  We describe the operation of an electrodynamic ion trap in which the electric
quadrupole field oscillates at two frequencies. This mode of operation allows
simultaneous tight confinement of ions with extremely different charge-to-mass
ratios, e.g., singly ionised atomic ions together with multiply charged
nanoparticles. We derive the stability conditions for two-frequency operation
from asymptotic properties of the solutions of the Mathieu equation and give a
general treatment of the effect of damping on parametric resonances.
Two-frequency operation is effective when the two species' mass ratios and
charge ratios are sufficiently large, and further when the frequencies required
to optimally trap each species are widely separated. This system resembles two
coincident Paul traps, each operating close to a frequency optimized for one of
the species, such that both species are tightly confined. This method of
operation provides an advantage over single-frequency Paul traps, in which the
more weakly confined species forms a sheath around a central core of tightly
confined ions. We verify these ideas using numerical simulations and by
measuring the parametric heating induced in experiments by the additional
driving frequency.
