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Journal of Physics A Mathematical and Theoretical

Systematic investigations on ion dynamics with noises in Paul trap

Ying-Xiang Wang, Sheng-Chen Liu, Lin Cheng, Liang-You Peng

Published Oct 25, 2023• 2 citations
Publisher

Abstract

Abstract Ions confined in a Paul trap serve as crucial platforms in various research fields, including quantum computing and precision spectroscopy. However, the ion dynamics is inevitably influenced by different types of noise, which require accurate computations and general analytical analysis to facilitate diverse applications based on trapped ions with white or colored noise. In the present work, we investigate the motion of ions in a Paul trap via the Langevin equation using both analytical and numerical methods, systematically studying three different types of noise: the white noise, the colored noise via the Ornstein–Uhlenbeck process and the Wiener process. For the white noise of the case, we provide a recursion method to calculate ion motion for a wide range of parameters. Furthermore, we present an analytical solution to the more realistic stochastic process associated with the colored noise, verified by the Monte Carlo simulation. By comparing the results of the colored noise with those of the white noise, and additionally considering another limit of noise parameters corresponding to the Wiener process, we summarize the effects of different noise types on the ion dynamics.

DOI
10.1088/1751-8121/ad0348

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Abstract Ions confined in a Paul trap serve as crucial platforms in various research fields, including quantum computing and precision spectroscopy. However, the ion dynamics is inevitably influenced by different types of noise, which require accurate computations and general analytical analysis to facilitate diverse applications based on trapped ions with white or colored noise. In the present work, we investigate the motion of ions in a Paul trap via the Langevin equation using both analytical and numerical methods, systematically studying three different types of noise: the white noise, the colored noise via the Ornstein–Uhlenbeck process and the Wiener process. For the white noise of the case, we provide a recursion method to calculate ion motion for a wide range of parameters. Furthermore, we present an analytical solution to the more realistic stochastic process associated with the colored noise, verified by the Monte Carlo simulation. By comparing the results of the colored noise with those of the white noise, and additionally considering another limit of noise parameters corresponding to the Wiener process, we summarize the effects of different noise types on the ion dynamics.
Based on the Langevin equation, a stochastic formulation is implemented to describe the dynamics of a trapped ion in a bath of ultracold atoms, including an excess of micromotion. The ion dynamics is described following a hybrid analytical-numerical approach in which the ion is treated as a classical impurity in a thermal bath. As a result, the ion energy's time evolution and distribution are derived from studying the sympathetic cooling process. Furthermore, the ion dynamics under different stochastic noise terms is also considered to gain information on the bath properties' role in the system's energy transfer processes. Finally, the results obtained from this formulation are contrasted with those obtained with a more traditional Monte Carlo approach.
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.
Linear Paul traps use radio frequency fields to confine ions for long times in deep potentials. The simplicity and versatility of these devices has led to their use in a wide variety of fields, from mass spectroscopy to quantum information. Within the past fifteen years, beginning with pioneering work at UConn, Paul traps have been combined with atom traps to form hybrid atom-ion traps. Hybrid traps serve as platforms for measuring ion-atom interaction rates, sympathetically cooling atomic and molecular ions, and studying the behavior of trapped ions. This work discusses simulations, analytical theory, and experiments in a hybrid trap that show a previously unknown dynamical effect in a population of trapped ions. The steady state trapped ion number in a Paul trap is non-monotonic with the loading rate of the Paul trap. This effect is universal to both Linear and 3D Paul traps and is likely to be found in other types of ion traps as well because it is the result of the Coulomb interaction in the presence of a trap. This behavior also affects the decay of ions from the trap. Understanding the loading and decay behavior from a saturated Linear Paul trap is important for the study of ion-atom interactions in a hybrid trap when the ions lack optical transitions.
Ion traps have been pivotal in opening new frontiers for the precision spectroscopy of stable ions. We report on the demonstration of an additional ion trap: the linear combined trap. This device is particularly well suited for trapping ions with unstable nuclei, due to its large range of stability parameters that facilitates external injection online to an accelerator. The motion of an ion in such a linear combined trap is investigated theoretically and experimentally. In the trap ions oscillate harmonically in the radial direction and move nearly harmonically between fixed boundaries along the longitudinal axis. The presence of a homogeneous magnetic field and the applied dc and rf electric fields, lead to a set of coupled Mathieu equations. Their approximate solutions exhibit motional frequencies, which are combinations of shifted macromotion frequencies and the cyclotron frequency. The dependence of these motional frequencies on the applied trapping fields was studied in detail. For the measurements we used small clouds of laser cooled B9e+ ions and crystallized B9e+ ions. The observed oscillation frequencies are compared to both the results of zeroth order analytical solutions and to numerical simulations in which the frequency spectrum was obtained from the ion orbits by using the fast fourier transform formalism. The various motional resonances were experimentally recorded by applying a weak dipole excitation field to one of the trap electrodes, and by simultaneously observing at resonance the changes in the fluorescence intensity. Depending on the detuning of the cooling laser the ions gain energy during the motional excitation process in such a way that they are shifted further in or out of optical resonance by the Doppler effect. This leads to either positive or negative ion motion signals at the various ion oscillation frequencies.
The motion of a particle in a finite-depth confining potential, such as that of an electromagnetically trapped atom or ion, is approximately harmonic at low energy and softens as the escape energy is approached. When the particle is driven by a stochastic force, the heating and escape times depend sensitively on the noise power spectrum. We study escape times of particles in softening potentials driven by power-law colored noise. We identify two nested conditions for weak noise: the particle must execute many oscillations before escape, and noise forces must be small relative to forces exerted by the trap so as not to tilt the potential appreciably. In this regime, we derive expressions for the mean escape time using stochastic averaging techniques and compare them with Monte Carlo simulations of the full equations of motion. We also examine the crossover to regimes where these conditions are not met and large quasi-static forces give rise to additional escape mechanisms. Applying these results to an experimental measurement of lifetimes in a microfabricated ion trap, we find that spectral color alone cannot account for the rapid escape rates observed, whereas quasi-static barrier tilting by a stray field of plausible magnitude could. These results are relevant to the heating and escape of trapped atoms due to noisy force fields, as well as to other nonlinear oscillators subject to colored noise.
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.