Extended Data Fig. 9: Detecting the gauge-invariant states.

We divide the atoms into double wells and then measure atom tunnelling within each two-site unit. a–d, The dynamics of tunnelling for four different experimental sequences, as sketched in the insets. Five different moments during the phase transition (t = 0, 30, 60, 90, 120 ms) are selected for detecting the gauge-invariant states. Error bars, s.d. We fit the data with a sinusoidal damping function, which has a period of 7.2 ms and an exponential decay constant of 96 ms. The amplitudes of the oscillations in a–d refer to \({A}_{|10\rangle }^{(1)}\), \({A}_{|01\rangle }^{(1)}\), \({A}_{|10\rangle }^{(2)}\) and \({A}_{|01\rangle }^{(2)}\), respectively. These amplitudes are then used for calculating the state probabilities, where the error bars are s.d.