Electric dipoles at ultralow temperatures

Any object with a net positive charge on one end and a net negative charge on the other end possesses an electric dipole moment. In ordinary classical electromagnetism this dipole moment is a vector quantity that can point in any direction, and is subject to electrical forces that are fairly straightforward to formulate mathematically. However, for a quantum mechanical object like an atom or molecule, the strength and orientation of the object’s dipole moment can depend strongly on the object’s quantum mechanical state. This is a subject that becomes relevant in low temperature molecular samples, where an ensemble of molecules can be prepared in a single internal state, as described elsewhere in this volume. In such a case, the mathematical description becomes more elaborate, and indeed the dipole-dipole interaction need not take the classical form given in textbooks. The description of this interaction is the subject of this chapter. We approach this task in three steps: first, we introduce the ideas of how dipoles arise in quantum mechanical objects; second, we present a formalism within which to describe these dipoles; and third, we give examples of the formalism that illustrate some of the basic physics that emerges. The discussion will explore the possible energy states of the dipole, the field generated by the dipole, and the interaction of the dipole with another dipole. We restrict the discussion to a particular “minimal realistic model,” so that the most important physics is incorporated, but the arithmetic is not overwhelming. Although we discuss molecular dipoles in several contexts, our main focus is on polar molecules that possess a Λ doublet in their ground state. These molecules are the most likely, among diatomic molecules at least, to exhibit their dipolar character at moderate laboratory field strengths. Λ-doubled molecules have another peculiar feature, namely, their ground states possess a degeneracy even in an electric field. This means that there is more than one way for such a molecule to align with the field; the two possibilities are characterized by different angular momentum quantum numbers. This degeneracy leads to novel properties of both the orientation of a single molecule’s dipole moment and the interaction between dipoles. In the examples we present, we focus on revealing these novel features.