For nonzero DM-SM coupling, a topological defect will manifest as a transient perturbation of the fine-structure constant. We denoted its magnitude and duration as δα and T, respectively. The corresponding perturbation of the relative cavity-atom frequency encoded in ri(t) depends on the atom sensitivity Kα and for two different clocks A and B can be written asEmbedded Image(6)

We defined normalized signals as Embedded Image. In the simple case involving two clocks, the goal of our data analysis was to determine the constraint on δα from Embedded Image and Embedded Image. This can be performed by cross-correlating Embedded Image and Embedded ImageEmbedded Image(7)where tBtA is the duration of the cross-correlated signals. For square events, the shape of their cross correlation is triangular with full width at half maximum equal to T and amplitudeEmbedded Image(8)

This gives the constraint on the transient variations in the fine-structure constant (see Eq. 1). For the quadratic coupling, Embedded Image, Eq. 3 directly follows from Eq. 1; note that the DM field inside a defect is Embedded Image (8). The DM field is assumed to be zero outside the topological defects.

The simplest model of DM topological defects requires at least two free parameters whose values have to be arbitrarily chosen; in our case, these parameters are the defect size, d, and the time between consecutive encounters with topological defects, T. We repeated our analysis for every combination of these two parameters within the ranges we accessed with our network. For given values of d and T, we cross-correlated the two Embedded Image and fit the cross correlation with the expected triangular shape. We repeated this fitting procedure for any accessible time and any time delay between the two laboratories. The maximal value of the fitted amplitude, A0, yields the noise level corresponding to the transient variations in α (Eq. 1), and strength of the DM-SM coupling (Eq. 3) (see black lines in Fig. 3). This approach is analogous to the VLBI technique used in radio astronomy.

To give a proper statistical interpretation of our results, we also calculated the 5 and 95% CLs (see the green and red lines, respectively, in Fig. 3). To calculate the CLs, we repeated (100 times) the above algorithm of maximal A0 determination, artificially shifting one of the Embedded Image in time, ensuring that no expected common physical signal is present [for every repetition, we shift Embedded Image by a different time]. Having this ensemble of 100 cross-correlation amplitudes (for a given d), we selected the five smallest and largest elements, which yields the 5 and 95% CLs. We treated the 95% CL as an actual constraint determined in this work; hence, the black line in the top right panel in Fig. 3 should statistically (i.e., without any common component) exceed the red line 95% CL one time per every 20 values in d (the step in d is 300 km). This is consistent with Fig. 3, which indicates that at present level of accuracy, we did not observe any signature of hypothetical DM.

In the dataset considered here, the time of overlap of at least three clocks was considerably smaller than the time of overlap of two clocks. Therefore, without losing the strength of the constraint, we could restrict our analysis to simultaneous comparison of two clocks at a given time. For more than two simultaneous Embedded Image, the approach presented here can be generalized in a straightforward manner. For any value of the model free parameters and for any value of the DM velocity, the expected DM signature can be simultaneously fit to all the cross correlations between the participating clocks. In our analysis, we chose ηT = 1, i.e., the lengths of the cross-correlated Embedded Image segments are equal to the event duration, which is optimal for this analysis. It should be mentioned that the sensitivity of the clocks to DM objects could drop by a factor of 2 for typical servo-loop settings when the DM object duration is comparable to one servo-loop cycle, Tc.

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