Cooling mechanical resonators to the quantum ground state from room temperature

Ground-state cooling of mesoscopic mechanical resonators is a fundamental requirement for testing of quantum theory and for implementation of quantum information. We analyze the cavity optomechanical cooling limits in the intermediate coupling regime, where the light-enhanced optomechanical coupling strength is comparable with the cavity decay rate. It is found that in this regime the cooling breaks through the limits in both the strong-coupling and the weak-coupling regimes. The lowest cooling limit is derived analytically under the optimal conditions of cavity decay rate and coupling strength. In essence, cooling to the quantum ground state requires $Q_{\mathrm{m}}>2.4n_{\mathrm{th}}$, with $Q_{\mathrm{m}}$ being the mechanical quality factor and $n_{\mathrm{th}}$ being the thermal phonon number. Remarkably, ground-state cooling is achievable starting from room temperature, when the mechanical $Q$-frequency product $Q_{\mathrm{m}}{\nu }_{\mathrm{m}}>1.5\times {10}^{13}$ Hz and both the cavity decay rate and the coupling strength exceed the thermal decoherence rate. Our study provides a general framework for optimizing the backaction cooling of mesoscopic mechanical resonators.

Figure 1

Time evolution of the mean phonon number ${\overline{N}}_b\left(t\right)$ for $G/{\omega }_{\mathrm{m}}=0.1$ (green dotted curve), $0.2$ (red solid curve), and $0.3$ (blue dashed curve). Other parameters are ${\Delta }^{\prime }=-{\omega }_{\mathrm{m}},\kappa /{\omega }_{\mathrm{m}}=0.5,{\gamma /{\omega }_{\mathrm{m}}=10}^{-5}$, and $n_{\mathrm{th}}={10}^3$. Inset: Sketch of the optomechanical system.

Figure 2

(a) Steady-state classical cooling limit $n_{\mathrm{s}}^{\left(1\right)}$, (b) quantum cooling limit $n_{\mathrm{s}}^{\left(0\right)}$, and (c) total cooling limit $n_{\mathrm{s}}$ as functions of the cavity decay rate $\kappa$ and the coupling strength $G$. The red dashed curves correspond to the boundary of the stability condition given by Eq. (1b). In panel (c), the vertical and horizontal lines denote the optimal $\kappa$ and $G$ given by Eqs. (5a) and (5b). Other parameters are ${\Delta }^{\prime }=-{\omega }_{\mathrm{m}},{\gamma /{\omega }_{\mathrm{m}}=10}^{-5}$, and $n_{\mathrm{th}}={10}^3$.

Figure 3

Cooling limits $n_{\mathrm{s}}$ (red circles and red solid curves), $n_{\mathrm{s}}^{\left(1\right)}$ (blue triangles and blue dashed curves), and $n_{\mathrm{s}}^{\left(0\right)}$ (green stars and green dotted curves) as functions of $\kappa$ for $G=G_{\mathrm{opt}}$ (a) and as functions of $G$ for $\kappa ={\kappa }_{\mathrm{opt}}$ (b). The circles, triangles, and stars are the numerical results, and the curves are the analytical results given by Eqs. (3a) and (3b). Other parameters are ${\Delta }^{\prime }=-{\omega }_{\mathrm{m}},{\gamma /{\omega }_{\mathrm{m}}=10}^{-5}$, and $n_{\mathrm{th}}={10}^3$. (c) $n_{\mathrm{opt}}$ as a function of the environmental temperature $T$ for $Q_{\mathrm{m}}{\nu =10}^{12}$ (red solid curve), $3\times {10}^{12}$ (blue dashed curve), ${10}^{13}$ (green dotted curve), $3\times {10}^{13}$ (purple dash-dotted curve), and ${10}^{14}$ (orange dash-dot-dotted curve).

Figure 4

(a) Contour plot of the ground-state region $\left(n_{\mathrm{s}}<1\right)$ as functions of $\kappa$ and $n_{\mathrm{th}}$. The shaded region corresponds to $n_{\mathrm{s}}<1$. The red dashed lines denote $\kappa =\gamma n_{\mathrm{th}}$ (left) and $\kappa =4{\omega }_{\mathrm{m}}$ (right), respectively. (b) Cooling limit $n_{\mathrm{s}}$ as a function of $\kappa$ for $n_{\mathrm{th}}={10}^3$. The red circles are the numerical results and the red solid curve corresponds to the analytical results given by Eqs. (3a) and (3b). Other parameters are ${\Delta }^{\prime }=-{\omega }_{\mathrm{m}},{\gamma /{\omega }_{\mathrm{m}}=10}^{-5}$, and ${G=G}_{\mathrm{opt}}\left(\kappa \right)$ [given by Eq. (9)].

Figure 5

Same as Fig. 4 except that the horizontal axes are $G$ and the value of the cavity decay rate is ${\kappa =\kappa }_{\mathrm{opt}}\left(G\right)$ [given by Eq. (12)]. In panel (a), the red dashed lines denote $G=\gamma n_{\mathrm{th}}$ (left) and $G=0.5{\omega }_{\mathrm{m}}$ (right), respectively.