Master-equation approach to optomechanics with arbitrary dielectrics

We present a master equation describing the interaction of light with dielectric objects of arbitrary sizes and shapes. The quantum motion of the object, the quantum nature of light, as well as scattering processes to all orders in perturbation theory are taken into account. This formalism extends the standard master-equation approach to the case where interactions among different modes of the environment are considered. It yields a genuine quantum description, including a renormalization of the couplings and decoherence terms. We apply this approach to analyze cavity cooling of the center-of-mass mode of large spheres. Furthermore, we derive an expression for the steady-state phonon numbers without relying on resolved-sideband or bad-cavity approximations.

Figure 1

Graphical illustration of the transition matrix ${\mathcal{T}}_{\mathbf{k},{\mathbf{k}}_0}\left(\widehat{r}\right)$ as an infinite series in the light-matter interaction. The first term denotes the direct interaction between the dielectric and light, the second one a process where a photon is virtually absorbed and reemitted, the third one a process where two intermediate photons are involved, etc.

Figure 2

Minimal phonon numbers for different detunings comparing the exact result (blue solid line) to the one with the cavity mode eliminated (red dashed line). Differences between the two solutions evolve as $\tilde{g}$ is increased. Only the regions where a steady state is attainable have been plotted; they all lie within the red-sideband regime. For blue detuning or an optomechanical coupling which is too strong, the system is heated and no steady state can be obtained. Upper-left panel shows the sideband-resolved regime with weak coupling: $\kappa =0.3{\tilde{\omega }}_{\mathrm{t}},\Gamma =0.03{\tilde{\omega }}_{\mathrm{t}},\tilde{g}=0.07{\tilde{\omega }}_{\mathrm{t}}$, Upper-right panel shows $\kappa =0.3{\tilde{\omega }}_{\mathrm{t}},\Gamma =0.03{\tilde{\omega }}_{\mathrm{t}},\tilde{g}=0.3{\tilde{\omega }}_{\mathrm{t}}$. Lower-left panel shows the bad-cavity limit for weak coupling: $\kappa =3{\tilde{\omega }}_{\mathrm{t}},\Gamma =0,\tilde{g}=0.1{\tilde{\omega }}_{\mathrm{t}}$. Lower-right panel shows the bad-cavity limit for strong coupling: $\kappa =3{\tilde{\omega }}_{\mathrm{t}},\Gamma =0,\tilde{g}=0.864{\tilde{\omega }}_{\mathrm{t}}$.

Figure 3

Schematic of setup. A dielectric sphere of a radius similar or larger than the cavity wavelength is trapped by optical tweezers providing a trapping frequency ${\tilde{\omega }}_t$. It is placed inside an optical cavity, where a second laser is used to optically manipulate the dielectric's center-of-mass degree of freedom.

Figure 4

Cavity-optomechanical parameters for different sizes of the object. Upper-left panel shows the optomechanical coupling $\tilde{g}$, upper-right panel shows the recoil heating of the center of mass $\Gamma$, lower-left panel shows the cavity decay rate $\kappa$, and lower-right panel shows the cooperativity $\mathcal{C}$.

Figure 5

Minimal phonon number attainable for different sphere sizes for experimental parameters given in main text.