Classical Stückelberg interferometry of a nanomechanical two-mode system

Stückelberg interferometry is a phenomenon that has been well established for quantum-mechanical two-level systems. Here, we present classical two-mode interference of a nanomechanical two-mode system, realizing a classical analog of Stückelberg interferometry. Our experiment relies on the coherent energy exchange between two strongly coupled, high-quality factor nanomechanical resonator modes. Furthermore, we discuss an exact theoretical solution for the double-passage Stückelberg problem by expanding the established finite-time Landau-Zener single-passage solution. For the parameter regime explored in the experiment, we find that the Stückelberg return probability in the classical version of the problem formally coincides with the quantum case, which reveals the analogy of the return probabilities in the quantum-mechanical and the classical version of the problem. This result qualifies classical two-mode systems at large to simulate quantum-mechanical interferometry.

Figure 1

Nanomechanical resonator and measurement scheme. (a) False color scanning electron micrograph of the $50\text{−}\mu \text{m}$-long, 270-nm-wide, and 100-nm-thick silicon nitride string (green) flanked by two adjacent gold electrodes in the oblique view. Arrows indicate the flexural mode polarizations out-of-plane (Out) and in-plane (In). The normalized amplitude of the respective mode is denoted $c_1\left(t\right)$ for out-of-plane polarization and $c_2\left(t\right)$ for in-plane polarization, as explained in the text. (b) Avoided mode crossing of sample A exhibiting a frequency splitting of $\mathrm{\Delta }/2\pi =22.6\mathrm{kHz}$ at the avoided crossing voltage $U_{\mathrm{a}}=U_{\mathrm{i}}+1.47\mathrm{V}=9.37\mathrm{V}$. The lower (out-of-plane) mode is excited at frequency ${\omega }_1\left(U_{\mathrm{i}}\right)/2\pi =7.560\mathrm{MHz}$ defined by the initialization voltage $U_{\mathrm{i}}$. An additional sweep voltage applies a triangular voltage ramp rising to a maximum of $U_{\mathrm{p}}$ and back to the readout voltage $U_{\mathrm{f}}=U_{\mathrm{i}}+0.2\mathrm{V}=8.1\mathrm{V}$, thus transgressing the avoided crossing twice. The sweep voltage also decouples the mode from the fixed-frequency drive, consequently inducing an exponential decay of the amplitude. (c) Time evolution of the sweep voltage beginning at $t=0$, increasing to $U_{\mathrm{p}}$ and returning to $U_{\mathrm{f}}$ after interval $\vartheta$. The mechanical signal power (green dashed line) is measured after a delay $\varepsilon$, and a fit (black dotted line) is used to extract its magnitude at time $t=\vartheta$. The measured return signal is normalized to the mechanical signal power at $t=0$.

Figure 2

Classical Stückelberg oscillations. Normalized squared return amplitude (left axis, blue dots) and theoretically calculated return probability (right axis, red line) vs inverse sweep rate for fixed peak voltages of $U_{\mathrm{p}}=2.5\mathrm{V}$ (a), $U_{\mathrm{p}}=3.5\mathrm{V}$ (b), $U_{\mathrm{p}}=4.5\mathrm{V}$ (c), and $U_{\mathrm{p}}=5.0\mathrm{V}$ (d) measured on sample A.

Figure 3

Comparison of experimental data and the theoretical model. (a) Color-coded normalized squared return amplitude vs inverse sweep rate and peak voltage measured on sample B. The dataset is not interpolated. (b) Color-coded theoretical return probability given by Eq. (9) vs inverse sweep rate and peak voltage for the equivalent data range. The theory is calculated with a single set of parameters, extracted from the avoided crossing illustrated in Appendix pp3 (Fig. 7), and it contains no free parameters.

