Spatial dispersion in atom-surface quantum friction

We investigate the influence of spatial dispersion on atom-surface quantum friction. We show that for atom-surface separations shorter than the carrier's mean free path within the material, the frictional force can be several orders of magnitude larger than that predicted by local optics. In addition, when taking into account spatial dispersion effects, we show that the commonly used local thermal equilibrium approximation underestimates by approximately 95% the drag force, obtained by employing the recently reported nonequilibrium fluctuation-dissipation relation for quantum friction. Unlike the treatment based on local optics, spatial dispersion in conjunction with corrections to local thermal equilibrium change not only the magnitude but also the distance scaling of quantum friction.

Figure 1

Quantum frictional force acting on an atom moving at constant velocity and parallel to a metallic surface described by the SCIB model. The low-velocity limit of the force [Eqs. (9)] $\overline{F}$ is plotted as a function of the atom-surface separation $z_a$. In all plots, the parameters ${\omega }_p=9$ eV, $\mathrm{\Gamma }=30$ meV, and $v_F/c=1/137$ are fixed to these same values. In order to emphasize the role of spatial dispersion and the nonequilibrium physics, the force $\overline{F}$ is normalized to its expression obtained using local optics and the LTE approximation, ${\overline{F}}_{\mathrm{local}}^{\mathrm{LTE}}$. In the low-velocity regime the normalized force does not depend on the velocity [see Eqs. (9) and (19)]. For atom-surface separations much larger than the electron's mean free path $z_a\gg \ell =v_F/\gamma$, the force approaches a value which is almost twice that of ${\overline{F}}_{\mathrm{local}}^{\mathrm{LTE}}$, recovering the result reported in Ref. [17]. For $z_a<\ell$, spatial dispersion and the non-LTE correction result in a substantial increase of the force, with a maximum enhancement at $z_a\sim 10{\lambda }_{\mathrm{TF}}$, where ${\lambda }_{\mathrm{TF}}=v_F/\left(\sqrt{3}{\omega }_p\right)$ is the Thomas-Fermi screening length. The curve is dotted for distances less than 10 Å, where our description might not be reliable (see text). The black dashed line shows the total asymptotic behavior for distances ${\lambda }_{\mathrm{TF}}\ll z_a<\ell$ given by the sum of the expressions in Eqs. (20). The inset shows the correction exclusively due to the nonequilibrium physics, i.e., ${\overline{F}}^J/{\overline{F}}^{\mathrm{LTE}}$, with spatial dispersion taken into account in both forces of the numerator and the denominator. The curve shows a larger contribution of the non-LTE correction in the nonlocal case than in the local limit. It also indicates that, for the parameters used here, nonlocality is the main source of the force enhancement observed for $z_a<\ell$.