Fringe contrast as a function of the Ramsey time for non-interacting 1/sqrt(2)(|0> + |2>) (black circles) and dipolar 1/sqrt(2)(|0> + |-1>) (blue squares) superpositions. Empty markers indicate measurements using a standard Ramsey sequence, and filled markers indicate measurements performed with the addition of a single spin-echo pulse. The non-interacting results were fitted using a Gaussian model for decoherence, and the dipolar results were fitted, assuming an exponential decay in the fringe contrast. The blue lines indicate the decay in the fringe contrast from MACE simulations, as described in the main text. Uncertainties in the fits and simulations are indicated by the shaded regions. The fringes observed for 1/sqrt(2)(|0> + |2>) with spin echo at T = 0.7 s are shown inset. Spin-echo fringes are given without normalization for all Ramsey times, as shown in Supplementary Figs. 2 and 3 . In addition, the other inset shows the wavefunction for the dipolar superposition 1/sqrt(2)(|0> + |-1>) as a function of time (phase, phi). The resultant dipole (white arrow) rotates around the quantization axis (vertical black line) at a frequency proportional to the difference in energy delta E between the states.
