Optomechanical cooling of levitated spheres with doubly resonant fields

Optomechanical cooling of levitated dielectric particles represents a promising new approach in the quest to cool small mechanical resonators toward their quantum ground state. We investigate two-mode cooling of levitated nanospheres in a self-trapping regime. We identify a structure of overlapping, multiple cooling resonances and strong cooling even when one mode is blue-detuned. We show that the best regimes occur when both optical fields cooperatively cool and trap the nanosphere, where cooling rates are over an order of magnitude faster compared to corresponding single-resonance cooling rates.

Figure 1

Schematic setup: a levitated nanosphere is trapped and cooled cooperatively by two optical modes. The optical potential for each mode is shown.

Figure 2

Upper panels: comparison between numerical optical cooling rates (without linearization) and an analytical expression [Eq. (5)] from linearized dynamics, showing excellent agreement. $R=0.5$. The curves corresponding to different values of ${\delta }_1\pm 2$ MHz are shifted relative to each other. At single-resonance $r2$, field 2 is resonant with the oscillator and is exclusively responsible for cooling; field 1 is resonant with the cavity (${\Delta }_1^x=0$) and traps the sphere. Subsequently $r2\pm$ appear: they are cooling resonances of field 2, split by field 1; conversely, $r1\pm$ are cooling resonances of field 1, split by field 2. $r2-,r1-,r1+$ overlap giving a broad region of very strong cooling. The $a1\pm$ are heating (Stokes) resonances of field 1. The $a1-$ can coincide with the cooling resonance ($r2+$). Here field 2 absorbs phonons as fast as field 1 emits them. Lower panels: unshifted cooling curves at the single-field cooling resonance $r2$ (i) and double-resonant cooling (ii), showing that the latter gives over an order of magnitude stronger cooling. Asterisks are numerical results and the curves analytical results, corresponding to curves marked (i) and (ii), respectively, in the upper panels.

Figure 3

Dimensionless cooling rates $\Gamma$ as a function of scaled (dimensionless) detunings for ${\epsilon }^2=24$, $\frac{\kappa }{2A}=1$, and $R=1/2$. The inner contour corresponds to $\Gamma =1/2$ (where $\Gamma =1$ would correspond to a maximal cooling rate $=\kappa /2$). The outer contour indicates $\Gamma =1/80$, with intermediate contours at $\Gamma =1/4$, 1/8, 1/20, and 1/40, respectively. The black contour denotes $\Gamma =0$ and indicates the boundary between cooling and heating.

Figure 4

Comparison between single-resonant (SR) cooling due to $r2$ and double resonant (DR) cooling, i.e., simultaneous cooling by $r1-$ and $r2-$, as a function of laser driving power. Double resonance yields two orders of magnitude greater cooling for strong driving. The inset shows a cooling maximum of $r2$ at weak driving ${\epsilon }^2\simeq 1/2$; here the system is on the edge of the sideband-resolved regime ${\omega }_M={\kappa }_A$. The upper axes give input laser powers corresponding to $A=3\times {10}^5$ Hz$=\kappa /2$, $m=9\times {10}^{-18}$ Kg, and $R=0.5$.