Ion Coulomb crystals: an exotic form of condensed matter

Abstract

Ion Coulomb crystals are ordered structures formed by laser-cooled ions in traps that are characterized by interparticle distances of several micrometers and energy scales on the order of $μ$eV. Their crystalline structure emerges from the interplay between Coulomb repulsion and the external confining potential, which can be readily tuned. Moreover, individual ions can be precisely manipulated with lasers and imaged via resonance fluorescence. These unusual and unique properties make ion crystals a powerful platform for studying phases of matter in the strongly correlated regime and at low temperatures where their dynamics is manifestly quantum mechanical. This review examines the theoretical framework and experimental characterization of ion Coulomb crystals from a condensed-matter perspective. We discuss their dynamical and thermodynamic properties in one, two, and three dimensions, and review recent investigations into their out-of-equilibrium behavior. We provide outlooks on future directions for exploring novel condensed matter phenomena with trapped ion crystals, as well as for exploiting these features for scientific and technical applications.

Figures (53)

• Figure 1: Schematics of the standard three-dimensional rf trap (Paul trap). The trap consists of a central ring electrode (named "ring" in the figure), and two so-called end cap electrodes ("Top/Bottom end-cap"). All three electrodes have hyperboloid-shaped surfaces surrounding the trap center. They produce an electrical quadrupole field when the voltage difference $\it{U_{\rm dc}+U_{\rm rf}}\cos(\Omega_{\rm rf}t)$ is applied between the ring and the end caps. The strength of the quadrupole field depends on the distances $\it{r_0}$ and $\it{z_0}$. A perfect quadrupole field is generated when $\it{r_0}$= $\sqrt{2}\it{z_0}$.
• Figure 2: Stability diagram for standard rf trap. Left: Confined motion of a charged particle can only happen when its stability parameters $a_{z,\rho}$ and $q_{z,\rho}$ lie within common areas of stable motion ($z$- and $\rho$-stable). Right: Zoom on the largest area of common stability (marked in red in left panel). The parameters $\beta_{j}$ represent the ratio between the main motional frequencies of the trapped particle along the $j$ direction ($j=z,\rho)$ and the rf frequency $\Omega_{\rm rf}/2$. Reproduced from Gosh:1995
• Figure 3: (Color online) Motion of a trapped particle in an rf trap. The plot shows the trajectory of a trapped ion along one of the principal directions ($x, y, x$) for $a_z=0.0$ and $q_z=0.3$. The oscillatory red (solid) curve represents the full solution to Eq. (\ref{['DiffEq_u1']}), while the dark blue (dashed) curve is the solution obtained by using the harmonic pseudopotential of Eq. (\ref{['eq:pseudopot']}). The difference between the two curves is the so-called micromotion at the oscillation frequency $\Omega_{\rm rf}$.
• Figure 4: Linear trap geometries. a) Illustration of an ideal electrode geometry with hyperbolic-shaped electrodes to achieve two-dimensional trapping in the $x-y$ plane by a perfect electrical quadrupole field. This ideal geometry is often used in quadrupole mass spectrometers. b) An example of a three-dimensional linear rf trap with cylindrically-shaped electrodes. Here, one realizes a confinement in the $x-y$ plane like in $a)$ by applying the same rf voltage to each of the three sections of the electrodes. Axial confinement along the $z$-axis is achieved through application of the same dc voltage $U_{\rm end}$ to all the orange-colored electrode sections. c) Linear trap based on rf electrodes shaped like blades. This design creates a smaller effective value of $r_0$, and hence achieves higher trap frequencies in the $x-y$ plane at a given rf voltage (see Eqs. (\ref{['eq:omegau']}) and (\ref{['eqs:stabaq1']})-(\ref{['eqs:stabaq4']}). The built-in anharmonicity of this geometry eventually limits its use to one-dimensional ion Coulomb crystals or to small structures of two or three dimensions. Reproduced from Splatt:2009, Poulsen:2012 and Schindler:2013.
• Figure 5: Stability diagram for a linear rf trap. The grey shaded area represents the stability region for the specific case of $U_\mathrm{dc}$=0, for which $a_x=a_y=a$ according to Eqs. (17) and (18), and $q_x=-q_y=q$. The area of the stability diagram is significantly larger than for the two-dimensional confinement of the quadrupole mass filter (the hatched areas A and B) and of the three-dimension stable trapping are of the standard rf trap (see Fig. \ref{['fig:Paul_trap_stability_diagram']}). Reproduced from Drewsen:1999