Fig. 3: Measurement of the periodic modulation of the critical atom number.

The error bars in the plots represent the standard error and the numbers inside the brackets represent one standard deviation uncertainty. The uncertainty in the radius of the atomtronic SQUID consists of the systematic uncertainty in the estimation of the radius, which is 3% of the radius and the statistical uncertainty. The radius for a and d is 4.82 (0.14)_sys (0.01)_stat µm, for b and e is 3.85 (0.11)_sys (0.02)_stat µm, for c and f is 2.891 (0.084)_sys (0.022)_stat µm. The modulation periods for a–c were measured with a damped sine function fitting. The transition point from the 0 to 1 winding number for d–f was determined with the fitting function \(1/(1+e^{-k({\Omega}_{\rm{ext}}-\frac{{\Omega}_{0}}{2})})\). Modulation periods (Hz) were found to be 5.99 (0.22), 9.52 (0.48), and 17.99 (0.64) for a–c, respectively. Ω₀/2π (Hz) were measured to be 5.70 (0.10), 8.96 (0.12), and 18.11 (0.35) for d–f, respectively.