Fig. 4: Rotational coherence and dipolar interactions in the magic trap.

a, Fringe contrast as a function of the Ramsey time for non-interacting \(\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert \hat{2}\right\rangle )\) (black circles) and dipolar \(\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert \bar{1}\right\rangle )\) (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 \(\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert \hat{2}\right\rangle )\) 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 \(\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert \bar{1}\right\rangle )\) as a function of time (phase, ϕ). The resultant dipole (white arrow) rotates around the quantization axis (vertical black line) at a frequency proportional to the difference in energy δE between the states. b, Coherence time in the presence of dipole–dipole interactions. We plotted the 1/e coherence time measured with spin echo as a function of the effective laboratory-frame dipole moment, tuned by changing the states used in the spin-echo sequence. The combinations used were (i) \(\frac{1}{\sqrt{2}}(\left\vert 1\right\rangle +\left\vert \bar{2}\right\rangle )\), (ii) \(\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert \bar{1}\right\rangle )\) and (iii) \(\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert 1\right\rangle )\). The fringe contrast as a function of time for each state combination is shown in the top-right inset. The wavefunctions are illustrated with (i) and (ii) yielding dipoles rotating around the quantization axis and (iii) resulting in a dipole that oscillates up and down. The bottom-left inset shows the coherence time plotted as a function of the inverse of the dipole–dipole interaction strength ∝ 1/d². Error bars in all plots indicate 1σ standard error in the measurements, and the 1σ uncertainties in the fits and simulations are shown by the shaded regions.