Research ArticleCONDENSED MATTER PHYSICS

An electromechanical Ising Hamiltonian

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Science Advances  24 Jun 2016:
Vol. 2, no. 6, e1600236
DOI: 10.1126/sciadv.1600236
  • Fig. 1 The electromechanical system.

    A false-color electron micrograph of the coupled mechanical resonators sustaining phonons in symmetric and asymmetric vibration modes. The measurements were performed at room temperature (300 K) and in a high vacuum (10−6 mbar), and the mechanical vibrations were detected using a laser interferometer, which was demodulated either in a spectrum analyzer or in a phase-sensitive detector using the heterodyne mixing setup detailed in the circuit schematic. Five signal generators were used, with two piezoelectrically activating the parametric resonances at 2ωn, two generating the reference signals for the phase-sensitive detectors at ωn, and one creating the coupling between the parametric resonances via parametric down-conversion when activated at Embedded Image.

  • Fig. 2 Spins and spin coupling.

    (A and B) The occupation probabilities of the parametric resonances are measured by periodically switching on and off their excitation with VS(2ωS) = 2.0 VPP and VA(2ωA) = 3.0 VPP, respectively, and measuring the resultant evolution of their in-phase and quadrature components of motion. The phase portraits reconstructed from this measurement, from 2000 samples, indicate that when a parametric resonance is activated, each thermally occupied mode at the origin evolves to one of the two available oscillating states (dashed ovals) with a π phase separation via a single trajectory. When the parametric resonance is deactivated, the oscillating mode returns to the origin via another trajectory, which does not overlap with the upward trajectory (21). (C and D) Phase portraits corresponding to the thermal fluctuations of the symmetric and asymmetric mode as a function of Embedded Image (black points) and 1.3 V (red and green points, respectively). (E and F) The phase portraits reconstructed from the in-phase component of one of the modes versus the quadrature component of the other mode and vice versa, from the data in (C) and (D), again as a function of Embedded Image (black points) and 1.3 VPP (blue points), reveal squashed distributions, implying that the motion of both modes has become correlated.

  • Fig. 3 Mechanical spin coupling.

    (A) The pulse sequence used to demonstrate the fundamentals of the Ising Hamiltonian in the electromechanical system and the corresponding qualitative evolution of the underlying potentials from both modes. The correlations generated by Embedded Image can be visualized as the synchronous motion of the balls in the harmonic potentials of the symmetric (red) and asymmetric (green) modes. As VS(2ωS) and VA(2ωA) are slowly ramped, the harmonic potentials start to evolve into double-well potentials where the thermal energy in the system (in addition to dissipation) can drive transitions between the two oscillation phases. Finally, when both modes are parametrically resonating, their vibrations are correlated (that is, they occupy the same potential minima) and frozen where the thermal fluctuations are too small to destroy the phase of the vibrations. (B) The temporal response of both modes to the above pulse sequence with VS(2ωS) = 1.6 VPP and VA(2ωA) = 2.1 VPP when Embedded Image and 800 μVPP with the mechanical spin orientations visualized by the arrows. (C) The correlation coefficient extracted from the temporal response of both modes as a function of Embedded Image (points) reveals a ferromagnetic state at large pump amplitudes, but as it is reduced to zero, the spins become disordered with a profile that is consistent with the Ising interaction between a pair of spins (line) as detailed in Materials and Methods.

  • Fig. 4 The Ising Hamiltonian replicated with electromechanical phonons.

    (A) The correlation coefficient extracted from the electromechanical system’s temporal response to the pulse sequence in Fig. 3A as a function of pump phase (points) with Embedded Image. (B) As the phase is adjusted, the correlated mechanical spins corresponding to a ferromagnetic state smoothly transition first to a disordered state without any coupling corresponding to random spin alignment (upper panel) and then to an anticorrelated state with the spins exhibiting antiferromagnetic ordering (lower panel), with this variation being consistent with the spin Ising interaction (line) as detailed in Materials and Methods.

  • Fig. 5 The electromechanical simulator.

    A conceptual image of an electromechanical Ising machine based on phonons confined in an array of mechanical elements each parametrically resonating to encode classical spin information (red corresponds to spin up and green corresponds to spin down) where two-mode squeezing is used to create coupling between elements. The parametric resonances are piezoelectrically activated and read out via the individual gate electrodes (orange) on each mechanical element where the ith element has a natural frequency ωi. The global gate electrode (yellow) located on the left clamping point of all the mechanical elements can enable coupling between any pair of mechanical spins with the application of a sum frequency pump. Using FDM, the pump can execute multiple degrees of coupling between large numbers of spins, potentially enabling the Ising Hamiltonian to be explored in a nontrivial configuration.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/6/e1600236/DC1

    I. Degenerate and nondegenerate parametric amplification

    II. The double-well potential

    III. Pump phase

    fig. S1. Experimentally measured degenerate and nondegenerate parametric amplification of both modes in the electromechanical system.

    fig. S2. The double-well potential underpinning a parametric resonance.

    fig. S3. The pump phase dependence of the two-mode squeezing.

  • Supplementary Materials

    This PDF file includes:

    • I. Degenerate and nondegenerate parametric amplification
    • II. The double-well potential
    • III. Pump phase
    • fig. S1. Experimentally measured degenerate and nondegenerate parametric amplification of both modes in the electromechanical system.
    • fig. S2. The double-well potential underpinning a parametric resonance.
    • fig. S3. The pump phase dependence of the two-mode squeezing.

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