Fundamental and second-subharmonic Autler-Townes splitting in classical systems

Abstract

The dynamic Stark effect and the Autler-Townes splitting (ATS) are hallmarks of driven two-level systems. We establish a direct correspondence between these quantum phenomena and the parametric normal mode splitting in coupled classical oscillators. This gives rise to a second-subharmonic ATS under a two-tone parametric drive. We find excellent agreement between the theory and the vibrations of a nanomechanical two-mode system, capturing both the fundamental and second-subharmonic ATS, and allowing quantitative extraction of the modal coupling irrespective of the degree of modal hybridization.

Figures (4)

• Figure 1: (a) Scanning electron micrograph of a SiN nanostring (green) flanked by two golden electrodes (yellow), each connected either to the drive and the detection circuitry. (b) Spectrogram of the IP and OOP frequencies of the nanostring as a function of the bias voltage $V_{dc}$ varied from –32 V to +32 V. The data are overlaid with a model of the uncoupled, tunable eigenfrequencies $\tilde{\omega}_i$ (dashed black lines), revealing two avoided crossing points at $V_{dc}\simeq \pm20$ V.
• Figure 2: (a) Unperturbed resonance of $\Omega_1$ at $V_{dc}=-22$V (blue), and split PNMS peaks (orange) under additional parametric perturbation at $\Omega_p\approx\delta\Omega$. (b) Sweeping $\Omega_p$ around $\delta\Omega$ reveals full PNMS branches, at $V_{dc}=-14$ V (left) and $-20$ V (right), encoding response amplitudes in color (blue to red). Solid black lines represent the fitted model (see Eq. (S47)) showing an excellent agreement with the experiment.
• Figure 3: Splitting width $2g$ and the corresponding scaled parametric coupling $\Lambda_\zeta=\sqrt{\zeta}\Lambda$ values across the bias voltage range (black points), fitted with Eq. \ref{['eq:Lambda_lambda_rel']} based on the ansatz $\lambda= \sqrt{c_\lambda} V_{dc}$ to obtain $c_\lambda$, see Sec. S8 supp.
• Figure 4: Experimental map of the quasi-energies at the avoided crossing point $V_{dc}\simeq -20$ V by sweeping $\Omega_p$, showing the fundamental ATS at $\Omega_p=\delta \Omega$ and the second-subharmonic ATS at $\delta \Omega/2$ (magnified in the inset). The splittings of $\Omega_1$ are clearly resolved, while those of $\Omega_2$ lie mostly under the noise level. The dashed black lines show the Floquet model for a two-tone drive.