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

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

R. Blümel, C. Kappler, W. Quint, HERBERT WALTHER

Published Jul 1, 1989• 267 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.40.808

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

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

Role of laser damping in trapped ion crystals

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

It is currently thought that the laser cooling of an ordered ion array, an ion crystal, is qualitatively the same as that of a single ion. We show that this assumption is strongly violated in recent experiments in Paul traps, due to ion micromotion. Our analysis demonstrates an unexpected laser damping instability that induces an ion-crystal order-chaos transition and also explains why ion crystals in Paul traps have been limited to less than 100 ions. We directly observe the effects of ion micromotion on the laser cooling of a single trapped ${\mathrm{Ba}}^{+}$ ion by a new technique.

Physical Review A•May 1, 1989•68 citations

Order and chaos in laser cooled trapped ions

Ralph G. DeVoe, John A. Hoffnagle, Luis G. Reyna +1 more

Several groups have recently used laser cooling to produce ordered arrays of ions (ion crystals) in rf quadrupole traps. We discuss two physical mechanisms which underlie and explain the behavior of ion crystals. First, we show that deterministic chaos can be responsible for phase transitions between the ordered (crystal) state and the disordered (cloud) state.1 The crystal—cloud transition is explained as an order—chaos transition. Chaos arises from the nonlinear interaction of Coulomb repulsion with the oscillating trap potential.

Quantum Electronics and Laser Science Conference•Apr 24, 1989•0 citations

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

Andreas W. Vogt

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.

Physical Review A•Feb 1, 1994•11 citations

Formation process of large ion Coulomb crystals in linear Paul traps

Liv Hornekær, Michael Drewsen

We report on the results on various aspects of the formation of ion Coulomb crystals in linear Paul traps, for plasmas containing from a few hundred to several thousand laser-cooled ${}^{24}{\mathrm{Mg}}^{+}$ ions. The presented studies include observations of changes in emitted fluorescence, changes in plasma aspect ratio, and the formation of shell structure, during the transition from the cloudlike to the crystalline state. The competition between rf heating and laser cooling is observed to result in a very dramatic transition from the cloudlike disordered state to a liquidlike state. After the liquid state is reached ordering into shell structure takes place as a continuous process at even higher plasma coupling parameters. This is in contrast to earlier observations of few-ion plasmas in hyperbolic rf traps.

Physical Review A•Jul 29, 2002•66 citations
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.
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.
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.
It is currently thought that the laser cooling of an ordered ion array, an ion crystal, is qualitatively the same as that of a single ion. We show that this assumption is strongly violated in recent experiments in Paul traps, due to ion micromotion. Our analysis demonstrates an unexpected laser damping instability that induces an ion-crystal order-chaos transition and also explains why ion crystals in Paul traps have been limited to less than 100 ions. We directly observe the effects of ion micromotion on the laser cooling of a single trapped Ba+{\mathrm{Ba}}^{+}Ba+ ion by a new technique.
Several groups have recently used laser cooling to produce ordered arrays of ions (ion crystals) in rf quadrupole traps. We discuss two physical mechanisms which underlie and explain the behavior of ion crystals. First, we show that deterministic chaos can be responsible for phase transitions between the ordered (crystal) state and the disordered (cloud) state.1 The crystal—cloud transition is explained as an order—chaos transition. Chaos arises from the nonlinear interaction of Coulomb repulsion with the oscillating trap potential.
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.
We report on the results on various aspects of the formation of ion Coulomb crystals in linear Paul traps, for plasmas containing from a few hundred to several thousand laser-cooled 24Mg+{}^{24}{\mathrm{Mg}}^{+}24Mg+ ions. The presented studies include observations of changes in emitted fluorescence, changes in plasma aspect ratio, and the formation of shell structure, during the transition from the cloudlike to the crystalline state. The competition between rf heating and laser cooling is observed to result in a very dramatic transition from the cloudlike disordered state to a liquidlike state. After the liquid state is reached ordering into shell structure takes place as a continuous process at even higher plasma coupling parameters. This is in contrast to earlier observations of few-ion plasmas in hyperbolic rf traps.