Influence of a surface in the nonretarded interaction between two atoms

In this work we obtain analytical expressions for the nonadditivity effects in the dispersive interaction between two atoms and a perfectly conducting surface of arbitrary shape in the nonretarded regime. We show that this three-body quantum-mechanical problem can be solved by mapping it onto a two-body electrostatic one. We apply the general formulas developed in this paper in several examples. First we rederive the London interaction as a particular case of our formalism. Then we investigate other interesting examples, such as the setup where two atoms lie inside a plane capacitor. Here we show that the nonadditivity is strikingly manifest since the planes lead to an exponential suppression of the interaction of the atoms. As a last example we deal with two atoms in the presence of a sphere, both grounded and isolated. We show that for realistic experimental parameters the nonadditivity may be relevant for the force in each atom.

Figure 1

Two atoms $A$ and $B$ in the presence of an infinite conducting plane. Here $\overline{A}$ is the image of $A$ and the image of $B$ is not represented; $R_{AB}$ is the distance between the atoms and ${\overline{R}}_{AB}$ is the distance between $B$ and the image of $A$, denoted by $\overline{A}$.

Figure 2

Two atoms $A$ and $B$ inside a perfectly conducting plate conductor. We choose the plane $z=0$ midway between the plates. Without any loss of generality, atom $A$ is put at $(0,0,z_A)$ and $B$ at $(x_B,0,z_B)$.

Figure 3

Nonadditive part of the interaction energy (normalized by the London interaction energy) as a function of the distance between atoms $A$ and $B$. The atoms are both equidistant to the plates. Here $R_{AB}$ is measured in units of $D$. The blue dashed curve shows the asymptotic behavior obtained via Eq. (50) while the red solid curve presents the exact result obtained through (49).

Figure 4

Nonadditive part of the force (normalized by the force between the atoms in vacuum) as a function of the distance between atom $A$ and atom $B$. The atoms are both equidistant to the plates. Here $R_{AB}$ is measured in units of $D$. The blue dashed curve shows the asymptotic behavior obtained via Eq. (50) while the red solid curve presents the exact result obtained through (49).

Figure 5

Two atoms near a conducting sphere. Here ${\mathbf{R}}_{AB}$ is parallel to the $\mathcal{Y}$ axis.

Figure 6

Nonadditive part of the $x$ component of the force exerted on atom $B$ (normalized by the London force) as a function of the distance between atom $B$ and the center of the sphere. Here ${\mathbf{R}}_{AB}$ remains always perpendicular to ${\mathbf{r}}_B$. The red solid curve is for two atoms separated by $d_A=0.002a$ and the blue dashed curve is for $d_A=0.003a$.

Figure 7

Ratio between the isolated sphere's and the grounded sphere's nonadditive contributions for the interaction energy. The horizontal axis stands for the distance of atom $A$ to the center of the sphere (in units of the radius of the sphere). Atom $B$ is farther from the sphere, separated by a distance $R_{AB}=0.002a$ from atom $A$ and collinear with it and the center of the sphere.