Quantum quench of Kondo correlations in optical absorption

Transcription

1 Quantum quench of Kondo correlations in optical absorption C. Latta 1, F. Haupt 1, M. Hanl 2, A. Weichselbaum 2, M. Claassen 1, W. Wuester 1, P. Fallahi 1, S. Faelt 1, L. Glazman 3, J. von Delft 2, H. E. Türeci 4,1 A. Imamoglu 1 1 Institute of Quantum Electronics, ETH-Zürich, CH-8093 Zürich, Switzerland, 2 Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, D München,Germany, 3 Sloane Physics Laboratory, Yale University, New Haven, CT 06520, USA 4 Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA To whom correspondence should be addressed; clatta@phys.ethz.ch, imamoglu@phys.ethz.ch The interaction between a single confined spin and the spins of a Fermionic reservoir leads to one of the most spectacular phenomena of many the Kondo correlations off[23]. Our experiments demonstrate that optical measurements on single body physics the Kondo effect[12, 13]. Here artificial atoms offer new perspectives on manybody phenomena previously studied exclusively we report the observation of Kondo correlations in optical absorption measurements on a single semiconductor quantum dot tunnel-coupled using transport spectroscopy. Moreover, they initiate a new paradigm for quantum optics where to a degenerate electron gas. In stark contrast many-body physics influences electric field and intensity correlations. to transport experiments[17 19], absorption of a single photon leads to an abrupt change in the system Hamiltonian and a quantum quench of Kondo correlations. By inferring the characteristic power law exponents from the experimental absorption line-shapes, we find a unique signature of the quench in the form of an Anderson orthogonality catastrophe[15, 16], originating from a vanishing overlap between the initial and final many-body wave-functions. We also show that the power-law exponents that determine the degree of orthogonality can be tuned by applying an external magnetic field which gradually turns Optical spectroscopy of single quantum dots (QD) has demonstrated its potential for applications in quantum information processing, particularly in the realization of single and entangled photon sources[1, 2], coherent spin qubits[3, 4] and a spin-photon interface[5, 6]. Even though recent experiments have established this system as a new paradigm for solid-state quantum optics, all of the striking experimental observations to date could be understood within the framework of single- or few-particle physics enriched by perturbative coupling to reservoirs involving either phonons, a degenerate electron gas [7, 8], or nuclear spins [9, 10]). We present differential transmission (DT)

2 2 both the perturbative and non-perturbative regimes of the Kondo effect in one absorption line shape, without having to change the FR (electron) temperature T FR. Experimental setup. The schematic of the QD sample we study is shown in Figure 1a: a gate voltage V g applied between a top Schottky gate and the degenerate electron gas allows us to tune the charging state of the QD[20]. Figure 1B shows the photoluminescence (PL) spectrum as a function of V g, where different discrete charging plateaux are clearly observable. In this paper we focus on the X plateau, for which the QD is singleelectron charged and the influence of the FR on the QD PL dispersion and linewidth is strongest. The X optical transition couples the initial configuration, containing on average one electron in the QD, to a final configuration, containing on average two electrons and a valence-band experiments[11] on single charge-tunable QDs that reveal optical signatures of the Kondo effect[12, 13]. In contrast to prior experiments[8, 14], the tunnel coupling between the QD and a nearby degenerate electron gas, which we refer to as the Fermionic reservoir (FR), is engineered to be so strong that the resulting exchange interactions cannot be treated within the framework of a perturbative system-reservoir theory: In the initial state, the system (QD spin) is maximally entangled with the FR, forming a singlet. The fundamentally new feature that differentiates the results we present from all prior transport based investigation of the Kondo effect[17 19], is the realization of a quantum quench of the local Hamiltonian; in our experiments, photon absorption abruptly turns the exchange interaction between the QD electron and the FR off, leading to the destruction of the correlated QD-FR singlet hole (a negatively charged trion). This transition can that otherwise acts as a local scattering potential for all FR electrons. As was shown by Anderson[15, 16], the be described within the framework of an excitonic Anderson model (EAM) [23, 24], depicted schematically in overlap between N-electron FR states with and without a Fig. 2C (and described explicitly in Supporting Online local scattering potential scales as N α with α > 0. This Material). It is parameterized by the energy ε of the reduced overlap, termed Anderson orthogonality catastrophe (AOC), leads to a power-law tail in absorption if the scattering potential is turned on or off by photon absorption. Here, we determine the AOC induced powerlaw exponents in absorption line shape that uniquely characterize the (quench of) Kondo correlations. Moreover, by tuning the applied laser frequency, we observe QD electron level with respect to the Fermi level, the onsite Coulomb repulsion U ee, the tunnel rate Γ between QD and FR, the half-bandwidth D of the FR, and the electron-hole Coulomb attraction U eh. The latter is relevant only in the final configuration, where it effectively lowers the electron level energy to ε U eh, thus ensuring the double occupancy of the electron level in the final

