Figure 1

A schematic of the nanoelectromechanical system under consideration. A superconducting microwave cavity of frequency ${\omega }_c$ mediates a coupling between $N$ nanomechanical resonators capacitively coupled to it. The $i$th nanomechanical resonator has resonant frequency ${\omega }_i$ and microwave-mechanical coupling strength $g_i$. The microwave cavity is driven by a linear drive of amplitude $\epsilon$ at a detuning from the cavity of $\delta$.Reuse & Permissions

Figure 2

The bifurcation diagram for $\omega =2$ and $\gamma =0.001$. In the shaded region there are no periodic orbits and there is one stable critical point. The Hopf bifurcation curve, which is in red, provides a partial boundary of this region. At the generalized Hopf GH (which is at $\delta =\sqrt{7}$ for $\omega =2$) the Hopf bifurcation changes from super- to subcritical. For $\delta <\sqrt{7}$ the Hopf bifurcation is subcritical and periodic orbits exist to the left of the Hopf curve. Also for $0<\delta <\sqrt{3}$ there are regions where periodic orbits exist to the left of the Hopf curve. The blue curves A-GH, BGK, CFK, DEK, KHCusp, KMcusp, etc., are saddle node bifurcations of periodic orbits creating a stable and an unstable periodic orbit existing to their right. (Only the first eight are shown.) The lozengelike dashed curves are also saddle node bifurcations of periodic orbits, this time destroying a stable and an unstable periodic orbit. (Once again only a sample are shown.) In the regions ABG(GH) and HMCusp there is one stable critical point and a pair of periodic orbits with opposite stability. In the region G(GH)HK there is one unstable critical point and one stable periodic orbit. In the region BCFG and the region to the left of MCusp there is one stable critical point and two pairs of periodic orbits with opposite stability. In the region FGK there is one unstable critical point and two stable periodic orbits and one unstable periodic orbit. In the region CDEF there is one stable critical point and three stable and three pairs of periodic orbits with opposite stability, etc.Reuse & Permissions

Figure 3

Bifurcation diagrams (a) $\omega =5$ and (b) $\omega =10$ and $\gamma =0.001$ with $\delta <0$ (blue detuning) showing the Hopf bifurcation (red) and saddle node bifurcations of periodic orbits. The labeling in (a) is similar to that in Fig. 2. For instance, in the region ABG(GH) there is one stable critical point and a pair of periodic orbits with opposite stability.Reuse & Permissions

Figure 4

The amplitudes ($N\left|A\right|=Nr$) of the periodic orbits of the system calculated from the amplitude equations as a function of $\sqrt{NG}\epsilon$ for $\omega =2$ and $\gamma =0.01,0.001,0.0001$ and various values of $\delta$. In (a) $\delta =-2$, (b) $\delta =-5$, (c) $\delta =-9$, and (d) $\delta =-10$. The unstable periodic orbits are given by dashed lines and the stable one are given by solid lines.Reuse & Permissions

Figure 5

The amplitudes $\left|\alpha \right|$ and $N\left|A\right|$ as $\epsilon$ is increased for $\omega =10$ and $\gamma =0.00001$ compared with the saddle node bifurcations that create them. (a) is the cavity amplitude $\left|\alpha \right|$, (b) $N\left|A\right|$, and (c) the bifurcation diagram.Reuse & Permissions

Figure 6

The contours of the function $F_r\left(Nr,\omega ,\delta \right)$ for $\omega =2$ plotted as a function of $\left(Nr,\delta \right)$, calculated using 10 terms in the sum of products of Bessel functions. Stable periodic orbits exist in green shaded regions. There are no periodic orbits in the regions enclosed by red lines where $F_r(N\left|A\right|,2,\delta )<0$.Reuse & Permissions

