Casimir-Polder forces: A nonperturbative approach

Within the frame of macroscopic QED in linear, causal media, we study the radiation force of Casimir-Polder type acting on an atom which is positioned near dispersing and absorbing magnetodielectric bodies and initially prepared in an arbitrary electronic state. It is shown that minimal and multipolar coupling lead to essentially the same lowest-order perturbative result for the force acting on an atom in an energy eigenstate. To go beyond perturbation theory, the calculations are based on the exact center-of-mass equation of motion. For a nondriven atom in the weak-coupling regime, the force as a function of time is a superposition of force components that are related to the electronic density matrix elements at a chosen time. Even the force component associated with the ground state is not derivable from a potential in the ususal way, because of the position dependence of the atomic polarizability. Further, when the atom is initially prepared in a coherent superposition of energy eigenstates, then temporally oscillating force components are observed, which are due to the interaction of the atom with both electric and magnetic fields.

Figure 1

(a) Transition frequency shift (solid and dotted lines) and (b) decay rate (solid and dotted lines) versus bare transition frequency for a two-level atom that is situated at distance $z_{\mathrm{A}}$ from a semi-infinite half space medium of complex permittivity according to Eq. (156) and whose transition dipole moment is perpendicular to the interface [${\omega }_P∕{\omega }_{\mathrm{T}}=0.75$, $\gamma ∕{\omega }_{\mathrm{T}}=0.01$; ${\omega }_{\mathrm{T}}^2{d}_{\mathrm{A}}^2∕\left(3\pi ħ{\epsilon }_0{c}^3\right)={10}^{-7}$; $z_A∕{\mathrm{\lambda }}_{\mathrm{T}}=0.0075$ (solid and dashed lines), $z_A∕{\mathrm{\lambda }}_{\mathrm{T}}=0.009$ (dotted and dot-dashed lines)]. For comparison, the approximate results obtained by using the bare frequencies in Eqs. (158, 159) are also displayed (dashed and dot-dashed lines).

Figure 2

The resonant part of the CP force $F_{11}^{\mathrm{r}}{\mathrm{\lambda }}_{\mathrm{T}}^4\times {10}^{-9}∕\left(3C\right)$ on a two-level atom that is situated at distance (a) $z_{\mathrm{A}}∕{\mathrm{\lambda }}_{\mathrm{T}}=0.0075$ and (b) $z_{\mathrm{A}}∕{\mathrm{\lambda }}_{\mathrm{T}}=0.009$ of a semi-infinite half space medium of complex permittivity according to Eq. (156) and whose transition dipole moment is perpendicular to the interface (solid lines). The parameters are the same as in Fig. 1. For comparison, both the perturbative result (dashed lines) and the separate effects of level shifting (dotted lines) and level broadening (dash-dotted lines) are shown.

Figure 3

The off-resonant part of the CP force $F_{11}^{\mathrm{r}}{\mathrm{\lambda }}_{\mathrm{T}}^4\times {10}^{-9}∕\left(3C\right)$ on a two-level atom that is situated at distance (a) $z_A∕{\mathrm{\lambda }}_{\mathrm{T}}=0.0075$ and (b) $z_{\mathrm{A}}∕{\mathrm{\lambda }}_{\mathrm{T}}=0.009$ of a semi-infinite half space medium of complex permittivity according to Eq. (156) and whose transition dipole moment is perpendicular to the interface (solid lines). The parameters are the same as in Fig. 1. For comparison, the perturbative result (dashed lines) is shown. The insets display the difference between the force with and without consideration of the level broadening (solid lines). For comparison, we show this difference in the case where the level shifts are ignored (dashed lines).