  We investigate the dynamical properties of nanolasers comprising a few
two-level emitters coupled to an optical cavity. A set of rate equations is
derived, which agree very well with a solution of the full master equation
model and makes it simple to investigate the properties of the system. Using a
linearized version of these rate equations, we can analytically express the
response of the nanolaser to a modulation of the pumping rate. These results
are compared to the modulation response obtained directly from the master
equation using a novel method. Using the rate equation method, we calculate the
modulation bandwidth and show that, contrary to conventional semiconductor
lasers, the nanolaser is typically over-damped and displays a dip in the
modulation bandwidth as the two-level systems become inverted. Both these
features can be traced back to the modeling of the emitters as two-level
systems that are incoherently pumped.
