We used ri(t) from four optical lattice atomic clocks located at NIST, Boulder, CO, USA (23, 24), at LNE-SYRTE, Paris, France (25, 26), at KL FAMO, Torun, Poland (27, 28), and at NICT, Tokyo, Japan (29, 30). During the measurement sessions, the reported fractional clock instabilities at 1 s were equal to 3 × 10−16, 1 × 10−15, 2 × 10−14, and 7 × 10−15, and the reported fractional optical cavity instabilities at 1 s were equal to 2 × 10−16, 8 × 10−16, 2 × 10−14, and 2 × 10−15, in the NIST, LNE-SYRTE, KL FAMO, and NICT laboratories, respectively. The measurements were performed from 9 January 2015 to 17 December 2015. The measurements were collected for 11, 24, 42, and 54 days in the NIST, LNE-SYRTE, KL FAMO, and NICT laboratories, respectively. Together, the laboratories collected data for 114 days. In our topological defect analysis, we used the longest consecutive overlapping records for each pair of clocks. Their length is equal to 3336 and 13,533 s for NIST and LNE-SYRTE and for LNE-SYRTE and KL FAMO, respectively. The constraints reported in Fig. 3 are meaningful only if they are referred to the lengths of the analyzed signal. In oscillating massive scalar field analysis, we used all our data spanning 114 days from all the laboratories. One clock cycle lasts 0.5 s in NIST, 1 s in LNE-SYRTE, 1.5 s in NICT, and 1.3 s in KL FAMO. The clock cycle duration restricts the size of the topological defects that we can detect with the network of unsynchronized clocks. In the case of oscillating scalar fields, the results are limited by the feedback servos’ time constants, which vary from 2 to 20 s depending on the clock.

In our oscillating field analysis, we weighted ri(t) from each clock with their standard deviations. All ri(t) are heavily affected by low-frequency cavity drifts. Before the analysis, we applied a high-pass filter to eliminate the low-frequency cavities’ instabilities. In the case of topological defect analysis, we applied an FIR high-pass Gaussian filter with frequency cutoff at 0.005 to 0.027 Hz to all ri(t). This choice of transmission characteristics increases the sensitivity to the considered range of events. In the oscillating massive scalar field analysis, the main contribution to the signal instabilities are cavity drifts in a much longer time scale. We filtered them by subtracting a second-order polynomial fit from continuous sections of ri(t). To verify the filter response, we added artificial harmonic oscillations with specified parameters to nonfiltered data, and we tested how the filter performs on different frequencies.

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