Ion Coulomb Crystals
After Doppler cooling, ions often form Coulomb crystals in the trap. This unusual state of matter allows an assembly of ions to be treated as a single quantum mechanical system. Coulomb crystals are used in many types of experiments with trapped ions because they have very well controlled properties, for example, the ions are well isolated from the environment and do not interact with each other except through the Coulomb interaction, which ensures that they remain far apart from each other. The structure is very stable and in some cases the quantum states of individual ions can be manipulated.
A short review of ion Coulomb crystals and their applications is given in reference .
Formation of ion Coulomb crystals
A single ion in a trap moves around the centre of the trap in the potential minimum created by the electric field set up by the electrodes. When it is cooled it slows down and the amplitude of its motion reduces till it effectively comes to rest at the centre of the trap.
If several ions are present in the trap, the ions interact with each other through the Coulomb interaction. Therefore, as they cool down they get closer to each other and their mutual repulsion limits how close to each other they will be once they have slowed down. Under these conditions they form a “non-neutral plasma” – a type of plasma that only has one sign of charge (positive in this case) because the confining potential takes the place of the other charge that would normally be present in a neutral plasma. This non-neutral plasma has a uniform density and forms an ellipsoidal shape with a sharp boundary.
When their residual kinetic energy is low enough, the ions form a so-called “ion Coulomb crystal” (ICC) where they are arranged in a regular pattern like a conventional crystal. The density of the Coulomb crystal is however much less than that of a conventional crystal, by a factor of up to 1015, because the Coulomb interaction is relatively strong. This density is actually less than that of the residual gas in the ultra-high vacuum in which the ion trap is located. The configuration adopted by an ICC depends on the details of the trap parameters but it is often a three-dimensional ellipsoid, as shown in Figure 1. The typical distance between ions in such crystals is 5 – 20 μm .
One particularly important configuration of an ICC is called an ion string (or chain). This is where the ions are all aligned along the axis of the trap and are therefore stationary, even in a Penning trap. The ions adopt this configuration when the trapping potential is very low so that the axial confinement is very weak compared to the strong radial confinement. A sequence of images of such ion strings is shown in Figure 2 below. The ion string can be treated as a quantum mechanical object with a size that is measured in up to hundreds of micrometres, which is unusually large for quantum systems. A string of N ions has a total of 3N degrees of freedom, N of which describe motion in the axial direction. The quantum mechanical state is therefore very complex for large strings. A linear string of ions is the configuration usually required for applications in quantum information processing.
Planar ion Coulomb crystals
In contrast to the case of ion strings, the ions will be forced into a two-dimensional flat crystal arrangement if the axial confinement is very strong compared to the radial confinement. An example of this is shown in Figure 3 (from the NIST ion trap group, reproduced in ). The ions form a triangular pattern in a single layer, perpendicular to the magnetic field. This configuration of ions has been used for quantum mechanical simulations of the behaviour of large numbers of interacting particles in a two-dimensional lattice. We have also published a proposal to use a crystal of 6 ions in a Penning tap for a demonstration of the 5-qubit code which may be used for error correction in quantum computing .
We have carried out several experiments with two ions in our Penning trap. Depending on the trap parameters, the ions can either line up along the axis of the trap (weak axial confinement) or in the radial plane perpendicular to that axis (strong axial confinement), in which case they rotate due to the presence of the magnetic field. These configurations are shown in Figure 4 together with their axial modes of oscillation. In reference  we show that we are able to sideband cool the axial motions of both of these configurations of the ions to the ground state. In future work in our linear RF trap, we will be working with two-ion strings to optimise quantum gates using the methods of optimal control.
 Thompson R C. Ion Coulomb crystals, Contemporary Physics. 2015;56: 63-79. doi: 10.1080/00107514.2014.989715.
 Mavadia S, Goodwin JF, Stutter G, Bharadia S, Crick D R, Segal D M, Thompson R C. Control of the conformations of ion Coulomb crystals in a Penning trap, Nature Communications. 2013;4: 2571. doi: 10.1038/ncomms3571.
 Goodwin J F, Brown B J, Stutter G, Dale H, Thompson R C, Rudolph T. Trapped-ion quantum error-correcting protocols using only global operations, Physical Review A. 2015;92(3): 032314. doi: 10.1103/PhysRevA.92.032314.
 Stutter G, Hrmo P, Jarlaud V, Joshi M K, Goodwin J F, Thompson R C. Sideband cooling of small ion Coulomb crystals in a Penning trap, Journal of Modern Optics. 2018;65(5-6): 549-559. doi: 10.1080/09500340.2017.1376719.