Velocity-dependent dipole forces on an excited atom

We present a time-dependent calculation of the velocity-dependent forces which act on an excited atomic dipole in relative motion with respect to ground state atoms of a different kind. Both its interaction with a single atom and with a dilute atomic plate are evaluated. In either case, the total force consists of a conservative van der Waals component and a nonconservative Röntgen component. On physical grounds, the former corresponds to the velocity-dependent recoil experienced by the excited atom in the processes of absorption and emission of the photons that it exchanges with the ground-state atoms on a periodic basis. The latter corresponds to the time-variation of the Röntgen momentum, which is also mediated by the periodic exchange of quasiresonant photons. We find that, at leading order, all these interactions are linear in velocity. In the nonretarded regime the van der Waals force dominates, being antiparallel to the velocity. On the contrary, in the retarded regime the velocity-dependent forces oscillate in space, van der Waals and Röntgen forces are of the same order in the atom-atom interaction, and the Röntgen component dominates in the atom-surface interaction.

Figure 1

(a) Sketch of the interaction of atom $A$ with an atom $B$. Atom $A$ moves at constant velocity $\mathbf{v}$ while atom $B$ remains at rest at a distance $R$ which varies in time. (b) Sketch of the interaction of an atom $A$ with a thin plate made of a random distribution of atoms of kind $B$, with numerical surface density $\sigma$. Atom $A$ flies with velocity $\mathbf{v}$ parallel to the plate at a height $d$. The wavy lines represent the pairs of photons exchanged by the atoms. Emission and absorption processes take place at different locations of atom $A$.

Figure 2

(a) Diagrammatic representation of Eq. (2) for $\langle W_A^E\left(T\right)\rangle$. Time runs along the vertical as indicated by the arrows. $\mathbf{R}$ is intended, generally, as a quantum operator. Only for both atoms at rest can $\mathbf{R}$ be treated in this picture as a classical variable. (b) Diagrammatic representation of Eq. (10) for ${\langle W_A^E\left(T\right)\rangle }^{\mathbf{v}}$ up to order ${(v/c)}^3$. The interatomic displacement here is a classical variable which depends on time. $\mathbf{v}(T-t^{″})$ is the displacement of atom $A$ during the lag between the emission of the first photon and the absorbtion of the second photon.