van der Waals interaction of excited media

The Casimir interaction between two media of ground-state atoms is well described with the help of the Lifshitz formula depending upon the permittivity of the media. We will show that this formula is in contradiction with experimental evidence for excited atoms. We calculate the Casimir force between two atoms if one or both of them are excited. We use methods of quantum electrodynamics specially derived for the problem. It enables us to take into account the excited-state radiation widths of atoms. Then we calculate the force between the excited atom and medium of ground-state atoms. The results are in agreement with the ones obtained by other authors who used perturbation theory or linear response theory. Generalization of our results to the case of the interaction between two media of excited atoms results in a formula, which is in not only in quantitative, but in qualitative contradiction with the Lifshitz formula. This contradiction disappears if the media of ground-state atoms are considered. Moreover, our result does not include the permittivity of the media. It includes a quantity which differs from the permittivity only for excited atoms. The main features of our results are as follows. The interaction is resonant, the force may be either attractive or repulsive depending on the resonant frequencies of the atoms of different media, and the value of the Casimir force may be several orders of magnitude lager than that predicted by the Lifshitz formula. The features mentioned here are in agreement with known experimental and theoretical evidence obtained by many authors for the interaction of a single excited atom with dielectric media.

Figure 1

Contour $c$.

Figure 2

Feynman’s diagrams.

Figure 3

Interacting media.

Figure 4

Normalized Casimir force calculated by means of quantum electrodynamics (53) (solid line) and the Lifshitz formula (56) (dashed line). (a) $T∕{\omega }_B=0.1$, ${\gamma }_B∕{\omega }_B=0.002$ and (b) $T∕{\omega }_B=0.08$, ${\gamma }_B∕{\omega }_B=0.002$.

Figure 5

Normalized Casimir force calculated by means of quantum electrodynamics (53) (solid line) and the Lifshitz formula (56) (dashed line). $T∕{\omega }_B=0.3$, ${\gamma }_B∕{\omega }_B=0.002$.

Figure 6

Normalized Casimir force calculated by means of quantum electrodynamics (53) (solid line) and the Lifshitz formula (56) (dashed line). ${\omega }_A∕{\omega }_B=0.9$, ${\gamma }_B∕{\omega }_B=0.002$.

Figure 7

Normalized Casimir potential for excited atom and cold media interaction calculated by means of (a) quantum electrodynamics (44) and (b) the Lifshitz formula (57). ${\gamma }_B∕{\omega }_B=0.002$.

Figure 8

Feynman’s diagram corresponding to Eq. (A1). (a) First term, (b) second term, (c) third term, and (d) fourth term. The solid line corresponds to $g^{0A}$, the dashed line corresponds to $D_{l{l}^{\prime }}^{0\nu {\nu }^{\prime }}$, and the dash-dotted line represents ${\rho }_0^A$.

Figure 9

Feynman’s diagram of the fourth order of the coherent channel.