Figure 7

The bifurcation diagram for two mechanical resonators for $N_1=N_2,\gamma =0.0001,\omega =2$, and $\delta =-1.5$. The in-phase solutions are stable outside the shaded central regions. The out-of-phase solutions are stable outside the slice near the horizontal axis. They are singular at $\Delta {\omega }_{21}=0$ and unstable for $\left|\Delta {\omega }_{21}\right|$ small, where they occur for very large values of $r_i$. In the unshaded regions both in-phase and out-of-phase solutions are stable, but have different basins of attraction. (b) shows the in-phase solution in $(r_1,r_2)$ space as $\Delta {\omega }_{21}$ is varied. $r_1+r_2$ remains approximately constant. (c) shows the out-of-phase solution in $(r_1,r_2)$ space as $\Delta {\omega }_{21}$ is varied.Reuse & Permissions

Figure 8

Transient unsynchronized motion for two nonidentical mechanical resonators for $N_1=N_2,\omega =2,\delta =-1.5,\gamma =0.0001,\sqrt{N_{1}G}\epsilon =2$, and $\Delta {\omega }_{21}=0.04$. Started near the in-phase solution, in the blue shaded region of Fig. 7, the transient unsynchronized motion is only temporary. Eventually solutions are trapped by an out-of-phase solution [$\phi$ mod ($2\pi )\to \pi$]. The variables are plotted against time. In (a) $r_1$ and $r_2$ are shown in solid lines, the collective variable $r$ is dashed, and $\phi$ mod($2\pi )$ is dotted. (b) is a plot of $\phi$ and (c) is a plot of the amplitude of $GN{\left|\alpha \right|}^2$.Reuse & Permissions

Figure 9

Transient unsynchronized motion of the in-phase solution for three sets of mechanical nonidentical resonators for $N_i$ equal, $\omega =2,\delta =-1.5,\gamma =0.0001,\sqrt{N_{i}G}\epsilon =2$, $\Delta {\omega }_{21}=0.04$, and $\Delta {\omega }_{21}=0.045$. Eventually solutions are trapped by an out-of-phase solution [${\phi }_i$ mod($2\pi )\to \pi$ here]. The variables are plotted against time. In (a) $r_i$ are shown in solid lines, the collective variable $r$ is dashed, and ${\phi }_i\mathrm{mod}\left(2\pi \right)$ are dotted. (b) is a plot of ${\phi }_i$ and (c) is a plot of the amplitude of $GN{\left|\alpha \right|}^2$.Reuse & Permissions

Figure 10

The bifurcation diagram for three mechanical resonators for $N_i$ equal, $\omega =2,\delta =-1.5,\gamma =0.0001$, and $\Delta {\omega }_{21}=5\Delta {\omega }_{32}$. The in-phase solution is stable outside the shaded central regions. The out-of-phase solutions are stable outside the gray regions. In the dark shaded central region the in-phase solution is unstable to both ${\phi }_1$ and ${\phi }_2$ and the motion is transiently unsynchronized, eventually setting on an out-of-phase solution. Once again the out-of-phase solutions tend to exist where at least some of the $r_i$ take larger values.Reuse & Permissions

Figure 11

Linearized quantum noise spectrum of the microwave cavity $S_{11}\left(\Omega \right)$ (a) approaching the Hopf bifurcation; and (b) continuing the linearization beyond the Hopf bifurcation. The magnitude of the normally ordered cavity spectrum at steady state $(1/2\pi ){\int }_{-\infty }^{\infty }{e^{}}^{-i\Omega \tau }{\langle \alpha \left(t\right)\alpha {(t+\tau )}^T\rangle }_{t\to \infty }d\tau$, the first diagonal element of $\mathbf{S}\left(\Omega \right)$, is plotted at the frequency $\Omega$ for varying driving amplitude $\epsilon$. Here we have set $\omega =10$, $\delta =-4$, $\gamma =0.001$, $N=1$, $G=1$, for which the Hopf bifurcation occurs at a driving strength of ${\epsilon }_h\approx 1.76$.Reuse & Permissions