Abstract
In a radiative Auger process, optical decay leaves other carriers in excited states, resulting in weak redshifted 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 solidstate quantum emitters. Here, we demonstrate the optical driving of the radiative Auger transition, linking fewbody 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 Λsystem. 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.
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Introduction
Nonradiative Auger processes have been observed in both atoms^{1} and solidstate quantum emitters^{2,3}. They play an important role in determining the efficiency of semiconductor lightemitting diodes and lasers^{4}. In the nonradiative Auger process, one electron reduces its energy by transferring it to a second electron that is promoted to a highenergy state. In the radiative Auger process (shakeup), in contrast, one electron makes an optical decay, creating a photon. Part of the photon energy is transferred to a second electron such that the radiative Auger emission is redshifted with respect to the main emission line. Both radiative and nonradiative Auger processes arise as a consequence of the Coulomb interactions between electrons in close proximity^{5,6,7}. Nonradiative Auger is a purely Coulomb scattering process. In contrast, radiative Auger involves both Coulomb scattering and electronphoton interactions. It can either be viewed as a higherorder scattering process or interpreted in terms of Coulombinduced admixtures of higher singleparticle states to the multielectron wave function^{7,8}. What appears to be an optical relaxation of one electron in the singleparticle picture involves, in fact, a sudden change of the manyparticle configuration.
Radiative Auger emission has been observed over a large spectral range: in the Xray emission of atoms^{9}; close to visible frequencies on donors in semiconductors^{10} and quantum emitters^{11,12}; and at infrared frequencies as shakeup lines in twodimensional systems^{13,14,15,16,17}. Furthermore, radiative Auger connects carrier dynamics to the quantum optical properties of the emitted photons^{11}, making it a powerful probe of multiparticle systems. Driving the fundamental transition between the electron ground state and an optically excited state is an important technique in quantum optics^{18,19}. In contrast, driving the radiative Auger transition has not been achieved, neither on atoms nor on solidstate systems. Success here would significantly increase the number of quantum optics techniques that can be employed.
We demonstrate driving the radiative Auger transition on an epitaxial GaAs quantum dot embedded in AlGaAs^{20,21}. The quantum dot forms a potential minimum and confines charge carriers, resulting in discrete energy levels like in an atom. Without optical illumination, a single electron is trapped in the conduction band of the quantum dot and occupies the lowest possible shell (the sshell, \(\lefts\right\rangle\)). Upon resonant excitation of the fundamental transition, a second electron is promoted from the filled valence band to the conduction band and a negative trion X^{1−} (\(\leftt\right\rangle\)) is formed. This trion consists of two electrons in the conduction band and one electronvacancy (hole) in the valence band. Figure 1a shows the possible optical decay paths: in the fundamental transition, one electron decays, removing the valence band hole while the other electron remains in the conduction band ground state \(\lefts\right\rangle\); in the radiative Auger process, the remaining electron is left in an excited state \(p\rangle\). The emitted photon is redshifted by the energy separation between \(p\rangle\) and \(\lefts\right\rangle\)^{5,11}. Figure 1b shows a typical emission spectrum from the trion decay. This spectrum is measured on resonantly driving the fundamental transition \(\lefts\right\rangle\)–\(\leftt\right\rangle\) at 384.7 THz (1.591 eV) with a narrowbandwidth laser^{11}. Redshifted by 3.2 THz (13.2 meV) from the fundamental transition, there is a weak satellite line that arises from the radiative Auger process.
