Optically driving the radiative Auger transition

In a radiative Auger process, optical decay leaves other carriers in excited states, resulting in weak red-shifted satellite peaks in the emission spectrum. The appearance of radiative Auger in the emission directly leads to the question if the process can be inverted: simultaneous photon absorption and electronic demotion. However, excitation of the radiative Auger transition has not been shown, neither on atoms nor on solid-state quantum emitters. Here, we demonstrate the optical driving of the radiative Auger transition, linking few-body Coulomb interactions and quantum optics. We perform our experiments on a trion in a semiconductor quantum dot, where the radiative Auger and the fundamental transition form a Λ-system. On driving both transitions simultaneously, we observe a reduction of the fluorescence signal by up to 70%. Our results suggest the possibility of turning resonance fluorescence on and off using radiative Auger as well as THz spectroscopy with optics close to the visible regime.

The level scheme to describe the two-laser experiments is shown in Supplementary Fig. 1. It consists of the electron ground state |s , an excited electron state |p , and the trion state |t . The laser driving the fundamental transition is labelled as ω 1 and the laser driving the radiative Auger transition is labelled as ω 2 . The corresponding Rabi frequencies are given by Ω 1 , Ω 2 and the detunings of the lasers from the corresponding transition are ∆ 1 , ∆ 2 . The spontaneous decay rates are the decay rate via the fundamental transition (Γ r ), the decay rate via radiative Auger (Γ A ), and the p-to-s decay rate (Γ p ). We simulate the system with a standard quantum optics approach. Making the dipole and the rotating-wave approximations, the Hamiltonian of the system is given by 1,2 : H = 2 [2(∆ 2 − ∆ 1 ) |p p| − 2∆ 1 |t t| +Ω 1 |t s| + Ω 2 |t p| + Ω 1 |s t| + Ω 2 |p t|]. (1) The Hamiltonian describes the coherent evolution of the system. The incoherent decay paths are taken into account by the Lindblad collapse operators for the spontaneous emission from the fundamental transition (L 1 = √ Γ r |s t|), the spontaneous radiative Auger emission (L 2 = √ Γ A |p t|)), the p-to-s relaxation (L 3 = Γ p |s p|)), and the p-shell dephasing (L 4 = √ γ p |p p|)). The dynamics of the system are described by the following master equation: where ρ is the density matrix. Using this equation, we determine the steady state of the system ( dρ dt = 0). The steady state occupation of the trion state is used for simulating the experiments as it is proportional to the fluorescence intensity.
This simulation fits well to our experimental results in Fig. 3 of the main text. We also use it to estimate the Rabi frequency Ω 2 and the dephasing γ p : when ∆ 2 is close to zero, the resonance fluorescence depends on Ω 2 . Auger transition form a Λ-system where both transitions can be driven by two independent lasers. The Rabi-frequency of the laser on the fundamental transition (ω1) is given by Ω1, the Rabi frequency of the laser on the radiative Auger transition (ω2) is given by Ω2. The corresponding laser detunings are ∆1 and ∆2, the corresponding spontaneous decay rates from the trion state |t are Γr (fundamental transition), ΓA (radiative Auger). The parameter Γp is the relaxation rate from the electron excited state |p to the electron ground state |s . (b) Comparison of the density matrix elements ρtt and Im(ρst) as a function of ∆2. The parameters are identical to those used to describe the deepest fluorescence dip shown in Fig. 3(b) of the main text.
Due to the small dipole moment of the radiative Auger transition, strong laser powers are required to achieve high values of Ω 2 . For the strongest laser power of ω 2 (increasing the power of ω 1 by a factor of ∼ 8 × 10 3 ), we estimate Ω 2 = 2π × 3.2 GHz from the simulation. Alternatively, one could estimate the ratio of the corre-50:50 1 � a g (2) 50:50 g (2) b 50:50 auto-correlation (radiative Auger) (d) Comparison between the auto-correlation of the resonance fluorescence (blue) and the cross-correlation between resonance fluorescence and radiative Auger emission (red, data from QD1). Both correlation-measurements (g (2) ) are performed with a single laser on the fundamental transition and show Rabi oscillations due to the strong driving (Ω1). The cross-correlation has a small offset given by the |p -to-|s relaxation time (τp = 1/Γp = 17 ps). This offset measures the finite time for which the Auger electron remains in the excited state after a radiative Auger process has occurred 3 . The origin of the relaxation Γp is probably a phonon-assisted decay 4 but further investigations are needed. (e) Simulation of the measurements shown in (d). (f ) Auto-correlation of the radiative Auger emission. Since the radiative Auger emission is relatively weak (count rates: 630 Hz on the first, 530 Hz on the second detector), a long integration time (∼ 50 h) is needed to resolve the Rabi oscillations in this measurement. The excitation power and Rabi frequency are slightly different with respect to the auto-and cross-correlation shown in (d).
sponding dipole moments by using the intensity ratio between resonance fluorescence and radiative Auger emission (∼ 50 : 1). Ω 2 could then be obtained by using this estimation together with the power saturation curve of the resonance fluorescence. We find that this method underestimates Ω 2 compared to the simulation. Since effects such as chromatic aberration make this second approach more prone to systematic errors, we always use the two-laser experiment and the corresponding simulation to determine Ω 2 . The dephasing term γ p is also estimated by simulating the two-laser experiment. We find that it mainly affects the width of the fluorescence dip. As explained in the main text, other parameters (Γ r , Γ p , Ω 1 ) are determined from independent measurements and kept fixed in the simulation.
There are two mechanisms that contribute to the fluorescence reduction when driving the Auger transition with ω 2 : a coherent part related to EIT/CPT and dark state formation 1 , an incoherent part due to a fast deexcitation channel from |t to |p via radiative Auger and from |p to |s by two-phonon emission 4 . The incoherent decay path is irrelevant in systems where the ground state lifetime is long 1 . To distinguish these two mechanisms we compare the density matrix element ρ tt (proportional to the overall fluorescence signal) to Im(ρ st ) (proportional to the susceptibility). The susceptibility determines the system's absorption 1 and is associated with the coherent contribution of the fluorescence reduction. In Supplementary Fig. 1(b) we plot ρ tt and Im(ρ st ) as a function of ∆ 2 . This comparison shows that the coherent contri- bution to the fluorescence reduction (EIT/CPT mechanism) is only part of the overall fluorescence reduction.

