Phonon-Laser Ultrasensitive Force Sensor

Developing nanomechanical oscillators for ultrasensitive force detection is very useful in exploring science. We report our achievement of ultrasensitive detection of the external force regarding the radiofrequency electric field by a nanosensor made of a single trapped ${}^{40}{\mathrm{Ca}}^{+}$ ion under injection locking, where squeezing is additionally applied to detection of the smallest force in the ion trap. The employed ion is confined stably in a surface electrode trap and works as a phonon laser that is very sensitive to the external disturbance. The injection locking drove the ion’s oscillation with phase synchronization, yielding the force detection with sensitivity of 347 $\pm 50\mathrm{yN}/\sqrt{\mathrm{Hz}}$. Further with 3-dB squeezing applied on the oscillation phase variance, we achieve a successful detection of the smallest force to be $86.5\pm 70.1$ yN.

Figure 1

Experimental system and scheme. (a) Sketch of the phonon laser of an ion trapped above the surface of the SET. The ${}^{40}{\mathrm{Ca}}^{+}$ ion with oscillating motion irradiated by red- and blue-detuned 397-nm laser beams is trapped by the potential $\mathrm{\Psi }(t)$ regarding a time-varying electric field. The electrodes are labelled with names for convenience of description in the text, where EE, SE, ME, AE and rf indicate, respectively, the endcap electrode, the steering electrode, the middle electrode, the axial electrode and the radio-frequency electrode. The injection signal applied on AE with frequency ${\omega }_i={\omega }_z$, locking the oscillation frequency, is to be detected. The squeezing, with frequency doubling of ${\omega }_i$, acts on EEs when required. The top-right inset is the level scheme of ${}^{40}{\mathrm{Ca}}^{+}$ for our purpose. (b) Signal processing scheme of the photon-arrival-time measurements. The applied injection signal and the squeezing signal are phase synchronized, with the latter of frequency doubling with respect to the former. The photon-arrival-time transited to voltage is measured by TAC. The ratio of the beam splitter for the PMT to CCD is set to be 7/3. (c) Signal and sequence diagram. The Injection signal contributes to locking the oscillation frequency/phase. The Squeezing signal is applied, when required, to suppress the variance of the oscillation phase. The Gate signal controls the measurement time $t_m$ containing a series of oscillation period $T$ and the TAC records the photon’s arrival time $t$ in each period. The Start signal comes from the PMT regarding the photon pulses. The Stop TTL is synchronized with injection locking signal and the Stop signal is produced by the photon pulses from the spontaneous emission. For clarity, the plotted pulses are not strictly to scale. The typical values: $t_m=10$ s, $T=5.38\mu \mathrm{s}$, and 0 $<t⩽T$.

Figure 2

Typical fitting cases for the accumulated photon counts. (a) $A=21.677\mu \mathrm{m}$ and $\phi =0.028$ are obtained under the injection voltage $V_i=9.5$ mV. (b) $A=24.462\mu \mathrm{m}$ and $\phi =0.042$ are acquired under the injection voltage $V_i=18.25$ mV. The dots are experimental data and solid curves are fitted by Eq. (2). Total photon counts $N=5.353\times {10}^4$ are recorded by 538 unit time intervals, where each measurement time is $t_m=10$ s and we set the parameter values as ${\sigma }_t=80t_r=0.8\mu \mathrm{s}$, $\alpha =4\times {10}^{-5}$, and $\beta =46$.

Figure 3

Detection sensitivity and smallest force measurement. (a) Oscillation amplitude as a function of the injection voltage, where the red dots represent the case without the injection voltage and the blue dots are experimental data. (b) Relative variance as a function of $g$ in the case of $\varphi$ = $\pi /2$. (c) Relative variance as a function of $\varphi$ in the case of $g=0.9$. In (b),(c) the variance without squeezing is normalized as unit, as represented by the dashed lines. (d) Experimental results of the lower bound of the force detection, where the blue dots are experimental data without squeezing and the red dots with 3-dB squeezing. The horizontal axis is with ${\log }_{10}$ scale for distinguishing the difference due to the applied squeezing. The values of force and amplitude variation are identified from the measured injection voltage, following the linear relationship in the text. In all the panels, the voltage regarding the injection signal is 18.25 mV. Further voltage of 125 mV is applied on the EEs to generate 3-dB squeezing. Each measurement takes 10 s and we make 50 measurements for each data point.