The Stark Accelerator/Decelerator viewed as a Biased Pendulum

Bretislav Friedrich
Fritz Haber Institute of the Max Planck Society, Berlin

Stark deceleration, along with buffer-gas cooling and photoassociation of cold atoms, ushered in the era of the ultra-cold in molecular physics. Stark decelerators (or accelerators) of polar molecules bear similarity with charged-particle accelerators used over the past 60 years in particle physics. However, since molecules possess states whose eigenenergy can both increase and decrease with the strength of the field they are subjected to, and since both acceleration and deceleration are of interest, the molecular case calls for a more general approach than the one which was adequate for the understanding of acceleration of charged particles.

The Stark accelerator/decelerator relies on time-dependent inhomogeneous electric fields. So far, these have been generated by linear switchable field arrays. First I Fourier-analyze the field produced by such a field array. This analysis reveals that the field consists of a number of partial waves traveling at distinct phase vlocities. Next I describe the kinematics of the field-molecule interaction and introduce the notion of a phase of a molecule in a traveling periodic accelerator/decelerator field. Then I introduce the Stark potential and force that act on a molecule and derive the molecule's equations of motion, both in terms of the laboratory-fixed coordinates and of the phase with respect to the traveling field. Then I discuss a special case of a Stark accelerator/decelerator, the first-harmonic accelerator/decelerator, whose equation of motion is isomorphic with that of a biased pendulum. Since the biased pendulum problem can be solved analytically (this will be shown in an interlude), valuable lessons about the accelerator/decelerator can be drawn from it. The dynamics of a first-harmonic accelerator/decelerator or, interchangeably, of a biased pendulum, will be subsequently presented and discussed in terms of phase diagrams. Finally, I'll consider the general properties of the velocity of the molecules in a phase-stable accelerator/decelerator. These properties reveal that the Stark accelerator/decelerator behaves like a flying accordion.