Derivation of recoil heating of a Rayleigh scatterer from the quantum fluctuations of the electromagnetic fields

We investigate the recoil heating phenomenon experienced by a dielectric spherical particle when it interacts with a propagating or standing plane wave within the Rayleigh regime. We derive the fluctuating force acting on the particle due to the fluctuations of the electromagnetic fields. For the case of the propagating wave, we show that the spectral density of the fluctuating force in the propagation direction is $7\ensuremath{\hbar}{\ensuremath{\omega}}_{0}{P}_{\mathrm{scat}}/5{c}^{2}$, while in the direction of the electric and magnetic fields it is $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}{P}_{\mathrm{scat}}/5{c}^{2}$ and $2\ensuremath{\hbar}{\ensuremath{\omega}}_{0}{P}_{\mathrm{scat}}/5{c}^{2}$, respectively, where ${P}_{\mathrm{scat}}$ denotes the power scattered by the particle, $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$ represents the energy of a photon, and $c$ is the speed of light. When the particle interacts with a standing wave, the longitudinal force fluctuation at the trapping position equals $2\ensuremath{\hbar}{\ensuremath{\omega}}_{0}{P}_{\mathrm{scat}}/5{c}^{2}$, while the transverse force fluctuations are similar to those in the case of a propagating wave.