Manifestation of classical nonlinear dynamics in optomechanical entanglement with a parametric amplifier

Cavity optomechanical systems involving an optical parametric amplifier (OPA) can exhibit rich classical and quantum dynamical behaviors. By simply modulating the frequency of the laser pumping the OPA, we find two interesting parameter regimes, with one of them enabling us to study quantum-classical correspondence in system dynamics, while there exist no classical counterparts of the quantum features for the other. For the detuning of the laser pumping around the mechanical frequency, as the parametric gain of OPA increases to a critical value, the classical dynamics of the optical or mechanical modes can experience a transition from the regular periodic oscillation to a period-doubling motion, in which case the light-mechanical entanglement can be well studied by the logarithmic negativity and can manifest the dynamical transition the classical nonlinear dynamics. In addition, the optomechanical entanglement shows a second-order transition characteristic at the critical parametric gain. For the laser detuning being about twice the mechanical frequency, the kind of normal mode splitting comes up in the laser detuning dependence of optomechanical entanglement, which is induced by the squeezing of the optical and mechanical hybrid modes and finds no classical correspondence. The OPA-assisted optomechanical systems therefore offer a simple way to study and exploit quantum manifestations of classical nonlinear dynamics.

Figure 1

(a) Sketch of the optomechanical system under two-field driving. The cavity is driven by an external driving laser with strength $E$ and detuning ${\mathrm{\Delta }}_0$, and the OPA inside the cavity is pumped by another laser field. (b) Long-time behavior of the cavity field strength $|\langle a(t)\rangle |$ and (c) the light-mechanical entanglement $E_N$ with $\tau =2\pi /\mathrm{\Omega }$ for a typical set of experimental parameters [61]: $(E,g,{\gamma }_m,\kappa ,{\mathrm{\Delta }}_0)/{\omega }_m=(6\times {10}^4,4\times {10}^{-6},{10}^{-6},0.2,1)$. The OPA is pumped with the parametric gain $\mathrm{\Lambda }/{\omega }_m=0.02$, the detuning $\mathrm{\Omega }/{\omega }_m=1.16$, and the parametric phase $\theta =0,\pi$. Note that the mechanical thermal noise is not taken into account so far (i.e., $n_m=0$); however, it can be detrimental for experimental generation of the optomechanical entanglement, see Sec. 5 for further discussions. For current consideration, the analytical approximation of the classical mean values (dash) shows nice agreement with the numerical solution.

Figure 2

${\left|\langle a\left(t\right)\rangle \right|}_{\text{max}}$ [(a)–(c)] and $E_{N,\text{max}}$ [(d)–(f)] in the long time limit as a function of $\mathrm{\Omega }/{\omega }_m$ for driving strengths $E/{\omega }_m=4\times {10}^4$ [(a), (d)], $E/{\omega }_m=6\times {10}^4$ [(b), (e)], and $E/{\omega }_m=8\times {10}^4$ [(c), (f)]. The numerical (analytical) results are presented by the blue solid (green dashed) lines. Other parameters are the same as in Figs. 1 and 1.

Figure 3

(a) The real parts of roots of $d\left(\mathrm{\Omega }\right)=0$ in the domain $\text{Re}\left(\mathrm{\Omega }\right)>0$, and (b) the imaginary parts of the roots of $d\left(\mathrm{\Omega }\right)=0$ as a function of driving $E/{\omega }_m$. Other parameters are the same as in Fig. 2.

Figure 4

(a) $E_{N,\text{max}}$ versus the parametric gain $\mathrm{\Lambda }/{\omega }_m$ and the detuning $\mathrm{\Omega }/{\omega }_m$ for $E/{\omega }_m=8.6\times {10}^4$. The black area denotes the classically unstable region, confirmed via the Routh-Hurwitz criterion. (b) $E_{N,\text{max}}$ and (c) ${\left|\langle a\left(t\right)\rangle \right|}_{\text{max}}$ versus $\mathrm{\Omega }/{\omega }_m$ for parametric gains $\mathrm{\Lambda }/{\omega }_m=0.01$ (green) and $\mathrm{\Lambda }/{\omega }_m=0.02$ (blue), as indicated by the dashed lines in (a). Other parameters are the same as in Figs. 1 and 1.