SUPPLEMENTARY NOTE 2: MODEL FOR CORRELATION MEASUREMENTS
Time-resolved correlation measurements (g (2)measurements) are used to determine the relaxation time τ p = 1/Γ p . The corresponding setups are shown in Supplementary Fig. 2(a-c). An auto-correlation of the resonance fluorescence from the fundamental transition and a cross-correlation between emission from the fundamental transition and radiative Auger emission are shown in Supplementary Fig. 2(d). As shown in Supplementary Fig. 2(e), the theoretical model fits well to the data. In these measurements, only a single laser at ω 1 is used. The system is described by Eqs. 1 and 2, with the parameter Ω 2 set to zero. We use the Quantum Toolbox in Python (QuTiP 6 ) to compute the steady state density matrix. With the resulting density matrix, we then compute the auto-and the cross-correlation. The auto-correlation is: and the cross-correlation is: In both cases, t is the time and τ is the time delay between two subsequently detected photons.â † describes the creation of a photon via decay into the s-shell (fundamental transition), andâ † A describes the creation of a photon via radiative Auger decay into the excited electron state, |p .

SUPPLEMENTARY NOTE 3: MAGNETIC FIELD DISPERSION OF THE EMISSION
The magnetic field dispersion of the radiative Auger emission is significantly stronger than that of the emission from the fundamental transition (see Supplementary Fig. 3). The reason is the different final state after the optical decay: the electron ground state |s (sshell) has a weak magnetic field dispersion and, in contrast, higher shells such as the excited state |p (p-shell) have a much stronger dependence on the magnetic field. Since the optical emission energy is given by the energy of the trion minus the energy of the final state, the strong magnetic field dispersion is transferred to the radiative Auger lines. The strong dispersion of the radiative Auger emission is an important feature allowing it to be distinguished unambiguously from phonon replicas. For a two-dimensional harmonic confinement potential, the magnetic field dispersions of the different shells form the Fock-Darwin spectrum 7 . The dispersion of the radiative Auger emission is, therefore, typically close to an inverted Fock-Darwin spectrum 3 . A model for the magnetic field dispersion has been developed in Ref. 3.

SUPPLEMENTARY NOTE 4: FURTHER MEASUREMENTS OF AUTLER-TOWNES SPLITTING
In Supplementary Fig. 4 we show additional measurements of the Autler-Townes splitting on QD1. The measurements are performed for different Rabi frequencies Ω 1 and the detuning ∆ 1 between laser and fundamental transition is varied. The quantum dot transitions are detuned from the fixed laser by applying a gate voltage, V g . The detuning from the fundamental transition is ∆V ·S s , where S s is the Stark-shift of the fundamental transition and ∆V g the difference in gate voltage. The Rabi frequencies at zero laser detuning are independently determined from a power saturation curve (red bars in Supplementary Fig. 4). They match the measured Autler-Townes splittings in the emission spectra. Furthermore, on detuning the quantum dot resonance from the laser (∆V g = 0), there is a small probability to excite the trion via the phonon sideband giving rise to a weak "diagonal" emission line. In the case of a red-detuned quantum dot (∆V g < 0), the laser has more energy than the quantum dot transition and the additional energy can be transferred to LA-phonons. In the case of a blue-detuned quantum dot, the laser energy is too small and the missing energy can be provided by phonon absorption.