Figure 1

Sketch of the cylindrical dynamic Kingdon trap. A superposition of ac and dc voltages is applied between a rectilinear wire and a surrounding cylindrical conductor resulting in dynamical trapping of a charged particle in the free space between the wire and the cylinder.Reuse & Permissions

Figure 2

Multiple-exposure Poincaré section for $\eta =5$ illustrating the organization of the phase space of the dynamic Kingdon trap around its limit cycle (full line).Reuse & Permissions

Figure 3

The stability function $S$, a measure of the phase-space area of stable islands, as a function of the control parameter $\eta$ for the cw-driven trap (a) and the kicked trap (b). The insets show that for both traps $S\left(\eta \right)$ vanishes twice in the interval $3⩽\eta ⩽10$ at $\eta$ values, called ${\eta }_3^{*}$ and ${\eta }_4^{*}$ (marked by arrows), indicating complete instability of the traps at these critical control parameters. The arrows at ${\eta }_N^R$, $N⩾5$, mark the $\eta$ values where the rotation frequency of the center of the primary trapping island is in resonance with (a) the frequency of the trap’s ac voltage and (b) the kick frequency of the trap.Reuse & Permissions

Figure 4

Illustration of the collapse of the primary trapping island of the cylindrical dynamic Kingdon trap in the vicinity of $\eta ={\eta }_3^{*}$. (a) $\eta =3.57$, (b) $\eta =3.613\approx {\eta }_3^{*}$, (c) $\eta =3.65$. A set of unstable period-3 fixed points $\left(\eta <{\eta }_3^{*}\right)$ (a) collapses at $\eta ={\eta }_3^{*}$ (b) squeezing the primary trapping island to zero phase-space area. The primary trapping island reemerges for $\eta >{\eta }_3^{*}$ (c).Reuse & Permissions

Figure 5

Collapse of the primary trapping island of the cylindrical dynamic Kingdon trap in the vicinity of $\eta ={\eta }_4^{*}$. (a) $\eta =4.41$, (b) $\eta =4.431\approx {\eta }_4^{*}$, (c) $\eta =4.44$. A set of unstable period-4 fixed points $\left(\eta <{\eta }_4^{*}\right)$ (a) collapses at $\eta ={\eta }_4^{*}$, (b) squeezing the primary trapping island to zero phase-space area. The primary trapping island reemerges for $\eta >{\eta }_4^{*}$ (c).Reuse & Permissions

Figure 6

Sketch of the spherical dynamic Kingdon trap. A superposition of ac and dc voltages is applied between two concentric ideally conducting shells. The resulting oscillating electric field produces dynamical trapping of a charged particle in the free space between the two shells.Reuse & Permissions

Figure 7

Island collapse for the spherical dynamic Kingdon trap at the $N=3$ resonance. (a) $\eta =3.13$ (immediately before the collapse point), (b) $\eta ={\eta }_3^{*}=\mathrm{3.161\hspace{0.17em}713\hspace{0.17em}2}\dots$ (at the collapse point), (c) $\eta =3.19$ (immediately after the collapse point). The scenario is reminiscent of the scenario exhibited by the cylindrical dynamic Kingdon trap at the $N=3$ resonance shown in Fig. 4a, 4b, 4c.Reuse & Permissions

Figure 8

Island collapse for the spherical dynamic Kingdon trap at the $N=4$ resonance. (a) $\eta =3.82$ (immediately before the collapse point), (b) $\eta ={\eta }_4^{*}=\mathrm{3.864\hspace{0.17em}074\hspace{0.17em}7}\dots$ (at the collapse point), (c) $\eta =3.9$ (immediately after the collapse point). The scenario is reminiscent of the scenario exhibited by the cylindrical dynamic Kingdon trap at the $N=4$ resonance shown in Fig. 5a, 5b, 5c.Reuse & Permissions

Figure 9

Sketch of the Paul trap. A superposition of ac and dc voltages is applied between an ideally conducting hyperbolic ring electrode and two ideally conducting hyperbolic end caps.Reuse & Permissions

Figure 10

Stability diagram of the Paul trap. The trap is globally stable if operated with a $\left(q,a\right)$ control parameter setting taken from inside of the stability region framed by the curves $c,d,e$, and $f$. The different shades of gray indicate control parameter combinations $\left(q,a\right)$ where two simultaneously stored ions in their lowest-energy state are $z$ aligned (light shade of gray, labeled “$Z$”), $xy$-aligned (darker shade of gray, labeled “$XY$”), and aligned at an angle with respect to the $z$ axis (dark shade of gray, labeled “$A_1$” and “$A_2$”). The areas $A_1$ and $A_2$ are not contiguous. Areas $C_1$ and $C_2$ separated by area $A_2$ are chaotic regions. The heavy lines labeled $N=3\left(XY\right)$ and $N=4\left(XY\right)$ correspond to $\left(q,a\right)$ parameter combinations where a two-ion crystal in the $xy$ plane experiences an $N=3$ $(N=3)$ resonance and breaks up. The heavy lines labeled $N=3\left(Z\right)$ and $N=4\left(Z\right)$ correspond to $\left(q,a\right)$ parameter combinations where a two-ion crystal aligned with the $z$ axis experiences an $N=3$ $(N=4)$ resonance and breaks up.Reuse & Permissions

Figure 11

Paul trap $N=3$ resonance scenario. $a=0.05$, and (a) $q=0.412$, (b) $q=0.4137\approx q^{*}$, and (c) $q=0.416$. In analogy to Figs. 4a, 4b, 4c the primary trapping island vanishes at $q^{*}$. In contrast to Figs. 4a, 4b, 4c the three large, stable elliptic islands stay intact before, during, and after the instability.Reuse & Permissions

Figure 12

Paul trap $N=4$ resonance scenario. $a=0.3$, and (a) $q=0.6055$, (b) $q=\mathrm{0.605\hspace{0.17em}654}\approx q^{*}$, and (c) $q=0.6057$. In analogy to Figs. 5a, 5b, 5c the primary trapping island vanishes at $q^{*}$. In contrast to Figs. 5a, 5b, 5c the four large, stable, elliptic islands stay intact before, during, and after the instability.Reuse & Permissions

Figure 13

Destruction of a two-ion crystal at the Paul trap $N=3$ instability at $a=0.25$, $q=0.5712\approx q^{*}$. (a) Phase-space portrait for $a=0.25$, $q=0.5712$ illustrating the destruction of the primary trapping island. (b) Composite phase-space portrait showing the primary trapping island at $a=0.25$, $q=q_1=0.56$ and $a=0.25$, $q=q_2=0.59$. Also shown is the phase-space trajectory of a $z$-aligned two-ion crystal dragged slowly from $q_1$ to $q^{*}$, breaking up at $q^{*}$ along the unstable manifolds of the fixed point at $q^{*}$ [see panel (a)] and thus revealing the presence of the $N=3$ instability at $q^{*}$.Reuse & Permissions

Figure 14

The limit cycle $r_L\left(t\right)$ at $\eta =5$ (full line) and two of its approximations. Long dashes: two-harmonics approximation $r_L^{\left(2\right)}\left(t\right)$. Short dashes: three-harmonics approximation $r_L^{\left(3\right)}\left(t\right)$. Inset: magnification of the limit cycles in the vicinity of $r=2.85$.Reuse & Permissions