Force sensitivity of multilayer graphene optomechanical devices

Introduction

Considerable effort has been devoted to developing mechanical resonators based on low-dimensional materials, such as carbon nanotubes1,2,3,4,5,6,7,8,9,10,11,12, semiconducting nanowires13,14,15,16,17,18,19,20,21,22, graphene23,24,25,26,27,28,29 and monolayer semiconductors30,31,32. The specificity of these resonators is their small size and their ultra-low mass, which enables sensing of force and mass with unprecedented sensitivities7,10. Such high-precision sensing capabilities hold promise for studying physical phenomena in new regimes that have not been explored thus far, for instance, in spin physics33, quantum electron transport34,35, light-matter interaction19and surface science36,37. However, the transduction of the mechanical vibrations of nanoscale mechanical systems into a measurable electrical or optical output signal is challenging. As a result, force and mass sensing is often limited by the imprecision in the measurement of the vibrations, and cannot reach the fundamental limit imposed by thermo-mechanical noise.

A powerful method to obtain efficient electrical readout of small resonators is to amplify the interaction between mechanical vibrations and the readout field using a superconducting microwave cavity27,28,29. Increasing the field in the cavity improves the readout sensitivity and eventually leads to dynamical back-action on the thermo-mechanical noise. This effect has been studied intensively on comparatively large micro-fabricated resonators, resulting for instance in enhanced optomechanical damping38,39, ground-state cooling of mechanical vibrations40,41 and displacement imprecision below the standard quantum limit42,43. Another phenomenon often observed when detecting and manipulating the motion of mechanical resonators is the induced heating that can occur through Joule dissipation and optical adsorption28,44. Heating is especially prominent in tiny mechanical resonators because of their small heat capacity. An additional difficulty in characterizing mechanical vibrations is related to the fluctuations of the mechanical resonant frequency, also called frequency noise, which are particularly sizable in small resonators endowed with high-quality factors Q10.

Here we study the force sensitivity of multilayer graphene mechanical resonators coupled to superconducting cavities. In particular, we quantify how the force sensitivity is affected by dynamical back-action, Joule heating and frequency noise upon increasing the number of pump photons inside the cavity. We demonstrate a force sensitivity of , of which ≈50% arises from thermo-mechanical noise and ≈50% from measurement imprecision. The force sensitivity tends to be limited by measurement imprecision and frequency noise at low pump power, and by optomechanical damping and Joule heating at high pump power.

Results

Thermal force noise and imprecision force noise

A fundamental limit of force sensing is set by the thermo-mechanical noise of the eigenmode that is measured. According to the fluctuation-dissipation theorem, the associated thermal force noise is white and is quantified by

where Tmode is the temperature of the mechanical eigenmode, and meffis its effective mass8,45. This force noise is transduced into a mechanical resonance with line width  and height  in the displacement spectrum (Fig. 1). Importantly, equation (1) shows that the low mass of graphene decreases the size of the thermo-mechanical force noise. However, a drawback of tiny resonators with high Q-factors is their tendency to feature sizable frequency noise that broadens the resonance and, therefore, increases the size of the force noise10,46.

Figure 1: Mechanical displacement and force sensitivity.

(a) Mechanical displacement spectrum Sz close to the mechanical resonance frequencyωm/2π. The total displacement spectral density  at ωm is the sum of the displacement noise  and the displacement imprecision . (b) Corresponding force sensitivity  (dark grey). The individual components are the thermal force noise  (dark yellow) and the imprecision force noise  (turquoise), given by equations (1) and (2), respectively. The quantum back-action noise is neglected for simplicity. For the plots most of the parameters are those of device B, but we estimate the mass assuming that the graphene flake is a single layer. Further we choose nadd=0.5,Tbath=0.015 K, and np=2·105 in a (see text).