Ground-state cooling of mechanical motion in the unresolved sideband regime by use of optomechanically induced transparency

We present a scheme for cooling mechanical motion to the ground state in an optomechanical system. Unlike standard sideband cooling, this scheme applies to the so-called unresolved sideband regime, where the resonance frequency of the mechanical mode is much smaller than the cavity linewidth. Ground-state cooling becomes possible when assuming the presence of an additional, auxiliary mechanical mode and exploiting the effect of optomechanically induced transparency. We first consider a system where one optical cavity interacts with two mechanical modes and show that ground-state cooling of the unresolved mechanical mode is possible when the auxiliary mode is in the resolved sideband regime. We then present a modified setup involving two cavity modes, where both mechanical modes are allowed to be in the unresolved sideband regime.

Figure 1

The photon number fluctuation spectrum (9) for both the (a) resolved and (b) unresolved sideband regime. Ground-state cooling is only possible in the resolved sideband regime ${\omega }_1>\kappa$, where $S\left[{\omega }_1\right]\gg S[-{\omega }_1]$.

Figure 2

Setup for the proposed cooling scheme. There are two optical drives, one at frequency ${\omega }_{d,0}$ (beam 0) and one at ${\omega }_{d,1}$ (beam 1). The off-resonant beam 0 and mechanical mode 2 will lead to squeezing of the photon number fluctuation spectrum, such that Stokes scattering from beam 1 is suppressed. This allows for ground-state cooling of mechanical mode 1 even though it is in the unresolved sideband regime.

Figure 3

The photon number fluctuation spectrum $S_{nn}\left[\omega \right]$ for the setup discussed here. The parameters are ${\omega }_1/\kappa =0.05$, ${\omega }_2/\kappa =5$, ${\omega }_2/{\gamma }_2={10}^5$, $C_{2,0}=100$, and $n_{\mathrm{th},2}=10$. The presence of an auxiliary mechanical mode 2 and the additional drive (beam 0) leads to a dip in the spectrum at $\omega =-{\omega }_1$. This gives a large asymmetry between $S_{nn}\left[{\omega }_1\right]$ and $S_{nn}[-{\omega }_1]$ and ground-state cooling of mechanical mode 1 is thus possible. The dashed curves are the approximate expressions (46) and (47), and the solid line is the expression in the second line of Eq. (32). We see that the approximations are accurate for $\omega \sim -{\omega }_1$ [Eq. (46)] and for $\omega \sim {\omega }_1$ [Eq. (47)].

Figure 4

The average phonon number $n_1$ of the low-frequency mechanical oscillator as a function of the coupling rate $|G_{1,1}|$. The solid line is the exact result in Eq. (67) and the dashed curve is the result in Eq. (68) from the Fermi golden rule approach. We have checked that Eq. (67) agrees very well with Eq. (66). The thermal phonon number is $n_{\mathrm{th},1}=500$, and the intrinsic mechanical $Q$ is ${\omega }_1/{\gamma }_1={10}^5$. The other parameters are the same as in Fig. 3. We observe that the two curves agree well for $|G_{1,1}|\ll |G_{2,0}|$, but for larger $|G_{1,1}|$ the exact result reaches a minimum given by Eqs. (69) and (70).

Figure 5

Setup for the modified cooling scheme. We now have two optical cavity modes, 0 and 1, with resonance frequencies ${\omega }_{c,0}$ and ${\omega }_{c,1}$. Since mechanical mode 2 couples the two cavity modes, the drive in cavity mode 0 at frequency ${\omega }_{d,0}$ will squeeze the photon number fluctuation spectrum in cavity mode 1. This enables the drive in cavity mode 1 at ${\omega }_{d,1}$ to cool mechanical mode 1 to the quantum ground state.

Figure 6

The photon number fluctuation spectrum $S_{nn}\left[\omega \right]$ for the modified setup. The parameters are ${\omega }_1/\kappa =0.01$, ${\omega }_2/\kappa =0.02$, ${\omega }_2/{\gamma }_2={10}^5$, ${\Delta }_{0,1}/\kappa =-2$, $C_{2,0}=2\times {10}^4$, and $n_{\mathrm{th},2}=100$. The optomechanical interaction with mechanical mode 2 leads to a dip at $\omega =-{\omega }_1$, enabling ground-state cooling of mechanical mode 1. The dashed curves are the approximate expressions (84) and (89), and the solid line is the expression in Eq. (80).

Figure 7

The average phonon number $n_1$ of the low-frequency mechanical oscillator as a function of the coupling rate $|G_{1,1}|$. The exact result in Eq. (66) (solid line) and the result from the Fermi golden rule approach (dashed line) agree very well, and the deviation between them is much smaller than 1 for all $|G_{1,1}|$. The parameters are $n_{\mathrm{th},1}=100$, $n_{\mathrm{th},2}=50$, ${\omega }_1/{\gamma }_1={\omega }_2/{\gamma }_2={10}^5$, ${\omega }_1/\kappa =0.01$, ${\omega }_2/\kappa =0.02$, ${\Delta }_{0,1}/\kappa =-2$, and $C_{2,0}=20000$. We note that ground-state cooling is predicted also from the exact result, even for $|G_{1,1}|\sim |G_{2,0}|$.