Figure 5

Long-time dynamics of the mechanical coordinate $\tilde{q}$, which displays the regular periodic behavior for $\mathrm{\Lambda }/{\omega }_m=0.03$, and the period-doubling motion for $\mathrm{\Lambda }/{\omega }_m=0.05$ with the two-field detuning being $\mathrm{\Omega }/{\omega }_m=1.16$ and the cavity driving strength being $E/{\omega }_m=8.6\times {10}^4$. Other parameters are the same as in Figs. 1 and 1.

Figure 6

(a) Asymptotic extreme values of the mechanical coordinate ${\tilde{q}}_{em}$ in the classical dynamics, (b) Lyapunov exponent, and (c) $E_{N,\text{max}}$ versus parametric gain $\mathrm{\Lambda }/{\omega }_m$. Typical bifurcation diagram is found in ${\tilde{q}}_{em}$. The green dash line indicates the critical point ${\mathrm{\Lambda }}_c/{\omega }_m=0.043$ at which the transition from the regular periodic to period-doubling motion occurs. The Lyapunov exponent is taken from the slope of the time-evolutional curve of $\text{ln}\left({\varepsilon }_{I_c}\right)$ for the time interval $[280\tau , 300\tau$] by using linear fitting, with $\vec{\varepsilon }(t=0)=({10}^{-10},{10}^{-10},{10}^{-10},{10}^{-10})$. Other parameters are the same as in Fig. 5. The red dashed lines indicate the parametric regime discussed in Figs. 7, 8, 9.

Figure 7

Phase space trajectories of (a) the mechanical mode and (b) the cavity mode for time intervals $[30\tau , 40\tau$] (green dashed lines), $[290\tau ,300\tau$] (blue lines). (c) Time dependence of the cavity intensity $I_c$ for the time interval $[290\tau , 300\tau$] in comparison with that for the interval $[30\tau , 40\tau$] (the inset). (d) Time dependence of $\text{ln}\left({\varepsilon }_{I_c}\right)$ in [0,$300\tau$]. Parameters are $\mathrm{\Lambda }/{\omega }_m=0.03, \vec{\varepsilon }(t=0)=({10}^{-10},{10}^{-10},{10}^{-10},{10}^{-10})$. Other parameters are the same as in Fig. 5.

Figure 8

Phase space trajectories of (a) the mechanical mode and (b) the cavity mode for time intervals $[30\tau , 40\tau$] (green dashed lines), $[290\tau ,300\tau$] (blue lines). (c) Time dependence of the cavity intensity $I_c$ for the time interval $[290\tau , 300\tau$] in comparison with that for the interval $[30\tau , 40\tau$] (the inset). (d) Time dependence of $\text{ln}\left({\varepsilon }_{I_c}\right)$ in [0,$300\tau$]. Parameters are $\mathrm{\Lambda }/{\omega }_m=0.043, \vec{\varepsilon }(t=0)=({10}^{-10},{10}^{-10},{10}^{-10},{10}^{-10})$. Other parameters are the same as in Fig. 5.

Figure 9

Phase space trajectories of (a) the mechanical mode and (b) the cavity mode for time intervals $[30\tau , 40\tau$] (green dashed lines), $[290\tau ,300\tau$] (blue lines). (c) Time dependence of the cavity intensity $I_c$ for the time interval $[290\tau , 300\tau$] in comparison with that for the interval $[30\tau , 40\tau$] (the inset). (d) Time dependence of $\text{ln}\left({\varepsilon }_{I_c}\right)$ in [0,$300\tau$], respectively. Parameters are $\mathrm{\Lambda }/{\omega }_m=0.05, \vec{\varepsilon }(t=0)=({10}^{-10},{10}^{-10},{10}^{-10},{10}^{-10})$. Other parameters are the same as in Fig. 5.

Figure 10

Short-time dynamics [(a)–(c)] and long-time dynamics [(d)–(f)] of $E_P$ for $\mathrm{\Lambda }/{\omega }_m=0.03$ [(a), (d)], 0.043 [(b), (e)], and 0.05 [(c), (f)]. Other parameters are the same as in Fig. 6.

Figure 11

$E_{N,\text{max}}$ as a function of the thermal temperature $T$ for $\mathrm{\Lambda }/{\omega }_m=0$ (the black lower line), 0.04 (the green middle line), and 0.06 (the blue upper line). $\mu$ is the entanglement to temperature ratio and the inset shows the renormalized $E_{N,\text{max}}$. Other parameters are the same as in Fig. 6.