Spectroscopy and Directed Transport of Topological Solitons in Crystals of Trapped Ions

We study experimentally and theoretically discrete solitons in crystalline structures consisting of several tens of laser-cooled ions confined in a radio frequency trap. Resonantly exciting localized, spectrally gapped vibrational modes of the soliton, a nonlinear mechanism leads to a nonequilibrium steady state of the continuously cooled crystal. We find that the propagation and the escape of the soliton out of its quasi-one-dimensional channel can be described as a thermal activation mechanism. We control the effective temperature of the soliton’s collective coordinate by the amplitude of the external excitation. Furthermore, the global trapping potential permits controlling the soliton dynamics and realizing directed transport depending on its topological charge.

Figure 1

Schematic depiction of discrete solitons and their propagation in a trapped Coulomb crystal. (a) The self-assembled crystal features reflection symmetry, and its energy minimum comes in two degenerate configurations: the “zigzag” and its mirror image “$\overline{\mathrm{zigzag}}$” (only one shown). (b) Realizing both configurations in one crystal requires an interface, a domain wall called a “kink” (left) or “$\overline{\mathrm{kink}}$” (right), which is a discrete soliton, carrying a topological charge of $\pm 1$, illustrated with a dashed red line. (c) In this Letter, we ask whether a resonant global excitation of radial vibrational kink modes can be rectified by the soliton and exploited to conditionally propagate it to the right or left in a noisy environment.

Figure 2

Spectroscopy and directed escape of a discrete soliton. (a) Spectroscopically resolving kink modes via its complete escape from the crystal. With a periodic excitation of amplitude $\epsilon =1.45\times {10}^{-3}$ (blue squares), we resolve a resonance at ${\omega }_d=2\pi \times (325.3\pm 0.2)\mathrm{kHz}$ with a width of $2\pi \times (4.8\pm 0.5)\mathrm{kHz}$. Strong excitation $\epsilon =1.74\times {10}^{-3}$ (red disks) leads to saturation and an additional weaker resonance at ${\omega }_d=2\pi \times (311\pm 0.5)\mathrm{kHz}$ of width $2\pi \times (4.4\pm 1.8)\mathrm{kHz}$, in agreement with the numerically derived frequencies of internal modes (see the text for details). Error bars represent binomial statistics, and the offset level of 0.1 originates from background gas collisions at $\epsilon =0$, leading to melting and recrystallization (background lifetime $\approx 3.2\mathrm{s}$). (b) CCD images of the fluorescence of 34 ions confined in a Paul trap. (I) The center features the $\overline{\mathrm{kink}}$. (II) The modulation of the radial confinement resonantly excites a localized high-frequency vibrational kink mode, and its nonlinear coupling to other modes results in the axial blurring of the oscillating $\overline{\mathrm{kink}}$. (III) Escape of the soliton via the right-hand side is witnessed as the left half of the configuration remains unaffected, while on the right the two opposing ion chains have to “slide” atop each other. Because of the long exposure time of the camera, $t_{\mathrm{CCD}}=150\mathrm{ms}$, both configurations are superimposed.

Figure 3

Resonant drive of the radial kink mode leads to a thermally activated escape out of the PN potential. (a) Experimental survival probability of the soliton as a function of $t_d$ for ${\omega }_d/(2\pi )=327\mathrm{kHz}$ and $\epsilon ={10}^{-3}\times \{1.15(\text{green triangles}),1.30(\text{black inverted triangles}),1.45(\text{blue squares}),\mathrm{and}1.74(\text{red disks})\}$. Solid lines represent exponential fits yielding the corresponding lifetimes of the kink, $\tau (\epsilon )=\{(544\pm 35),(248\pm 16),(71\pm 5),(23\pm 2)\}\mathrm{ms}$, respectively. Error bars represent the $1\sigma$ confidence interval. The residual kink loss rate for $\epsilon =0$ is subtracted based on a calibration measurements. (b) Lifetime of the kink in dependence on $\epsilon$ derived from panel (a), fitted with an overdamped Kramers’ model [81] for symmetric confinement (see the text for details), indicated by the gray solid line. The experiment control parameter $\epsilon$ is numerically found to be linearly related to the mean kinetic energy of the ions, leading to an effective temperature. The gray shaded region represents $1\sigma$ uncertainty of the fit. We extract a related barrier height of $(26.5\pm 1.0)k_B{T}_D$.

Figure 4

Experimental transport directionality (TD) conditional on the topological charge: $\pm 1\widehat{=}\{\mathrm{kink},\overline{\mathrm{kink}}\}$. (a) While the kink escapes with close to zero TD in the experiment, the $\overline{\mathrm{kink}}$ reveals the broken symmetry tunable by nonlinear terms of the confining potential. Additionally, the TD depends on $\epsilon$ (symbols as in Fig. 3). Error bars give $1\sigma$ standard deviation. (b) The trapping potential for the soliton (schematically depicted) depends on its topological charge. It remains approximately left-right symmetric for the kink and is strongly asymmetric for the $\overline{\mathrm{kink}}$. The trap depth for the whole Coulomb crystal amounts to $10^4\times V_{\mathrm{kink}}$.