Photons at the radiative Auger frequency have insufficient energy to excite the fundamental transition \(\lefts\right\rangle\)–\(\leftt\right\rangle\). Figure 1c shows how the trion state \(\leftt\right\rangle\) still can be excited with a laser at the Auger transition. The missing energy is provided by the electron, which initially occupies the excited state \(p\rangle\). However, driving the radiative Auger transition is experimentally challenging for two main reasons: first, there is a fast nonradiative relaxation from the excited singleelectron state \(p\rangle\) back to \(\lefts\right\rangle\)^{11,22}, and the state \(p\rangle\) is not occupied at thermal equilibrium. Second, the dipole moment of the radiative Auger transition is small. Therefore, it is difficult to achieve high Rabi frequencies on driving the transition, plus the radiative Auger emission is very weak and hard to distinguish from the backreflected laser light.
Results
We perform a twolaser experiment revealing optical driving of the radiative Auger transition. The fundamental transition \(s\rangle\)–\(\leftt\right\rangle\) (at ~ 1.591 eV) is driven with one laser (labelled by ω_{1}) while the radiative Auger transition (at ~ 1.578 eV) is simultaneously driven with a second laser (labelled by ω_{2}). This Λconfiguration has the following advantages: First, on driving \(\lefts\right\rangle\)–\(t\rangle\) with ω_{1}, there is a small chance of initializing the system in state \(p\rangle\) via the radiative Auger emission. Additionally, driving the \(p\rangle\)–\(\leftt\right\rangle\)transition with ω_{2}, while transferring population to \(\leftt\right\rangle\) with ω_{1} also leads to a finite occupation of \(p\rangle\). Second, the small dipole matrix element of the radiative Auger transition is compensated by using high power for ω_{2}. The high power causes a high laser background when detecting the fluorescence from the radiative Auger transition. Instead, we tune the second laser over the Auger transition while measuring just the fluorescence originating from the fundamental transition \(\lefts\right\rangle\)–\(\leftt\right\rangle\). Figure 1d shows the result of this twolaser experiment. We observe a strong reduction in fluorescence on addressing the transition \(p\rangle\)–\(\leftt\right\rangle\) which is characteristic of twocolour excitation of a Λconfiguration. Our approach has a conceptual similarity to the driving of weak phonon sidebands of mechanical resonators resulting in optomechanically induced transparency^{23,24}.
Autler–Townes splitting in singlelaser experiments
We consider initially the situation where the fundamental transition (\(\lefts\right\rangle\)–\(\leftt\right\rangle\)) is strongly driven by a single laser. If radiative Auger and fundamental transition form a Λsystem, one would expect an Autler–Townes splitting in the radiative Auger emission. Figure 2a shows the corresponding level scheme including the dressed states \(\frac{1}{\sqrt{2}}(\leftN+1,s\right\rangle \pm \leftN,t\right\rangle )\) and \(\frac{1}{\sqrt{2}}(\leftN,s\right\rangle \pm \leftN1,t\right\rangle )\), where N is the photon number. The dressedstate splitting leads to the Mollow triplet in the resonance fluorescence^{19,25,26}. For a decay into a third level, the Autler–Townes splitting^{27,28} in the emission is expected to be Ω_{1}. Figure 2b shows the radiative Auger emission of one quantum dot (QD1). In this measurement, the laser is on resonance with the fundamental transition. The Rabi frequency (Ω_{1} = 2π × 31.9 GHz, red bar in Fig. 2b) is estimated independently by measuring the fluorescence intensity as a function of laser power (Supplementary Fig. 3b). We observe an Autler–Townes splitting that agrees well with this Rabi frequency. For this quantum dot, we also observe an additional weak emission appearing on the low energy side of the spectrum when using high Rabi frequencies (Fig. 2b and Supplementary Fig. 4). We speculate that this emission is connected to optical coupling between \(p\rangle\) and an excited trion state, \(\left{t}^{* }\right\rangle\). Figure 2c shows radiative Auger emission from a second quantum dot (QD2). For this quantum dot, we measure the radiative Auger emission as a function of detuning between the quantum dot transition and the laser (see Supplementary Fig. 4 for the corresponding measurement on QD1). On applying a gate voltage ΔV_{g}, the quantum dot transition \(\lefts\right\rangle\)–\(\leftt\right\rangle\) is detuned from the fixed laser by Δ_{1} = ΔV_{g} ⋅ S_{s} via the quantumconfined Stark shift. S_{s} parameterizes the Stark shift of the fundamental transition. At zero detuning, the observed Autler–Townes splitting again agrees with the Rabi frequency obtained from a power saturation curve (Ω_{1} = 2π × 67.7 GHz).
