Dynamics of an unbalanced two-ion crystal in a Penning trap for application in optical mass spectrometry

In this paper, the dynamics of an unbalanced two-ion crystal comprising the “target” and the “sensor” ions confined in a Penning trap along the magnetic-field axis has been studied. First, the low amplitude regime is addressed. In this regime, the overall potential including the Coulomb repulsion between the ions can be considered harmonic and the axial, magnetron, and reduced-cyclotron modes split up into the so-called stretch and common modes, that are generalizations of the well-known “breathing” and “center-of-mass” motions of a balanced crystal made of two ions. By using optical detection to measure the frequencies of the modes of the crystal, and of the sensor ion on its own, in the quantum regime of motion, it will be possible to determine the target ion's free-cyclotron frequency. The nonharmonicity of the Coulomb interaction is also discussed since this causes large systematic effects, which are minimized due to the high sensitivity of the optical detection method when the crystal is cooled to the ground state of motion in the Penning trap.

Figure 1

Left: Original shape of a Penning trap (revolution hyperboloid). The yellow electrode is referred to as the ring, and the red ones are known as endcaps. The so-called characteristic distance of the trap is given by $d_0^2=\left(z_0^2+{\rho }_0^2/2\right)/2$. Right: Normal modes of a single ion stored in a Penning trap. The black solid line depicts the overall motion.

Figure 2

Electric potentials as seen by the ion at $z=-d/2$ (colored in cyan) along the trap axis. The repulsion term is due to the presence of a second (pink-colored) ion.

Figure 3

Eigenvalues and eigenvectors for the two-ion crystal in the axial degree of freedom. The solid lines correspond to the axial frequencies in units of the axial frequency of the sensor ion. Note that the stretch frequency is almost constant for large values of $\mu$; the same happens for the common mode when $\mu$ is small. The dashed lines correspond to the ions' amplitude ratios (target over sensor).

Figure 4

(a) Balanced crystal moving in the breathing (magnetron and reduced-cyclotron) modes. (b) Balanced crystal moving in the center-of-mass modes. (c) Unbalanced crystal moving in the stretch mode (magnetron and reduced cyclotron). (d) Unbalanced crystal moving in the center-of-mass mode.

Figure 5

Eigenvalues and eigenvectors of the reduced-cyclotron motion of the two-ion crystal. The solid lines correspond to the frequencies in units of the cyclotron frequency of the sensor ion ${\omega }_{cs}$, whereas the dashed lines correspond to the amplitude ratios (target over sensor). The inset shows a zoomed-in version around $\mu =1$.

Figure 6

Eigenvalues and eigenvectors of the magnetron motion of the two-ion crystal. The solid lines show the frequencies normalized to the cyclotron frequency of the sensor ion ${\omega }_{cs}$. The dashed lines, on the other hand, show the amplitude ratios (target over sensor).