Fig. 2: Trapped-ion implementation of the system-meter Hamiltonian \({\hat{H}}_{{\mathcal{SM}}}\).

a Ion string with N system ions (white) illuminated by four laser beams in a double Mølmer–Sørensen configuration. As described in the text this generates \({\hat{H}}_{{\mathcal{SM}}}\) [see Eq. (2)] with transverse Ising Hamiltonians \(\hat{{H}^{\prime}}\) (6) and \(\hat{H}\) (3), and the meter variable \(\hat{P}\) representing the COM motion. The meter variable \(\hat{X}\) is read by driving one, or potentially several ancilla ions (red) with a laser (red beam) tuned to the red motional COM sideband (see text). Homodyne detection of the scattered light to read \(\hat{X}\), and thus revealing \(\hat{H}\) in the photocurrent \(I(t) \sim {\langle \hat{H}\rangle }_{{\rm{c}}}\) [see Eq. (5)]. b Level scheme of a pair of ions sharing the COM phonon mode, illustrating one of the elementary processes contributing to the Ising term \(-{\sum }_{i < j}{J}_{ij}{\hat{\sigma }}_{i}^{x}{\hat{\sigma }}_{j}^{x}\otimes {\mathbb{I}}\) in second order in η. c Level scheme showing the corresponding third-order processes contributing to \(-\vartheta {\sum }_{i < j}{J}_{ij}{\hat{\sigma }}_{i}^{x}{\hat{\sigma }}_{j}^{x}\otimes \hat{P}\) (see text).