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Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика

Efficient computational scheme for ion dynamics in RF-field of Paul trap

Vladimir S. Melezhik

Published Dec 15, 2019• 4 citations

Abstract

We have developed an efficient computational scheme for integration of the classical Hamilton equations describing the ion dynamics confined in the radio-frequency field of the Paul trap. It has permitted a quantitative treatment of cold atom-ion resonant collisions in hybrid atom-ion traps with taking into account unremovable ion micromotion caused by the radio-frequency fields (V.S. Melezhik et. al., Phys. Rev. A100, 063406 (2019)). The important element of the hybrid atom-ion systems is the electromagnetic Paul trap confining the charged ion. The oscillating motion of the confined ion is defined by two frequencies of the Paul trap. It is the frequency of the order of 100 kHz due to the constant electric field and the radio-frequency of about 1-2 MHz defined by the alternating electromagnetic field of the ion trap. The necessity to accurately treat the ion motion in the combined field with two time scales defined by these two very different frequencies has demanded to develop the stable computational scheme for integration of the classical Hamilton equations for the ion motion. Moreover, the scheme must be stable on rather long time-interval of the ion collision with the cold atom ∼ 10 × 2/ defined by the atomic trap frequency ∼ 10 kHz and in the moment of the atom-ion collision when the Hamilton equations are strongly coupled. The developed numerical method takes into account all these features of the problem and makes it possible to integrate the system of coupled quantum-semiclassical equations with the necessary accuracy and quantitatively describes the processes of atomic-ion collisions in hybrid traps, including resonance effects.

DOI: 10.22363/2658-4670-2019-27-4-378-385

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We have developed an efficient computational scheme for integration of the classical Hamilton equations describing the ion dynamics confined in the radio-frequency field of the Paul trap. It has permitted a quantitative treatment of cold atom-ion resonant collisions in hybrid atom-ion traps with taking into account unremovable ion micromotion caused by the radio-frequency fields (V.S. Melezhik et. al., Phys. Rev. A100, 063406 (2019)). The important element of the hybrid atom-ion systems is the electromagnetic Paul trap confining the charged ion. The oscillating motion of the confined ion is defined by two frequencies of the Paul trap. It is the frequency of the order of 100 kHz due to the constant electric field and the radio-frequency of about 1-2 MHz defined by the alternating electromagnetic field of the ion trap. The necessity to accurately treat the ion motion in the combined field with two time scales defined by these two very different frequencies has demanded to develop the stable computational scheme for integration of the classical Hamilton equations for the ion motion. Moreover, the scheme must be stable on rather long time-interval of the ion collision with the cold atom ∼ 10 × 2/ defined by the atomic trap frequency ∼ 10 kHz and in the moment of the atom-ion collision when the Hamilton equations are strongly coupled. The developed numerical method takes into account all these features of the problem and makes it possible to integrate the system of coupled quantum-semiclassical equations with the necessary accuracy and quantitatively describes the processes of atomic-ion collisions in hybrid traps, including resonance effects.

We investigate confinement-induced resonances in atom-ion quantum mixtures confined in hybrid traps for small atom-ion mass ratios. Specifically, we consider an ion confined in a time-dependent radio-frequency Paul trap with linear geometry, while the atom is constrained to move into a quasi-one-dimensional optical waveguide within the ion trap. We evaluate the impact of the ion intrinsic micromotion on the resonance position. Thus, we solve the atom-ion dynamics semiclassically, namely, the atom dynamics is governed by the three-dimensional time-dependent Schr\"odinger equation, whereas the ion motion is described by the classical Hamilton equations. We find that the energy of the ion provided by the oscillating radio-frequency fields can affect the resonance position substantially. Notwithstanding, the peculiar phenomenology of those resonances regarding perfect transmission and reflection is still observable. These findings indicate that the intrinsic micromotion of the ion is not detrimental for the occurrence of the resonance and that its position can be controlled by the radio-frequency fields. This provides an additional means for tuning atom-ion interactions in low spatial dimensions. The study represents an important advancement in the scattering physics of compound atomic quantum systems in time-dependent traps.

