Time fractals and discrete scale invariance with trapped ions

We show that a one-dimensional chain of trapped ions can be engineered to produce a quantum mechanical system with discrete scale invariance and fractal-like time dependence. By discrete scale invariance we mean a system that replicates itself under a rescaling of distance for some scale factor, and a time fractal is a signal that is invariant under the rescaling of time. These features are reminiscent of the Efimov effect, which has been predicted and observed in bound states of three-body systems. We demonstrate that discrete scale invariance in the trapped ion system can be controlled with two independently tunable parameters. We also discuss the extension to $n$-body states where the discrete scaling symmetry has an exotic heterogeneous structure. The results we present can be realized using currently available technologies developed for trapped ion quantum systems.

Figure 1

Bound-state wave functions. Plot of the normalized wave functions for the first 12 even-parity bound states for the case $\alpha =2, \beta =1, J_0=-1$, and $V_0=-30$. We plot the region $r>0$. All quantities are in dimensionless lattice units.

Figure 2

Time fractals. The amplitude $A\left(t\right)$ is displayed over the range from $t=0$ to 80 in the upper left, $t=0$ to 160 in the upper right, $t=0$ to 320 in the lower left, and $t=0$ to 640 in the lower right. All quantities are in dimensionless lattice units.