3 3 configuration. A Hartree-Fock estimate from the PL data in Figure 1b yields U eh U ee + 4 mev. Absorption line shape. The inset of Figure 2a shows high resolution laser absorption spectroscopy on the same QD across the X single electron charging plateau (see Supporting Online Material). Here, we parameterize V g in terms of ε, normalized and shifted such that ε = 1 2 U ee for the gate voltage where the absorption contrast is maximal. Instead of the usual linear dc-stark shift of the absorption peak that is characteristic of charge-tunable QDs, we find a strongly non-linear ε-dependent shift of the X transition energy[7, 8], which measures the energy difference between the final and initial ground states. This energy shift arises from a renormalization of the initial state energy [22] due to virtual tunneling between the singly-occupied QD and FR (analogous to the Lamb shift of atomic ground states). The final trion state energy, on the other hand, is hardly affected by virtual tunneling processes, due to large U eh U ee. As depicted in Figure 2c and explained in its caption, this renormalizationinduced red-shift of the initial state is strongest at the plateau edges and leads to an ε-dependent blue-shift of the optical resonance frequency. The latter can be used to determine the EAM model parameters: U ee =7.5 mev, Γ = 0.7 mev, and D = 3.5 mev. NRG calculations for Figure 3a shows on a log-log scale, the blue tail of the normalized absorption line shapes, where the applied laser frequency is larger than the transition energy, for the four values of gate voltage indicated by arrows in inset of Figure 2a. Inset to Figure 3a compares the full unnormalized absorption line shapes for the identical gate voltages in linear scale; the red absorption tail allows us to determine the temperature of the FR as T FR = 180 mk= 15.6µeV (see Supporting Online Material). The strong variation of the peak absorption strength in Figure 3a inset is a consequence of the exponential dependence of the Kondo temperature T K (ε) = ΓDe [1 (2ε/U ee+1) 2 ](πu ee /8Γ) on the gate voltage ε. For this QD, T K varies between 24µeV and 464µeV [27]. All line shapes carry the signatures of an optical interference effect induced by the sample structure (causing some lineshapes to become negative for small red detunings), and of independently measured fluctuations in gate voltage; both effects have been taken into account in the calculated line shapes (see Supporting Online Material). Calculating the line shapes by NRG (solid lines) without any further fit parameters, we find remarkable agreement with experiment for all four lineshapes depicted in Fig. 3a, demonstrating the validity of the excitonic Anderson Model[23] for the coupled QD-FR system. Perturbative regime. To obtain better insight the transition energy (Figure 2a, solid blue line) give excellent agreement with the experimental data (blue symbols). into the underlying physics, we first consider the absorption line shape for blue detunings satisfying ν >

