Cavity Cooling of a Levitated Nanosphere by Coherent Scattering

We report three-dimensional (3D) cooling of a levitated nanoparticle inside an optical cavity. The cooling mechanism is provided by cavity-enhanced coherent scattering off an optical tweezer. The observed 3D dynamics and cooling rates are as theoretically expected from the presence of both linear and quadratic terms in the interaction between the particle motion and the cavity field. By achieving nanometer-level control over the particle location we optimize the position-dependent coupling and demonstrate axial cooling by two orders of magnitude at background pressures of $6\times {10}^{-2}\mathrm{mbar}$. We also estimate a significant ($>40\mathrm{dB}$) suppression of laser phase noise heating, which is a specific feature of the coherent scattering scheme. The observed performance implies that quantum ground state cavity cooling of levitated nanoparticles can be achieved for background pressures below $1\times {10}^{-7}\mathrm{mbar}$.

Figure 1

Different paradigms for cavity cooling of a levitated nanosphere. (a) Cavity cooling by coherent scattering from an optical tweezer is based on dipole radiation being emitted into an empty cavity, giving the best performance for a particle placed at the intensity minimum of the cavity mode. (b) In standard dispersive optomechanics, an external laser drives both the cavity and the scattering. Optimal cooling is at the largest intensity gradient of the cavity mode.

Figure 2

Setup for cooling by coherent scattering. An optical tweezer is formed by a laser at frequency ${\omega }_{\mathrm{tw}}$ that is tightly focussed by a microscope objective (MO) inside a vacuum chamber (vac). It levitates a nanoparticle at the center of a high-finesse Fabry-Pérot cavity. Its linear polarization is set by a half wave plate ($\lambda /2$). A weak locking beam is derived from the tweezer laser and drives the cavity resonantly at frequency ${\omega }_2$, allowing ${\omega }_{\mathrm{tw}}$ and ${\omega }_2$ to be stably locked relative to the cavity frequency. Four independent detection schemes (I)–(IV) monitor the particle motion and the cavity field (see main text for details; PBS: polarizing beam splitter; ${\omega }_{\mathrm{het}}$: heterodyne demodulation frequency). Inset: The particle is trapped at a position $x_0$ relative to a cavity antinode. Maximal cavity cooling of the $x$ motion by coherent scattering occurs for $x_0=\lambda /4$, i.e., at a cavity node.

Figure 3

Polarization dependent cavity cooling. Shown are the noise power spectra (NPS) measured with direct detection (I) for a particle located at $x_0=\lambda /8$, to couple all three directions of motion, and for three different tweezer polarizations as illustrated on the right panel. The red arrow indicates the polarization. The sketch also indicates the transverse optical tweezer potential (grey ellipse) and the dipole emission (red ellipses). NPS in each panel have been obtained along the tweezer axis ($z$, blue) and in its transverse directions ($x$, red; $y$, green). Cooling measurements are performed with a tweezer detuning close to the mechanical frequency ($\mathrm{\Delta }=2\pi \times 300\mathrm{kHz}$, bright color). Measurements at large detuning ($\mathrm{\Delta }=2\pi \times 4\mathrm{MHz}$, dark color) serve as reference for no cooling (see main text). (a) At $\theta =0$ no cooling is observed, because polarization along the cavity axis suppresses scattering into the cavity. (b) At $\theta =\pi /4$ full 3D cavity cooling by coherent scattering is observed, since the cavity axis does not coincide with a principal axis of the optical tweezer. Cooling both broadens the spectra and reduces the overall area, while the mechanical frequency is shifted due to an optical spring. (c) For $\theta =\pi /2$ scattering into the cavity is maximal, as is the cooling along the cavity axis ($x$) and the tweezer axis ($z$).

Figure 4

Position dependent cavity cooling. Shown are relative coherent scattering powers $P/P_0$ (top), mechanical damping rates ${\gamma }_{\mathrm{eff}}$ (middle) and inverse cooling factors $T_{\mathrm{eff}}/T_0$ (bottom) for different particle positions $x_0$ along the cavity axis and at background pressures of $p=4\mathrm{mbar}$ (left) and 0.06 mbar (right). Top panel (a),(d): Coherent scattering into the cavity mode. The black line is a fit to the data following the cavity standing wave. The scattering is minimal (maximal) for a particle placed at the node $x_0=\lambda /4$ (antinode $x_0=0$) of the cavity field. Middle panel (b),(e): The damping ${\gamma }_{\mathrm{eff}}$ of the nanoparticle motion is obtained via the width (FWHM) of the NPS for the $x$ axis (red) and $z$ axis (blue). Bright colors indicate measurements with cavity cooling ($\mathrm{\Delta }/2\pi =400\mathrm{kHz}$), dark colors without cooling ($\mathrm{\Delta }/2\pi =4\mathrm{MHz}$). The grey line shows the theoretically predicted gas damping ${\gamma }_{\mathrm{gas}}$, which agrees with the damping observed in the absence of cooling. As expected, maximal damping along the $x$ ($z$) direction is obtained for minimal (maximal) coherent scattering powers at $x_0=\lambda /4$ ($x_0=0$), as predicted by our theoretical model (solid line; see main text). Bottom panel (c),(f): The effective mode temperatures $T_{\mathrm{eff}}$ are obtained by NPS integration (see main text). As expected for both directions, maximum damping implies maximal cooling. Purely linear coupling would result in a maximum temperature of $T_{\mathrm{eff}}/T_0=1$ (grey line). A theoretical model that also includes quadratic coupling matches the data very well without free parameters (dashed lines).