Trapping of Topological-Structural Defects in Coulomb Crystals

We study experimentally and theoretically structural defects which are formed during the transition from a laser cooled cloud to a Coulomb crystal, consisting of tens of ions in a linear radio frequency trap. We demonstrate the creation of predicted topological defects (“kinks”) in purely two-dimensional crystals and also find kinks which show novel dynamical features in a regime of parameters not considered before. The kinks are always observed at the center of the trap, showing a large nonlinear localized excitation, and the probability of their occurrence saturates at $\sim 0.5$. Simulations reveal a strong anharmonicity of the kink’s internal mode of vibration, due to the kink’s extension into three dimensions. As a consequence, the periodic Peierls-Nabarro potential experienced by a discrete kink becomes a globally confining potential, capable of trapping one cooled defect at the center of the crystal.

Figure 1

Coulomb crystals, here with 31 ions (${\omega }_z/{\omega }_y\approx 1.34$). (a) Top: A CCD image of the fluorescence light of individual ions trapped and laser cooled in a zigzag configuration. Bottom: Circles indicate the fitted positions, while crosses depict the expected positions at the given trapping parameters, derived by MD simulations. (b) Same as in (a); however, the ions in the center region form an extended kink defect (dashed box). While the ions to the left of the center form a zigzag structure, as in (a), the positions on the right are mirrored about the $x$ axis.

Figure 2

Coulomb crystals containing 50 ions, similar to those depicted in Fig. 1; however, for reduced asymmetry of the radial confinement, ${\omega }_z/{\omega }_y\approx 1.05$. (a) CCD images of two crystals obtained for identical experimental parameters. Top: Pure zigzag featuring the minimal energy configuration. Bottom: Two central ions clearly deviate and show a blurred extension in the radial direction. As in Fig. 1b, the right parts of the crystals are close to identical while the left part of the lower crystal depicts a flipped “zagzig” structure, separated by the two central ions. (b) Integrated numerical results for the structure depicted in (a)-bottom projected onto the crystal plane. (c) Numerically derived components of ion oscillations in the localized, low-frequency normal mode of the kink in (perpendicular to) the crystal plane are depicted on the left (right).

Figure 3

Experimentally derived probability for the occurrence of a pure zigzag and a zigzag with a single kink, as depicted in Fig. 2, formed during a nonadiabatic cooling-induced transition from a cloud to a crystal, plotted versus the number of ions. With 27 ions and more, the minimum energy configuration is a zigzag structure (blue circles). Starting with 46 ions, we observe an increasing probability for a zigzag structure with one kink (red rhombi) at the center of the crystal. No simultaneous occurrence of two or more defects is observed. The probability for observing one kink saturates at $\sim 0.5$ (dashed line) for up to 56 ions. The error bars represent the statistical error ($1\sigma$) based on the number of attempts.

Figure 4

Simulation of the effective PN potential energy of a kink (see Fig. 2), (a) dependent on its distance from the center of the Coulomb crystal. The potential is symmetric about the trap center and reaches its maximum roughly at the final values shown (dependent on the total ion number). (b) The total potential barrier for the kink to escape its trap by reaching the sides is given in dependence of the number of ions, fitted by a quadratic function (blue line).