Figure 2: Motional modes and Ising couplings.

Transverse²⁶ vibrational modes are used in the experiment to generate Ising couplings according to equation (2). (a) Raman sideband spectrum of vibrational normal modes along transverse X-direction for nine ions, labelled by their index m. The two highest frequency modes at v₁ (CM mode) and ('tilt' mode) occur at the same position independent of the number of ions. The dotted and the dashed lines show beatnote detunings of μ≈v₁+30 KHz and μ≈v₁+63 KHz used in the experiment for N=9 and N=2 ions respectively. Carrier transition, weak excitation of transverse -Y and axial -Z normal modes and higher order modes are faded (light grey) for clarity. (b) Theoretical Ising-coupling pattern (equation (2)) for N=9 ions and uniform Raman beams. The main contribution follows from the uniform CM mode, with inhomogeneities given by excitation through the other nearby modes (particularly the tilt mode). Here J_1,1+r∼1/r^0.35 (r≥1), as found out empirically. For larger detunings, the range of the interaction falls off even faster with distance, approaching the limit J_i,j∼1/|i−j|³ for μ≫v₁⁵.