Breaking the optomechanical cooling limit by two drive fields on a membrane-in-the-middle system

We present a theoretical scheme for ground-state cooling of a mechanical resonator in a membrane-in-the-middle optomechanical system (OMS) driven by two red-detuned drive fields. The details of dynamical evolution of OMS are provided, and the effect of system conditions on cooling results are systematically studied. Most importantly, the setups with two drives are found to be capable of achieving better cooling results than the theoretical cooling limit with single cavity. Even an improvement by one order of thermal phonon number is possible with proper combination of the cavity damping rate and drive intensity.

Figure 1

Model of optomechanical cooling with the membrane-in-the-middle system. The two continuous-wave drive fields with red detuning ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\omega }_m$ are used to cool the same mechanical resonator (the membrane inside the cavity). A more efficient cooling effect than that with only one cavity can be obtained by choosing proper conditions.

Figure 2

Comparisons of the quadratures of two cavities $X_{c_1}\left(t\right)$, and $X_{c_2}\left(t\right)$, the displacement of mechanical resonator $X_m\left(t\right)$ predicted with the nonlinear dynamical equations (5) (red lines) and our linearized equations (9) (blue lines). The parameters are ${\kappa }_2/{\kappa }_1=5, g_m/{\kappa }_1={10}^{-5}, {\omega }_m/{\kappa }_1=50, {\gamma }_m/{\kappa }_1={10}^{-3}, {\mathrm{\Delta }}_1/{\kappa }_1=45, {\mathrm{\Delta }}_2/{\kappa }_1=55, E_1/{\kappa }_1=4.5\times {10}^6, E_2/{\kappa }_1=5.5\times {10}^6$, and $J/{\kappa }_1=1$.

Figure 3

Evolution of thermal phonon number in a MIM system under different drive intensities $E/{\kappa }_1$. The evolution with the corresponding single-cavity setup (denoted by red dashed line) is presented for comparison. In (a) the final phonon number with the same cavity damping rate as in the MIM system, i.e., ${\kappa }_1={\kappa }_2$ (blue dash-dotted line), is lower than the one achieved with the single-cavity setup, while in (b) it is higher than the one with single cavity. With the proper choice of the second cavity damping rate, e.g., ${\kappa }_2/{\kappa }_1=1.5$ in (a) and ${\kappa }_2/{\kappa }_1=4$ in (b) (black solid line), the final phonon number can be always lower than the one of the single-cavity system. The other parameters are chosen as $g_m/{\kappa }_1={10}^{-5}, {\mathrm{\Delta }}_1/{\kappa }_1={\mathrm{\Delta }}_2/{\kappa }_1={\omega }_m/{\kappa }_1=100, {\gamma }_m/{\kappa }_1={10}^{-3}$, and $n_{th}=100$.

Figure 4

Relations between the cooling ratio $n_{m,f}/{\mathrm{\Gamma }}_m{n}_{th}$ and the second cavity damping rate ${\kappa }_2/{\kappa }_1$ or the tunneling strength $J/{\kappa }_1$. In each figure, three cases with different mechanical frequencies ${\omega }_m/{\kappa }_1=50$, 100, and 200 are displayed. From (a) and (b), we find that for the fixed effective drive intensity $(g_m/{\omega }_m)\times (E/{\kappa }_1)$, there is an optimum damping rate, ${\kappa }_2/{\kappa }_1$, for which the cooling ratio is the smallest. From (c) and (d), we conclude that tunneling will not help improve the cooling effect. The other parameters are chosen as $g_m/{\kappa }_1={10}^{-5}, {\mathrm{\Delta }}_1/{\kappa }_1={\mathrm{\Delta }}_2/{\kappa }_1={\omega }_m/\kappa , {\gamma }_m/{\kappa }_1={10}^{-3}$, and $n_{th}=100$.

Figure 5

Relation between the cooling ratio $n_{m,f}/\left({\mathrm{\Gamma }}_m{n}_{th}\right)$ and the effective drive intensity $(g_m/{\omega }_m)\times (E/{\kappa }_1)$ with different mechanical frequency ${\omega }_m/{\kappa }_1=20$, 100, 500. The dash-dotted lines represent the relations in the case of single cavity, which are used for comparison. The pink dashed line is the cooling limit of the single cavity with ${\omega }_m/{\kappa }_1\to \infty$. It is obvious that the cooling effect can be efficiently improved by a coupled cavity system. In particular, the improvement is significant with large mechanical resonator such as the one with ${\omega }_m/{\kappa }_1=100$ or 500, and the cooling ratio with it can be even lower than the cooling limit of the single cavity. The system parameters are set as $g_m/{\kappa }_1={10}^{-5}, {\mathrm{\Delta }}_1/{\kappa }_1={\mathrm{\Delta }}_2/{\kappa }_1={\omega }_m/\kappa , {\gamma }_m/{\kappa }_1={10}^{-3}$, and ${\kappa }_2/{\kappa }_1=4$.

Figure 6

Cooling ratio $n_{m,f}/\left({\mathrm{\Gamma }}_m{n}_{th}\right)$ under different conditions. (a) ${\kappa }_1={\kappa }_2=\kappa$. The circle boundary lines indicate that the cooling ratio mainly depends on the quadratic sum of the two effective drive intensities $J_{E_{1\left(2\right)}}=(g_m/{\omega }_m)\times (E_{1\left(2\right)}/\kappa )$. The cooling ratio approaches the limit 1, which is the same as that with single cavity. (b) ${\kappa }_2\ne {\kappa }_1, E_1=E_2=E$. The cooling limit can be very small with the high effective drive intensity associated with optimum cavity damping rate ${\kappa }_2/{\kappa }_1$. (c) ${\kappa }_2\ne {\kappa }_1, E_1\ne E_2$. With the appreciated choice of the second cavity damping rate ${\kappa }_2/{\kappa }_1$, the cooling rate can be significantly reduced by increasing the second drive intensity. However, increasing the first drive intensity will not help improve the cooling effect at all; in contrast, it may worsen the scenario. The parameters, which are not indicated in the figures, are chosen as $g_m/{\kappa }_1={10}^{-5}, {\mathrm{\Delta }}_1/{\kappa }_1={\mathrm{\Delta }}_2/{\kappa }_1={\omega }_m/{\kappa }_1$, and ${\gamma }_m/{\kappa }_1={10}^{-3}$.