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Physical Review A

Higher-order resonances in a Stark decelerator

Sebastiaan Y. T. van de Meerakker, Nicolas Vanhaecke, Hendrick L. Bethlem, Gerard Meijer

Published May 26, 2005• 58 citations
Publisher

Abstract

The motion of polar molecules can be controlled by time-varying inhomogeneous electric fields. In a Stark decelerator, this is exploited to select a fraction of a molecular beam that is accelerated, transported, or decelerated. Phase stability ensures that the selected bunch of molecules is kept together throughout the deceleration process. In this paper an extended description of phase stability in a Stark decelerator is given, including higher-order effects. This analysis predicts a wide variety of resonances that originate from the spatial and temporal periodicity of the electric fields. These resonances are experimentally observed using a beam of OH $(^{2}\ensuremath{\Pi}_{3∕2},v=0,J=3∕2)$ radicals passing through a Stark decelerator.

DOI
10.1103/physreva.71.053409

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The concept of phase stability in a Stark decelerator ensures that polar molecules can be accelerated, guided, or decelerated without loss; molecules within a certain position and velocity interval are kept together throughout the deceleration process. In this paper the influence of the transverse motion on phase stability in a Stark decelerator is investigated. For typical deceleration experiments---i.e., for high values of the phase angle ${\ensuremath{\phi}}_{0}$---the transverse motion considerably enhances the region in phase space for which phase stable deceleration occurs. For low values of ${\ensuremath{\phi}}_{0}$, however, the transverse motion reduces the acceptance of a Stark decelerator and unstable regions in phase space appear. These effects are quantitatively explained in terms of a coupling between the longitudinal and transverse motion. The predicted longitudinal acceptance of a Stark decelerator is verified by measurements on a beam of OH $(X\phantom{\rule{0.2em}{0ex}}^{2}\ensuremath{\Pi}_{3∕2},J=3∕2)$ radicals passing through a Stark decelerator.

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The motion of polar molecules can be controlled by time-varying inhomogeneous electric fields. In a Stark decelerator, this is exploited to select a fraction of a molecular beam that is accelerated, transported, or decelerated. Phase stability ensures that the selected bunch of molecules is kept together throughout the deceleration process. In this paper an extended description of phase stability in a Stark decelerator is given, including higher-order effects. This analysis predicts a wide variety of resonances that originate from the spatial and temporal periodicity of the electric fields. These resonances are experimentally observed using a beam of OH (2\ensuremathΠ3∕2,v=0,J=3∕2)(^{2}\ensuremath{\Pi}_{3∕2},v=0,J=3∕2)(2\ensuremathΠ3∕2​,v=0,J=3∕2) radicals passing through a Stark decelerator.
The concept of phase stability in a Stark decelerator ensures that polar molecules can be accelerated, guided, or decelerated without loss; molecules within a certain position and velocity interval are kept together throughout the deceleration process. In this paper the influence of the transverse motion on phase stability in a Stark decelerator is investigated. For typical deceleration experiments---i.e., for high values of the phase angle ${\ensuremath{\phi}}_{0}$---the transverse motion considerably enhances the region in phase space for which phase stable deceleration occurs. For low values of ${\ensuremath{\phi}}_{0}$, however, the transverse motion reduces the acceptance of a Stark decelerator and unstable regions in phase space appear. These effects are quantitatively explained in terms of a coupling between the longitudinal and transverse motion. The predicted longitudinal acceptance of a Stark decelerator is verified by measurements on a beam of OH $(X\phantom{\rule{0.2em}{0ex}}^{2}\ensuremath{\Pi}_{3∕2},J=3∕2)$ radicals passing through a Stark decelerator.
The motion of polar molecules can be controlled by time-varying inhomogeneous electric fields. In a Stark decelerator, this is exploited to accelerate, transport, or decelerate a fraction of a molecular beam. When combined with a trap, the decelerator provides a means to store the molecules for times up to seconds. Here, we review our efforts to produce cold molecules via this technique. In particular, we present a new generation Stark decelerator and electrostatic trap that selects a significant part of a molecular beam pulse that can be loaded into the trap. Deceleration and trapping experiments using a beam of OH radicals are discussed.
Stark deceleration relies on time-dependent inhomogeneous electric fields which repetitively exert a decelerating force on polar molecules. Fourier analysis reveals that such fields, generated by an array of field stages, consist of a superposition of partial waves with well-defined phase velocities. Molecules whose velocities come close to the phase velocity of a given wave get a ride from that wave. For a square-wave temporal dependence of the Stark field, the phase velocities of the waves are found to be odd-fraction multiples of a fundamental phase velocity $\ensuremath{\lambda}∕\ensuremath{\tau}$, with $\ensuremath{\lambda}$ and $\ensuremath{\tau}$ the spatial and temporal periods of the field. Here we study explicitly the dynamics due to any of the waves as well as due to their mutual perturbations. We first solve the equations of motion for the case of single-wave interactions and exploit their isomorphism with those for the biased pendulum. Next we analyze the perturbations of the single-wave dynamics by other waves and find that these have no net effect on the phase stability of the acceleration or deceleration process. Finally, we find that a packet of molecules can also ride a wave which results from an interference of adjacent waves. In this case, small phase stability areas form around phase velocities that are even-fraction multiples of the fundamental velocity. A detailed comparison with classical trajectory simulations and with experiment demonstrates that the analytic ``wave model'' encompasses all the longitudinal physics encountered in a Stark decelerator.
With a Stark decelerator, beams of neutral polar molecules can be accelerated, guided at a constant velocity, or decelerated. The effectiveness of this process is determined by the six-dimensional (6D) volume in phase space from which molecules are accepted by the Stark decelerator. Couplings between the longitudinal and transverse motion of the molecules in the decelerator can reduce this acceptance. These couplings are nearly absent when the decelerator operates such that only every third electric-field stage is used for deceleration, while extra transverse focusing is provided by the intermediate stages. For many applications, the acceptance of a Stark decelerator in this so-called s=3s=3s=3 mode significantly exceeds that of a decelerator in the conventionally used (s=1)(s=1)(s=1) mode. This has been experimentally verified by passing a beam of OH radicals through a $2.6\text{\ensuremath{-}}\mathrm{m}$-long Stark decelerator. The experiments are in quantitative agreement with the results of trajectory calculations, and can qualitatively be explained with a simple model for the 6D acceptance. These results imply that the 6D acceptance of a Stark decelerator in the s=3s=3s=3 mode of operation approaches the optimum value, i.e., the value that is obtained when any couplings are neglected.
Stark deceleration is a technique that uses time-varying inhomogeneous electric fields to decelerate polar molecules for various molecular beam and trapping experiments. New ring-geometry Stark decelerators with continuously varying voltages offer a method to produce a more intense source of molecules in a technique called traveling-wave Stark deceleration. However, this type of deceleration is more experimentally challenging than the more typically used crossed-pin geometry decelerators with pulsed voltages. Here, we present an experimental realization of a ring-geometry Stark decelerator using either continuously varying or discrete voltages. Pulsed-ring Stark deceleration using discrete voltages is easier to implement and, under certain circumstances, is more efficient than traveling-wave Stark deceleration. A comparison of experimental and simulated results between traveling-wave and pulsed-ring Stark deceleration is presented along with a simple model for determining when each mode is more efficient.