Optomechanical laser cooling with mechanical modulations

We theoretically study the laser cooling of optomechanical cavities when the mechanical resonance frequency and damping depend on time. In the regime of weak optomechanical coupling we extend the theory of laser cooling using an adiabatic approximation. We discuss the modifications of the cooling dynamics and compare it with numerical simulations in a wide range of modulation frequencies.

Figure 1

An optomechanical setup, consisting of an optical cavity coupled to a harmonically supported mechanical element by radiation forces, experiences modulations of the mechanical properties, i.e., periodic changes of the mechanical frequency $\nu \left(t\right)$ and damping rate $\gamma \left(t\right)$. The cavity is pumped by an external laser at rate $\Omega$, while photons leak out of the cavity at rate $2\kappa$. For certain parameters, the light scattering leads to cooling of the mechanical motion, which is influenced by the mechanical modulation.

Figure 2

The rates $A_{\pm }$ for a sinusoidal modulation. The blue (dark gray) and red (light gray) lines represent the time averages $A_{-}$ and $A_{+}$, Eq. (34), respectively. The black curves show the same rates before averaging at different times $t=0,T/4,T/2$, and $3T/4$ of a modulation period. (a) The fast modulation $\omega =\nu /2$ generates rapidly oscillating sidebands displaced by multiples of $\omega$ from the main resonance at $\Delta =\pm \nu$. The sidebands have a nonvanishing temporal mean as shown by the colored curves. (b) Slow modulations with $\omega =\nu /100$ essentially move the resonance peaks along the momentary oscillator frequency. Other parameters are $\widehat{\nu }=0.2\nu ,\kappa =0.05\nu$.

Figure 3

Comparison between the analytic approximation [Eq. (40), solid line] and numerical results (points) for sinusoidal modulation. In the inset, the numerically calculated time curve $\langle m\rangle \left(t\right)$ is shown for selected values of $\Delta$. Parameters are $\widehat{\nu }=0.1\nu ,\kappa =0.05\nu ,\omega =\nu /2,\gamma =0,\overline{m}=0$.

Figure 4

Mean phonon number $\langle m\rangle$, Eq. (40), for a pulse $f\left(t\right)$, shown in inset (a), comprising several harmonics. It is compared to the unmodulated case (dashed curve). Inset (b) displays the mean vibrational occupation on a logarithmic scale, showing that the modulation increases the minimal achievable temperature. The parameters are $\widehat{\nu }=0.5\nu ,\kappa =0.07\nu ,\gamma =0,\overline{m}=0,\omega =\nu /6$.

Figure 5

(a) Mean vibrational occupation number as a function of the modulation frequency $\omega$ and damping $\gamma$. For each point, the detuning was optimized in order to find the minimal occupation number. Missing lines between the points mark regions where the system is instable. (b) Same as in (a), but showing the cooling rate ${\Gamma }_{\mathrm{cool}}$. Parameters are $\widehat{\nu }=0.2\nu ,\widehat{\gamma }=0.2{\gamma }_0,\kappa =0.1\nu$, and $\eta =0.25$ at resonance.