallel and 2-D SQUID and SQIF Arrays', IEEE Transactions on Applied Superconductivity 32, 1600205 (2022). DOI: 10.1109/TASC.2021.3133202, https://doi.org/10.1109/TASC.2021.3133202; arXiv:2108.11059, https://arxiv.org/abs/2108.11059
  4. M. A. Galí Labarias, K.-H. Müller and E. E. Mitchell, 'Modeling the Transfer Function of Two-Dimensional SQUID and SQIF Arrays with Thermal Noise', Physical Review Applied 17, 064009 (2022). DOI: 10.1103/PhysRevApplied.17.064009, https://doi.org/10.1103/PhysRevApplied.17.064009; arXiv:2106.11439, https://arxiv.org/abs/2106.11439
  5. M. A. Galí Labarias and E. E. Mitchell, 'Numerical Study of SQUID Array Responses Due to Asymmetric Junction Parameters', IEEE Transactions on Applied Superconductivity 33, 1–4 (2023). DOI: 10.1109/TASC.2023.3236255, https://doi.org/10.1109/TASC.2023.3236255; arXiv:2211.13833, https://arxiv.org/abs/2211.13833
  6. V. K. Kornev, I. I. Soloviev, N. V. Klenov, T. V. Filippov, H. Engseth and O. A. Mukhanov, 'Performance advantages and design issues of SQIFs for microwave applications', IEEE Transactions on Applied Superconductivity 19, 916–919 (2009). DOI: 10.1109/TASC.2009.2019589, https://doi.org/10.1109/TASC.2009.2019589
  7. C. D. Tesche and J. Clarke, 'dc SQUID: Noise and optimization', Journal of Low Temperature Physics 29, 301–331 (1977). DOI: 10.1007/BF00655097, https://doi.org/10.1007/BF00655097