Cavity optomechanics is showing promise for studying quantum mechanics in large systems. However, smallness of the radiation-pressure coupling is a serious hindrance. Here we show how the charge tuning of the Josephson inductance in a single-Cooper-pair transistor can be exploited to arrange a strong radiation pressure -type coupling g0 between mechanical and microwave resonators. In a certain limit of parameters, such a coupling can also be seen as a qubit-mediated coupling of two resonators. We show that this scheme allows reaching extremely high g0. Contrary to the recent proposals for exploiting the non-linearity of a large radiation pressure coupling, the main non-linearity in this setup originates from a cross-Kerr type of coupling between the resonators, where the cavity refractive index depends on the phonon number. The presence of this coupling will allow accessing the individual phonon numbers via the measurement of the cavity. PACS numbers: 42.50.Wk,81.07.Oj,73.23.Hk,85.25.Cp arXiv:1311.3802v2 [cond-mat.mes-hall] 7 Apr 2014

We consider the response of a nanomechanical resonator interacting with an electromagnetic cavity via a radiation pressure coupling and a cross-Kerr coupling. Using a mean field approach we solve the dynamics of the system, and show the different corrections coming from the radiation pressure and the cross-Kerr effect to the usually considered linearized dynamics.Comment: 6 pages, 9 figure

The experimental observation of quantum phenomena in mechanical degrees of freedom is difficult, as the systems become linear towards low energies and the quantum limit, and thus reside in the correspondence limit. Here we investigate how to access quantum phenomena in flexural nanomechanical systems which are strongly deflected by a voltage. Near a metastable point, one can achieve a significant nonlinearity in the electromechanical potential at the scale of zero point energy. The system can then escape from the metastable state via macroscopic quantum tunneling (MQT). We consider two model systems suspended atop a voltage gate, namely, a graphene sheet, and a carbon nanotube. We find that the experimental demonstration of the phenomenon is currently possible but demanding, since the MQT crossover temperatures fall in the millikelvin range. A carbon nanotube is suggested as the most promising system.

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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...

The exact solution to a velocity-dependent forced quantum anharmonic oscillator is de-rived by using the integral oprators and iteration method. The time development of the dis-placement and momentum operators of the anharmonic oscillator is given. These operators are represented as a Laplace transform and a subsequent inverse Laplace transform of suitable functionals.

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