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The Journal of Chemical Physics

Kinetic theory of radio frequency quadrupole ion traps. I. Trapping of atomic ions in a pure atomic gas

Larry A. Viehland, Douglas E. Goeringer

Published Apr 28, 2004• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.1691405

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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.
The immersion of a single ion confined by a radiofrequency (RF) trap in an ultracold atomic gas extends the concept of buffer gas cooling to a new temperature regime. The steady-state energy distribution of the ion is determined by its kinetics in the RF field rather than the temperature of the buffer gas. Moreover, the finite size of the ultracold gas facilitates the observation of back-action of the ion onto the buffer gas. We numerically investigate the system's properties depending on atom–ion mass ratio, trap geometry, differential cross-section and non-uniform neutral atom density distribution. Experimental results are well reproduced by our model considering only elastic collisions. We identify excess micromotion to set the typical scale for the ion energy statistics and explore the applicability of the mobility collision cross-section to the ultracold regime.
We provide a comprehensive theoretical framework for describing the dynamics of a single trapped ion interacting with a neutral buffer gas, thus extending our previous studies on buffer-gas cooling of ions beyond the critical mass ratio [B. H\"oltkemeier et al., Phys. Rev. Lett. 116, 233003 (2016)]. By transforming the collisional processes into a frame, where the ion's micromotion is assigned to the buffer-gas atoms, our model allows us to investigate the influence of nonhomogeneous buffer-gas configurations as well as higher multipole orders of the radio-frequency trap in great detail. Depending on the neutral-to-ion mass ratio, three regimes of sympathetic cooling are identified which are characterized by the form of the ion's energy distribution in equilibrium. We provide analytic expressions and numerical simulations of the ion's energy distribution, spatial profile, and cooling rates for these different regimes. Based on these findings, a method for actively decreasing the ion's energy by reducing the spatial expansion of the buffer gas arises (forced sympathetic cooling).
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)
We apply kinetic theory to the problem of evaporative cooling of a dilute collisional gas in a trap. Assuming ``sufficient ergodicity'' (phase-space distribution only a function of energy) and s-wave collisions with an energy-independent cross section, an equation for the evolution of the energy distribution of trapped atoms is derived for arbitrary trap shapes. Numerical integration of this kinetic equation demonstrates that during evaporation the gas is accurately characterized by a Boltzmann distribution of atom energies, truncated at the trap depth. Adopting the assumption of a truncated Boltzmann distribution, closed expressions are obtained for the thermodynamic properties of the gas as well as for the particle and energy loss rates due to evaporation. We give analytical expressions both for power-law traps and for a realistic trapping potential (Ioffe quadrupole trap). As an application, we discuss the evaporative cooling of trapped atomic hydrogen gas. \textcopyright{} 1996 The American Physical Society.
We study the quantum dynamics of an ion confined in a radio-frequency trap in interaction with either a Bose or spin-polarized Fermi gas. To this end, we derive quantum optical master equations in the limit of weak coupling and the Lamb-Dicke approximations. For the bosonic bath, we also include the so-called ``Lamb-shift'' correction to the ion trap due to the coupling to the quantum gas as well as the extended Fr\"ohlich interaction within the Bogolyubov approximation that have been not considered in previous studies. We calculate the ion kinetic energy for various atom-ion scattering lengths as well as gas temperatures by considering the intrinsic micromotion and we analyze the damping of the ion motion in the gas as a function of the gas temperature. We find that the ion's dynamics depends on the quantum statistics of the gas and that a fermionic bath enables to attain lower ionic energies.
An ion in a radiofrequency (rf) trap sympathetically cooled by a simultaneously trapped neutral buffer gas exhibits deviations from thermal statistics caused by collision-induced coupling of the rf field to the ion motion. For a uniform density distribution of the buffer gas, the energy distribution of the ion can be described by Tsallis statistics. Moreover, runaway heating of the ion occurs if the buffer gas particles are sufficiently heavy relative to the ion. In typical experiments, however, ultracold buffer gases are confined in traps resulting in localised, non-uniform density distributions. Using a superstatistical approach, we develop an analytical model for an ion interacting with a localised buffer gas. We demonstrate theoretically that limiting collisions to the centre of the ion trap enables cooling at far greater mass ratios than achievable using a uniform buffer gas, but that an upper limit to the usable mass ratio exists even in this case. Furthermore, we analytically derive the functional form of the energy distribution for an ion interacting with a buffer gas held in a harmonic potential. The analytical distribution obtained is found to be in excellent agreement with the results of numerical simulations.