A two-frequency ion trap confining ions with widely different charge-to-mass ratios

We describe the theory of two-frequency operation of an ion trap and solve the equations of motion for two species of ions with molecular mass, charge: $M_A,+1$ and $M_B, +33$ respectively, where $M_A = \SI{138}{amu}$ is an isotope of barium and $M_B = \SI{1.4e6}{amu}$, e.g., a large protein or molecular complex. The quadrupole electric field is created by RF radiation with angular frequencies $\omega_1$ and $\omega_2$ (with $\omega_2 = 100 \omega_1$). For such very different charge-to-mass ratios, and radio-frequencies, the heavy ions (molecular mass $M_B$) are confined most strongly by the field at the lower frequency $\omega_1$, and trapping of ions of atomic mass $M_A$ arises from the field at $\omega_2$. Thus we obtain a superposition of two almost independent Paul traps whose centres can be made coincident or moved apart. Importantly the effective spring constants can be adjusted to be the same for both species so that all the ions interact strongly. This allows efficient sympathetic cooling of the heavy ions by laser-cooled atomic ions as we demonstrate by numerical simulations for one particular example. This approach can be extended to charged particles with more dissimilar masses.