Cooling in the single-photon strong-coupling regime of cavity optomechanics

In this Rapid Communication we discuss how red-sideband cooling is modified in the single-photon strong-coupling regime of cavity optomechanics where the radiation pressure of a single photon displaces the mechanical oscillator by more than its zero-point uncertainty. Using Fermi's golden rule we calculate the transition rates induced by the optical drive without linearizing the optomechanical interaction. In the resolved-sideband limit we find multiple-phonon cooling resonances for strong single-photon coupling that lead to nonthermal steady states including the possibility of phonon antibunching. Our study generalizes the standard linear cooling theory.

Figure 1

Multiple cooling resonances. Steady-state phonon number $\overline{n}=\langle {\widehat{b}}^{†}\widehat{b}\rangle$ (a) for $g/{\omega }_M=0.5$ (red solid), $g/{\omega }_M=0.25$ (green dashed) and $g/{\omega }_M=0.1$ (blue dash-dotted) as well as ${\omega }_M/\kappa =4$ and (b) for ${\omega }_M/\kappa =0.5$ (red solid), ${\omega }_M/\kappa =2$ (green dashed) and ${\omega }_M/\kappa =4$ (blue dash-dotted) as well as $g/{\omega }_M=0.5$. Results from the set of rate equations (4) are shown as lines and those from the quantum master equation (8) as dots. The other parameters are ${\omega }_M/\gamma =1000$, $n_{\mathrm{th}}=1$, and $\Omega /\kappa =0.2$.

Figure 2

Validity of the rate equation approach and phonon antibunching. (a) Mean photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ (blue crosses) and mean phonon number $\overline{n}=\langle {\widehat{b}}^{†}\widehat{b}\rangle$ (red solid) as well as (b) phonon number fluctuations $F=\langle {\widehat{b}}^{†}{\widehat{b}}^{†}\widehat{b}\widehat{b}\rangle /{\langle {\widehat{b}}^{†}\widehat{b}\rangle }^2$ as a function of the drive strength $\Omega$ for a detuning $\tilde{\Delta }/{\omega }_M=-2$. Results from the set of rate equations (4) are shown as lines and those from the quantum master equation (8) are shown as dots and crosses. The other parameters are ${\omega }_M/\kappa =4$, $g/{\omega }_M=0.5$, and ${\omega }_M/\gamma =1000$.

Figure 3

Nonthermal steady states. (Left) Phonon number $\overline{n}=\langle {\widehat{b}}^{†}\widehat{b}\rangle$ (top) and phonon number fluctuations $F=\langle {\widehat{b}}^{†}{\widehat{b}}^{†}\widehat{b}\widehat{b}\rangle /{\langle {\widehat{b}}^{†}\widehat{b}\rangle }^2$ (bottom) as a function of detuning $\Delta$ for $n_{\mathrm{th}}=1$ (solid red) and $n_{\mathrm{th}}=10$ (dashed blue). (Right) Phonon number $\overline{n}$ (top) and phonon number fluctuations $F$ (bottom) as a function of the thermal phonon number $n_{\mathrm{th}}$ for $\tilde{\Delta }=-{\omega }_M$ (solid red) and $\tilde{\Delta }=-2{\omega }_M$ (dashed blue). The other parameters are ${\omega }_M/\kappa =4$, $g/{\omega }_M=0.5$, $\Omega /\kappa =0.2$, and ${\omega }_M/\gamma =1000$. Thin black lines show results of the linear model (11) and (12) for $\Omega \lambda /\kappa =0.2$.