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Physical Review A

Quantum theory of the stability region of an ion in a Paul trap

Wang Kelin, Feng Mang, Juhao Wu

Published Aug 1, 1995• 14 citations
Publisher

Abstract

The stability regions of an ion in a two-dimensional Paul trap are studied by a quantum-mechanical treatment, which is very important for the experiment when the temperature is fairly low. The result is compared with a classical one published before by Paul [Rev. Mod. Phys. 62, 531 (1990)]. It is shown that their properties are quite similar and their differences are related to the quantum effect, which cannot be included within a classical description. All of these properties are discussed in detail.

DOI
10.1103/physreva.52.1419

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Addendum to “Quantum theory of the stability region of an ion in a Paul trap”

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The authors call attention to a previous work [Kelin Wang, Mang Feng, and Juhao Wu, Phys. Rev. A 52, 1419 (1995)] for quantum-mechanically studying the stability region of the Paul trap, where the calculation does not work for the case of zero dc voltage. But the ions can be confined experimentally at zero dc voltage which is understandable via the Mathieu equation. We present a quantum-mechanical study for such a case as a supplement.

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Characteristics of single ion motion and mechanisms of trapping ions in the second stability region of a Paul trap are discussed. The mean kinetic energy and mean width of the spatial distribution of an ion cloud in that region are calculated. Similarities and differences of the results between the first and second stability regions are emphasized.

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Motivated by recent developments in ion trap design and fabrication, we investigate the stability of the motion of an ion in asymmetrical, planar versions of the linear Paul trap. The equations of motion of an ion in such a trap are generally coupled due to a nonzero relative angle θ between the principal axes of RF and DC fields, invalidating the assumptions behind the standard stability analysis for symmetric Paul traps. Using numerical methods, we obtain stability diagrams for the coupled system for various values of θ, generalizing the standard q-a stability diagrams. We then use multi-scale perturbation theory to obtain approximate formulas for the boundaries of the primary stability region and check our formulas against results from numerical analysis. Our results show that while the primary stability region is quite robust to changes in θ, a secondary stability region is highly variable, joining the primary stability region at the special case of θ=45°, which results in a significantly enlarged stability region for this particular angle. We conclude that while the standard stability diagrams for classical, symmetric Paul traps are not entirely accurate for asymmetric surface traps (or for other types of traps with a relative angle between the RF and DC axes), they are “safe” in the sense that operating conditions deemed stable according to standard stability plots are in fact stable for asymmetric traps, as well. By ignoring the coupling in the equations, one only underestimates the size of the primary stability region.

arXiv:1109.2160•Journal of Applied Physics•Oct 1, 2012•20 citations

Quantum stability of an ion in a Paul trap revisited

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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.

arXiv:1703.09463•Molecular Physics•Apr 10, 2017•0 citations

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

Achyut Khadka

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.

•Jan 1, 2013•0 citations

Quantum mechanical uncertainties and exact transition probabilities for time dependent quadratic Hamiltonian with special results for Paul ion trap

Gal Harari, Y. Ben-Aryeh, A. Mann

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.

•May 12, 2013•0 citations
The stability regions of an ion in a two-dimensional Paul trap are studied by a quantum-mechanical treatment, which is very important for the experiment when the temperature is fairly low. The result is compared with a classical one published before by Paul [Rev. Mod. Phys. 62, 531 (1990)]. It is shown that their properties are quite similar and their differences are related to the quantum effect, which cannot be included within a classical description. All of these properties are discussed in detail.
Motivated by recent developments in ion trap design and fabrication, we investigate the stability of the motion of an ion in asymmetrical, planar versions of the linear Paul trap. The equations of motion of an ion in such a trap are generally coupled due to a nonzero relative angle θ between the principal axes of RF and DC fields, invalidating the assumptions behind the standard stability analysis for symmetric Paul traps. Using numerical methods, we obtain stability diagrams for the coupled system for various values of θ, generalizing the standard q-a stability diagrams. We then use multi-scale perturbation theory to obtain approximate formulas for the boundaries of the primary stability region and check our formulas against results from numerical analysis. Our results show that while the primary stability region is quite robust to changes in θ, a secondary stability region is highly variable, joining the primary stability region at the special case of θ=45°, which results in a significantly enlarged stability region for this particular angle. We conclude that while the standard stability diagrams for classical, symmetric Paul traps are not entirely accurate for asymmetric surface traps (or for other types of traps with a relative angle between the RF and DC axes), they are “safe” in the sense that operating conditions deemed stable according to standard stability plots are in fact stable for asymmetric traps, as well. By ignoring the coupling in the equations, one only underestimates the size of the primary stability region.
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 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.
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.