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Journal of Atomic and Molecular Physics

Periodical motion of an ion-cloud in an anharmonic Paul trap

Hai Kuo

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Interaction between an external detecting field and an ion-cloud in an actual Paul trap is described by the Duffing equation.In this paper,we reported an exactly periodical solution,which reveals the dependence of the trap parameters and external field on the ion-cloud's periodical motion.The result shows that under the periodicity condition the frequency of ion's motion is one third of the frequency of detecting field,the motion amplitude is positively correlated to the frequency and amplitude of the detecting field and the trap frequency respectively,and is negatively correlated with the anharmonic parameters of the Paul trap.

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Interaction between an external detecting field and an ion-cloud in an actual Paul trap is described by the Duffing equation.In this paper,we reported an exactly periodical solution,which reveals the dependence of the trap parameters and external field on the ion-cloud's periodical motion.The result shows that under the periodicity condition the frequency of ion's motion is one third of the frequency of detecting field,the motion amplitude is positively correlated to the frequency and amplitude of the detecting field and the trap frequency respectively,and is negatively correlated with the anharmonic parameters of the Paul trap.
In an experiment using a Paul trap, we create a H2+{\mathrm{H}}_{2}^{+}H2+​ ion cloud by electron ionization of the background gas at ${10}^{\ensuremath{-}9}$-mbar residual pressure. Exciting the ions parametrically at twice the frequency of the secular motion of ions in the rrr or zzz direction, we observe a narrow resonance at some distance from the motional resonance center if the amplitude of the exciting field exceeds a threshold value. The threshold value decreases with increasing ion number. Since the narrow resonance does not shift with ion number, we interpret it as a collective resonance of the center of mass of the ion cloud. The resonance shape exhibits the typical form of a driven anharmonic oscillator. The conclusions drawn from the experiments are supported by detailed analytical and numerical computations.
Collective motions and anharmonic oscillations induced by a mixture of two ion species with different masses in a rf ion trap are reported. The absorption spectra of the probing rf power by the resonant oscillations of ion clouds have been observed for several ion species with various mixing ratios. It is shown that significant nonlinear effects manifest themselves strongly when the mass difference is in a certain range. One of such nonlinear effects, the first observation of significant hysteresis in the absorption spectrum of the probing rf field, is understood to be due to the anharmonic force term of the third order induced by the interactions of two ion clouds. The term has been estimated by fitting the observed absorption profile to the approximate solution of Duffing's equation.
Excitation of the motional frequencies of clouds of H2+ and N2+ ions, confined in a Paul trap, by an additionally applied radiofrequency quadrupole field leads to the observation of parametric resonances that are predicted to occur at frequencies 2ωz/n, n=1,2,3…, where ωz is the axial secular frequency. In the case of clouds of N2+ ions resonances up to n=10 are detected. They can be explained by parametric instabilities of the center-of-mass motion in the axial direction. The fractional resonances are observable only if the excitation field surpasses a critical strength. We find an odd-even staggering of the thresholds.
Anharmonicity of the quantized motional states of ions in a Paul trap can be utilized to address the state-to-state transitions selectively and control the motional modes of trapped ions coherently and adiabatically [Zhao and Babikov, Phys. Rev. A 77, 012338 (2008)]. In this paper we study two sources of the vibrational anharmonicity in the ion traps: the intrinsic Coulomb anharmonicity due to ion-ion interactions and the external anharmonicity of the trapping potential. An accurate numerical approach is used to compute energies and wave functions of vibrational eigenstates. The magnitude of the Coulomb anharmonicity is determined and shown to be insufficient for successful control. In contrast, anharmonicity of the trapping potential allows one to control the motion of ions very efficiently using the time-varying electric fields. Optimal control theory is used to derive the control pulses. One ion in a slightly anharmonic trap can be easily controlled. In the two- and three-ion systems the symmetric stretching mode is dark and cannot be controlled at all. The other two normal modes of the three-ion system can be controlled and used, for example, to encode a two-qubit system into the motional states of ions. A trap architecture that allows the necessary amount of vibrational anharmonicity to be achieved is proposed.