Purcell factors and spontaneous emission decay rates in a linear gain medium

We introduce a theory to model spontaneous emission rates and the Purcell factor for linear gain media, including coupled loss and gain cavity systems. We demonstrate why the usual Fermi's golden rule fails and show the impact of a non-local gain term. We show how to model such effects with a standard Maxwell calculation, which also exploits the power of quasinormal modes. As an application of the theory, we show how one can use gain compensation to improve the Purcell factors of gold-dimer plasmon modes by over one million fold.