Laser-assisted motional-mode spectroscopy in a Penning trap and the generalized invariance theorem

In this paper we measure all the motional-mode frequencies (common and stretch) of a balanced and an unbalanced two-ion Coulomb crystal in a 7-tesla Penning trap using an electric-dipole transition in 40Ca+. The three common modes of the balanced crystal were obtained using a laser-pulse measurement technique developed in a previous work for a single ion, whereas the stretch mode frequencies were measured in a stationary state while applying laser cooling. Due to the Coulomb interaction, and the heating of the crystal, nonlinearities appear in the oscillators, which are treated analytically and numerically to obtain the experimental frequency values. The generalized invariance theorem has been tested yielding a relative shift of -1.0 x 10-6 with a relative uncertainty of 4 x 10-6. Single-ion frequency measurements are used to assess its validity.