Observation of Three-Dimensional Long-Range Order in Small Ion Coulomb Crystals in an rf Trap

Abstract

Three-dimensional long-range ordered structures in smaller and near-spherically symmetric Coulomb crystals of 40 Ca + ions confined in a linear rf Paul trap have been observed when the number of ions exceeds ∼1000 ions. This result is unexpected from ground state molecular dynamics (MD) simulations, but found to be in agreement with MD simulations of metastable ion configurations. Previously, three-dimensional long-range ordered structures have only been reported in Penning traps in systems of ∼50,000 ions or more. [3,4,5,6,7,8,9,10,11] and most recently dusty plasmas [12]. In nature, Coulomb crystals are presently expected to exist in exotic dense astrophysical objects [13].Theoretically, it has been found that the thermodynamic properties of infinite OCPs of a single species, are fully characterized by the coupling parameter [14]where Q is the charge of the particles, a ws is the Wigner Seitz radius defined from the zero temperature particle density n 0 by 4πa 3 ws /3 = 1/n 0 . Furthermore, a liquidsolid transition to a body centered cubic (bcc) structure is expected to occur for Γ ∼ 170 [15,16]. For finite OCPs simulations have shown that the situation is more complex, and the properties will depend both on the size and shape of the ion plasma, since surface effects cannot be neglected [17,18,19,20,21].Ion Coulomb crystals, which for more than a decade have been realized with laser-cooled ion plasmas confined by electromagnetic fields in Penning traps [4,5,9] or in radio-frequency (rf) traps [3,6,7,8,10], offer an excellent opportunity to study finite size effects of OCPs under various conditions. The Coulomb crystal structures studied range from one-dimensional (1D) long cylindrical crystals [3,6,7,8] over 2D thin planar crystals [4] to 3D spheroidal crystals [4,5,6,9]. The 3D spheroidal ion Coulomb crystals reported in Refs. [6,7] are composed of concentric ion shells formed under the influence of the surface of the Coulomb crystals. Simulations indicate that the ions form a near-2D hexagonal short-range ordered structure within each shell [18]. Similarly, shell and short-range order have recently been observed in dusty plasma experiments [12].Observations of three-dimensional long-range order in Coulomb crystals have previously only been reported in the case of ∼50,000 or more laser-cooled ions in a Penning trap [4,5,9]. In contrast to Penning traps, in rf traps Coulomb crystals undergo strong quadrupole deformations at the frequency of the applied rf field due to the so-called micro-motion [22] of the ions. Since this motion is known to produce heating [22,23], it has not been obvious that three-dimensional long-range order could be obtained in such traps.In this Letter, we present observations of long-range structure in Coulomb crystals of 40 Ca + ions confined in a linear rf Paul trap. By varying the number of ions in near-spherically symmetric crystals, we have shown that bcc structures indeed can be observed in such traps even with the number of ions being below a thousand.The linear Paul trap used i...

Optimization of segmented linear Paul traps and transport of stored particles

“…From the fact that linear traps with optimized electrode shapes have been shown to load large crystals of ions [30], we would try to improve the loading rate by reducing non-harmonic contributions to the potential. Especially for larger q-values when the trap drive power is chosen relatively high, non-linear resonances have been observed [31].…”

Optically induced structural phase transitions in ion Coulomb crystals

“…Such a state, as shown in Fig. 1(a), resembles the experimentally observed structures [40,41]. Based on this configuration, full MD simulations are then performed for up to 10ms under the influence of the forces described in Eq.…”

“…For medium sized crystals often employed in experiments [26,27] (∼ 10 3 − 10 5 ions), the structure is generally not very stable, and thermally induced transitions between a large variety of states including metastable bcc and fcc structures and incommensurable crystallite formations can be observed [40,41]. While structural stability can be dramatically increased using two-species crystals [21,23], means to control and manipulate the structures of single-species crystals are highly wanted, not only for applications in quantum information science, but also for exploiting Coulomb crystals as simulators of solid state physics, like structural transitions of iron under extreme pressure [42] of relevance for geophysics and thin-film growth [43] of importance for nanotechnology.…”

“…(1) are slow, and typically last a few hours on a single CPU core for 10ms of temporal evolution for 1000 ions. This is to be compared to ion numbers of order 10 4 used in experiments and time scales of order seconds on which crystal structure transitions have been observed [40,41]. It is consequently not feasible to simulate the experimental observations on such long timescales and in particular to observe the formation of crystals from hot ion ensembles through cooling.…”

“…Instead, we follow the approach of Refs. [40,41] and initialize the simulations with an idealized structure consisting of 1000 ions, where a finite core (typically 125 ions) is fixed in an ideal bcc or fcc structure and the remaining ions form a surrounding shell minimizing the harmonic and Coulomb energies. For such a structure the ion density is constant through-out the crystal, with a value given by [9] …”