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Physical Review A

Analytic wave model of Stark deceleration dynamics

K. B. Gubbels, Gerard Meijer, Břetislav Friedrich

Published Jun 12, 2006• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.73.063406

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•Jan 1, 2019•2 citations
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 \ensuremathλ∕\ensuremathτ\ensuremath{\lambda}∕\ensuremath{\tau}\ensuremathλ∕\ensuremathτ, with \ensuremathλ\ensuremath{\lambda}\ensuremathλ and \ensuremathτ\ensuremath{\tau}\ensuremathτ 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.
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.
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.
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.
Abstract 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 than in the more typically used crossed-pin geometry Stark decelerators. However, this new technique, traveling-wave Stark deceleration, is more experimentally challenging due to the electronic requirements. A third method, pulsed-ring Stark deceleration, uses ring-geometry electrodes with high-voltage pulses to decelerate molecules. The discrete voltages make it easier to implement, and it is more efficient in certain situations due to an increased longitudinal potential gradient. Here, we present an experimental realization of a ring-geometry Stark decelerator using both continuously varying and discrete voltages. Using the pulsed-ring method, we demonstrate a newly accessible low-velocity regime for moderate peak voltages on ring electrodes. This opens up the opportunity to efficiently decelerate various molecular species using a medium length decelerator. 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.
Physicists have observed all the elementary particles predicted by the Standard Model. These particles, however, account for only 5% of the total mass-energy of the Universe. In the quest to solve the mystery of missing matter and to extend our knowledge about nature, a large number of experimental research programs are devoted to find a glimpse of New Physics. Table-top high precision measurements on molecules can contribute to these efforts in a manner complementary to high-energy experiments at colliders. In this thesis, we describe the operation of a traveling-wave Stark decelerator. This device is designed and constructed in order to slow down heavy polar molecules for various future explorations. Deceleration is achieved by manipulation of the molecules in controllable electric fields. We demonstrate in an experiment the successful operation of the Stark decelerator by slowing down a beam of SrF molecules to one-third of their initial velocity, thus removing almost 90% of the initial kinetic energy. We also reveal limiting conditions for this process. These results are of importance for many exciting experiments using heavy molecules in the near future. One relevant and promising example of such experiments is the search for a permanent electric dipole moment of the electron.