Modes of oscillation in radiofrequency Paul traps

Abstract

We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general threedimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the ('Floquet-Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small 'peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.numerical study of a small crystal in a linear Paul trap, and conclude the paper in section 6 with comments on directions for further applications and research. Overview of Paul trappingThe first crystallization of a cloud of charged aluminum microparticles has been reported in [42]. Wigner crystals of 2 to approximately 100 trapped ions in a Paul trap were reported in [9,10] and further investigated in [43] both experimentally and numerically. The simulations included rf trapping, Coulomb interaction, laser cooling and random noise. Depending on trap parameters, the ions were found to equilibrate either as an apparently chaotic cloud or in an ordered structure. The latter is defined as the 'crystal' solution when it is a simple limit cycle, with the ions oscillating at the rf frequency about well-defined average points. The transition between the two phases has been investigated, it was shown that both phases can coexist and hysteresis in the transition has been observed [43].The motion of two ions in the Paul trap has been investigated in detail in various publications [44][45][46][47][48][49][50][51][52][53]. In addition to the aforementioned phases, frequency-locked periodic attractors (where the nonlinearity pulls the motional frequencies into integral fractions of the external rf frequency) were found in numerical simulations and experiments. These solutions are different from the crystal in that the ions move in extended (closed) orbits in the trap, whose period is a given multiple of the rf period. However, many of these frequency-locked solutions are unstable, especially those of a large period, and perturbations such as those coming from the nonlinearity of the laser-cooling mechanism tend to destroy them.Despite the large amplitude motion, the frequency-locked solutions, being periodic, are of course not chaotic. However, even in the presence of cooling, some solutions in the ion trap may behave chaotically for exponentially long times. The authors of [54] suggest that, eventually, all trajectories settle at frequency-locked attractors, at least for two ions at a = 0. Numerical simulations and experiments with more ion...

Precise Experimental Investigation of Eigenmodes in a Planar Ion Crystal

“…while in the axial z direction, the micromotion is negligible [34]. In the experiment, the resonance fluorescence of Doppler cooled 40 Ca + ion crystals near 397 nm is imaged on a CCD camera [37].…”

Prospects for atomic clocks based on large ion crystals

“…However the framework we have used follows that given in [14] and should allow such effects to be included given a particular design. The effect will be to give a spatial dependence to Λ s and Λ rf .…”

Investigation of two-frequency Paul traps for antihydrogen production

“…The validity of the pseudopotential approximation is discussed in detail in Refs. [28,29]. The pseudopotential for particle B in (5) can be arrived at by alternately considering the limiting cases where q B 1,2 go to zero.…”

“…[30], and application of Floquet theory to Paul traps in Refs. [28,29,31]. Here we simply state that the boundary of stability regions may be found by identifying parameters for which the solution x(τ ) has periodicity T or 2T .…”