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Electromagnetically induced transparency (EIT) cooling has established itselfas one of the most widely used cooling schemes for trapped ions during the pasttwenty years. Compared to its alternatives, EIT cooling possesses importantadvantages such as a tunable effective linewidth, a very low steady statephonon occupation, and applicability for multiple ions. However, existinganalytic expression for the steady state phonon occupation of EIT cooling islimited to the zeroth order of the Lamb-Dicke parameter. Here we extend suchcalculations and present the explicit expression to the second order of theLamb-Dicke parameter. We discuss several implications of our refined formulaand are able to resolve certain difficulties in existing results.

Steady state phonon occupation of EIT cooling: higher order calculations