4 4 max(t FR, T K ), where a perturbative description is possible. All four line shapes in Fig. 3a impressively show the by a constant factor ξ for all three ε values yields a scaling collapse of nearly identical quality, provided 1 ξ 5. ν 1 dependence expected for the perturbative regime. This reflects the fact that T K, being a crossover scale, is The origin of the ν 1 tail can be understood as arising from a two-step process, where a virtual excitation of a valence-band electron to the conduction band is followed by excitation of an electron-hole pair in the FR via an effective spin-exchange coupling. The energy of the electron-hole pair is equal to ν, ensuring energy conservation. The lowest-energy electron in the FR, which can contribute to this process, has an energy ν (we took the Fermi energy to be 0). Assuming a constant density of states around the Fermi level of the FR, the effective bandwidth of electrons contributing to this twostep absorption process at a detuning ν is equal to ν. The electron-hole pair generation therefore contributes an additional phase space factor ν to the Lorentzian broadened QD trion transition [23], such that the overall lineshape has approximately a 1/ν tail. defined only up to an arbitrary pre-factor of order one. Strong-coupling Kondo regime. In the limit T FR < ν < T K, a perturbative description of the lineshape is no longer valid. In the initial configuration, the exchange interaction between QD and FR induces a Kondo screening cloud that forms a singlet with the QD spin. This acts as a scattering potential that induces strong phase shifts for those low-energy fermionic excitations whose energies are within T K from the Fermi level. In the final configuration after photon absorption, the QD has two electrons in a local singlet state. Therefore the Kondo screening cloud, and the scattering potential for FR electrons constituted by it, disappear in the long time limit: the corresponding ground state wave-function is a tensor product of the local singlet and free electronic states, with only weak phase shifts. Since the initial and Scaling collapse. For gate voltages such that the final FR phase shifts differ (as depicted schematically in initial ground state is a Kondo singlet, the absorption line shape, normalized to its value at ν = T K, is expected [23] to be a universal function of ν/t K in the regime T FR ν U ee. To confirm this prediction, Fig. 3c shows A(ν)/A(T K (ε)) as a function of ν/t K for three of the line shapes of Fig. 3a (but omitting the black one, for which the coupled system is in the mixed valence regime). A striking scaling collapse is evident [28]. Multiplying T K Fig. 2d), the FR does not remain a spectator during the X transition: instead, the transition matrix element between the ground states of the initial and final configurations is vanishingly small. This leads to an AOC which manifests itself by transforming a delta-function resonance (of an uncoupled QD) into a power-law singularity [15] of the form ν η, where the exponent η characterizes the extent of AOC. For T FR ν T K, the

5 5 absorption lineshape of the X transition is expected to show an analogous power-law singularity. The exponent η is predicted [23, 24] to range between 0 and 0.5 (assuming no magnetic field), with η 0.5 being characteristic for a Kondo-correlated initial state and an uncorrelated final state. This lineshape modification is a consequence of a redistribution of the optical oscillator strength, associated with the fact that the FR wave-function in the Kondo correlated initial state has finite overlap with a range of final states consisting of electron-hole pair excitations out of a non-interacting FR. If T FR T K and the optical detuning is reduced below T K, the lineshape is predicted to smoothly cross over from the perturbative 1/ν tail to the strong-coupling 1/ν 0.5 power law just discussed. This crossover is illustrated in Fig. 3b (dashed lines) by NRG calculations, performed at T FR = 0 for the four ε-values of Fig. 3a: Remarkably, despite drastic differences in the ν > T K tails due to different values of T K (ε), all four lineshapes show similar power-law exponents around η 0.5 for ν T K, in accordance with predictions [23] based on Hopfield s rule (see Supplementary Information). For nonzero temperature, however, the 1/ν 0.5 power law is cut off and saturates once ν decreases past T FR (Fig. 3b, solid lines), because of thermal averaging over initial states with excitation energies T FR. Magnetic field-tuning of Kondo correlations. A direct extraction of the 1/ν 0.5 power law from the measured data is difficult due to the small accessible experimental window T FR < ν < T K. Nevertheless, we are able to determine the power-law exponent corresponding to our data accurately by using the fact that the detailed form of the line shape sensitively depends on the exponent η which can be tuned by an external magnetic field[23]. Figure 4a shows the measured absorption line shape of a second QD at B ext = 0 Tesla (black squares) and the line shapes of the circularly polarized blue (blue dots) and red (red triangles) Zeeman/trion transition at B ext = 1 Tesla in a linear scale. We have determined the parameters of this QD to be U ee = 7.5 mev, Γ = 1 mev, D = 6.5 mev and U eh = 3/2U ee ; the data were taken at T FR = 15.6µeV for ε/u ee = 0.43 where T K = 140µeV. While the peak contrast at B ext = 1Tesla for the blue (red) trion transition increases (decreases) by a factor 2, the area under the absorption curve increases (decreases) by less than 20%; this observation proves that the change in the line shape is predominantly due to a line narrowing (broadening) associated with a increase (decrease) of the power-law exponent η. Conversely, the small change in the area under the absorption curve despite having B ext = 2.5T FR demonstrates that the initial state is a Kondo singlet that suppresses the electron Zeeman splitting. The inset to Fig. 4a shows the B ext dependence of the peak absorption contrast for the blue and red trion transitions; while the agrement with the predictions of NRG is excellent for B ext 1.5Tesla, the

