Nonlinear nanomechanical resonators for quantum optoelectromechanics

We present a scheme for tuning and controlling nanomechanical resonators by subjecting them to electrostatic gradient fields, provided by nearby tip electrodes. We show that this approach enables access to a regime of optomechanics where the intrinsic nonlinearity of the nanoresonator can be explored. In this regime, one or several laser-driven cavity modes coupled to the nanoresonator and suitably adjusted gradient fields make it possible to control the motional state of the nanoresonator at the single-phonon level. Some applications of this platform have been presented previously [S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, New J. Phys. 14, 023042 (2012); S. Rips and M. J. Hartmann, Phys. Rev. Lett. 110, 120503 (2013)]. Here we provide a detailed description of the corresponding setup and its optomechanical coupling mechanisms together with an in-depth analysis of possible sources of damping or decoherence and a discussion of the readout of the nanoresonator state.

Figure 1

Section of a deflected rod. The local strain $u_{xx}$ depends on the transverse coordinate $\tilde{y}$ and on the local curvature $y^{\prime \prime }$ and determines the energy density $\frac{1}{2}Y{u}_{xx}^2$.

Figure 2

Resonator with electrodes modeled by point charges $q,q^{\prime }$. The field profile (red) leads to an inverted parabola for the dielectric potential (blue) around the equilibrium position.

Figure 3

(a) Microtoroid cavity with nearby CNT (not to scale). (b) NEMS chip with CNT resonator and electrodes. (c) Schematic top view of the NEMS chips with the CNT oscillating in the $\widehat{z}$-$\widehat{y}$ plane. (d) Schematic cross-sectional view of the NEMS chip and the cavity rim (yellow). The NEMS chip is mounted in the evanescent cavity field at a distance $d$ to the cavity rim surface. The nanoresonator is displaced from the closest point to allow for an optomechanical coupling that is linear in the resonator's deflection.

Figure 4

Relative alignment of electrode coordinates $(r^{\prime },{\phi }^{\prime },z^{\prime })$ and waveguide coordinates $(r,\phi ,z)$. The misalignment is determined by a small angle $\theta$.

Figure 5

Different schemes for resonant interactions. (a) Interaction between mechanical excitations and detuned photons entering the cavity. (b),(c) Interaction between a classical gradient field and mechanical excitations.

Figure 6

Output spectrum characterizing a motional state of a single nanoresonator, where only the lowest few states are populated. Here $P_0=0.0391$, $P_1=0.9137$, $P_2=0.0430$, $P_3=0.0024$, $P_4=0.0006$, $P_5=0.0003$, and $P_j<0.0003$ for $j>5$. Due to the parity of $\mathcal{X}$, there are only sideband groups at frequencies that correspond to processes where an odd number of phonons is scattered. The sidebands drop quickly in intensity with growing distance to the main line. The fine structure of the $n-m=\pm 1$ groups gives information about the level population of the resonator. The frequency axis is labeled in terms of the mechanical transition frequencies ${\delta }_{nm}$, and a peak at $\omega -{\omega }_{\mathrm{L}}={\delta }_{mn}$ with ${\delta }_{mn}>0$ (${\delta }_{mn}<0$) corresponds to anti-Stokes (Stokes) photon scattering events that occur at a rate proportional to the occupation of the respective initial mechanical level. In this example the level spacings correspond to a 1-$\mu \text{m}$-length $(10,0)$ CNT that is softened so that ${\omega }_{\mathrm{m}}/2\pi =5.23\mathrm{M}\mathrm{Hz}$, which leads to ${\delta }_{10}\approx {\lambda }^{\prime }=209\mathrm{k}\mathrm{Hz}$. Three “active” state-preparation lasers with detunings ${\Delta }_1\approx {\delta }_{10}$, ${\Delta }_2\approx {\delta }_{12}$, and ${\Delta }_3\approx {\delta }_{23}$ induce optomechanical couplings $g_{\mathrm{m},1}=g_{\mathrm{m},2}=g_{\mathrm{m},3}=20.9\mathrm{k}\mathrm{Hz}$, respectively; for the parameters considered (cf. Sec. 5) this value can be achieved keeping the intracavity absorption in the microwatt range [25]. The resulting optomechanical interactions with the nanobeam dominate over the intrinsic mechanical decoherence rates and determine the broadening of the levels involved (see Sec. 10); the remaining relevant parameters are ${\kappa }_{\mathrm{ex}}/\kappa =0.1$, ${\kappa }_1/2\pi ={\kappa }_2/2\pi ={\kappa }_3/2\pi =52.3$ kHz, ${\omega }_{\mathrm{m}}/\gamma =5\times {10}^6$, and $T=20$ mK.