An electromechanical Ising Hamiltonian

The phonons localized in a mechanical resonator can be electrically manipulated to emulate the Ising Hamiltonian.

Degenerate parametric amplification can be activated in the electromechanical system by piezoelectrically modulating the spring constant of either mode at twice their natural frequency via (2 ) in Fig. 1 namely in equation 1. This process amplifies the thermally driven random displacement fluctuations of either mode as shown in fig. S1, A and B and the corresponding gain is accompanied by a reduction in their damping rates. Once the phonon generation rate from this process exceeds the rate of loss from either mode, it results in parametric resonance which yields an oscillation with a bistable phase (16).
Nondegenerate parametric amplification can also be activated in the electromechanical system by piezoelectrically modulating the spring constant at the sum frequency of both modes via ( ) in Fig. 1 namely in equation 1. In contrast to the degenerate parametric amplification, this process simultaneously amplifies the random displacement fluctuations of both modes as shown in fig. S1C and is also accompanied by a reduction in their damping rates (15).
fig. S1. Experimentally measured degenerate and nondegenerate parametric amplification of both modes in the electromechanical system. (A-C) The thermal motion of the symmetric and asymmetric modes as they are degenerately and nondegenerately amplified when the electromechanical system's spring constant is piezoelectrically modulated at either twice the frequency of the symmetric or asymmetric mode or at their sum frequency respectively. The logarithmic spectral displacement density scale (mHz -0.5 ) is identical in all figures.

II. The double-well potential
The Hamiltonian in equation 5, describing a single degenerate parametric resonator excited on resonance, can be plotted via the canonically conjugate coordinates and with ω 0 = 1, β = 1 and Γ = 1 as shown in fig. S2. This plot reveals a symmetric double well potential underpinning the parametric resonance where the two minima correspond to the two stable oscillation phases, separated by a quasi-energy , which are used to encode mechanical spins namely σ (37).

III. Pump phase
The correlations generated in the two-mode squeezed states from the nondegenerate parametric down-conversion can be tuned via the pump phase as observed in other physical implementations (24). Here this effect is employed to control the sign of the coupling constant between the two mechanical spins as detailed in equation 2 and Fig. 4A. To understand the nature of this phenomenon, the polarity of the correlations as a function of are measured from the thermal Langevin force driven displacement fluctuations of both modes whilst they are nondegenerately amplified (15). At each pump phase, the correlation coefficients from the cross-quadratures : and : are extracted as detailed in fig. S3A. This reveals that a correlated state with unity correlation coefficient (upper panels in fig. S3B) can be tuned to an uncoupled or random configuration with zero correlation coefficient (middle panels in fig. S3B) and to an anti-correlated state with a correlation coefficient of -1 (lower panels in fig. S3B). These observations indicate that by conditioning the two-mode thermal squeezed states via enables the creation of a ferromagnetic or an anti-ferromagnetic state on-demand in the electromechanical Ising simulator. portraits reconstructed from the cross-quadratures of the symmetric and asymmetric modes with the pump phase detailed via the color coding in A. As the pump phase is adjusted, the two-mode squeezed states can be continuously tuned from a positively correlated to an anti-correlated state via a random state.