6 6 blue trion contrast exhibits oscillations for higher B ext ; this is most likely a consequence of the modification of the FR density of states at high fields in Faraday configuration. Fig. 4b shows the normalized line shape in a log-log plot together with the results of the NRG calculation (solid lines): η values that we determine from these plots range from 0.31 (red trion) to 0.66 (blue trion), and prove the sensitivity of the measured line shapes to the AOC determined power-law exponents. These experiments unequivocally demonstrate the magnetic field tuning of the AOC exponent for the first time. The remarkable agreement between our experimental data depicted in Figs.2-4 and the NRG calculations clearly demonstrate Kondo correlations between a QD electron and the electrons in a FR, as predicted by the excitonic Anderson model. The optical probe of these correlations unequivocally show the signatures of Anderson orthogonality physics associated with the quantum quench of Kondo correlations. Our experiments establish the potential of single optically active QDs in investigating many-body physics. In addition, they pave the way for a new class of quantum optics experiments where the influence of simultaneous presence of non-perturbative coherent cavity/laser coupling and Kondo correlations on electric field and photon correlations could be investigated.

7 7 [1] Michler, P. et al., A Quantum Dot Single-Photon Turnstile Device. Science, 290, (2000). [2] Dousse, A. et al., Ultrabright source of entangled photon pairs. Nature, 466, (2010). [3] Press, D., Ladd, T. D., Zhang, B., Yamamoto, Y., Complete quantum control of a single quantum dot spin using ultrafast optical pulses. Nature, 456, (2008). [4] Kim, D., Carter, S. G., Greilich, A., Bracker, A., Gammon, D., Ultrafast optical control of entanglement between two quantum dot spins. arxiv: [5] Yilmaz, S. T., Fallahi, P., Imamoglu, A. Quantum- Dot-Spin Single-Photon Interface. Phys. Rev. Lett., 105, (2010) [6] Claassen, M., Türeci, H., Imamoglu, A., Solid-State Spin-Photon Quantum Interface without Spin-Orbit Coupling. Phys. Rev. Lett., 104, (2010) [7] Dalgarno, P. A., et al., Optically Induced Hybridization of a Quantum Dot State with a Filled Continuum. Phys. Rev. Lett., 100, (2008) [8] Kleemans, N. A. J. M. et al., Many-body exciton states in self-assembled quantum dots coupled to a Fermi sea. Nature Physics, 6, , (2010) [9] Latta, C. et al., Confluence of resonant laser excitation and bidirectional quantum-dot nuclear-spin polarization. Nature Physics, 5, (2009) [10] Xu, X. et al., Optically controlled locking of the nuclear field via coherent dark-state spectroscopy. Nature, 459, , (2009) [11] Högele, A. et al., Voltage-Controlled Optics of a Quantum Dot. Phys. Rev. Lett., 93, (2004). [12] Kondo, J., Resistance Minimum in Dilute Magnetic Alloys. Progress of Theoretical Physics, 32, 37 (1964). [13] Kouwenhoven, L. P., Glazman, L., Revival of the Kondo effect. Physics World, 14, 33 (2001). [14] Atatüre, M. et al., Quantum-Dot Spin-State Preparation with Near-Unity Fidelity. Science 312, (2006). [15] Mahan, G. Many-Particle Physics, Kluwer Academic(New York), [16] Anderson, P. W., Infrared Catastrophe in Fermi Gases with Local Scattering Potentials. Phys. Rev. Lett., 18, (1967) [17] Goldhaber-Gordon, D. et al., Kondo effect in a singleelectron transistor. Nature, 391, (1998). [18] Cronenwett, S. M., Oosterkamp, T. H., Kouwenhoven, L. P., A Tunable Kondo Effect in Quantum Dots. Science, 281, 540 (1998). [19] van der Wiel, W. G. et al., The Kondo Effect in the Unitary Limit. Science, 289, , (2000) [20] Warburton, R. J. et al., Optical emission from a chargetunable quantum ring. Nature, 