Nonlinear response of a driven vibrating nanobeam in the quantum regime

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.

The dissipative quantum Duffing oscillator: A comparison of Floquet-based approaches

“…Exemplarily we consider the dynamics of the Duffing oscillator near the one-photon resonance, where the oscillator dynamics is described analytically. As in [20,21,22,23] we obtain that for weak dissipation the amplitude of the oscillations displays an antiresonance rather than a resonance. We find a characteristic asymmetry of the antiresonance lineshape.…”

“…For comparison we show the result obtained by Peano et al [22]. We see that both curves show an exact antiresonance, but the dip position is slightly different and our result shows an asymmetric line shape.…”

“…The condition for an exact antiresonance, A = 0, is ρ 00 y (1) 00 = −ρ 11 y (1) 11 . As the result of [22] is given in lowest order without back transformation, the dip position occurs at the one-photon resonance, when the driving is such that sin η 0 2 = cos η 0 2 = 1/ √ 2. This corresponds to resonance in [22] due to ρ 00 = ρ 11 = 1 2 and with y (+1)…”

“…It was also shown that the response of the Duffing oscillator displays antiresonant dips and resonant peaks [20,21,22,23] depending on the frequency of the driving field, originating from special degeneracies of the eigenenergy spectrum of the nonlinear oscillator [20]. While the antiresonances persist in the presence of a weak Ohmic bath, for strong damping the nonlinear response turns to a resonant behaviour, namely the one of a linear oscillator at a shifted frequency [22]. Finally, recently Nakano et al [24] looked at the composed qubit-DO dynamics during read-out process.…”

“…We find a characteristic asymmetry of the antiresonance lineshape. In contrast to [20,21,22,23], our analytic results are obtained without applying a rotating wave approximation (RWA) on the Duffing oscillator. The paper is organized as follows: In section 2 we introduce the Hamiltonian of the non-dissipative Duffing oscillator and the two Floquet based approximation schemes to treat it.…”