One-Loop Dominance in the Imaginary Part of the Polarizability: Application to Blackbody and Noncontact van der Waals Friction

Phenomenologically important quantum dissipative processes include blackbody friction (an atom absorbs counterpropagating blueshifted photons and spontaneously emits them in all directions, losing kinetic energy) and noncontact van der Waals friction (in the vicinity of a dielectric surface, the mirror charges of the constituent particles inside the surface experience drag, slowing the atom). The theoretical predictions for these processes are modified upon a rigorous quantum electrodynamic treatment, which shows that the one-loop “correction” yields the dominant contribution to the off-resonant, gauge-invariant, imaginary part of the atom’s polarizability at room temperature, for typical atom-surface interactions. The tree-level contribution to the polarizability dominates at high temperature.

Figure 1

The Feynman diagram for the ac Stark shift involves the absorption or emission of two laser photons by the atom (a). A tree-level imaginary part [cut of the diagram, see (b)] is generated only if the absorbed laser photon happens to be at resonance with regard to a transition of the atom to an excited state [see Eq. (12)].

Figure 2

The radiative correction to the ac Stark shift involves an additional virtual photon loop (green). The imaginary part (cut of the diagram) is generated when the virtual photon becomes real, i.e., when the laser photon has the same energy as the spontaneously emitted photon. The different insertions of the radiative photon in time-ordered perturbation theory lead to the diagrams in (a)–(c).

Figure 3

In velocity gauge, the seagull term leads to additional diagrams with a two-photon vertex. As in Fig. 2, the different insertions of the radiative photons in time-ordered perturbation theory lead to the diagrams in (a)–(c).

Figure 4

Theoretical predictions (a)–(c) for the attenuation time ${\tau }_{\mathrm{BB}}$ (equal to the ratio of atomic mass to ${\eta }_{\mathrm{BB}}$) are displayed for blackbody radiation friction. For ${\mathrm{CaF}}_2$ van der Waals friction (d)–(f), the coefficient ${\gamma }_0$ is defined in Eq. (17). The dashed lines in (a)–(c) and (f) are obtained with the tree-level term given in Eq. (12).