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Quantum Theory of Ion Motion and Laser Cooling in a Radio Frequency Quadrupole Trap.

Ann Laurie Wells

Published Dec 1, 1984• 0 citations
OpenAlex

Abstract

The problem of ion motion and laser cooling in a radio frequency quadrupole trap is considered. It is shown by a simple numerical calculation that for ion motion in the absence of laser cooling the effect of the time varying potential v(R-bar)cos(Omega x t) can be approximated by a time independent effective potential whenever the frequency of oscillation of the electric field, Omega is sufficiently greater than a resonant frequency, (Omega)squared = (sq. root of (2) times eVo)/M were e is the electronic charge, Vo is the maximum potential applied to the trap electrodes, and M is the mass of the ion. Equations for the Hamiltonian matrix elements in the presence of laser cooling, in the Lambe-Dicke limit are derived and studied to determine the cooling transitions. The equations of motion for the density matrix in the Lambe-Dicke limit are derived using a simple representation of spontaneous emission. (Theses) (Author)

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arXiv:1004.5202•Physical Review A•Sep 9, 2010•23 citations
The problem of ion motion and laser cooling in a radio frequency quadrupole trap is considered. It is shown by a simple numerical calculation that for ion motion in the absence of laser cooling the effect of the time varying potential v(R-bar)cos(Omega x t) can be approximated by a time independent effective potential whenever the frequency of oscillation of the electric field, Omega is sufficiently greater than a resonant frequency, (Omega)squared = (sq. root of (2) times eVo)/M were e is the electronic charge, Vo is the maximum potential applied to the trap electrodes, and M is the mass of the ion. Equations for the Hamiltonian matrix elements in the presence of laser cooling, in the Lambe-Dicke limit are derived and studied to determine the cooling transitions. The equations of motion for the density matrix in the Lambe-Dicke limit are derived using a simple representation of spontaneous emission. (Theses) (Author)
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 laser cooling of two ions that are trapped in a harmonic potential and interact by Coulomb repulsion. Sideband cooling in the Lamb-Dicke regime is shown to work analogously to sideband cooling of a single ion. Outside the Lamb-Dicke regime, the incommensurable frequencies of the two vibrational modes result in a quasicontinuous energy spectrum that significantly alters the cooling dynamics. The cooling time decreases nonlinearly with the linewidth of the cooling transition, and the effect of dark states which may slow down the cooling is considerably reduced. We show that cooling to the ground state is also possible outside the Lamb-Dicke regime. We develop the model and use quantum Monte Carlo calculations for specific examples. We show that a rate equation treatment is a good approximation in all cases.
A kinetic theory based on the Boltzmann equation is developed for the trapping of atomic ions in a radio-frequency quadrupole ion trap containing enough neutral atoms that ion-neutral collisions cannot be ignored. The collisions are treated at the same level of sophistication and detail as is used to deal with the time- and space-dependent electric fields in the trap. As a result, microscopic definitions are obtained for the damping and stochastic forces that originate from such collisions. These definitions contrast with corresponding phenomenological terms added ad hoc in previous treatments to create damped Mathieu and Langevin equations, respectively. Furthermore, the theory indicates that either collisional cooling or heating of the ions is possible, depending upon details of the ion-neutral mass ratios and interaction potential. The kinetic theory is not dependent on any special assumptions about the electric field strengths, the ion-neutral interaction potentials, or the ion-neutral mass ratio. It also provides an ab initio way to describe the ion kinetic energies, temperatures, and other properties by a series of successive approximations.
We study the probability distribution of an atomic ion being laser-cooled in a radio-frequency Paul trap, in terms of Hamiltonian action-angle coordinates appropriate for the time-dependent potential. Close to the effective minimum in many Paul traps, the periodically-driven potential is approximately quadratic in the coordinates. We find that despite the microscopic nonequilibrium forces, a stationary thermal-like exponential distribution in the action is reached after coarse-graining the angles, with a mean action corresponding to the Doppler cooling limit. However, residual electric fields can push the ion from the origin of the quadrupole potential and set it into a large-amplitude oscillation. Above a threshold amplitude of such excess micromotion, the action distribution for motion expanded about the driven oscillation broadens and becomes distinctly nonthermal. We find that by a proper choice of the laser detuning the distribution can be made exponential again, with a mean action close to the Doppler cooling limit. This is important for quantum information processing and other applications, and in particular the derived approach can be immediately applied to crystals of trapped ions in planar configurations, where the driven motion of ions is unavoidable.
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.
We analyze the possibility of cooling ions with a single laser beam, due to the coupling between the three components of their motion induced by the Coulomb interaction. For this purpose, we numerically study the dynamics of ion clouds of up to 140 particles, trapped in a linear quadrupole potential and cooled with a laser beam propagating in the radial plane. We use molecular dynamics simulations and model the laser cooling by a stochastic process. For each component of the motion, we systematically study the dependence of the temperature on the anisotropy of the trapping potential. Results obtained using the full radio-frequency (rf) potential are compared to those of the corresponding pseudopotential. In the rf case, the rotation symmetry of the potential has to be broken to keep ions inside the trap. Then, as for the pseudopotential case, we show that the efficiency of the Coulomb coupling to thermalize the components of motion depends on the geometrical configuration of the cloud. Coulomb coupling appears to be not efficient when the ions organize as a line or a pancake and the three components of motion reach the same temperature only if the cloud extends in three dimensions.