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Journal of Physics B Atomic Molecular and Optical Physics

Theory of laser evaporative cooling of trapped negative ions. I. Harmonically bound ions and RF traps

A. Crubellier

Published Oct 28, 1990• 11 citations
Publisher

Abstract

A theoretical description of a specific method of laser cooling of negative ions is presented based on evaporation through photodetachment. The ideal case of harmonically bound ions and the case of RF traps are considered in the weak space charge limit. Evolution equations of the ion cloud in the quasistatic regime are derived, in which eventually other cooling or heating processes could be inserted. These equations are analytically solved in a particular case where the laser is tuned slightly below photodetachment threshold with a gradually decreasing absolute value of the detuning. Numerical results concerning the evolution of ion cloud temperature and density are given and discussed for either s or p photodetached electrons. The influence of laser beam geometry and in particular the existence, beside the previously considered 'Doppler' cooling, of 'geometrical' cooling is investigated. In Paul traps, RF heating might impose a preliminary cooling by another method.

DOI
10.1088/0953-4075/23/20/020

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A theoretical description of a specific method of laser cooling of negative ions is presented based on evaporation through photodetachment. The ideal case of harmonically bound ions and the case of RF traps are considered in the weak space charge limit. Evolution equations of the ion cloud in the quasistatic regime are derived, in which eventually other cooling or heating processes could be inserted. These equations are analytically solved in a particular case where the laser is tuned slightly below photodetachment threshold with a gradually decreasing absolute value of the detuning. Numerical results concerning the evolution of ion cloud temperature and density are given and discussed for either s or p photodetached electrons. The influence of laser beam geometry and in particular the existence, beside the previously considered 'Doppler' cooling, of 'geometrical' cooling is investigated. In Paul traps, RF heating might impose a preliminary cooling by another method.
We report the first cooling of atomic anions by laser radiation. O^{-} ions confined in a linear Paul trap were cooled by selectively photodetaching the hottest particles. For this purpose, anions with the highest total energy were illuminated with a 532 nm laser at their maximal radial excursion. Using laser-particle interaction, we realized a both colder and denser ion cloud, achieving a more than threefold temperature reduction from 1.15 to 0.33 eV. Compared with the interaction with a dilute buffer gas, the energy-selective addressing and removal of anions resulted in lower final temperatures, yet acted 10 times faster and preserved twice as large a fraction of ions in the final state. An ensemble of cold negative ions affords the ability to sympathetically cool any other negative ion species, enabling or facilitating a broad range of fundamental studies from interstellar chemistry to antimatter gravity. The technique can be extended to any negative ion species that can be neutralized via photodetachment.
Laser cooling of a single trapped ion in a Paul trap is discussed theoretically in the Lamb-Dicke limit, with full consideration of the time dependence of the trapping potential. Resulting mean kinetic energies are defined as time averages over one period of the micromotion and are compared with final temperatures expected from the laser-cooling treatment with harmonic traps. For laser-atom detunings close to the micromotion frequency the results differ significantly from those expected for a harmonic trap potential. A physical interpretation is given and simple formulas are derived for the strong confinement case.
Several groups have recently used laser cooling to produce ordered arrays of ions, ion crystals, in rf quadrupole ion traps (Paul traps). Laser cooling has been understood using previous theories which ignore ion micromotion and replace the oscillating rf trap potential with a static pseudopotential. We show that in recent experiments, ion micromotion can exceed one wavelength of the cooling radiation and that the resulting Doppler shifts can transform laser cooling into laser heating. Changes in trap voltage, ion position, and laser intensity can all reverse the sign of laser cooling. This, together with ideas of deterministic chaos, can lead to a reinterpretation of experiments on ion crystals. The recognition that effects previously attributed to an experimental artifact, rf heating, may actually be due to a failure of laser cooling opens the door, on the one hand, to precise theories of phase transitions and, on the other, to improved cooling techniques that can permit the creation of larger ion crystals.
An experimental study of sideband laser cooling of trapped ions in miniature radio-frequency traps is presented. In+{\mathrm{In}}^{+}In+ ions are laser cooled by exciting the narrow intercombination line ${5s}^{2}{}^{1}{S}_{0}\ensuremath{\rightarrow}5s5p{}^{3}{P}_{1}$, which shows resolved sidebands due to the vibrational motion of the ions in the trap. The influence of the micromotion at the frequency of the oscillating trapping field is investigated. Heating processes through coupling with the trapping field occur for certain regions of negative laser detuning. A method of bichromatic sideband cooling is introduced that combines efficient cooling from high temperatures with the possibility of reaching low mean vibrational quantum numbers 〈n〉<1〈n〉<1〈n〉<1 by using narrow optical transitions. This technique allows recording of high-resolution excitation spectra of the cooling transition in the Lamb-Dicke regime and thus determination of 〈n〉〈n〉〈n〉. Single ions as well as two-ion Coulomb crystals have been cooled to temperatures below $100 \ensuremath{\mu}\mathrm{K}$, and population of the ground state of motion to more than 50% over long time scales is demonstrated. These values are close to the lower limit for the detection of vibrational excitation with the present method.
We study the stochastic dynamics of a particle in a periodically driven potential. For atomic ions trapped in radio-frequency Paul traps, noise heating and laser cooling typically act slowly in comparison with the unperturbed motion. These stochastic processes can be accounted for in terms of a probability distribution defined over the action variables, which would otherwise be conserved within the regular regions of the Hamiltonian phase space. We present a semiclassical theory of low-saturation laser cooling applicable from the limit of low-amplitude motion to large-amplitude motion, accounting fully for the time-dependent and anharmonic trap. We employ our approach to a detailed study of the stochastic dynamics of a single ion, drawing general conclusions regarding the nonequilibrium dynamics of laser-cooled trapped ions. We predict a regime of anharmonic motion in which laser cooling becomes diffusive (i.e., it is equally likely to cool the ion as it is to heat it), and can also turn into effective heating. This implies that a high-energy ion could be easily lost from the trap despite being laser cooled; however, we find that this loss can be counteracted using a laser detuning much larger than Doppler detuning.
The nonlinear dynamics of large ion clouds ($N\ensuremath{\ge}256$ ions) trapped in radio-frequency (rf) traps and coupled to laser cooling give rise to a bistable behavior of the temperature. Numerical simulations of the free evolution of a large three-dimensional spherical cloud is used to characterize how the oscillating field amplitude and the ion number control the rf-heating rate and the switching between the two stable states. We show that the heating rate does not significantly depend on the ion number but strongly depends on the oscillating field. This exhibits the major role played by the density of the ensemble. The competition between the radio-frequency heating and the laser cooling is also discussed. They are used to design scenarios to take advantage of the threshold effect to detect and quantify perturbations by an intruder.