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We show how this principle can be extended to measuring individual mechanical resonator modes in an array of nanobeams.
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Let us consider a single mode of oscillations of a linear mechanical resonator.
As well, by using this coupling method, it is possible to excite a larger number of optical modes, some of which have higher Qs and larger optomechanical coupling to the mechanical resonator.
In order to confirm the applicability of the semi-empirical formula, twenty different sizes of clamped clamped beam Si mechanical resonators having the first-mode resonant frequency of 5 20 MHz are designed and fabricated from a silicon-on-insulator (SOI) wafer, and their vibration characteristics in a vacuum environment are measured by a network analyzer.
The motion of the mechanical resonator is described by the Hamiltonian (H_{r}=omega_{r}b^{dagger}b), with (omega_{r}) as the frequency of the fundamental vibration mode of the resonator and b as the corresponding annihilation operator.
The resulting mechanical resonator is shown in Figure 3a.
We take the NV-mechanical resonator interaction strength (eta=200mbox{ kHz}), eigen frequency of the mechanical resonator (omega_{r}=1mbox{ MHz}) and the number of spin is (N=10mbox000).
(a) Schematic set up of a hybrid NV and mechanical resonator system.
A mechanical resonator with frequency (omega_{r}) is attached with a magnet.
Therefore, the resonator modes are not subject of the phase mixing phenomenon (Mager and Klimushkin, 2006).
We neglect the interaction between the mechanical resonator and the 14N nuclear spins since it is three orders of magnitude smaller than that between the NV center and the mechanical resonator.
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