Figure 1

Schematic of the system under consideration. A nonlinear nanomechanical resonator (NR) of frequency ${\omega }_m$ is coupled to a superconducting microwave resonator of frequency ${\omega }_c$. The nanomechanical oscillator is driven parametrically at frequency $2{\omega }_m$ and the microwave cavity is driven at two tones ${\omega }_1$ and ${\omega }_2$. The annihilation operators for the microwave and nanomechanical modes are $a$ and $b$, respectively. In terms of these operators, the coupling takes the form $a^{†}a(b+b^{†})$. The superconducting microwave resonator is modeled as a lumped element LC circuit, where $L_T$ and $C_T$ are, respectively, the inductance and capacitance of the tank circuit.Reuse & Permissions

Figure 2

Physical schematic of the system under consideration (not to scale). A nonlinear nanomechanical oscillator of frequency ${\omega }_m$ is capacitively coupled to a superconducting microwave resonator of frequency ${\omega }_c$. A theoretical schematic of the system is shown in Fig. 1.Reuse & Permissions

Figure 3

(a) Radial and (b) angular components of the nanomechanical oscillator amplitude of the semiclassical fixed points. The fixed points are functions of two parameters: the scaled normalized parametric pumping $\xi {\kappa }^{\text{'}}$ and the scaled normalized nanomechanical dissipation rate $\xi {\gamma }^{\text{'}}$ [see Eq. (27)]. There are three different classes of fixed points: the origin ($r_{b_0}^2=0$) exists for all parameter values (the undefined angular component is not plotted), the unit circle pair ($r_{b_0}^2=1$) exists for $\varphi >0$, and the off-circle pair exists for $0<\varphi <1$. The colors of the plot indicate the stability: Green indicates stable fixed points; checkered red indicates unstable fixed points; and striped blue indicates fixed points that may or may not be stable, dependent on two other parameters (see Fig. 5). Note that the apparent discontinuity in crossing the $\xi {\kappa }^{\text{'}}$ axis is not real since to cross means a flipping of the sign $\xi {\gamma }^{\text{'}},$ and hence the detuning $\omega$ (which hits a pole in the parameters as the detuning crosses $0$), and which also flips the sign of $\xi {\kappa }^{\text{'}}$. It is best to consider positive and negative detuning separately.Reuse & Permissions

Figure 4

Two-dimensional projection of the phase diagram of the semiclassical microwave cavity-nanomechanical system. The qualitatively different regions of semiclassical fixed-point structures are plotted against the scaled normalized parametric pumping $\xi {\kappa }^{\text{'}}$ and the scaled normalized nanomechanical dissipation rate $\xi {\gamma }^{\text{'}}$. The existence and positions of the fixed points depend upon only these two parameters. The colors of the regions indicate if the region contains a stable fixed point: Green indicates the presence of a stable fixed point, checkered red indicates the absence of a stable fixed point, and striped blue indicates the case that depends on an additional two parameters (see Fig. 5). The origin is always a semiclassical fixed point. It is stable in regions $B$ and $G$ and may be stable or unstable, including undergoing Hopf bifurcations, in regions $A$, $E$, and $G$. The fixed-point pair on the nanomechanical unit circle exists in regions $C$, $D$, $E$, $F$, $G$, $H$, $I$, $K$, and $L$ and is stable in regions $C$, $D$, $E$, $F$, and $G$. The off-circle fixed points exist in regions $C$, $E$, $G$, $I$, and $K$, but are always unstable. A fuller explanation of the semiclassical dynamics is given in the text. The upper dotted line shows the location in parameter space of the systems shown in Figs. 7 and 9; the lower dotted line shows the location in parameter space of the systems shown in Figs. 8 and 10. The lines are close to the $x$ axis for ease of quantum-numerical simulation, as explained in Sec. 4.Reuse & Permissions

Figure 5

Two-dimensional phase diagram of the semiclassical microwave-cavity-nanomechanical system at a particular value of the scaled normalized parametric pumping $\xi {\kappa }^{\text{'}}=-0.8$ and the scaled normalized mechanical dissipation rate $\xi {\gamma }^{\text{'}}=0.3$. This phase diagram shows the extra two dimensions present at a particular point in region $A$ in the two-dimensional projection of Fig. 4 of the full four-dimensional phase space. These extra two dimensions are the normalized microwave-cavity dissipation rate ${\mu }^{\text{'}}$ and the normalized microwave-cavity-nanomechanics coupling $\xi$. The colors of the regions indicate the stability of the fixed point at the origin: Green indicates stable and checkered red indicates unstable. For large values of either the mechanical dissipation rate or the normalized coupling, the origin is generally stable.Reuse & Permissions

