Resonances in dissipative optomechanics with nanoparticles: Sorting, speed rectification, and transverse cooling

The interaction between dielectric particles and a laser-driven optical cavity gives rise to both conservative and dissipative dynamics, which can be used to levitate, trap, and cool nanoparticles. We analytically and numerically study a two-mode setup in which the optical potentials along the cavity axis cancel, so that the resulting dynamics is almost purely dissipative. For appropriate detunings of the laser drives, this dissipative optomechanical dynamics can be used to sort particles according to their size, to rectify their velocities, and to enhance transverse cooling.

Figure 1

(a) A subwavelength nanoparticle of mass $m$ is confined to the axis ($z$) of an optical cavity but can move freely along it. It is optomechanically coupled to two laser-driven cavity modes. The drive strengths are denoted by ${\Omega }_{1,2}$ and the detunings from the cavity frequencies are given by ${\Delta }_{1,2}$. The decay of the cavity is $\kappa$. (b) Effective damping force for one drive detuned to the red (see main text for the parameters used). For small velocities, the effective force exerted on the nanoparticle depends linearly on the velocity (dashed line) while resonances appear for larger velocities. (c) By detuning one drive to the red and the other to the blue with an additional relative detuning ${\Delta }_1=-{\Delta }_2+\delta$, the competing damping (dotted light) and heating (dotted dark) give rise to an effective dissipative force (black) with finite stationary velocities. Remarkably, since the force has a negative slope around these points, the stationary points are stable and all particles end up with the same final speed (determined by the detunings).

Figure 2

(a) Velocity as a function of time for various nanosphere sizes ranging from $r=46\mathrm{nm}$ (dark) to $r=54\mathrm{nm}$ (light). (b) Position as a function of time for the same nanospheres. All particles are eventually trapped and the distance traveled $z_{\mathrm{f}}$ is inversely proportional to the particle mass $z_{\mathrm{f}}\propto 1/r^3$ (inset). (c) Magnification of the point where a particle is trapped in the small reminiscent optical potential. The result from the effective force Eq. (7) (dotted), the analytic result from Eq. (6) (light) and the full numerical data (dark) are in excellent agreement up to the point where the particle is trapped. (d) The amplitude of the cooling mode $|a_1|/|\alpha |$ as function of time for $r=50\mathrm{nm}$. The regime of strongly enhanced cooling in (a) corresponds to a sharp resonance in the optical response.

Figure 3

Velocity as a function of time for particles of size $r=50\mathrm{nm}$ for various initial velocities ranging from $v_0=0.02\mathrm{m}/\mathrm{s}$ to $v_0=0.16\mathrm{m}/\mathrm{s}$ in a two-mode setup with one damping and one amplifying mode. Particles with velocities below $v=0.1\mathrm{m}/\mathrm{s}$ are accelerated while particles with larger velocities are decelerated to reach the same stationary velocity $v=0.1\mathrm{cm}/\mathrm{s}$.

Figure 4

Laser cooling of a $50-\mathrm{nm}$ nanoparticle and $\dot{z}\left(0\right)=0.25\mathrm{m}/\mathrm{s}$ with (light) and without (black) a compensating potential. Without the compensating mode, the particle oscillates rapidly and is quickly trapped so that this type of laser cooling becomes much less efficient. Even with a compensating potential, the particle is eventually trapped, but the range of strong exponential damping is much larger. For parameters, see main text.