  In quantum dots (QDs) the Auger recombination is a non-radiative scattering
process in which the optical transition energy of a charged exciton (trion) is
transferred to an additional electron leaving the dot. Electron tunneling from
a reservoir is the competing process that replenishes the QD with an electron
again. Here, we study the dependence of the tunneling and Auger recombintaion
rate on the applied electric field using high-resolution time-resolved
resonance fluorescence (RF) measurements. With the given p-i-n diode structure
and a tunnel barrier between the electron reservoir and the QD of $45\,$nm, we
measured a tunneling rate into the QD in the order of ms$^{-1}$. This rate
shows a strong decrease by almost an order of magnitude for decreasing electric
field, while the Auger emission rate decreases by a factor of five in the same
voltage range. Furthermore, we study in detail the influence of the Auger
recombination and the tunneling rate from the charge reservoir into the QD on
the intensity and linewidth of the trion transition. Besides the well-known
quenching of the trion transition, we observe in our time-resolved RF
measurements a strong influence of the tunneling rate on the observed
linewidth. The steady-state RF measurement yields a broadened trion transition
of about $1.5\,$GHz for an Auger emission rate of the same order as the
electron tunneling rate. In a non-equilibrium measurement, the Auger
recombination can be suppressed, and a more than four times smaller linewidth
of $340\,$MHz ($1.4\,$${\mu}$eV) is measured.
