Quantum Nondemolition Measurement of a Nonclassical State of a Massive Object

By coupling a macroscopic mechanical oscillator to two microwave cavities, we simultaneously prepare and monitor a nonclassical steady state of mechanical motion. In each cavity, correlated radiation pressure forces induced by two coherent drives engineer the coupling between the quadratures of light and motion. We, first, demonstrate the ability to perform a continuous quantum nondemolition measurement of a single mechanical quadrature at a rate that exceeds the mechanical decoherence rate, while avoiding measurement backaction by more than 13 dB. Second, we apply this measurement technique to independently verify the preparation of a squeezed state in the mechanical oscillator, resolving quadrature fluctuations 20% below the quantum noise.

Popular Summary

Quantum mechanics exquisitely describes the behavior of microscopic systems, but an ongoing challenge is exploring the applicability of quantum mechanics to systems with larger sizes and masses. For macroscopic mechanical systems in particular, witnessing states of motion that have no analog in classical physics has direct applications in areas of quantum-enhanced sensing and quantum information processing. However, it is challenging to design a mechanical system that is simultaneously weakly coupled to its classical environment and yet easily accessible by a control and measurement apparatus. As light transfers momentum to whatever it impinges upon, coherent light fields serve as the intermediary between the fragile mechanical states and our inherently classical world by exerting radiation pressure forces and extracting mechanical information. Here, we use microwave light to stabilize a nonclassical steady state of motion while independently, continuously, and nondestructively monitoring it.

The residual quantum fluctuations of a mechanical resonator can be further reduced to create a squeezed state of motion. In this study, we generate a mechanical squeezed state by exploiting the quantum interference between the radiation pressure forces of two microwave pumps operating at roughly 10 GHz and 30 mK. Similarly, we use a second pair of coherent pumps to monitor a single quadrature of the motion. Here, the measurement backaction is coherently cancelled, realizing a quantum nondemolition measurement. Thus, by coupling the motion of an aluminum membrane to two microwave cavities, we separately prepare and measure a mechanical squeezed state and demonstrate the first quantum nondemolition measurement of squeezed mechanical quadrature fluctuations. These fluctuations do not exist in the classical world.

We expect that our findings will pave the way for quantum-enhanced sensing.

Figure 1

Device description and experimental setup. (a) False-color optical micrograph of the aluminum device (in gray) on a sapphire substrate (blue). Centered at the bottom of the micrograph is a mechanically compliant vacuum gap capacitor. The capacitor’s electrode is split into two plates, each shunted by a different coil inductor, giving rise to two microwave resonances. (b) Frequency space diagram. Except for the mechanical linewidth, all the frequencies and linewidths are to scale. The microwave cavities have Lorentzian densities of states (DOS) of width ${\kappa }_1/2\pi =1.7\mathrm{MHz}$ and ${\kappa }_2/2\pi =2.1\mathrm{MHz}$, centered at ${\omega }_1/2\pi =8.89\mathrm{GHz}$ and ${\omega }_2/2\pi =9.93\mathrm{GHz}$. These resonance frequencies are tuned by the motion of the top plate of the capacitor, at the mechanical frequency ${\mathrm{\Omega }}_m/2\pi =14.98\mathrm{MHz}$. The red and blue dashed lines indicate the four mechanical sideband frequencies at which the cavities are driven, ${\omega }_{1,2}^{\pm }={\omega }_{1,2}\pm {\mathrm{\Omega }}_m$. (c) The circuit is placed on the cold stage of a cryogenic refrigerator (base temperature $T=30\mathrm{mK}$). Up to four strong microwave drives and one weak microwave probe are inductively coupled to the cavities via a single port. The reflected signals and the noise emitted by the cavities are amplified at low temperature and demodulated at room temperature.

Figure 2

Quantum nondemolition measurement. (a) Measurement schematic. A cooling drive of strength ${\mathrm{\Gamma }}_2^{-}=2\pi \times 4.87\mathrm{kHz}=529{\mathrm{\Gamma }}_m$ is applied on the lower mechanical sideband of the control cavity (${\omega }_2^{-}={\omega }_2-{\mathrm{\Omega }}_m$). Two drives of equal strength ${\mathrm{\Gamma }}_1^{-}={\mathrm{\Gamma }}_1^{+}$ are applied close to the mechanical sidebands of the measurement cavity. Their frequencies can be tuned to ${\omega }_1^{\pm }={\omega }_1\pm {\mathrm{\Omega }}_m$ to perform a single mechanical quadrature measurement [QND, in gray in (b)–(d)] or detuned by many mechanical linewidths away from that optimum to measure both mechanical quadratures [non-QND, in red in (b)–(d)]. (b) Mechanical noise spectra (normalized, background subtracted), for ${\mathrm{\Gamma }}_1^{-}/{\mathrm{\Gamma }}_2^{-}=0.9$. (c),(d) Mechanical occupancy extracted from the measured spectra of the two-drives measurement (c) and cooling drive (d), for both the non-QND case (in red) and the QND case (in gray), as a function of the measurement strength ${\mathrm{\Gamma }}_1^{-}/{\mathrm{\Gamma }}_2^{-}$.