405, , (2000) [21] Govorov, A. O., Karrai, K. Warburton, R. J., Kondo excitons in self-assembled quantum dots. Phys. Rev. B, 67, (2003). [22] Anderson, P. W., Localized Magnetic States in Metals Phys. Rev., 124, (1961) [23] Türeci, H. E. et al., Shedding light on non-equilibrium dynamics of a spin coupled to fermionic reservoir. arxiv: [24] Helmes, R. W., Sindel, M., Borda, L., von Delft, J., Absorption and emission in quantum dots: Fermi surface effects of Anderson excitons. Phys. Rev. B, 72, (2005). [25] Goldhaber-Gordon, D. et al., From the Kondo Regime to the Mixed-Valence Regime in a Single-Electron Transistor. Phys. Rev. Lett. 81, 5225 (1998) [26] Hopfield, J. J., Comments on Sol. St. Phys., II, 2 (1969). [27] T K looses significance for the black curve, for which the QD-FR system is in the mixed-valence regime. [28] Deviations from scaling for ν < T FR are expected, but masked by an insufficiently small signal-to-noise ratio of the experimental data.

8 8 Figure 1: A single quantum dot strongly coupled to a fermionic reservoir. (a) Band structure of the device. The QDs are separated by a 15 nm tunnel barrier from a n ++ - doped GaAs layer (Fermi sea). A voltage V g applied between the electron gas and a semi-transparent NiCr gate on the sample surface controls the relative value of the QD single-particle energy levels with respect to the Fermi energy E F. (b) Low temperature (4 K) photoluminescence spectrum of a single QD as a function of V g. The interaction of the QD electron with the Fermi sea leads to a broadening of the photoluminescence lines at the plateau edges (yellow arrows) and indirect recombinations of a QD hole and a Fermi sea electron (red arrow). Indirect transitions are identified by the stronger V g dependence of the transition energy compared to direct transitions.

9 9 Figure 2: The gate voltage dependence of the peak absorption strength of the negatively charged exciton X, measured at 180 mk. (a) Inset: absorption as a function of the gate voltage. Main figure: Experimental data (symbols) for the ε-dependence of the shift in the resonance energy E transition (blue, left axis) and the absorption contrast (red, right axis) are well reproduced by NRG calculations (solid lines) for the following parameters: U ee = 7.5 mev, Γ = 0.7 mev, D = 3.5 mev, U eh = 11 mev, T FR = 180 mk. (b) Lower panel: NRG results for the occupancy of the QD electron level in the initial and final ground states. (c) Schematic of the energy renormalization process: The initial configuration (bottom) features a single electron in the QD, whose energy is lowered by virtual tunneling between QD and FR. Since virtual excitations with energy E contribute a shift proportional to Γ/ E, the total shift (involving a sum over all possible E), is strongest near the edges of the X plateau. Toward the right edge (ε near 0), the dominant contribution comes from virtual tunneling of the QD electron into the FR (as depicted); toward the left edge (ε near U ee ), it comes from virtual tunneling of a FR electron into the QD (not depicted). In the final configuration (top), the QD contains two electrons and a hole. The electron-hole Coulomb attraction U eh effectively lowers the QD electron level energy to ε U eh. This raises the energy costs E for virtual excitations by U eh U ee (which is Γ), so that final state energy renormalization is negligible. The renormalization of the transition energy (red arrow), probed by a weak laser, is thus mainly due to initial state energy renormalization. (d) Cartoon for Anderson orthogonality: the Kondo cloud (bottom) and local singlet (top) of the initial and final configurations produce strong or weak phase shifts, respectively.

