  Penning traps, with their ability to control planar crystals of tens to
hundreds of ions, are versatile quantum simulators. Thermal occupations of the
motional drumhead modes, transverse to the plane of the ion crystal, degrade
the quality of quantum simulations. Laser cooling using electromagnetically
induced transparency (EIT cooling) is attractive as an efficient way to quickly
initialize the drumhead modes to near ground-state occupations. We numerically
investigate the efficiency of EIT cooling of planar ion crystals in a Penning
trap, accounting for complications arising from the nature of the trap and from
the simultaneous cooling of multiple ions. We show that, in spite of
challenges, the large bandwidth of drumhead modes (hundreds of kilohertz) can
be rapidly cooled to near ground-state occupations within a few hundred
microseconds. Our predictions for the center-of-mass mode include a cooling
time constant of tens of microseconds and an enhancement of the cooling rate
with increasing number of ions. Successful experimental demonstrations of EIT
cooling in the NIST Penning trap [E. Jordan, K. A. Gilmore, A. Shankar, A.
Safavi-Naini, M. J. Holland, and J. J. Bollinger, "Near ground-state cooling of
two-dimensional trapped-ion crystals with more than 100 ions", (2018),
submitted.] validate our predictions.
