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Physical Review A

Quantum motion of a charged particle in a Paul trap

Fuli Li

Published Jun 1, 1993• 14 citations
Publisher

Abstract

A numerical method to solve the Schr"odinger equation for a charged particle in a Paul trap is suggested. The quantum secular motion approximation is examined by comparing the exact numerical solution with the approximate result. The stability of quantum motion of the particle is discussed.
DOI
10.1103/physreva.47.4975

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Physical Review Letters•Jun 10, 1991•65 citations
Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.
This work is devoted to the investigation of possibility of controlling of ions motion inside Paul trap. It has been shown that by proper selection of the parameters of controlling electric fields, stable localization of ions inside Paul trap is possible. Quantum consideration of this problem is reduced to the investigation of the Mathieu-Schrodinger equation. It has been shown that quantum consideration is appreciably different from classical one that leads to stronger limitations of the values of the parameters of stable motion. Connection between the problem under study and the possibility of experimental observation of quantum chaos has been shown.
We present a theoretical study of trapping an electron in a two-frequency Paul trap, motivated by future hybrid architecture consisting of a doubly charged calcium ion (Ca2+^{2+}2+) and an electron. By isolating the electron from ionic-electron interactions, we formulate the equations of motion for the electron in the time-dependent trap potentials and cast them in the Hill form. Stability and dynamics are then analysed in terms of the dimensionless q-parameters, which encode the applied voltages, drive frequencies, trap geometry, and the particle's charge-to-mass ratio. Under the combined action of the two radio-frequency fields, the trapped electron's classical motion is shown to be approximated well by a slow secular oscillation modulated by two distinct motion sidebands originating from the low- and high-frequency RF drives. The approximation method presented here enables a quick evaluation of the relation between the device-design q-parameters and the electron motion. It offers a crucial foundation for developing physical models of electron-ion interactions in electron-ion traps.
We study the quantum stability of the dynamics of ions in a Paul trap. We revisit previous results that showed that quantum trajectories did not have the same region of stability as their classical counterpart, contrary to what is obtained from a Floquet analysis of the motion in the periodic trapping field. Using numerical simulations of the full wave-packet dynamics, we confirm that the classical trapping criterion is fully applicable to quantum motion, when considering both the expectation value of the position of the wave packet and its width.
The quantum motion of a charged particle in a quadrupole radio-frequency trap is solved exactly in terms of the classical trajectories. Thus the quantum stability regions are exactly given in terms of the stability regions for the associated Mathieu equations, and quantum trapping is demonstrated inside these stability area. These exact results enable us to test a commonly believed «static effective potential approximation» for a particle in a rapidly oscillating time-periodic potential: our results exhibit serious limitations to this approximation. As a subproduct of our approach, we solve the eigenvalue problem for the Floquet operator of our system. This exactly solvable system should be a good starting point for the study of quantum instabilities under small perturbations Le mouvement quantique d'une particule chargee dans un piege quadripolaire de radio-frequence est exactement resoluble en termes des solutions du mouvement classique. Il en resulte que les regions de stabilite quantique sont exactement les regions de stabilite pour les equations de Mathieu correspondantes, et on demontre le piegeage quantique a l'interieur de ces regions. Ces resultats exacts permettent de tester «l'approximation de potentiel effectif statique» communement admise pour une particule dans un potentiel periodique dans le temps rapidement oscillant, et mettent en evidence de serieuses limitations a cette approximation. Comme sous produit a cette approche, on resout l'equation aux valeurs propres pour l'operateur de Floquet de ce systeme. Ce probleme exactement resoluble devrait etre un bon point de depart pour l'etude de la stabilite quantique sous de petites perturbations
We have carried out a molecular-dynamics simulation of hundreds of ions confined in a Paul trap. The simulation includes the trapped particles' micromotion and interparticle Coulomb interactions. We have implemented a random walk in velocity to bring the secular motion to a given temperature which we have numerically measured. When the coupling \ensuremath{\Gamma} is large the ions form concentric shells which undergo a quadrupole oscillation at the rf frequency, while the ions within a shell form a 2D hexagonal lattice. Ion clouds at 5 mK show no rf heating for qz{\mathit{q}}_{\mathit{z}}qz​\ensuremath{\lesssim}0.6, whereas rapid heating is seen for qz{\mathit{q}}_{\mathit{z}}qz​=0.8.