Twolaser experiments
We now consider the experiments with the second laser (labelled as ω_{2}) at the radiative Auger transition. Figure 3a shows the corresponding level scheme. We set ω_{1} to a modest Rabi frequency (Ω_{1} = 2π × 0.08 GHz) compared to the decay rate of the trion (Γ_{r} = 2π × 0.50 GHz). The frequency of the radiative Auger transition is estimated from the trion emission spectrum (Fig. 1b). We sweep the frequency ω_{2} and simultaneously monitor the resonance fluorescence intensity from the fundamental transition. Figure 3b shows this measurement for different powers of the laser on the Auger transition. On increasing the power of ω_{2} to several orders of magnitude higher than the power of ω_{1}, there is a pronounced dip in the fluorescence intensity. This intensity dip appears precisely when the laser frequency ω_{2} matches the radiative Auger transition (\(p\rangle\)–\(\leftt\right\rangle\)) and is characteristic for a Λsystem that is driven with two lasers. We estimate the Rabi frequency Ω_{2} driving \(p\rangle\)–\(\leftt\right\rangle\) by simulating the resonance fluorescence intensity as a function of Δ_{2} (see Supplementary Note 1 for the quantum optics simulation). In this simulation, we keep the decay rate from \(p\rangle\) to \(\lefts\right\rangle\) (Γ_{p} ~ 2π × 9.3 GHz) fixed to the value that we determine from independent auto and crosscorrelation measurements^{11} (Supplementary Fig. 2d). The value for Ω_{2} can then be determined by a corresponding fit to the twolaser experiment. Additionally, we fit a constant pure dephasing, γ_{p}, for the state \(p\rangle\) which leads to an additional broadening of the fluorescence dip. We estimate γ_{p} ~ 2π × 8.8 GHz from the fit and a Rabi frequency of Ω_{2} = 2π × 3.2 GHz (ω_{2}) for the strongest fluorescence dip. Note that additional excitationinduced dephasing via phonons is expected to be weak for such Rabi frequencies^{29,30}.
In Fig. 3c, we plot the minimum of the resonance fluorescence dip as a function of Ω_{2}. The Λsystem model with two driving lasers fits this data set well. For the highest value of Ω_{2}, we achieve a reduction of the resonance fluorescence intensity by up to 70%. The intensity reduction is limited by the power that we can reach in our optical setup. The measurement shows that resonance fluorescence can be switched on and off by using the radiative Auger transition. In our system, part of the fluorescence dip is due to the reduction of the overall absorption via the formation of a dark state. This effect is related to electromagnetically induced transparency (EIT)^{31} and coherent population trapping (CPT)^{32,33}. An additional reduction of the signal comes from the fact that there is a fast decay rate Γ_{p} from state \(p\rangle\) to \(\lefts\right\rangle\). Thus, after the laserinduced transition from state \(\leftt\right\rangle\) to \(p\rangle\), the system quickly decays to the ground state \(\lefts\right\rangle\). This deexcitation channel reduces the population of the trion state and, therefore, the fluorescence intensity. We can distinguish the contributions of the two mechanisms by our quantum optics simulation. The density matrix element ρ_{tt} (occupation of state \(\leftt\right\rangle\)) is proportional to the overall fluorescence intensity. The term Im(ρ_{st}) (coherence between the states \(\lefts\right\rangle\) and \(\leftt\right\rangle\)) is proportional to the absorption and reflects the coherent part of the intensity reduction. The contribution of both mechanisms is comparable for the parameter regime in which we operate (Supplementary Fig. 1b).