Figure 4

Exemplary classical Stückelberg oscillations of sample B. Line cuts of Figs. 3 and 3. (a) Normalized squared return amplitude (left axis, blue dots) and theoretically calculated return probability given by Eq. (9) (right axis, red line) vs inverse sweep rate for a fixed peak voltage $U_{\mathrm{p}}=3.3\mathrm{V}$. (b) The same quantities as above but plotted as a function of peak voltage for a fixed inverse sweep rate $1/\beta =51.6\mu \mathrm{s}/\mathrm{V}$. Blue dots are joined by blue dashed lines for illustration reasons.

Figure 5

Nanoelectromechanical system. (a) False color scanning electron micrograph of the $50\text{−}\mu \text{m}$-long, 270-nm-wide, and 100-nm-thick silicon nitride string (green) in the oblique view. The adjacent $1\text{−}\mu \text{m}$-wide gold electrodes (yellow) are processed on top of the silicon nitride layer. Arrows indicate the flexural mode polarizations out-of-plane (Out) and in-plane (In). (b) Electrical transduction setup. The arbitrary function generator (AFG) ramp voltage and the dc tuning voltage are added via a summation amplifier and then combined with the rf drive using a bias tee. The microwave readout is bypassed by the second capacitor, acting as a ground path for the microwave cavity.

Figure 6

Dielectric frequency tuning. Color-coded frequency spectrum of sample B as a function of applied dc tuning voltage. The resonance frequency of the $55\text{−}\mu \text{m}$-long resonator's fundamental out-of-plane oscillation (Out) increases quadratically as a function of dc voltage. The resonance frequency of the corresponding in-plane mode (In) decreases quadratically. Tuning both modes into resonance, they exhibit a pronounced avoided crossing indicated by the black dashed rectangle. This particular region is displayed in Fig. 7. The additional resonances in the spectrum originate from a different mechanical resonator, which is coupled to the same microwave cavity.

Figure 7

Calibration of the conversion factor. (a) Avoided crossing region of sample B. A sweep voltage equal to zero corresponds to the initialization point $U_{\mathrm{i}}=10.4\mathrm{V}$. The two modes exhibit a frequency splitting of ${\mathrm{\Delta }}_{\mathrm{B}}/2\pi =6.3\mathrm{kHz}$ at the avoided crossing voltage $U_{\mathrm{a}}=U_{\mathrm{i}}+1.96\mathrm{V}=12.36\mathrm{V}$. The gap in the upper branch (red) results from a signal detection efficiency of the particular mode polarization below the noise level. (b) Frequency difference of the two modes (black) and averaged slope of the linearized frequency tuning illustrated by a green dash-dotted line.

Figure 8

Temperature fluctuations. Initialization voltage shift vs inverse sweep rate for the dataset depicted in Fig. 4 $\left(U_{\mathrm{p}}=3.3\mathrm{V}\right)$ of the main text. Each point corresponds to the measurement of the normalized squared return amplitude for a given inverse sweep rate. The first measurement is performed at $1/\beta =100\mu \mathrm{s}/\mathrm{V}$, representing the initialization at resonance frequency ${\omega }_{\mathrm{i}}\left(U_{\mathrm{i}}\right)/2\pi$ for voltage $U_{\mathrm{i}}=10.4\mathrm{V}$. The implemented feedback loop regulates the initialization voltage in order to compensate for the temperature fluctuations of the mechanical resonance. Consequently, the voltage shift illustrates the fluctuations of the ambient temperature.

Figure 9

Classical Stückelberg oscillations. (a) Normalized squared return amplitude (left axis, blue dots) and theoretically calculated return probability given by Eq. (9) of the main text (right axis, red line) vs inverse sweep rate for a fixed peak voltage $U_{\mathrm{p}}=3.85\mathrm{V}$. (b) Same quantities as above but plotted as a function of peak voltage for a fixed inverse sweep rate $1/\beta =60\mu \mathrm{s}/\mathrm{V}$. Blue dots are joined by blue dashed lines for illustration reasons.