Figure 6

Nanomechanical components of several trajectories through the semiclassical phase space on (a) the stable side and (b) the unstable side of a supercritical Hopf bifurcation of the origin. Here we are labeling the sides based on the fixed point at the origin, which is a stable attractor in (a) and an unstable node in (b). The parameters for this trajectory are a scaled normalized parametric pumping of $\xi {\kappa }^{\text{'}}=-0.8$, a scaled normalized mechanical dissipation rate of $\xi {\gamma }^{\text{'}}=0.3$, a normalized microwave-cavity dissipation rate of ${\mu }^{\text{'}}=0.3$, and a normalized microwave-cavity-nanomechanics coupling of $\xi =2.4$. This places the system in region $A$ of Fig. 4 and the (a) green and (b) checkered red regions of Fig. 5. Qualitatively similar behavior is found for other system parameter values lying in these same regions. Trajectories that start both close to and away from the origin are shown; the trajectories start as white or light blue at the initial time and move toward solid blue such that the darker blue is the approximation of the steady state. Trajectories are attracted to the origin in (a) and away from the origin toward a stable attracting limit cycle that has formed in (b). The raw couplings giving the parameters mentioned are $\omega =3,\hspace{0.5em}\chi =0.1,\hspace{0.5em}\kappa =-1,\hspace{0.5em}g=2.86,\hspace{0.5em}\mu =0.9$, and $\gamma =0.38$.Reuse & Permissions

Figure 7

(a) Microwave and (b) nanomechanical components of some semiclassical trajectories for positive detuning ($\omega >0$) through a linear sweep of increasing $\xi {\kappa }^{\text{'}}$ [(i) $\xi {\kappa }^{\text{'}}=-4$, (ii) $\xi {\kappa }^{\text{'}}=-1$, and (iii) $\xi {\kappa }^{\text{'}}=2]$ by changing only the parametric pumping. The semiclassical fixed points are marked with a green circle if stable or a red cross if unstable. The trajectories start as white or light blue and move toward blue such that the darker blue is the better approximation of the steady state. The trajectories are clearly attracted to the stable semiclassical fixed points. In terms of the phase diagram of Fig. 4, the linear sweep of (i)-(ii)-(iii) is along the upper dotted line $\xi {\gamma }^{\text{'}}=0.2$; there is qualitatively similar behavior for all systems in the same region of this diagram. The axes here are scaled as the original parameters $(x_a^{\text{'}},y_a^{\text{'}})=\frac{\sqrt{2}\hspace{0.5em}\eta }{\sqrt{\xi }}(x_a,y_a)$ and $(x_b^{\text{'}},y_b^{\text{'}})=\eta (x_a,y_a)$, so that they can be compared to the quantum Wigner functions in Fig. 9. The raw couplings giving the parameters mentioned are $\omega =2,\hspace{0.5em}\chi =1,\hspace{0.5em}g=1.16,\hspace{0.5em}\mu =0.2$, and $\gamma =0.07$.Reuse & Permissions

Figure 8

Equivalent of Fig. 7 for negative detuning: (a) microwave and (b) nanomechanical components of some semiclassical trajectories for negative detuning ($\omega <0$) through a linear sweep of increasing $\xi {\kappa }^{\text{'}}$ [(i) $\xi {\kappa }^{\text{'}}=-4$, (ii) $\xi {\kappa }^{\text{'}}=-1$, and (iii) $\xi {\kappa }^{\text{'}}=2]$ by changing only the parametric pumping. The semiclassical fixed points are marked with a green circle if stable or a red cross if unstable. The trajectories start as white or light blue and move toward blue such that the darker blue is the better approximation of the steady state. The trajectories are clearly repelled by the unstable semiclassical fixed points, out to an outer limit cycle. In terms of the phase diagram of Fig. 4, the linear sweep of (i)-(ii)-(iii) is along the lower dotted line $\xi {\gamma }^{\text{'}}=-0.2$; there is qualitatively similar behavior for all systems in the same region of this diagram. The axes here are scaled as the original parameters $(x_a^{\text{'}},y_a^{\text{'}})=\frac{\sqrt{2}\hspace{0.5em}\eta }{\sqrt{\xi }}(x_a,y_a)$ and $(x_b^{\text{'}},y_b^{\text{'}})=\eta (x_a,y_a)$ so that they can be compared to the quantum Wigner functions in Fig. 10. The raw couplings giving the parameters mentioned are $\omega =-2,\hspace{0.5em}\chi =1,\hspace{0.5em}g=1.16,\hspace{0.5em}\mu =0.2$, and $\gamma =0.07$.Reuse & Permissions

