Space-Time Crystals of Trapped Ions

Spontaneous symmetry breaking can lead to the formation of time crystals, as well as spatial crystals. Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. We also study the effects of finite temperatures on the persistent rotation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.

Figure 1

Schematic of creating a space-time crystal. (a) A possible structure of a space-time crystal. It has periodic structures in both space and time. The particles rotate in one direction even at the lowest energy state. (b), Ultracold ions confined in a ring-shaped trapping potential in a weak magnetic field. The mass and charge of each ion are $M$ and $q$, respectively. The diameter of the ion ring is $d$, and the magnetic field is $B$. (c), Examples of the pseudopotentials ($V_{\mathrm{ext}}$) for a ${}^9\mathrm{Be}^{+}$ ion in a quadrupole ring trap (solid curve) and a linear octupole trap (dashed curve) along the $x$ or $y$ axis.

Figure 2

The energy levels and rotation frequencies of trapped ions. (a) The energy levels of identical bosonic ions as a function of the magnetic flux $\alpha$. The quantum number $n_1$ is labeled on each curve. The angular frequency of the persistent rotation as a function $\alpha$ is shown in (b) for an even number of fermionic ions, and (c) for bosonic ions. (a) and (c) are also applicable to an odd number of fermionic ions. (d) The angular frequency of the persistent rotation of a bosonic ion ring in a constant magnetic field $B_0>0$ as a function of its normalized diameter $d/d_0$.

Figure 3

The temperature dependence of the persistent rotation of trapped ions. (a) The average angular frequency of the persistent rotation of identical bosonic ions as a function of the temperature. From top down, the magnetic flux increases from $-0.45$ to 0.45. (b) The characteristic temperature (left axis) and the characteristic frequency (right axis) of the persistent rotation of an ion (or electron) ring consisting 100 identical ions (or electrons) as a function of the diameter. From top down, the trapped particles are electrons, ${}^9\mathrm{Be}^{+}$ ions, and ${}^{24}\mathrm{Mg}^{+}$ ions.