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Physical Review A

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

J. W. Emmert, M. G. Moore, R. Blümel

Published Sep 1, 1993• 26 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.48.r1757

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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.
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.
Contrary to the results of standard pseudopotential analysis we predict the existence of an alignment transition region in the parameter space of the Paul trap in which the orientation of two-ion crystals with average angular momentum zero is different from x-y or z alignment. This prediction is based on an improved pseudopotential derived by keeping terms up to second order in the micromotion amplitudes. Our analytical predictions are supported by numerical evidence and can be checked experimentally with existing setups.
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.
We theoretically investigate the process of splitting two-ion crystals in segmented Paul traps, i.e. the structural transition from two ions confined in a common well to ions confined in separate wells. The precise control of this process by application of suitable voltage ramps to the trap segments is non-trivial, as the harmonic confinement transiently vanishes during the process. This makes the ions strongly susceptible to background electric field noise, and to static offset fields in the direction of the trap axis. We analyze the reasons why large energy transfers can occur, which are impulsive acceleration, the presence of residual background fields and enhanced anomalous heating. For the impulsive acceleration, we identify the diabatic and adiabatic regimes, which are characterized by different scaling behavior of the energy transfer with respect to time. We propose a suitable control scheme based on experimentally accessible parameters. Simulations are used to verify both the high sensitivity of the splitting result and the performance of our control scheme. Finally, we analyze the impact of trap geometry parameters on the crystal splitting process.