Qubit-Assisted Transduction for a Detection of Surface Acoustic Waves near the Quantum Limit

We demonstrate ultrasensitive measurement of fluctuations in a surface-acoustic-wave (SAW) resonator using a hybrid quantum system consisting of the SAW resonator, a microwave (MW) resonator, and a superconducting qubit. The nonlinearity of the driven qubit induces parametric coupling, which up-converts the excitation in the SAW resonator to that in the MW resonator. Thermal fluctuations of the SAW resonator near the quantum limit are observed in the noise spectroscopy in the MW domain.

Figure 1

Characterization of the SAW-qubit-MW hybrid system. (a) Schematic of the hybrid system. A SAW resonator defined by Bragg mirrors (red) and a MW resonator (cyan) are coupled to each other via a transmon qubit with an interdigitated transducer (green). The square with a cross represents a Josephson junction. Port 1 is an external feed line for the MW resonator and ports 2 and 3 are those for the SAW resonator having a spatial mode shown in yellow. (b) Qubit spectrum obtained with two-tone spectroscopy. The squared change of the reflection coefficient, $|\mathrm{\Delta }{S}_{11}{|}^2$, of the MW resonator at the resonance is plotted as a function of the qubit drive frequency. The blue dots represent the data, and the red lines are the Lorentzian fits. (c) [(d)] Resonance frequency of the MW (SAW) resonator (blue dots) as a function of the MW (SAW) drive power. In each panel, the red dotted line indicates the resonance frequency of the bare resonator, and the green dashed line indicates that of the dispersively shifted resonator.

Figure 2

Calibration of the photon and phonon numbers in the resonators. (a)[(b)] ac Stark shift $\mathrm{\Delta }{\omega }_{q,m}$ ($\mathrm{\Delta }{\omega }_{q,s}$) of the qubit as a function of the MW (SAW) drive power. The corresponding mean intraresonator photon (phonon) number is indicated on the right axis.

Figure 3

Parametrically induced coupling and conversion. (a) Energy diagram of the hybrid system. (b) Optimization of the parametric conversion efficiency $\tilde{\eta }$ as a function of the qubit drive powers applied at port 2. The drive frequencies ${\omega }_{d1}$ and ${\omega }_{d2}$ are varied between panels: The single photon detuning $\delta {\omega }_p\equiv ({\omega }_{d1}-{\omega }_m+{\omega }_f)/2\pi$ are set to 0, 20, 40, 60, and 80 MHz from left to right, while the two-photon detuning ${\omega }_s+{\omega }_{d1}+{\omega }_{d2}-{\omega }_m$ is kept at 0. The white dashed line in the second panel from the right is a contour on which the ac Stark shift of the qubit $|f⟩$ state is constant. The red circle indicates the optimized condition used in (c). (c) Conversion from the surface acoustic waves to the microwaves (blue circles with an error bar). The horizontal axis expresses the input phonon flux to the SAW resonator, and the vertical axis represents the output photon flux from the MW resonator. The black line depicts the maximum conversion efficiency ($\tilde{\eta }=1$) attainable with the external couplings of the resonators in the present device. The red line is a linear fit to the data. The saturation effect observed at high phonon flux is due to the shift of the SAW resonator frequency [see Fig. 1].

Figure 4

Thermal noise power spectral density (PSD) in the SAW resonator as a function of the parametric-drive detuning $\delta {\omega }_p={\omega }_s+{\omega }_{d1}+{\omega }_{d2}-{\omega }_m$. Here, we sweep ${\omega }_{d2}$, while fixing ${\omega }_{d1}$. The SAW phonon noise is observed via the parametric up-conversion into the MW photon noise and plotted in the unit of phonon number fluctuation. The blue circles and the red squares represent the data taken at the nominal bath temperatures $T_{\mathrm{bath}}$ of 10 and 80 mK, respectively. The solid curves are the Lorentzian fits. The backgrounds indicated by the dashed lines are the noise floors in the microwave measurement. The apparent difference in the background level is due to the temperature dependence of the conversion efficiency.