Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Nonlinear resonances and the melting of Wigner crystals in a Paul trap

Andreas W. Vogt

Published Feb 1, 1994• 11 citations
Publisher

Abstract

This article discussed the melting of ``crystals'' of two laser-cooled ions in a Paul trap. It presents computer simulations which reproduce melting at some realistic trap parameters in a model which includes laser-ion interaction but no collisions with residual gas molecules, highlighting the importance of spontaneous-emission fluctuations. Crystal fragility at certain trap voltages coincides with low-order nonlinear resonances, an aspect not mentioned in earlier discussions of the system.

DOI
10.1103/physreva.49.r657

Related papers

Prediction of a deterministic melting transition of two-ion crystals in a Paul trap

J. W. Emmert, M. G. Moore, R. Blümel

With the help of detailed molecular-dynamics calculations we prove the existence of an instability region I embedded in the single-ion Mathieu stability region M of Paul-trap control parameters a and q. Two-ion crystals do not exist in I. A two-ion crystal created outside of I can be dragged into I by an adiabatic change of a and q. A deterministic order-chaos transition occurs at the point of entry into I. This type of melting transition does not depend on the presence of laser noise and occurs even in the absence of collisions with rest-gas atoms. The prediction can be checked with existing experimental setups.

Physical Review A•Sep 1, 1993•26 citations

Chaos and order of laser-cooled ions in a Paul trap

R. Blümel, C. Kappler, W. Quint +1 more

We present recent experimental and theoretical results on the behavior of two-, three-, and four-ion crystals close to the Mathieu instability. In particular, we show that the crystals are stable until the Mathieu instability is reached, i.e., for the parameter space investigated, there is no ``melting'' of the crystals. Laser and rf heating are studied in detail. A simple model of chaotic rf heating as well as a classification of the ion dynamics into four characteristic regimes are presented.

Physical Review A•Jul 1, 1989•267 citations

Inhibition of laser cooling in Paul traps

Ralph G. DeVoe, John A. Hoffnagle, Richard G. Brewer

Several groups have recently used laser cooling to produce ordered arrays of ions, ion crystals, in rf quadrupole ion traps (Paul traps). Laser cooling has been understood using previous theories which ignore ion micromotion and replace the oscillating rf trap potential with a static pseudopotential. We show that in recent experiments, ion micromotion can exceed one wavelength of the cooling radiation and that the resulting Doppler shifts can transform laser cooling into laser heating. Changes in trap voltage, ion position, and laser intensity can all reverse the sign of laser cooling. This, together with ideas of deterministic chaos, can lead to a reinterpretation of experiments on ion crystals. The recognition that effects previously attributed to an experimental artifact, rf heating, may actually be due to a failure of laser cooling opens the door, on the one hand, to precise theories of phase transitions and, on the other, to improved cooling techniques that can permit the creation of larger ion crystals.

Annual Meeting Optical Society of America•Jan 1, 1988•0 citations

Search for superradiance of two trapped ions

Ralph G. DeVoe, Richard G. Brewer

Laser-cooled trapped ion crystals offer a unique opportunity to study the interactions between atoms. We have previously developed a microscopic planar Paul trap1 which is small and strong enough to bring the ion separation into the sub-micron region where quantum collective effects become important. A search is underway for the superradiant coupling of two atoms which manifests itself as a variation of the spontaneous emission rate with the ion-ion distance. We have produced an ion crystal with a separation of 1.1 microns in an 80 micron radius planar Paul trap. At this distance, equal to 2.2 wavelengths of the 493 nm Ba + transition, superradiant effects should alter the decay rate by about 10%. The two-ion crystal is imaged with sub-micron resolution by a new diffraction-limited vacuum-compatible microscope. The spontaneous emission rate is measured by a high repetition-rate transient spectrometer with an error of less than 1% for a single ion in a few minutes. Efficient extinction of the exciting laser light is provided by a new nonlinear electrooptic sideband switch.

Annual Meeting Optical Society of America•Jan 1, 1993•0 citations

Non-stationary Coulomb crystals in linear Paul traps

Michael Drewsen, Inger S. Jensen, Niels Kjærgaard +4 more

Ion Coulomb crystals form when trapped ions are cooled below a certain critical temperature. In radiofrequency (RF) traps, such crystals are usually stationary (i.e. non-rotating) with shapes similar to those of cold, homogeneously charged liquids when the number of ions is sufficient for three-dimensional structures to appear. ‘Dynamically stable’ structures can, however, also be created either by directly applying a torque to an initially stationary crystal or by choosing trap parameters just within the stability region where stationary crystals will melt due to parametric resonances between the RF trap field and ion plasma modes.

