Lifetimes of metastable ion clouds in a Paul trap: Power-law scaling

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 ${\overline{\tau }}_m$ of the metastable cloud follows a power law, ${\overline{\tau }}_m\sim {(\gamma -{\gamma }_c)}^{-\beta }$, where ${\gamma }_c$ is the value of the damping constant at which the transition occurs. The exponent $\beta$ 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 $q, {\gamma }_c$ scales approximately like ${\gamma }_c=Cln[ln\left(N\right)]+D$ 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.

Figure 1

Square displacement $\langle x^2\left(\tau \right)\rangle$ of a 100-particle cloud for $q=0.2, a=0.02$, and $\gamma =8.81\times {10}^{-4}>{\gamma }_c=8.47\times {10}^{-4}$. An initial transient (thermalization stage) is followed by a plateau of length ${\tau }_m$ (metastable state), which ultimately transitions into a state of constant $\langle x^2\left(\tau \right)\rangle$ (flat line; crystalline state).

Figure 2

Critical value ${\gamma }_c(N,q=0.2,a=0.02)$ of the damping constant $\gamma$ [see (1)] as a function of $N$ at which the cloud $\to$ crystal transition occurs (red, closed circles). The best-fitting power law (blue, solid line), log law (green, solid line), and the iterated log law (red, solid line) are also shown. Only the iterated log law, according to (2), provides a satisfactory fit.

Figure 3

Average time ${\overline{\tau }}_m$ spent in the metastable state versus the distance $\gamma -{\gamma }_c$ from the critical point ${\gamma }_c$. (a) $q=0.15$, (b) $q=0.20$, and (c) $q=0.25$ for the spherical case $a=q^2/2$. Shown are $N=100,200,500$ (blue, red, green, respectively). (d) $q=0.20$, where $a=0$ (oblate, red), $a=q^2/2$ (spherical, blue), and $a=4q^2/5$ (prolate, green) for $N=100$. The exponents $\beta$ and the goodness of the fit, tested according to ${\chi }^2$ statistics, are available in Table 1.