Ion traps and the memory effect for periodic gravitational waves

The Eisenhart lift of a Paul trap used to store ions in molecular physics is a linearly polarized periodic gravitational wave. A modified version of Dehmelt’s Penning trap is, in turn, related to circularly polarized periodic gravitational waves, sought in inflationary models. Similar equations also govern the Lagrange points in celestial mechanics. The explanation is provided by anisotropic oscillators.

Figure 1

In a weak linearly polarized periodic (LPP) wave, (3.4), the transverse coordinate $\mathit{X}(U)$ oscillates in a bounded “bow tie”-shaped domain. The initial conditions are ${\dot{X}}^{+}(U=0)={\dot{X}}^{-}(U=0)=0$ (at rest for $U=0$), at initial position $X^{+}(U=0)=1$, $X^{-}(U=0)=0$.

Figure 2

(a) In a sufficiently weak circularly polarized gravitational wave (3.5) the transverse trajectory of a particle initially at rest remains confined in a toroidal region. (b) For a strong wave the trajectory becomes unbounded. The initial conditions are ${\dot{X}}^{+}(U=0)={\dot{X}}^{-}(U=0)=0$ and $X^{+}(U=0)=1$, $X^{-}(U=0)=0$.

Figure 3

In the circularly polarized periodic gravitational wave (3.5) the trajectory unfolded into “time” (in heavy blue) winds about the guiding center (dotted in red). If the wave is weak, $A_0<2$, the trajectory remains bounded, projecting to the plane consistently with Fig. 2.

Figure 4

Motion in a three-dimensional Paul trap for (a) $a=0.1$ (axial symmetry) (b) $a=0$ (periodic profile). The initial conditions $X^{+}(0)=0$, ${\dot{X}}^{+}(0)=1$, $X^{-}(0)=5$, ${\dot{X}}^{-}(0)=0$, $z(0)=0$, $\dot{z}(0)=1$.

Figure 5

Trajectory of a charged particle in a Penning trap (a) in three dimensions. (b) its projection on the $Y^{\pm }$ plane. The initial conditions are $Y^{+}(0)=1.0$, ${\dot{Y}}^{+}(0)=0.0$, $Y^{-}(0)=0$, ${\dot{Y}}^{-}(0)=1.0$, $z(0)=0$, $\dot{z}(0)=0.2$.

Figure 6

In the fine-tuned case (5.7) and for initial conditions $Y^{+}(0)=1.0$, ${\dot{Y}}^{+}(0)=0.0$, $Y^{-}(0)=0$, ${\dot{Y}}^{-}(0)=1.0$, $z(0)=0$, $\dot{z}(0)=0.2$, the $2d$ projection (b) of the $3d$ trajectory (a) spirals outward with expanding radius. The $z$ coordinate oscillates with frequency ${\omega }_z={\omega }_c/\sqrt{2}$.

Figure 7

Motions around a Lagrange point are bounded for $\mu =0.02<1/27$ and unbounded for $\mu =0.4>1/27$.