Nonlinear response of a driven vibrating nanobeam in the quantum regime

TL;DR

This paper analytically explores the nonlinear quantum response of a driven nanobeam near buckling instability, revealing multiphonon transitions and bath-induced shifts between resonant and antiresonant behaviors.

Contribution

It develops a quantum mechanical model for a driven nanobeam near buckling, including dissipation, and derives analytical solutions for multiphonon resonances and their lineshapes.

Findings

Identification of multiphonon transitions at avoided level crossings

Derivation of analytical lineshape expressions for resonances

Discovery of bath-induced transition from resonance to antiresonance

Abstract

We analytically investigate the nonlinear response of a damped doubly clamped nanomechanical beam under static longitudinal compression which is excited to transverse vibrations. Starting from a continuous elasticity model for the beam, we consider the dynamics of the beam close to the Euler buckling instability. There, the fundamental transverse mode dominates and a quantum mechanical time-dependent effective single particle Hamiltonian for its amplitude can be derived. In addition, we include the influence of a dissipative Ohmic or super-Ohmic environment. In the rotating frame, a Markovian master equation is derived which includes also the effect of the time-dependent driving in a non-trivial way. The quasienergies of the pure system show multiple avoided level crossings corresponding to multiphonon transitions in the resonator. Around the resonances, the master equation is solved analytically using Van Vleck perturbation theory. Their lineshapes are calculated resulting in simple expressions. We find the general solution for the multiple multiphonon resonances and, most interestingly, a bath-induced transition from a resonant to an antiresonant behavior of the nonlinear response.