Realization of Translational Symmetry in Trapped Cold Ion Rings

We crystallize up to 15 ${{}^{40}\mathrm{Ca}}^{+}$ ions in a ring with a microfabricated silicon surface Paul trap. Delocalization of the Doppler laser-cooled ions shows that the translational symmetry of the ion ring is preserved at millikelvin temperatures. By characterizing the collective motion of the ion crystals, we identify homogeneous electric fields as the dominant symmetry-breaking mechanism at this energy scale. With increasing ion numbers, such detrimental effects are reduced. We predict that, with only a ten-ion ring, uncompensated homogeneous fields will not break the translational symmetry of the rotational ground state. This experiment opens a door towards studying quantum many-body physics with translational symmetry at the single-particle level.

Figure 1

(a) Optical image of the trap electrodes illuminated with yellow light. The trap consists of three circular electrodes and eight static voltage electrodes in a plane. The outer radius of the three circular electrodes are 125, 600, and $1100\mu \mathrm{m}$, respectively. The gap between the innermost and the next circular electrode is $15\mu \mathrm{m}$, and the gaps between other electrodes are $25\mu \mathrm{m}$. The whole electrode pattern possesses a diameter of 6 mm, outside of which is ground. (b) Cross-sectional schematics of the fabricated trap. The resistivity of the doped silicon is $<0.005\mathrm{ohm}\mathrm{cm}$. The thickness of the Ag, Si, and ${\mathrm{SiO}}_2$ layers are 100 nm, $250\mu \mathrm{m}$, and $70\mu \mathrm{m}$, respectively. Electrical vias through the ${\mathrm{SiO}}_2$ layer are coated with an 800 nm layer of gold. (c) Scanning electron microscope image of the trap cross section.

Figure 2

(a)–(c) Images of ion crystals composed of four ions (a) and ten ions (b),(c) for different total dipole fields $E_y$. (d) Image of a delocalized ten-ion ring when the total external dipole field is close to zero (exposure time 200 ms). In (a)–(d), the scale bars are $20\mu \mathrm{m}$. The fluorescence inhomogeneity of the images is caused by the size of the Gaussian cooling beam of $\sim 70\mu \mathrm{m}$ full width at half maximum [30].

Figure 3

Dependence of the tangential trapping frequency ${\omega }_t$ on the total external dipole field strength $E_y$ (a) and ion number $N$ (b), respectively. Error bars are smaller than the sizes of the data points. The curves correspond to the calculated collective tangential frequencies using the electric potential model considering only a homogeneous electric field and the Coulomb repulsion of the ions confined to a ring. The agreement between the calculated results and the experimental data confirms that the homogeneous electric fields are the dominant symmetry-breaking mechanism.

Figure 4

The lowest potential energy configurations of a ten-ion crystal as a function of the position of the final ion (marked in red). The original equilibrium configuration is shown in (a). When the final ion moves from right to left, the nine other ions move into a new minimum configuration (b), and the crystal eventually recovers the original equilibrium configuration (c). (d) The calculated potential energy of ten ions as a function of the final ion position angle $\theta$, where the difference between the minimum and the maximum is the rotational energy barrier $V_B$. The marked points correspond to the equilibrium configurations shown in (a)–(c), respectively. $E_y=-2.0\mathrm{V}/\mathrm{m}$.

Figure 5

Total external electric field strength (a) and corresponding rotational energy barrier $V_B/k_B$ (b) at which the ion crystals are observed to delocalize as a function of the ion number. The errors of the electric fields in (a) are $\pm 0.1\mathrm{V}/\mathrm{m}$. The gray line in (b) denotes the measured tangential temperature $\sim 3\mathrm{mK}$ of the ions, which is close to $V_B/k_B$, and suggests that the delocalization occurs when the thermal energy of the ions is large enough to overcome the rotational energy barrier.

Figure 6

(a)–(c) Image of eight-ion rings in the localization-delocalization transition regime. (d)–(f) Corresponding simulated angular distribution $p(\theta )$ of the ions using the Langevin equation at 3 mK. The dipole electric fields $E_y$ and the corresponding rotational energy barriers $V_B/k_B$ are $-1.37$, $-1.22$, and $-1.07\mathrm{V}/\mathrm{m}$ and 13, 5.6, and 2.2 mK for (a) and (d), (b) and (e), and (c) and (f), respectively. As in Fig. 2, reduced fluorescence on the extrema of the $\mathit{x}$ axis is due to the Gaussian profile of the detection beam incident from the positive $y$ direction.