Radiation Pressure Cooling as a Quantum Dynamical Process

One of the most fundamental problems in optomechanical cooling is how small the thermal phonon number of a mechanical oscillator can be achieved under the radiation pressure of a proper cavity field. Different from previous theoretical predictions, which were based on an optomechanical system’s time-independent steady states, we treat such cooling as a dynamical process of driving the mechanical oscillator from its initial thermal state, due to its thermal equilibrium with the environment, to a stabilized quantum state of higher purity. We find that the stabilized thermal phonon number left in the end actually depends on how fast the cooling process could be. The cooling speed is decided by an effective optomechanical coupling intensity, which constitutes an essential parameter for cooling, in addition to the sideband resolution parameter that has been considered in other theoretical studies. The limiting thermal phonon number that any cooling process cannot surpass exhibits a discontinuous jump across a certain value of the parameter.

Figure 1

(a) A setup to perform optomechanical cooling. The quantum noises from the environment accompany the cavity and mechanical dissipations. (b) An ideal cooling process that transforms a thermal state (mixed) of the mechanical resonator to a coherent state (pure). The three-dimensional plots represent the corresponding Wigner functions.

Figure 2

Time evolutions of the thermal phonon number $n_m$ and the corresponding cavity photon number $⟨{\widehat{a}}^{†}\widehat{a}⟩$ for the OMSs with $g/\kappa ={10}^{-5}$, ${\gamma }_m/\kappa ={10}^{-3}$, and $\mathrm{\Delta }/{\omega }_m=1$. In (a) and (c) we set ${\omega }_m/\kappa =10$ and $n_{\text{th}}=100$ for the parameter $J=(g/{\omega }_m)(E/\kappa )$; in (b) and (d) we take $J=1.0$ and $n_{\text{th}}=1$. The insets in (a) and (b) describe the transitional regimes between cooling and heating, approximately $J=2.5$ and ${\omega }_m/\kappa =4$, respectively. The different amplitudes in (c) come from the different ratios $E/{\omega }_m$ for the values of $J$. For clarity only two plots to be stabilized are given in (d).

Figure 3

Stabilized thermal phonon numbers achievable for the OMSs with $g/\kappa ={10}^{-5}$ and ${\gamma }_m/\kappa ={10}^{-3}$. In (a) and (b) the initial thermal phonon number $n_{\text{th}}$ is 100, and in (c) and (d) the systems are assumed to be with $n_{\text{th}}=0$. The drive intensity is set at $E=8\times 10^5\kappa$ in (a) and (c). The detuning $\mathrm{\Delta }$ equals to ${\omega }_m$ in (b) and (d). In (d) the dashed curve is $n_{m,f}={\kappa }^2/(4{\omega }_m^2)$, corresponding to ${\kappa }^2/(16{\omega }_m^2)$ in [16, 17] due to a different expression for the cavity damping term. The dot-dashed curve obtained with $E/\kappa =10^6$ is the previous prediction for the strong-coupling regime [20].

Figure 4

System parameter relations that determine the optimum cooling for a general OMS under the BS coupling resonance $\mathrm{\Delta }={\omega }_m$ and with a fixed ratio ${\mathrm{\Gamma }}_m={\gamma }_m/\kappa ={10}^{-3}$. The dashed curve represents the cooling limit obtained with ${\omega }_m/\kappa \to \infty$. The ending dot for each curve is the point of the optimum cooling for a fixed ${\omega }_m/\kappa$. These curves asymptotically tend to the cooling limit (the dashed curve) with increased ${\omega }_m/\kappa$. The tendency of the lower $J$ is shown in the inset, with all curves stuck together as viewed with the range of the obtained result.