Tension-induced nonlinearities of flexural modes in nanomechanical resonators

Abstract

We consider the tension-induced non-linearities of mechanical resonators, and derive the Hamiltonian of the flexural modes up to the fourth order in the position operators. This tension can be controlled by a nearby gate voltage. We focus on systems which allow large deformations u(x) h compared to the thickness h of the resonator and show that in this case the third-order coupling can become non-zero due to the induced dc deformation and offers the possibility to realize radiationpressure-type equations of motion encountered in optomechanics. The fourth-order coupling is relevant especially for relatively low voltages. It can be detected by accessing the Duffing regime, and by measuring frequency shifts due to mode-mode coupling.PACS numbers: 85.85.+j Recent progress in fabricating nanomechanical resonators has shown how these systems can be used for ultrasensitive measurements of mass, force and charge [1][2][3][4]. Within the couple of past years these systems have also entered the quantum realm [5] as superpositions of vibration states and zero-point vibrations have been measured. Even though such measurements can be performed in a regime where the elastic properties of the resonators could essentially be considered as linear, the extension to non-linear conditions is well within reach of the current experimental techniques.In this paper we consider the generic non-linearities of the resonators, how these show up in measurements, and how they arise when the resonators are manipulated electronically. In general, the effect of non-linearites is twofold: on one hand they modify in an amplitudedependent way the resonant frequency of a given normal mode (Duffing self-non-linearity); on the other, they introduce a coupling between normal modes. Such nonlinearities show up in the presence of strong external driving, which allows to control the coupling of different modes or to detect their occupation numbers.Motivated by the recent advances in fabricating graphene and carbon nanotube resonators [4, 6], we concentrate especially on the regime of thin resonators where the mechanical deformation can be large compared to the resonator thickness. In this case, the major source of non-linearity is the tension induced by the deformation itself. Starting from the mechanical energy of the deformations, we derive the generic Hamiltonian of the flexural modes, including non-linearities up to the fourth order in the vibration amplitudes. In contrast to the results discussed in Refs. [7,8],where it is not taken into account, we explicitly consider the dc deformation of the resonator. This additional aspect creates an asymmetry in our system which leads to a cubic non-linearity. The dc deformation, dictating the strength of the non-linearity, is driven by a nearby gate voltage as in Fig. 1. Concentrating first on the Duffing self-non-linearity of the modes, this then allows us to derive the voltage depen- dence of the Duffing constant and show that it changes sign for a certain value of voltage that depends on mode index and the...

Tunable Micro- and Nanomechanical Resonators

“…The nonlinear intrinsic coupling between the flexural–flexural, torsional–torsional and flexural–torsional modes of a microcantilever was experimentally reported in [ 351 ]. The direct bending-induced nonlinearities can be identified to facilitate the precise tuning of nanomechanical resonators [ 352 ].…”

Electromechanical quantum simulators

“…The required degree of anharmonicity for the vibrating oscillators to be defined as qubits deserves a separate transmission line resonator transmon (B) voltage bias voltage bias discussion. On one hand, anharmonic contributions to the mechanical vibrational eigenstates can be experimentally implemented through the use of intrinsic mechanical nonlinearities 26,27 , or by using static external electric fields [27][28][29][30] . On the other hand, the required anharmonic shift to achieve a reliable qubit behavior amounts to about 1 MHz at least (see, e.g., the effect of this value on the gate fidelities, reported in App.…”

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