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

The interplay between electron tunneling and Auger emission in a single quantum emitter weakly coupled to an electron reservoir