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Physical Review A

Universal heating curve of damped Coulomb plasmas in a Paul trap

J. D. Tarnas, Yunseong Nam, R. Blümel

Published Oct 4, 2013• 12 citations
Publisher

Abstract

A charged-particle cloud in a Paul trap is a nonlinear system that exhibits deterministic heating, i.e., conversion of energy provided by the radio-frequency (rf) drive of the trap into the disordered, gaslike motions of the trapped particles. In the presence of weak damping clouds reach an equilibrium state in which heating and cooling balance. For large damping, a cloud may collapse into a crystal. While the rf heating power of a given cloud depends sensitively on its particle number, its equilibrium size, the damping, and the control parameters of the trap, we find that scaling the heating power of the cloud with respect to its associated Coulomb-decoupled, non-interacting-gas state, and its equilibrium size with respect to its crystal size, results in a single, universal heating curve, independent of particle number and trap control parameters.

DOI
10.1103/physreva.88.041401

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A charged-particle cloud in a Paul trap is a nonlinear system that exhibits deterministic heating, i.e., conversion of energy provided by the radio-frequency (rf) drive of the trap into the disordered, gaslike motions of the trapped particles. In the presence of weak damping clouds reach an equilibrium state in which heating and cooling balance. For large damping, a cloud may collapse into a crystal. While the rf heating power of a given cloud depends sensitively on its particle number, its equilibrium size, the damping, and the control parameters of the trap, we find that scaling the heating power of the cloud with respect to its associated Coulomb-decoupled, non-interacting-gas state, and its equilibrium size with respect to its crystal size, results in a single, universal heating curve, independent of particle number and trap control parameters.
We have carried out a molecular-dynamics simulation of hundreds of ions confined in a Paul trap. The simulation includes the trapped particles' micromotion and interparticle Coulomb interactions. We have implemented a random walk in velocity to bring the secular motion to a given temperature which we have numerically measured. When the coupling \ensuremath{\Gamma} is large the ions form concentric shells which undergo a quadrupole oscillation at the rf frequency, while the ions within a shell form a 2D hexagonal lattice. Ion clouds at 5 mK show no rf heating for qz{\mathit{q}}_{\mathit{z}}qz​\ensuremath{\lesssim}0.6, whereas rapid heating is seen for qz{\mathit{q}}_{\mathit{z}}qz​=0.8.
We investigate the energy dynamics of noncrystallized (melted) ions, confined in a Paul trap. The nonperiodic Coulomb interaction experienced by melted ions forms a medium for nonconservative energy transfer from the radio-frequency (rf) field to the ions, a process known as rf heating. We study rf heating by analyzing numerical simulations of noncrystallized ion motion in Paul trap potentials, in which the energy of the ions' secular motion changes at discrete intervals, corresponding to ion-ion collisions. The analysis of these collisions is used as a basis to derive a simplified model of rf heating energy dynamics, from which we conclude that the rf heating rate is predominantly dependent on the rf field strength. We confirm the predictability of the model experimentally: Two trapped 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ ions are deterministically driven to melt, and their fluorescence rate is used to infer the ions' energy. Under these experimental conditions, we investigate which Doppler cooling parameters are required for efficient recrystallization, and compare our results to simulations.
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.
The nonlinear dynamics of large ion clouds ($N\ensuremath{\ge}256$ ions) trapped in radio-frequency (rf) traps and coupled to laser cooling give rise to a bistable behavior of the temperature. Numerical simulations of the free evolution of a large three-dimensional spherical cloud is used to characterize how the oscillating field amplitude and the ion number control the rf-heating rate and the switching between the two stable states. We show that the heating rate does not significantly depend on the ion number but strongly depends on the oscillating field. This exhibits the major role played by the density of the ensemble. The competition between the radio-frequency heating and the laser cooling is also discussed. They are used to design scenarios to take advantage of the threshold effect to detect and quantify perturbations by an intruder.
It is well known that ions stored in a Paul trap, one of the most versatile tools in atomic, molecular, and optical (AMO) physics, may undergo a transition from a disordered cloud state to a geometrically well-ordered crystalline state, the Wigner crystal. In this paper we predict that close to the transition, the average lifetime ${\overline{\ensuremath{\tau}}}_{m}$ of the metastable cloud follows a power law, ${\overline{\ensuremath{\tau}}}_{m}\ensuremath{\sim}{(\ensuremath{\gamma}\ensuremath{-}{\ensuremath{\gamma}}_{c})}^{\ensuremath{-}\ensuremath{\beta}}$, where ${\ensuremath{\gamma}}_{c}$ is the value of the damping constant at which the transition occurs. The exponent $\ensuremath{\beta}$ depends on the trap control parameter qqq, but is independent of both the number of particles NNN stored in the trap and the trap control parameter aaa, which determines the shape (oblate, prolate, or spherical) of the ion cloud. In addition, we find that for given aaa and $q, {\ensuremath{\gamma}}_{c}$ scales approximately like ${\ensuremath{\gamma}}_{c}=Cln[ln(N)]+D$ as a function of NNN, where CCC and DDD are constants. Our predictions may be tested experimentally with equipment already available at many AMO laboratories. In addition to their importance in AMO trap physics, we also discuss possible applications of our results to other periodically driven many-particle systems, such as, e.g., particle accelerator beams, and, based on our trap results, conjecture that power laws characterize the phase transition to the Wigner crystal in all such systems.
We study the stochastic dynamics of a particle in a periodically driven potential. For atomic ions trapped in radio-frequency Paul traps, noise heating and laser cooling typically act slowly in comparison with the unperturbed motion. These stochastic processes can be accounted for in terms of a probability distribution defined over the action variables, which would otherwise be conserved within the regular regions of the Hamiltonian phase space. We present a semiclassical theory of low-saturation laser cooling applicable from the limit of low-amplitude motion to large-amplitude motion, accounting fully for the time-dependent and anharmonic trap. We employ our approach to a detailed study of the stochastic dynamics of a single ion, drawing general conclusions regarding the nonequilibrium dynamics of laser-cooled trapped ions. We predict a regime of anharmonic motion in which laser cooling becomes diffusive (i.e., it is equally likely to cool the ion as it is to heat it), and can also turn into effective heating. This implies that a high-energy ion could be easily lost from the trap despite being laser cooled; however, we find that this loss can be counteracted using a laser detuning much larger than Doppler detuning.