Figure 3

Tomography of a mechanical squeezed state. (a) Measurement schematic. A pair of drives at ${\omega }_2^{\pm }={\omega }_2\pm {\mathrm{\Omega }}_m$ cools the mechanical mode to a squeezed bath, while a pair of drives at ${\omega }_1^{\pm }={\omega }_1\pm {\mathrm{\Omega }}_m$ measures the generalized mechanical quadrature ${\widehat{X}}_{\mathrm{\Phi }}$, given by the tunable phase $\mathrm{\Phi }$, allowing for the tomographic measurement of the squeezed state. (b) Normalized noise spectra of the QND measurement (background subtracted), for the squeezed and antisqueezed quadratures (${\mathrm{\Gamma }}_2^{-}/{\mathrm{\Gamma }}_2^{+}=0.07$) in blue and yellow, respectively, compared to a spectrum measured without squeezing (${\mathrm{\Gamma }}_2^{+}=0$) in gray. (c) Measured quadrature variances as a function of phase for ${\mathrm{\Gamma }}_2^{+}=0$ in gray and ${\mathrm{\Gamma }}_2^{-}/{\mathrm{\Gamma }}_2^{+}=0.07$ in green. The spectra in (b) correspond to the blue and yellow dots. (d) Squeezed and antisqueezed quadrature variances, respectively, in blue and yellow, as a function of the squeezing strength ${\mathrm{\Gamma }}_2^{-}/{\mathrm{\Gamma }}_2^{+}$. The solid lines are the theoretical predictions with no free parameters. The dashed lines are the theoretical predictions including a drift of the measurement phase of 35 deg during the acquisition of each spectrum (see Appendix pp5). The inset is a zoom-in for the low squeezing strength data, on a linear scale, where we can place the variance measured without squeezing (gray dot). In all figures, the black dotted line is the vacuum limit.

Figure 4

Mechanical mode shape. (a)–(c) Finite element simulation of the out-of-plane displacement of the first three mechanical modes of the aluminum membrane, and their respective measured resonance frequencies. The mode used in the main text is shown in (b).

Figure 5

Mechanical ringdown time. We measure the mechanical ringdown time ${\tau }_m$ as a function of the strength of a cooling drive on the control cavity, at a frequency ${\omega }_2^{-}={\omega }_2-{\mathrm{\Omega }}_m$, scattering light and damping the mechanical oscillator at the rate ${\mathrm{\Gamma }}_2^{-}$. The solid line is the theoretical prediction from the total damping, ${\mathrm{\Gamma }}_{\mathrm{tot}}={\mathrm{\Gamma }}_m+{\mathrm{\Gamma }}_2^{-}$. We measure an intrinsic mechanical ringdown time of ${\tau }_m=17.3\mathrm{ms}$, corresponding to a relaxation rate ${\mathrm{\Gamma }}_m=1/{\tau }_m=2\pi \times 9.2\mathrm{Hz}$.

Figure 6

Driven responses for the QND and non-QND measurement. Data corresponding to the measurement setup in Fig. 2: the pump scattering rates are ${\mathrm{\Gamma }}_2^{-}=2\pi \times 4.87\mathrm{kHz}=529{\mathrm{\Gamma }}_m$ and ${\mathrm{\Gamma }}_1^{-}={\mathrm{\Gamma }}_1^{+}=0.9{\mathrm{\Gamma }}_2^{-}$. (a), (b) Reflection coefficient $R$ around each cavity resonance. (c) Transmission $T^{\prime }$ from the upper mechanical sideband of the red measurement drive to the lower mechanical sideband of the blue measurement drive, as a function of the input frequency. A signal sent at $\omega$ is received at $\omega -{\omega }_1^{-}+{\omega }_1^{+}$. (d) Transmission $T$ from the upper mechanical sideband of the red cooling drive to the upper mechanical sideband of the red measurement drive, as a function of the input frequency. A signal sent at $\omega$ is received at $\omega -{\omega }_2^{-}+{\omega }_1^{-}$. All the solid lines are from the same numerical simulation.

Figure 7

Driven responses for the tomography of the mechanical squeezed state. Data corresponding to the measurement setup in Fig. 3: the pump scattering rates are ${\mathrm{\Gamma }}_2^{-}=2\pi \times 15.11\mathrm{kHz}=1643{\mathrm{\Gamma }}_m$, ${\mathrm{\Gamma }}_1^{-}/{\mathrm{\Gamma }}_2^{-}=0.48$, and ${\mathrm{\Gamma }}_2^{+}/{\mathrm{\Gamma }}_2^{-}=0.07$. (a), (b) Reflection coefficient $R$ around each cavity resonance. (c) Transmission $T^{\prime }$ from the upper mechanical sideband of the red measurement drive to the lower mechanical sideband of the blue measurement drive, as a function of the input frequency. A signal sent at $\omega$ is received at $\omega -{\omega }_1^{-}+{\omega }_1^{+}$. (d) Transmission $T$ from the upper mechanical sideband of the red cooling drive to the upper mechanical sideband of the red measurement drive, as a function of the input frequency. A signal sent at $\omega$ is received at $\omega -{\omega }_2^{-}+{\omega }_1^{-}$. All the solid lines are from the same numerical simulation.

Figure 8

System phase stability. (a) Measurement schematic. The device is driven by four drives, at ${\omega }_{1,2}^{\pm }={\omega }_{1,2}\pm {\mathrm{\Omega }}_m$. A probe is applied at $\omega \approx {\omega }_2$ and received at $\omega -{\omega }_2^{-}+{\omega }_1^{-}\approx {\omega }_1$. (b) Phase of the transmission coefficient as a function of time. (c) Histogram of the time trace measured in (b).