Signatures for a Classical to Quantum Transition of a Driven Nonlinear Nanomechanical Resonator

We seek the first indications that a nanoelectromechanical system (NEMS) is entering the quantum domain as its mass and temperature are decreased. We find them by studying the transition from classical to quantum behavior of a driven nonlinear Duffing resonator. Numerical solutions of the equations of motion, operating in the bistable regime of the resonator, demonstrate that the quantum Wigner function gradually deviates from the corresponding classical phase-space probability density. These clear differences that develop due to nonlinearity can serve as experimental signatures, in the near future, that NEMS resonators are entering the quantum domain.

Figure 1

Isolated driven Duffing resonator with $\hslash =0.2$. Wigner functions (left) and classical distributions (right) of the initial coherent state and its evolution at three later times, where $T=2\pi /\omega$. A square of area $\hslash$ is shown at the bottom right corner of (a). Full animation can be downloaded from [30].

Figure 2

As in Fig. 1, but for a driven Duffing resonator with $\hslash =0.1$, coupled to a heat bath at $T_{\mathrm{env}}=0$, with damping constant $\gamma =0.01$. The stable solution towards which all initial classical points flow is encircled in (d). Full animation can be downloaded from [30].

Figure 3

As in Fig. 1, but for a driven Duffing resonator with $\hslash =0.1$, coupled to a heat bath at $k_B{T}_{\mathrm{env}}=2\hslash \Omega$, with damping constant $\gamma =0.001$. The initial state is centered on the separatrix so that classically it splits towards both of the stable solutions. Full animation can be downloaded from [30].

Figure 4

Quantum (left) and classical (right) probability densities $P(x)$ for the phase-space distributions in Fig. 3d. The $y$ axes are scaled differently for better visualization.

Figure 5

As in Figs. 3d, 4, only that the classical calculation (right) is for $k_B{T}_{\mathrm{env}}=17\hslash \Omega$, yielding similar phase-space distributions (a) and probability densities (b). Both distributions in (a) are left unscaled.