The measurements so far were performed with ω_{1} on resonance (Δ_{1} = 0). We repeat the twolaser experiments while detuning ω_{1} from the fundamental transition. Figure 3d shows the fluorescence intensity for positive, zero, and negative detuning Δ_{1}. For nonzero detuning, the fluorescence dip is asymmetric as a function of Δ_{2}. The asymmetry is an important result as it cannot be explained by a rate equation description, but depends on the quantum coherence in the master equation model (Supplementary Note 1). The full dependence of the resonance fluorescence intensity as a function of Δ_{1} and Δ_{2} is plotted in Fig. 3e. This data set fits well to the corresponding quantum optics simulation in Fig. 3f using the parameters from the previous measurements.
Discussion
Upon the optical transition of a carrier, radiative Auger leaves other carriers in an excited orbital state, and the emitted photon is redshifted. We show here that this process can be inverted: radiative Auger exists in absorption and the corresponding transition can be optically driven. In both emission and absorption, the process has conceptual similarities to phonon scattering. For radiative Auger emission, the electronic configuration is left in an excited state, for the phonon sideband, the lattice configuration^{34,35}. We demonstrate that the resonance fluorescence can be strongly reduced by addressing the radiative Auger transition: a modulated laser on the radiative Auger transition could be used for fast optical gating of the emitter’s absorption. As an outlook, we suggest that an effective coupling between orbital states, split by frequencies in the THz band, can be created by two lasers at optical frequencies. The idea here is to establish a Ramanlike process: the lasers are equally detuned from their resonances, and an exciton is not created. This scheme facilitates control of the orbital degree of freedom with techniques that have been developed for manipulating spinstates^{33,36}. Further quantum optics experiments with radiative Auger photons are conceivable: by using a twocolour Ramanscheme^{37}, it might be possible to create deterministically highly excited shakeup states that are also subject of recent theoretical investigations^{38}. Adding a third laser with a THzfrequency at the transition \(\lefts\right\rangle\)–\(p\rangle\)^{22} might even allow closecontour driving schemes^{39}. In analogy to experiments on spins^{40}, the radiative Auger process could lead to an entanglement between the frequency of the emitted photon and the orbital state of the Auger electron.
Methods
For all our measurements, the quantum dot sample is kept in a liquid helium bath cryostat at 4.2 K. The quantum dots used in this work are GaAs quantum dots in AlGaAs grown by molecular beam epitaxy. Their decay rates Γ_{r} (typically in the range 2π × (0.5 − 0.6) GHz) were determined by lifetime measurements using pulsed resonant excitation^{21}. The decay rate of the radiative Auger transition, Γ_{A} ~ Γ_{r}/100, is estimated by comparing its emission intensity to the fundamental transition. QD1 is identical to the second quantum dot in Ref. ^{21}. The quantum dots presented in this work have a stronger radiative Auger emission compared to other IIIV quantum dots^{11} indicating a stronger dipole moment of the radiative Auger transition. We use radiative Auger lines where the final state of the Auger electron, \(p\rangle\), is a quantum dot pshell. In particular, we investigate the transition associated with the lower pshell (p_{+}) for QD1 and the higher pshell (p_{−}) for QD2. We can assign further emission lines to the corresponding higher electronic shells by measuring the magnetic field dispersion of the emission spectrum^{11} (Supplementary Fig. 3a).
To excite the quantum dots, we use a tunable diode laser with a narrow bandwidth ( ~ 100 kHz) far below the quantum dot linewidth. Resonant excitation is not necessary to observe the radiative Auger emission: aboveband excitation is also effective^{7,12}. It is also possible to observe radiative Auger on systems that suffer from much more charge noise than ours^{12}. However, resonant excitation has the advantage that no continuum states are excited making it easier to identify all emission lines, and low charge noise makes resonant excitation a lot easier to perform. For this work, resonant excitation is crucial to optically address a single radiative Auger transition. To suppress the reflected excitation laser, we use a crosspolarization scheme^{41}.