Journal of Physics B Atomic Molecular and Optical Physics•Jan 23, 2003•15 citations

Controlled melting of a Wigner ion crystal with atomic resolution

Yong Xu, W.-D. Zhao, YunLei Hou +7 more

The solid-liquid phase transition (melting) is a long-standing problem for which a complete understanding is challenging because of the strong many-body interactions. A trapped-ion Wigner crystal is a suitable platform for a controlled study of this phase transition with its unprecedented resolution of individual atomic ions. Here we study the melting dynamics of a linear chain of $^{174}\mathrm{Yb}^{+}$ ions, which is initially cooled to the Doppler temperature and is then periodically heated and detected by a focused laser beam. We achieve controlled melting of the ion chain and observe unusual partial melting of long chains (coexistence of solid and liquid phases). We also observe a counterintuitive nonuniform energy distribution over the chain under localized heating. These results are further confirmed by molecular dynamics simulations to rule out other heating mechanisms.

Physical Review A•Dec 28, 2020•5 citations
This article discussed the melting of ``crystals'' of two laser-cooled ions in a Paul trap. It presents computer simulations which reproduce melting at some realistic trap parameters in a model which includes laser-ion interaction but no collisions with residual gas molecules, highlighting the importance of spontaneous-emission fluctuations. Crystal fragility at certain trap voltages coincides with low-order nonlinear resonances, an aspect not mentioned in earlier discussions of the system.
With the help of detailed molecular-dynamics calculations we prove the existence of an instability region I embedded in the single-ion Mathieu stability region M of Paul-trap control parameters a and q. Two-ion crystals do not exist in I. A two-ion crystal created outside of I can be dragged into I by an adiabatic change of a and q. A deterministic order-chaos transition occurs at the point of entry into I. This type of melting transition does not depend on the presence of laser noise and occurs even in the absence of collisions with rest-gas atoms. The prediction can be checked with existing experimental setups.
We present recent experimental and theoretical results on the behavior of two-, three-, and four-ion crystals close to the Mathieu instability. In particular, we show that the crystals are stable until the Mathieu instability is reached, i.e., for the parameter space investigated, there is no ``melting'' of the crystals. Laser and rf heating are studied in detail. A simple model of chaotic rf heating as well as a classification of the ion dynamics into four characteristic regimes are presented.
Several groups have recently used laser cooling to produce ordered arrays of ions, ion crystals, in rf quadrupole ion traps (Paul traps). Laser cooling has been understood using previous theories which ignore ion micromotion and replace the oscillating rf trap potential with a static pseudopotential. We show that in recent experiments, ion micromotion can exceed one wavelength of the cooling radiation and that the resulting Doppler shifts can transform laser cooling into laser heating. Changes in trap voltage, ion position, and laser intensity can all reverse the sign of laser cooling. This, together with ideas of deterministic chaos, can lead to a reinterpretation of experiments on ion crystals. The recognition that effects previously attributed to an experimental artifact, rf heating, may actually be due to a failure of laser cooling opens the door, on the one hand, to precise theories of phase transitions and, on the other, to improved cooling techniques that can permit the creation of larger ion crystals.
Laser-cooled trapped ion crystals offer a unique opportunity to study the interactions between atoms. We have previously developed a microscopic planar Paul trap1 which is small and strong enough to bring the ion separation into the sub-micron region where quantum collective effects become important. A search is underway for the superradiant coupling of two atoms which manifests itself as a variation of the spontaneous emission rate with the ion-ion distance. We have produced an ion crystal with a separation of 1.1 microns in an 80 micron radius planar Paul trap. At this distance, equal to 2.2 wavelengths of the 493 nm Ba + transition, superradiant effects should alter the decay rate by about 10%. The two-ion crystal is imaged with sub-micron resolution by a new diffraction-limited vacuum-compatible microscope. The spontaneous emission rate is measured by a high repetition-rate transient spectrometer with an error of less than 1% for a single ion in a few minutes. Efficient extinction of the exciting laser light is provided by a new nonlinear electrooptic sideband switch.
Ion Coulomb crystals form when trapped ions are cooled below a certain critical temperature. In radiofrequency (RF) traps, such crystals are usually stationary (i.e. non-rotating) with shapes similar to those of cold, homogeneously charged liquids when the number of ions is sufficient for three-dimensional structures to appear. ‘Dynamically stable’ structures can, however, also be created either by directly applying a torque to an initially stationary crystal or by choosing trap parameters just within the stability region where stationary crystals will melt due to parametric resonances between the RF trap field and ion plasma modes.
The solid-liquid phase transition (melting) is a long-standing problem for which a complete understanding is challenging because of the strong many-body interactions. A trapped-ion Wigner crystal is a suitable platform for a controlled study of this phase transition with its unprecedented resolution of individual atomic ions. Here we study the melting dynamics of a linear chain of 174Yb+^{174}\mathrm{Yb}^{+}174Yb+ ions, which is initially cooled to the Doppler temperature and is then periodically heated and detected by a focused laser beam. We achieve controlled melting of the ion chain and observe unusual partial melting of long chains (coexistence of solid and liquid phases). We also observe a counterintuitive nonuniform energy distribution over the chain under localized heating. These results are further confirmed by molecular dynamics simulations to rule out other heating mechanisms.