Universal critical phenomena of the cloud --> crystal phase transition in the Paul trap: Powerlaws

Daniel Weiss, Yunseong Nam, R. Blümel

Daniel Weiss, Yunseong Nam, R. Blümel

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 $q$, but is independent of both the number of particles $N$ stored in the trap and the trap control parameter $a$, which determines the shape (oblate, prolate, or spherical) of the ion cloud. In addition, we find that for given $a$ and $q, {\ensuremath{\gamma}}_{c}$ scales approximately like ${\ensuremath{\gamma}}_{c}=Cln[ln(N)]+D$ as a function of $N$, where $C$ and $D$ 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.

Mário J. de Oliveira

J. D. Tarnas, Yunseong Nam, R. Blümel

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.

David A. Lavis, Roman Frigg

HERBERT WALTHER

In the paper experiments dealing with the formation of ordered ion structures in Paul traps are reviewed. The transition from an ion cloud to a crystal-like structure corresponds to a chaos-order transition. Furthermore recent experiments in polarization gradient cooling of ions in a quadrupole ring trap are described. The results show that it should be possible to cool the trapped ions into their vibrational ground state.