Recoil heating of a dielectric particle illuminated by a linearly polarized plane wave within the Rayleigh regime

We investigate the recoil heating phenomenon experienced by a dielectric spherical particle when it interacts with a linearly polarized plane wave within the Rayleigh regime. We derive the fluctuating force acted upon the particle arising from the fluctuations of the electromagnetic fields. Our derivations reveal that the spectral density of the fluctuating force along the propagation direction is $7\mathrm{\hslash }{\omega }_0{P}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}}\mathrm{/}5c^2$. Meanwhile, along the direction of the electric and magnetic fields, it is $\mathrm{\hslash }{\omega }_0{P}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}}\mathrm{/}5c^2$ and $2\mathrm{\hslash }{\omega }_0{P}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}}\mathrm{/}5c^2$, respectively. Here, $P_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}}$ denotes the power scattered by the particle, $\mathrm{\hslash }{\omega }_0$ represents the energy of a photon, and $c$ is the speed of light. Recoil heating imposes fundamental limitations in levitated optomechanics, constraining the minimum temperatures achievable in cooling processes, the coherence time of the system, and the sensitivity of force measurements.