Thermomechanical Two-Mode Squeezing in an Ultrahigh-$Q$ Membrane Resonator

We realize a quantum-compatible multimode interaction in an ultrahigh $Q$ mechanical resonator via a reservoir-mediated parametric coupling. We use this interaction to demonstrate nondegenerate parametric amplification and thermomechanical noise squeezing, finding excellent agreement with a theoretical model of this interaction over a large dynamic range. This realization of strong multimode nonlinearities in a mechanical platform compatible with quantum-limited optical detection and cooling makes this a powerful system for nonlinear approaches to quantum metrology, transduction between optical and phononic fields, and the quantum manipulation of phononic degrees of freedom.

Figure 1

(a) The resonator consists of a high-stress silicon nitride membrane deposited on a silicon substrate. Distinct eigenmodes of the membrane resonator (representative eigenfunctions shown) with frequencies ${\omega }_{i,j}$ are coupled via a substrate-mediated interaction. This two-mode interaction can be controlled by actuating the substrate to an amplitude $X_S$ at frequencies close to ${\omega }_i+{\omega }_j$. (b) Experimental schematic: Mechanical motion of the membrane is optically detected in a Michelson interferometer, with the two membrane modes ($i,j$) distinguished by phase-sensitive lock-in detection. The eigenfrequency of a third, high-$Q$ mechanical mode is continuously monitored and acts as a mechanical “thermometer.” The resonator modes are actively frequency stabilized to this thermometer mode by photothermal feedback. In the presence of this feedback, the frequency stability of each resonator mode is better than 1 ppb over 1000 seconds. ECDL–External cavity diode laser; AOM–Acousto-optic modulator; PID–Proportional-Integral-Derivative controller.

Figure 2

(a) Parametric amplification of a membrane mode due to actuation of the substrate. The vertical line indicates the threshold for parametric instability. The solid green line indicates the thermomechanical amplitude of the membrane mode. The dashed gray line shows the detection noise floor. (b) In the absence of the parametric drive, large amplitude oscillations of either membrane mode ($x_i$) result in increased dissipation (and a lower quality factor $Q_j$) of the other mode due to up-conversion of excitations into the substrate. The variation of the normalized dissipation ($Q_j/Q_{j,0}$), shown above for various pairs of coupled modes, is well described by a characteristic length scale $\xi$ (see text). (c) The linear dependence of the length scale $\xi$ [extracted from data such as shown in (b)] vs the threshold amplitude for parametric instability, as predicted by the two-mode model. See Supplemental Material [30] for details of the various modes depicted above.

Figure 3

Phase-sensitive amplification, $G_j(\varphi )$, vs the phase of the pump excitation. Data shown correspond to normalized pump amplitudes $\mu =X_S/X_{S,\text{th}}=0,0.021,0.038,0.042,0.086,0.13$ (red, blue, green, black, cyan, orange). For these data, the threshold for parametric self-oscillation corresponds to $X_S=40\mathrm{fm}$, the signal and idler modes are driven to $35(k_{B}T/m{\omega }_j^2{)}^{1/2}$ and $400(k_{B}T/m{\omega }_i^2{)}^{1/2}$, respectively, corresponding to $\eta =14.4$. Inset: Estimate of the pump amplitude from fits to these data agree with the actual pump amplitude to within 5%.

Figure 4

Steady-state thermomechanical two-mode squeezing. Left: Phase space distributions of the quadratures ${\alpha }_i,{\alpha }_j$ in the absence (blue) and presence (red) of the pump field, showing the emergence of correlations, i.e., noise squeezing, due to nondegenerate parametric amplification. Right: The standard deviations of the cross-quadratures $x_a,y_b$ (red, blue), (amplified) and $x_b,y_a$ (red, blue) (squeezed) vs pump amplitude. The shaded curves indicate the no-free-parameter prediction of our noise squeezing model based on independently measured parameters of our system. The green trace represents the expected degree of squeezing taking into account a finite measurement duration (see Supplemental Material [30]).