10 10 Figure 3: The absorption line shape A(ν). a The blue tail of A(ν)/A(ν max), plotted versus the laser detuning ν on a log-log scale. The experimental data were taken at an electron temperature of T FR = 180 mk for the four values of gate voltage (ε) indicated by arrows in Fig. 2a; the corresponding Kondo temperatures T K (ε) are indicated by vertical lines in matching colors. NRG results (solid lines), obtained using the parameters from the fit in Fig. 2a, are in remarkable agreement with experiment. Inset: the measured full (unnormalized) absorption line shape in linear scale for the identical ε values. b, NRG results for T = T FR and T = 0; the latter show the ν 0.5 behaviour expected in the strong-coupling regime, T ν T K. c, The rescaled lineshape A(ν)/A(ν max) versus ν/t K shows a universal scaling collapse characteristic of Kondo physics.

11 11 Figure 4: Magnetic field dependence of the absorption. (a) The absorption line shapes for a second QD with similar parameters (see text) for ε = 0.43 at B ext = 0 Tesla and B ext = 1 Tesla for the blue/red trion transition. The magnetic field changes the strength of the AOC and the lineshape. The small peak appearing at ν 80µeV in the red trion absorption is due to the incomplete suppression of the laser polarization that couples to the blue trion transition. Inset: the peak absorption contrast showing good agreement with the NRG calculations for B 1.5Tesla. (b) The normalized absorption line shape in a log-log plot. These measurements pin the value of η(b ext = 0) to a direct signature of a Kondo singlet in the absorption lineshape. In addition they demonstrate the tunability of an orthogonality exponent.

12 Supplementary material: Quantum quench of Kondo correlations in optical absorption C. Latta 1, F. Haupt 1, M. Hanl 2, A. Weichselbaum 2, M. Claassen 1, P. Fallahi 1, S. Faelt 1, L. Glazman 3, J. von Delft 2, H.E. Türeci 4,1 and A. Imamoglu 1 1 Institute of Quantum Electronics, ETH-Zürich, CH-8093 Zürich, Switzerland, 2 Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, D München,Germany, 3 Sloane Physics Laboratory, Yale University, New Haven, CT 06520, USA 4 Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA To whom correspondence should be addressed; clatta@phys.ethz.ch or imamoglu@phys.ethz.ch Methods 1 Electron temperature All experiments were carried out in a dilution refrigerator with a base temperature of 20 mk in the mixing chamber. From fits to experimental data, we find, that the (relevant) electron temperature is around 180 mk. The optical transitions were probed by focussing a weak singlemode (intensity and frequency stabilized) laser on a single quantum dot (QD). We recorded the intensity transmitted through the sample with a silicon photo diode. In order to increase the signal-to-noise ratio, a lock-in technique was used where the gate voltage was modulated[1]. 1

