The field-induced interaction between non-resonant magnetic dipoles

The interaction between two quantum dipoles exists in many different physical and chemical systems, and can be obtained either by the direct generalization from its classical counterpart, or by considering the mediation of the quantized electromagnetic field. However, these two methods do not always coincide with each other exactly. In particular, the interaction between two non-resonant dipoles has not yet been derived based on the field mediation. Here we make a general derivation for the magnetic dipole-dipole interaction based on the field mediation. By properly taking into account the field dynamics, our result naturally gives the interaction for non-resonant dipoles with different frequencies, and even the counter-rotating interaction terms. It is shown that the interactions propagated from dipoles different to each other vary due to their frequency difference, and the total interaction strength is their average. This result provides a general way to obtain the field-induced dipole-dipole interaction in many physical and chemical systems.