Precise estimation of the coupling strength between two nanomechanical modes from four Ramsey fringes

Abstract

We experimentally determine the coupling strength between two strongly coupled nanomechanical modes using a Ramsey-inspired technique optimized for signals as short as four fringes. The method is applied to precisely probe the change of the coupling rate induced by a modification of the microwave-cavity readout field. It opens a pathway towards sensing electrostatic field fluctuations approaching single-charge resolution.

Figures (12)

• Figure 1: Experimental realization. (a) Scanning electron micrograph of a typical silicon nitride string resonator (blue) with the two gold electrodes nearby. (b) Simplified measurement setup. See main text for details. (c) Color plot of amplitude spectra measured as a function of DC voltage, showing an avoided crossing of magnitude $\Omega_0$ between the OOP and the IP mode. The two orange circles denote the DC voltage used for initialization and for tuning on resonance, respectively. $\Delta_0$ is the frequency difference of the two mechanical modes at $U_{\text{i}}$.
• Figure 2: Implementation of the Ramsey interferometry. (a) Ramsey pulse sequence with the corresponding first harmonic Magnus-based correction of the leading (ii) and the trailing (iv) edges for frequency-sweep. Note we introduce a finite voltage offset $U_r-U_i\neq 0$ that allows the readout state $\boldsymbol{a}_r$ to decay, whereas the initial state $\boldsymbol{a}_i$ is continuously driven. (b) Ringdown measurement of the IP mode after completing the Ramsey sequence for two different waiting times $t_\mathrm{w}$. The exponential fits extrapolate back to $t_0=0$ to compensate the mechanical damping during the evolution time and are used to convert the signal into a return probability Seitner2016Seitner2017.
• Figure 3: State preparation. (a) Ramsey signal (return probability of the state $\boldsymbol{a}_i$) for two different initialization protocols. Measurement time captures approximately $2$ periods. Data has been averaged over $30$ measurements. Grey (blue) datapoints bar are obtained with a sinusoidal soft ramp (corrected ramp, c.f. \ref{['fig:02']}). Solid grey (blue) lines indicate fits. (b) Fast Fourier transform of the short, zero-padded Ramsey signals. Grey (blue) line corresponds to the initialization via the soft (corrected) ramp, respectively.
• Figure 4: Iterative adaptive spectroscopy for short and finite signals. (a) Frequency estimation for each iteration step $m$. The $m=1$ datapoint was obtained from a two-period Ramsey signal as in Fig. \ref{['fig:03NEW']}. Blue (orange) symbols refer to the unprocessed (processed) data. Error bars indicate standard deviation of the $30$ measurements. (b) Ramsey signal (return probability of the state $\boldsymbol{a}_i$) sampling $n=4$ fringes at $m=8$ before (blue) and after data processing (orange). (c) Fast Fourier transform of the zero-padded data shown in (b). Orange (blue) arrow marks position of the local maximum indicating frequency estimate $\bar{\Omega}_0^{(8)}$ in (a).
• Figure 5: Gradient field sensing. (a) Statistical frequency distributions of $30$ Ramsey measurements at iteration $m=8$ for cavity pump power $P_{\mathrm{c}}=22\dBm$. Blue and orange bars refer to unprocessed ($n=2$) and processed ($n=4$) data, respectively. Dotted lines indicate average $\bar{\Omega}_0^{(8)}$. (b) Statistical frequency distributions of $30$ Ramsey measurements after increasing the cavity pump power to $P_{\mathrm{c}}=22.5\dBm$, probed at iteration $m=3$.