  We theoretically study a nanospaser system, which consists of a spherical
silver nanoparticle embedded inside a sphere composed of dye molecules. The
gain of the system, dye molecules, are described by a three-level model, where
the transition frequency between the lowest two energy levels is close to the
surface plasmon frequency of the nanosphere. Contrary to a two-level model of
spaser, the three-level model takes into account finite relaxation time between
the high energy levels of the gain medium. These relaxation processes affect
both the spaser threshold and the number of generated plasmons in the
continuous wave regime. While for a two-level model of a spaser the number of
generated plasmons has a linear dependence on the gain, for the three-level
model this dependence becomes quadratic.
