  Electromagnetically induced transparency (EIT) cooling has established itself
as one of the most widely used cooling schemes for trapped ions during the past
twenty years. Compared to its alternatives, EIT cooling possesses important
advantages such as a tunable effective linewidth, a very low steady state
phonon occupation, and applicability for multiple ions. However, existing
analytic expression for the steady state phonon occupation of EIT cooling is
limited to the zeroth order of the Lamb-Dicke parameter. Here we extend such
calculations and present the explicit expression to the second order of the
Lamb-Dicke parameter. We discuss several implications of our refined formula
and are able to resolve certain mysteries in existing results.
