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Time-dependent perturbation theory approach to the quantum motion of a particle in a Paul trap

Achyut Khadka

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

The main objective of this study is to investigate the quantum mechanical lifetime of a trapped ion in a Paul trap. The motivation for this research is that there are several applications of trapped ion devices such as mass spectrometers, atomic clocks, and quantum computers. Active research is presently ongoing in the subject of an ion trapping by electric and magnetic fields. There are many classical studies of the characteristics of trapped ions, however there have been relatively few done quantum mechanically. It might be expected that purely quantum effects could play an important role in describing the trapped ions. So, those are the main factors behind the research on this topic. Time-dependent perturbation theory was used in this study which is to find the quantum mechanical lifetime of trapped ion in a Paul trap because time-dependent perturbation theory allows describing the complex system with the help of a simple system that can be calculated easily. It is shown that the lifetime of a trapped ion depends on its transition frequency and the transition between two allowed quantum states of the ion. The transition frequency of an ion has to be very close to the applied frequency of alternating current (AC). The results show that the lifetime of a trapped ion can be adjusted as desired for experimentation.

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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.

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The main objective of this study is to investigate the quantum mechanical lifetime of a trapped ion in a Paul trap. The motivation for this research is that there are several applications of trapped ion devices such as mass spectrometers, atomic clocks, and quantum computers. Active research is presently ongoing in the subject of an ion trapping by electric and magnetic fields. There are many classical studies of the characteristics of trapped ions, however there have been relatively few done quantum mechanically. It might be expected that purely quantum effects could play an important role in describing the trapped ions. So, those are the main factors behind the research on this topic. Time-dependent perturbation theory was used in this study which is to find the quantum mechanical lifetime of trapped ion in a Paul trap because time-dependent perturbation theory allows describing the complex system with the help of a simple system that can be calculated easily. It is shown that the lifetime of a trapped ion depends on its transition frequency and the transition between two allowed quantum states of the ion. The transition frequency of an ion has to be very close to the applied frequency of alternating current (AC). The results show that the lifetime of a trapped ion can be adjusted as desired for experimentation.
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.
Large quantum uncertainties for the Paul trap are obtained near the stability border. These results are important as a trapped ion can be stored only as long as the fluctuations are smaller than the trap dimensions. Exact transition probabilities for the forced Paul trap and harmonic oscillator are also obtained based on a simplified version of the propagator for time-dependent harmonic oscillator. The results have important implications, e.g., the methods of mass spectrometry can be applied only in the region of small fluctuations, which can be calculated exactly by our method. The classical trajectory of a general forced harmonic oscillator is also calculated which may exhibit chaotic behavior.
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.
We investigate the quantum motion of two ions stored in a Paul trap and interacting with a time-periodic laser field. In the pseudopotential approximation and large detuning condition, we find that the relative motion is independent of the laser field, but the exact centre-of-mass motion is closely related to the laser field. By adjusting the laser intensity and frequency, we can well control the quantum motion of the centre-of-mass. We illustrate some physical properties described by the centre-of-mass states, such as the squeezed coherent property, the widths and heights of the wavepackets of probability density, the classical-quantum correspondence, the resonance ladders of expectation energy and the transition probabilities between time-dependent quantum levels.
Single trapped ions trapped in Paul traps correspond to ideal candidates for realization of extremely accurate optical atomic clocks and practical studies of the light–atom interactions and nonlinear mechanical dynamics. These systems benefit from both, the superb isolation of the ion from surrounding environment and excellent control of its external and internal\ndegrees of freedom, at the same time, which makes them exquisite platforms for experimental studies and applications of light matter interaction at its most fundamental level. The exceptional degree of control of single or few ion's state enabled in past decade number of major advancements in the applications from the fields of experimental quantum information\nprocessing and frequency metrology, including recent realization of scalable Shor's\nalgorithm, fractional uncertainties of the frequency measurements close to 10-18 level, or simulations of complex quantum many-body effects. These results, together with the rapid advancements in the production of low-noise segmented micro-traps, promise prompt access to long-desired regimes of quantum optomechanics and further development and applications\nof optical atomic clocks.
Trapped-ion devices are being investigated for transportable applications, including the transfer of exotic charged particles between laboratories and the development of portable quantum technologies. To date, ion transport has been demonstrated only in magnetic (Penning) traps, whereas radiofrequency (Paul) traps are the predominant platform in quantum science. Here, we report the first transport of a few thousand ions confined in a linear Paul trap over distances of several kilometers, with transport durations of 15 and 45 min. These results establish a foundation for mobile trapped-ion quantum devices capable of operating while in motion and may enable the transport of exotic ions in Paul traps between experimental facilities.