Magnetic dipole-dipole interaction induced by the electromagnetic field

We give a derivation for the indirect interaction between two magnetic dipoles induced by the quantized electromagnetic field. It turns out that the interaction between permanent dipoles directly returns to the classical form; the interaction between transition dipoles does not directly return to the classical result, yet returns in the short-distance limit. In a finite volume, the field modes are highly discrete and both the permanent and transition dipole-dipole interactions are changed. For transition dipoles, the changing mechanism is similar with the Purcell effect, since only a few number of nearly resonant modes take effect in the interaction mediation; for permanent dipoles, the correction comes from the boundary effect: if the dipoles are placed close to the boundary, the influence is strong; otherwise, their interaction does not change too much from the free space case.

Figure 1

Demonstration for the boundary condition of Eq. (17). The influence of the periodic boundary condition can be equivalently replaced by the point “charge” lattice in free space, which is similar to the method of image charges.

Figure 2

Numerical estimation for the dipole-dipole interaction strength in a finite volume [Eq. (19)] comparing with the free space result. Here, as an example, we choose ${\widehat{\mathrm{e}}}_{\mathbf{r}}=(1,2,3)/\sqrt{14}$ and set the two dipoles parallel to each other in the direction ${\widehat{\mathrm{e}}}_{1,2}=(cos\varphi ,0,sin\varphi )$. When $r/L$ is large, the correction from the boundary effect is significant. It turns out Eq. (19) converges quite fast, and only very few “image” terms are needed ($|n_{x,y,z}|\lesssim 2$) to get a precise enough result, which means only the nearest image dipoles are important.