Advances in research in fields such as quantum information and quantum chemistry require subtle methods for trapping particles, including ions, neutral atoms, and molecules. Here, we propose a hybrid ion-trapping method that combines a Paul trap with optical tweezers. This trap incorporates both the deep-potential characteristic of the Paul trap and the micromotion-free nature of the optical dipole trap. By modulating the optical dipole trap synchronously with the radio-frequency voltage of the Paul trap, the alternating electrical force in the trap center is fully counteracted, and the micromotion temperature of a cold trapped ion can reach the nanokelvin scale, while the trap depth exceeds 300 K. These features will allow cold collisions between an ion and an atom in the

s

-wave regime and stably trap the produced molecular ion in the cold hybrid system. This provides a unique platform for probing the interactions between the ions and the surrounding neutral particles and enables the investigation of new reaction pathways and reaction products in the cold regime.

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 (Ca

^{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.

Ultracold atom-ion systems have been a topic of interest for atomic physicists studying chemical reactions and since recently, the cold ion community (ion trap quantum computation and simulation). They have been looking at the possibility of using an ultracold atom gas to sympathetically cool ions since intrinsic motional modulation i.e micromotion is an inherent cause of decoherence in coherent applications of cold ions. Interest is also piqued by the possibility of using this hybrid system for studying impurity physics and to better understand ion-neutral reactions aimed at creation of molecular ions. In this thesis, we aim to study the effect of ion micromotion in atom-ion collision. As a prelude, we treat the 1D collision of a particle in a harmonic trap (ion) and a free particle (atom) using different numerical schemes. This system is of interest in its own right due to the mixed 0D-1D dimensionality. Atom-ion potential is simpliﬁed to a zero range potential all through out the work. Next we deal with a similar problem but with the trapped particle in a time dependent harmonic trap identical to an ion Paul trap. Finally we extend the study of micromotion to a model system in 3D with an ion in a 3D spherical Paul trap and a heavy atom at the trap centre. We discuss the effect micromotion has on potential applications of such a system, like a quantum phase gate.

We propose a way for sympathetic cooling of ions in an electromagnetic Paul trap: it implies the use for this purpose of cold buffer atoms in the region of atom-ion confinement-induced resonance (CIR). The problem is that the unavoidable micromotion of the ion and the long-range nature of its interaction with the environment of colder atoms in a hybrid atomic-ion trap prevent its sympathetic cooling. We show that the destructive effect of ion micromotion on its sympathetic cooling can, however, be suppressed in the vicinity of the atom-ion CIR. Here, the resonant blocking of the complete approach of an atom with an ion during a collision also blocks the enhancement of its micromotion. We investigate the effect of sympathetic cooling around CIRs in atom-ion and atom-atom confined collisions within the quasiclassical-quantum approach using the $\mathrm{Li}\ensuremath{-}{\mathrm{Yb}}^{+}$ and Li-Yb confined systems as an example. In this approach, the Schr\"odinger equation for a cold light atom is integrated simultaneously with the classical Hamilton equations for a hotter heavy ion or atom during collision. We have found the region near the atom-ion CIR where the sympathetic cooling of the ion by cold atoms is possible in a hybrid atom-ion trap. We also show that it is possible to improve the efficiency of sympathetic cooling in atomic traps by using atomic CIRs.

In the development of atomic, molecular, and optical (AMO) physics, atom-ion hybrid systems are characterized by the presence of a new tool in the experimental AMO toolbox: atom-ion interactions. One of the main limitations in state-of-the-art atom-ion experiments is represented by the micromotion component of the ions’ dynamics in a Paul trap, as the presence of micromotion in atom-ion collisions results in a heating mechanism that prevents atom-ion mixtures from undergoing a coherent evolution. Here, we report the design and the simulation of a novel ion trapping setup especially conceived of for integration with an ultracold atoms experiment. The ion confinement is realized by using an electro-optical trap based on the combination of an optical and an electrostatic field, so that no micromotion component will be present in the ions’ dynamics. The confining optical field is generated by a deep optical lattice created at the crossing of a bow-tie cavity, while a static electric quadrupole ensures the ions’ confinement in the plane orthogonal to the optical lattice. The setup is also equipped with a Paul trap for cooling the ions produced by photoionization of a hot atomic beam, and the design of the two ion traps facilitates the swapping of the ions from the Paul trap to the electro-optical trap.