To determine the relaxation rate Γ_{p} ( ~2π × 9.3 GHz) from \(p\rangle\) to \(\lefts\right\rangle\), we make use of a technique developed in Ref. ^{11}: on driving \(\lefts\right\rangle\)–\(\leftt\right\rangle\) (Ω_{2} = 0), we measure an autocorrelation of the resonance fluorescence from the fundamental transition and compare it to the crosscorrelation between resonance fluorescence from the fundamental transition and radiative Auger emission. The corresponding measurement setups are shown in Supplementary Fig. 5a, b. To resolve the auto and crosscorrelations with high time resolution, we use two superconducting nanowire singlephoton detectors (SingleQuantum) with a timing jitter below 20 ps (FWHM) in combination with correlation hardware (Swabian Instruments).
Compared to the autocorrelation, the crosscorrelation has a small time offset when a radiative Auger photon is followed by a photon from the fundamental transition (Supplementary Figs. 2d, e). This time scale corresponds to the relaxation time, τ_{p} = 1/Γ_{p}, describing the relaxation from \(p\rangle\) to \(\lefts\right\rangle\). The relaxation time appears in the crosscorrelation: when a radiative Auger event is detected by the first detector, there is an additional waiting time of τ_{p} before the excited Auger electron relaxes to the ground state and the system can be optically reexcited. Therefore, it takes longer before a second photon is detected. The additional waiting time is only present for the crosscorrelation. For the autocorrelation, the system decays directly to the ground state \(\lefts\right\rangle\) and there is no additional waiting time.
Finally, we also measure the autocorrelation of the radiative Auger emission (see Supplementary Fig. 5c for the setup). The measurement is shown in Supplementary Fig. 5e. We observe a pronounced antibunching at zero delays proving the singlephoton nature of the radiative Auger photons. Going beyond the results in Ref. ^{11}, we observe the Rabi oscillation from strongly driving the transition \(\lefts\right\rangle\)–\(\leftt\right\rangle\) in the photonstatistics of the radiative Auger photons from the transition \(p\rangle\)–\(\leftt\right\rangle\).
Data availability
The data that supports this work is available from the corresponding author upon reasonable request.
Code availability
The code that has been used for this work is available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Krzysztof Gawarecki and Philipp Treutlein for fruitful discussions. We acknowledge financial support from Swiss National Science Foundation project 200020_175748, NCCR QSIT and Horizon 2020 FETOpen Project QLUSTER. LZ, GNN, AJ acknowledge support from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie SkłodowskaCurie grant agreement numbers 721394 / 4PHOTON (LZ), 861097 / QUDOTTECH (GNN), 840453 / HiFig (AJ). JR, AL and ADW gratefully acknowledge financial support from the grants DFH/UFA CDFA0506, DFG TRR160, DFG project 383065199, EU Horizon 2020 Grant No. 861097, and BMBF  QR.X KIS6QK4001.
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J.R., L.Z., A.D.W. and A.L. grew the sample. C.S., L.Z., G.N.N. fabricated the sample. L.Z., M.C.L., J.R. and A.L. designed the sample. L.Z., C.S., G.N.N. and M.C.L. carried out the experiments. M.C.L., C.S., P.M., D.E.R., L.Z., G.N.N., A.J. and R.J.W. analysed the data and developed the theory. C.S. and M.C.L. prepared the various figures. MCL initiated and conceived the project. M.C.L. and R.J.W. wrote the manuscript with input from all the authors.
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Spinnler, C., Zhai, L., Nguyen, G.N. et al. Optically driving the radiative Auger transition. Nat Commun 12, 6575 (2021). https://doi.org/10.1038/s41467021268758
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DOI: https://doi.org/10.1038/s41467021268758
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