Entangling levitated nanoparticles by coherent scattering

We show how entanglement between two optically levitated nanoparticles can be generated and detected by coherent scattering of tweezer photons into a single cavity mode. Triggered by the detection of a Stokes photon, the tweezer detuning is switched from the blue to the red; entanglement is then verified by the conditioned anti-Stokes photon flux, which oscillates with the mechanical beat frequency. The proposed setup is realizable with near-future technology and opens the door to experimental observation of nonclassical center-of-mass correlations between two or more levitated nanoscale objects.

Figure 1

Two nanoparticles, trapped by optical tweezers of powers $P_1, P_2$ are weakly coupled to an optical cavity of line width $\kappa$. Their harmonic motion, described in transverse direction by the ladder operators $a_{1,2}$ and frequencies ${\omega }_{1,2}$, is initially cooled to the ground state. Setting the tweezer detuning to $\mathrm{\Delta }=({\omega }_1+{\omega }_2)/2$ (blue sideband) a Stokes photon, coherently scattered into the intracavity field $b$, may then be detected in the output mode (photon detector 1). This effectively maps the two particles onto an entangled state. Triggered by the Stokes photon detection, the detuning is reversed to $\mathrm{\Delta }=-({\omega }_1+{\omega }_2)/2$ and the appearance of an anti-Stokes photon is measured as a function of time. Entanglement is verified by observing that this flux $I\left(t\right)$, oscillating with the mechanical beat frequency, exceeds a time-dependent threshold determined by the Rayleigh scattering rates ${\gamma }_j^{\mathrm{sc}}$. The beam splitter includes a filter directing off-resonant photons to detector 2, thus removing them from the feedback loop; possible occurrences of a second Stokes photon are thus discarded.

Figure 2

Flux of anti-Stokes scattered photons (black solid line) conditioned on the detection of a Stokes scattered photon for two Si spheres ($\varrho =2336$ kg/${\mathrm{m}}^3, \chi =2.4$) of 10 nm radius, assuming an initial ground-state population of 95%. Measuring a photon flux outside the shaded region indicates that entanglement has been generated by the measurement of the Stokes scattered photon. The boundary of this region was calculated numerically from (4) (red line) and approximated analytically with (10) (blue dashed line). The tweezer and cavity parameters are $2\pi /k=1560$ nm, $P_j=1.5$ W, $w_j=720$ nm, $\ell =12$ mm, $w=30$ $\mu \mathrm{m}, \kappa /2\pi =318$ kHz, yielding the coupling frequencies $g_j=61$ kHz, the trapping frequency $\overline{\omega }/2\pi =785$ kHz, and the decoherence rates ${\gamma }_j^{\mathrm{sc}}=145$ Hz for a mechanical detuning of $\delta {\omega }_{\mathrm{m}}=32$ kHz. The residual gas pressure is assumed to be below ${10}^{-9}$ mbar.