Macroscopic quantum effects in a strongly driven nanomechanical resonator

We investigate the nonlinear response of a vibrating suspended nanomechanical beam on external periodic driving. The amplitude of the fundamental transverse mode behaves like a weakly damped quantum particle in a driven anharmonic potential. Upon using a Born-Markovian master equation, we calculate the fundamental mode amplitude for varying driving frequencies. In the nonlinear regime, we observe resonances which are absent in the corresponding classical model. They are shown to be associated with resonant multiphonon excitations. Furthermore, we identify resonant tunneling in a dynamically induced bistable effective potential.

Figure 1

Amplitude $A$ of the expectation value $⟨x\left(t\right)⟩$ for varying the driving frequency $\omega$. Parameters are $k_{B}T=0.1\hslash {\omega }_0$, $\beta =0.1$, $f=0.1\hslash {\omega }_0∕x_0$, $\gamma =0.005{\omega }_0$. Dashed line: results of the classical Duffing oscillator at $T=0$ with the remaining parameters being the same. Inset: amplitude $A$ of the classical Duffing oscillator for varying driving frequencies. The driving strength $f$ increases from bottom to top.

Figure 2

Amplitude $A$ for varying the driving frequency $\omega$ and for different choices of (a) the nonlinearity coefficient $\beta$, (b) driving intensity $f$, (c) temperature $T$, and (d) damping strength $\gamma$. The other parameters are kept fixed according to $\beta =0.1$, $f=0.1\hslash {\omega }_0∕x_0$, $k_{B}T=0.1\hslash {\omega }_0$, and $\gamma =0.005{\omega }_0$, respectively. Dashed line in (a) and (d): same as in Fig. 1.

Figure 3

Average energies ${\overline{E}}_i$ (top), quasienergy levels ${\epsilon }_{\alpha ,k}$ (middle), and amplitude $A$ of the fundamental mode for varying driving frequencies $\omega$. The remaining parameters are $\beta =0.1$, $f=0.1\hslash {\omega }_0∕x_0$, and $\gamma =0.005{\omega }_0$.

Figure 4

Decay rate $\Gamma$ for the slow dynamics of the amplitude $A$ for approaching the steady state. Symbols: Solution from the numerical iteration of the Born Markovian master equation, solid line: Smallest nonzero eigenvalue of the rate matrix of an improved Floquet Markovian master equation. Inset: time-resolved dynamics of $A$ for $\omega =1.167{\omega }_0$ (solid line). The dashed line shows a fit to an exponential $e^{-\Gamma t}$. Here, $\beta =0.1$, $f=0.1\hslash {\omega }_0∕x_0$, $k_{B}T=0.1\hslash {\omega }_0$, and $\gamma =0.005{\omega }_0$.