Theory and simulation of ion Coulomb crystal formation in a Penning trap

Abstract

Ion Coulomb crystals (ICC) are formed by laser-cooled ions in both radio-frequency and Penning traps. In radio-frequency traps the crystals are generally stationary. In Penning traps ICC always rotate. The frequency of rotation is often set by an applied rotating wall drive which forces the crystal to rotate at the same frequency as the drive. In the absence of any applied rotating or oscillating fields, ICC in a Penning trap can be in stable equilibrium with a range of rotation frequencies. The density and shape of the crystal adjusts with the rotation frequency to ensure that equilibrium is reached. Here we show that the parameters of the radial laser cooling beam determine the rotation frequency of a small crystal in a Penning trap when no driving fields are present. We demonstrate, using an approximate theoretical treatment and realistic simulations, that the crystal rotation frequency is independent of the number of ions and the trap parameters, so long as the crystal radius remains smaller than the cooling laser beam waist. As the rotation frequency increases, the crystal eventually becomes a linear string, at which point it is no longer able to adjust its density. Instead, a small amplitude vibration in the zigzag mode of oscillation manifests itself as a rotation of the crystal at a fixed frequency that depends only on the applied trap potential.

Sideband cooling of small ion Coulomb crystals in a Penning trap

“…Doppler cooling beams propagate both along and perpendicular to the direction of the magnetic field to ensure that all of the motional modes of the crystal are cooled [18,27]. The radial cooling beam is offset from the trap centre, resulting in an intensity gradient across the crystal and giving coarse control of the crystal rotation frequency [28,29]. Note that good control of the radial beam position and intensity is paramount for the stability of the crystal.…”

Ion Coulomb crystals

“…In these experiments the rotation frequency of the ion crystal in the laboratory frame is not known, because there is no rotating wall applied. It can take a range of values, depending on the size, position and frequency of the laser beam, and this can be checked for consistency with the observed structures [24,49]. Note that for a small number of ions it is not strictly appropriate to describe the crystal shape as a spheroid.…”

Perpendicular laser cooling with a rotating-wall potential in a Penning trap

“…Doppler laser cooling in a Penning trap is complicated by the fact that the ion crystals rotate, producing large, coherent Doppler shifts [14][15][16][17]. A theoretical treatment of this complication for a many-ion crystal (or a cold non-neutral plasma) and the resulting minimum attainable temperatures were discussed more than two decades ago [15].…”