Basins of Attraction of a Nonlinear Nanomechanical Resonator

Abstract

We present an experiment that systematically probes the basins of attraction of two fixed points of a nonlinear nanomechanical resonator and maps them out with high resolution. We observe a separatrix which progressively alters shape for varying drive strength and changes the relative areas of the two basins of attraction. The observed separatrix is blurred due to ambient fluctuations, including residual noise in the drive system, which cause uncertainty in the preparation of an initial state close to the separatrix. We find a good agreement between the experimentally mapped and theoretically calculated basins of attraction. DOI: 10.1103/PhysRevLett.99.207201 PACS numbers: 85.85.+j, 05.45.ÿa In the last few years the dimensions of mechanical devices have been scaled deep into the submicrometer regime, which resulted in the increased detection sensitivity of extremely small physical quantities [1,2] and in an enhancement of the significance of nonlinear dynamics in such devices [3]. Deliberately operating the system in the nonlinear regime can improve precision of some experimental measurements on nanoscale. For example, a nonlinear resonator can be employed to suppress amplifier noise in an oscillator circuit [4], noise-induced switching between two stable states in a nonlinear beam resonator enables precision measurement of the resonant frequency [5], and the sensitivity of a resonator for mass detection can be improved when the resonator is driven into a region of nonlinear oscillations [6]. Finally, in a Josephson junction, which is dynamically similar to a mechanical resonator in the nonlinear regime, the bistable state of the nonlinear system can be used as a bifurcation amplifier to perform a nondissipative, low-backaction measurement of the phase across the junction [7]. When nanomechanical devices reach the quantum-limited regime [8], a nanomechanical version of such an amplifier could be used for a similar sensitive low-back-action measurement of the state of a quantum mechanical resonator. The nonlinear response of nanomechanical resonators could also be used to detect transition from the classical to quantum regime [9,10]. It is therefore important to understand nonlinear dynamics of these systems well, so that we can fully realize their potential in expanding our experimental capabilities.Although some work has been done with parametric systems [11,12], the majority of nonlinear nanoscale systems that have been studied are directly driven [5,13,14] and the dominant nonlinearity in the restoring force is cubic, also known as Duffing nonlinearity. When a system is driven strongly, the Duffing nonlinearity causes the resonance response curve to become asymmetric. When the resonance is pulled far enough to one side, hysteretic behavior is observed as two stable states appear in the system [15]. The stable states, known as ''attractors'' or ''fixed points'', correspond to the points in state space to which the system converges with time. For each attractor, a set of initial states that dynamically evo...

Nonlinear Dynamics of Nanomechanical Resonators

“…One can then map the regions of the phase space of initial conditions into the two so-called basins of attraction of the two possible stable solutions, where the unstable solution lies along the separatrix, or border line between the two basins of attraction. These basins of attraction were mapped out in a recent experiment using a suspended platinum nanowire by Kozinsky et al [41]. If one additionally considers the existence of random noise, which is always the case in real systems, then the separatrix becomes fuzzy and it is possible to observe thermally activated switching of the resonator between its two possible solutions.…”

Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Duffing model

“…Furthermore, the simple Duffing expression enables the theoretic and, with the MEMS/NEMS close implementation, the model experimental study of subtle dynamic properties like dynamical switching [14][15][16] and memory effects [17]. * Electronic address: eddy.collin@grenoble.cnrs.fr Due to the fundamental issue behind non-linear dynamics (the physics of chaos [1]) and the broad panel of applications in micro/nano mechanics, which can even be extended to the quantum-limited nano-mechanical device [18], it is important to understand the nature of these mechanical non-linearities [16]. The most commonly discussed cases are non-linear actuation with an electrostatic drive [19,20], and non-linear constituents (with i.e.…”

Jahn-Teller instability in dissipative quantum systems

“…This is a distinctly different kind of bifurcation from that discussed recently for the damped nanomechanical Duffing oscillator [15], which only involves a single degree of freedom. In the case considered here, the bifurcation results in steady-state correlations between the state of the oscillator and the two-level system.…”