Stability and dynamics of ion rings in linear multipole traps

Trapped singly charged ions can crystallize as a result of laser cooling. The emerging structure depends on the number of particles and on the geometry of the trapping potential. In linear multipole radio frequency traps, the geometry of the radial potential can lead to the formation of single-ring structures. We analyze the conditions and stability of single rings as a function of the number of poles. The rings form tubes for a large number of ions and sufficiently small trap aspect ratios. In these structures the arrangement of the ions corresponds to a triangular lattice folded onto a cylinder. The stability of the tubular structures is numerically studied for different lattice constants and their normal-mode spectrum is determined.

Figure 1

Schematic representation of a linear octupole radio frequency trap. The trap consists of eight parallel rods that are situated on a ring with equal radial spacing. Each rod is separated into three segments. An oscillating voltage is applied to the adjacent parts of the central segments of the electrodes and a static voltage to the outer segments of the electrodes [6].

Figure 2

Equilibrium configurations of $N=12$ ions in an octupole ($k=4$) trap, determined by simulated annealing. The panels on the left show the charge distribution in the radial, $x-y$, plane for different values of $c$, the panels on the right show the corresponding distribution on the plane $x-z$. In (a) and (b) the value $c=10$ was taken. The plots in (c) and (d) are evaluated for $c=8.5$ and exhibit a double ring structure.

Figure 3

Equilibrium configuration of 17 ions in a linear multipole trap for $k=4$ and $c=10$, determined by simulated annealing. The ions form two rings with equal number $M=8$ of ions, while one ion is located in the plane $z=0$ between both rings, forming a localized defect.

Figure 4

Equilibrium configuration of 70 ions in an octupole trap ($k=4$) for $c=4$, corresponding to a tubular arrangement. The positions have been found by simulated annealing. The inner rings are composed by 12 ions per ring, while at the two edges the rings are composed by 10 ions each.

Figure 5

Critical parameters $R_{\mathrm{crit}}$ (a) and $c_{\mathrm{crit}}$ (b) at the transition point from a single- to a double-ring structure as a function of the particle number for an octupole trap ($k=4$). The curves are evaluated from the analytical expressions in Eqs. (18, 19, 20, 21).

Figure 6

Top panel: sketch of the ring array at uniform distance $d/2$ along the $z$ axis. The lower panel shows the ions’ distribution in two adjacent rings: this structure is repeated along the chain. An example of the chosen elementary cell is given by the pair of ions labeled with $\sigma$. The other labels are $j=0,...,N_R-1$ for the number of cells in two adjacent rings and $l=0,...,N_A-1$ for the pair of rings.

Figure 7

Stability diagram of ion tubes with six ions per ring ($N_R=6$) in an octupole trap ($k=4$), here in the parameter space $d$ and $a_1$. The shaded areas indicate the regions where the ion tubes are stable. The value of tube radius depends on both $d$ and $a_1$ through Eq. (25). The inset displays a zoom where the parameters of the plots in Fig. 8 are shown.

Figure 8

Spectra of 400 rings of Ca${}^{+}$ ions in an octupole trap for different trap parameters. Each ring is composed by six ions, the distance between rings is $10\mu$m. The first row, (a)–(d), shows the spectrum for $a_1=3.4\times {10}^9$ J m${}^{-6}$ (upper point in the inset of Fig. 7), the second row, (e)–(h) for $a_1=8.4\times {10}^8$ J  m${}^{-6}$ (central point in Fig. 7), and the third row (i)–(l) for $a_1=2.7\times {10}^8$ J m${}^{-6}$ (lower point in Fig. 7). Each column corresponds to a value of $k_1$: from left to right $k_1=0,\pm 1,\pm 2,\pm 3$. Because of the specific symmetry of the crystal, for $k_1=N_R/2$ (when $N_R$ is even) each frequency is twofold degenerate at a fixed value of $k_2$.

Figure 9

Density profiles in (a) a multipole trap with $k=5$ and (b) a quadrupole trap ($k=2$) for different values of the parameter $\gamma$.