Two-Mode Thermal-Noise Squeezing in an Electromechanical Resonator

An electromechanical resonator is developed in which mechanical nonlinearities can be dynamically engineered to emulate the nondegenerate parametric down-conversion interaction. In this configuration, phonons are simultaneously generated in pairs in two macroscopic vibration modes, resulting in the amplification of their motion. In parallel, two-mode thermal squeezed states are also created, which exhibit fluctuations below the thermal motion of their constituent modes as well as harboring correlations between the modes that become almost perfect as their amplification is increased. The existence of correlations between two massive phonon ensembles paves the way towards an entangled macroscopic mechanical system at the single phonon level.

Figure 1

(a) An electron micrograph of the GaAs based coupled electromechanical resonators incorporating a buried Si-doped layer that is confined within a shallow mesa (purple) where the mechanical elements are integrated with Au Schottky gates (orange) to form the piezoelectric transducers [22]. The vibration of the modes is detected via the partially depicted laser interferometer from the right beam, i.e., not the center of mass with the parametric down-conversion being piezoelectrically activated by applying a pump voltage to both electrodes on the left beam [22]. The pumping generator (${\omega }_P$) and the local oscillators (${\omega }_S$ and ${\omega }_A$) are all continuously synchronized (gray dashed line) to ensure phase locked measurements. Also shown are the vibration profiles for the symmetric and asymmetric modes extracted from a finite element analysis. (b) The output noise from the interferometer, measured in a spectrum analyzer, reveals the two modes via their picometer order thermal vibrations. (c) The on-resonance thermal motion of both modes projected in phase space. Also shown is the electrical noise in the measurement setup (black points) acquired in an off-resonance configuration [22].

Figure 2

(a),(b) The thermal motion of the symmetric and asymmetric modes, respectively, is amplified when the electromechanical resonator’s spring constant is pumped at the sum frequency of the constituent modes. (c) The resultant gain of both modes referenced to their bare thermal motion (solid lines) is accompanied with a narrowing of their respective power bandwidths $\mathrm{\Delta }\omega /2\pi$ (points) as a function of the pump intensity. At the largest pump amplitudes, the quality factor of both modes converges to $\sim {10}^5$ before undergoing nondegenerate parametric resonance.

Figure 3

(a),(b) The phase portraits of the symmetric and asymmetric modes exhibit phase-conserving nondegenerate parametric amplification in response to the pump phonons undergoing parametric down-conversion when their intensity is increased from 0 to $0.25V_{\text{rms}}$ in increments detailed in panel (e). (c),(d) The cross quadratures of the above data take only specific values in phase space indicating the existence of correlations between the modes where the squeezing is enhanced as the pump intensity is increased. (e) The two-mode thermal squeezed states are quantified, in the rotated axis depicted in the insets to Figs. 3 and 3, via the standard deviations of their phase-space distributions (points) where the gain is calibrated with respect to the narrowest bare thermal distribution from $Y_A$ (dashed line) and is consistent with the theoretical trends derived from the above Hamiltonian (solid lines) as detailed in the Supplemental Material [22]. Note that the uncertainties associated with the extracted standard deviations are 2 orders of magnitude smaller and cannot be shown [22].

Figure 4

(a) The absolute correlation coefficient between various combinations of components from both modes extracted with a pump amplitude of $0.1V_{\text{rms}}$ as a function of delay time $\tau$. (b) The corresponding correlation coefficient matrix at $\tau =0$ reveals finite off-diagonal elements that verify the intertwined nature of the two-mode thermal squeezed states. (c) The variation of the off-diagonal elements of the correlation matrix as a function of pump intensity (points) and the corresponding theoretical response (solid line).