Figure 9

Density plots of Wigner functions for the (a) microwave cavity and (b) nanomechanical factor spaces for positive detuning ($\omega >0$) through a linear sweep of increasing $\xi {\kappa }^{\text{'}}$ [(i) $\xi {\kappa }^{\text{'}}=-4$, (ii) $\xi {\kappa }^{\text{'}}=-1$, and (iii) $\xi {\kappa }^{\text{'}}=2]$ by changing only the parametric pumping. The Wigner function $W(x,y)$ is plotted as the color against (a) two quadrature operators $x_a$ and $y_a$ of the microwave-cavity field and (b) the position and momentum quadratures (though not scaled as such) $x_b$ and $y_b$ of the nanomechanical oscillator. The quantum steady state shows clear evidence of the semiclassical bifurcations it undergoes; particular comparison can be made to the semiclassical trajectories of Fig. 7. However, the quantum fluctuations wash out the sharp transitions. In particular, the semiclassical bifurcations between regions $E$, $A$, $G$, and $C$ of the semiclassical phase diagram in Fig. 4 are not discernible. However, the signature of the overall transition from the $x$-axis-aligned density in region $F$ to the $y$-axis-aligned density in region $D$ is clearly visible. In terms of the phase diagram of Fig. 4, the linear sweep of (i)-(ii)-(iii) is along the upper dotted line $\xi {\gamma }^{\text{'}}=0.2$. The raw couplings giving the parameters mentioned are $\omega =2,\hspace{0.5em}\chi =1,\hspace{0.5em}g=1.16,\hspace{0.5em}\mu =0.2$, and $\gamma =0.07$.Reuse & Permissions

Figure 10

Equivalent of Fig. 9 for negative detuning: density plots of Wigner functions for the (a) microwave cavity and (b) nanomechanical factor spaces for negative detuning ($\omega <0$) through a linear sweep of increasing $\xi {\kappa }^{\text{'}}$ [(i) $\xi {\kappa }^{\text{'}}=-4$, (ii) $\xi {\kappa }^{\text{'}}=-1$, and (iii) $\xi {\kappa }^{\text{'}}=2]$ by changing only the parametric pumping. The Wigner function $W(x,y)$ is plotted as the color against (a) two quadrature operators $x_a$ and $y_a$ of the microwave cavity field and (b) the position and momentum quadratures (though not scaled as such) $x_b$ and $y_b$ of the nanomechanical oscillator. Compared with Fig. 8, the major and minor axes of symmetry of the phase-diffused rings are seen to be the same as the semiclassical limit cycles. In terms of the phase diagram of Fig. 4, the linear sweep of (i)-(ii)-(iii) is along the lower dotted line $\xi {\gamma }^{\text{'}}=-0.2$. The raw couplings giving the parameters mentioned are $\omega =-2,\hspace{0.5em}\chi =1,\hspace{0.5em}g=1.16,\hspace{0.5em}\mu =0.2$, and $\gamma =0.07$.Reuse & Permissions

Figure 11

Quantum entanglement, as measured by the logarithmic negativity, for positive detuning ($\omega >0$) (solid black plot) and negative detuning ($\omega <0$) (dashed black plot) through a linear sweep of increasing $\xi {\kappa }^{\text{'}}$ from $\xi {\kappa }^{\text{'}}=-4$ to 2 by changing only the parametric pumping. The four vertical lines show the positions of the semiclassical bifurcations. Referring to the phase diagram of Fig. 4, these are (from left to right) the $F$-$H$–$E$-$I$ boundary (dashed red line), the $E$-$I$–$A$-$J$ boundary (solid red line), the $A$-$B$-$J$–$C$-$G$-$K$ boundary (solid blue line), and the $C$-$G$-$K$–$D$-$L$ boundary (dashed blue line). For positive detuning, the quantum entanglement is peaked just to one side of the semiclassical bifurcation center. The quantum entanglement for negative detuning is remarkably flat. In terms of Fig. 4, the linear sweep is along the upper dotted line $\xi {\gamma }^{\text{'}}=0.2$ for positive detuning and the lower dotted line $\xi {\gamma }^{\text{'}}=-0.2$ for negative detuning. The raw couplings giving the parameters mentioned are $\omega =\pm 2,\hspace{0.5em}\chi =1,\hspace{0.5em}g=1.16,\hspace{0.5em}\mu =0.2$, and $\gamma =0.07$.Reuse & Permissions

Figure 12

Purity of the steady-state density matrix for positive detuning ($\omega >0$) (solid black plot) and negative detuning ($\omega <0$) (dashed black plot) through a linear sweep of increasing $\xi {\kappa }^{\text{'}}$ from $\xi {\kappa }^{\text{'}}=-4$ to 2 by changing only the parametric pumping. The four vertical lines show the positions of the semiclassical bifurcations. Referring to the phase diagram of Fig. 4, these are (from left to right) the $F$-$H$–$E$-$I$ boundary (dashed red line), the $E$-$I$–$A$-$J$ boundary (solid red line), the $A$-$B$-$J$–$C$-$G$-$K$ boundary (solid blue line), and the $C$-$G$-$K$–$D$-$L$ boundary (dashed blue line). For positive detuning, the purity is reduced as the separation of the Wigner function density increases. The purity of the steady-state density matrix for negative detuning is remarkably flat, as is its entanglement (cf. Fig. 11). In terms of Fig. 4, the linear sweep is along the upper dotted line $\xi {\gamma }^{\text{'}}=0.2$ for positive detuning and the lower dotted line $\xi {\gamma }^{\text{'}}=-0.2$ for negative detuning. The raw couplings giving the parameters mentioned are $\omega =\pm 2,\hspace{0.5em}\chi =1,\hspace{0.5em}g=1.16,\hspace{0.5em}\mu =0.2$, and $\gamma =0.07$.Reuse & Permissions