13 For red detunings, such that ν < T, the number of electron-hole pairs that could provide the energy necessary for FR-assisted laser absorption scales exponentially with ν due to Fermi statistics: as a consequence, the absorption lineshape shows an exponential tail whose slope gives us the electron temperature T. For very large red detunings, the dominant line broadening is due to spontaneous emission and we recover the associated Lorentzian tail. To determine T, we actually fit the NRG lineshape for red detunings, taking into account optical interference (as described in Section 2 below). Figure S1: For ν < T, the red side of the absorption resonance has a strong dependence on the temperature T. 2 Influence of optical interference on measured lineshapes All the measured lineshapes carry the signatures of an optical interference effect induced by the sample structure[2]. The experimental situation is depicted in Fig. 2A. The laser field is incident onto the QD through the top gate, and the light transmitted at the other side of the structure is detected. The response of the QD to the laser field ise QD =χ(ν)e L, wheree L is the laser field 2

14 at the position of the quantum dot, whileχ(ν) = χ (ν)+iχ (ν) is the susceptibility of the QD with an absorptive partχ (ν) and a dispersive partχ (ν). The total field at the detector position has three components: E tot = E L e iπ/2 +E QD,f +E QD,b. (S1) The first term is the laser field at the detector position, which features a Gouy phase (an additional phase shift occurring in the propagation of focused Gaussian beams) of π/2 relative to the QD field as it propagates from the QD position to the detector position. E QD,f = χ(ν)e L is the field of photons scattered from the QD into the forward direction. E QD,b = re iφ E QD,f is the field of photons scattered from the QD into the backward direction, which are then reflected from the sample surface (combination of top gate and dielectric interface) and thereby are redirected into the forward direction. Here r is the reflectivity at the sample surface, and the additional phase factor fore QD,b has the formφ = 2πn 2L, wherelis the QD-surface distance, λ n the refractive index of the sample (GaAs) andλthe laser wavelength. Now consider the measured intensity S(ν) at the detector. Since the laser intensity is constant in time and frequency, it contributes only an offset. Hence S(ν) is given by the interference of the field E L e iπ/2 from the laser photons and the field E QD (1+re iφ ) from the scattered QD photons: S(ν) Re ( iχ(ν)(1+re iφ ) ) (S2) Thus, the reflection at the sample surface causes a mixing of the absorptive and the dispersive part ofχ(ν). For our sample structure this interference effect would modify a Lorentzian lineshape as depicted in Fig. 2A: The mixing of the absorptive and dispersive part ofχ(ν) leads to a shift of the absorption peak to blue laser detunings and causes a negative signal for small red detunings. In order to compare NRG calculations with experimental data, we proceed as follows: Standard NRG techniques for calculating (nonequilibrium) spectral functions A(ν) (see section 5 3

15 Figure S2: A)Experimental situation. B) Effect of the interference on the measured lineshape. below) give us the absortive part of the susceptibility, χ (ν) A(ν). We use Kramers-Kronig relation to calculate the dispersive partχ (ν), and then use Eq. (S2) to obtain the expected signal S(ν) for a given sample geometry. The parameters are: L = 90 nm,λ = 904 nm andr = Gate voltage fluctuations It is important to include gate voltage fluctuations in the lineshape simulation. In contrast to conventional quantum dot samples, even small gate voltage fluctuations, originating from voltage fluctuations at the output of the function generator and charge fluctuations in the QD environment, have an impact on the absorption lineshapes due to the strong non-linear dependence of the transition energy on the gate voltage. In the plateau center, where a small change in gate voltage corresponds to a negligible change in energy, fluctuations do not affect the lineshape. However, at the plateau edges, even small fluctuations cause measurable energy shifts of the resonance. After having determined the temperature from the red tail of the lineshape in the plateau center, we find that the lineshapes at the plateau edges are modified by gate voltage fluctuations. We take these fluctuations into account by convoluting the expected signal S(ν) 4

16 with a Gaussian, Ã(ν) = dxs(ν +x)p(x,ε), p(x,ε) = e (x/σ(ε))2 /2 2πσ(ε), (S3) with a gate-voltage dependent width σ(ε). The form of the latter, shown in Fig. 3, follows the dispersion curve depicted in Fig. 2A and reflects the enhanced sensitivity of the peak of the absorption resonance to gate voltage fluctuations at the edges. We find that the magnitude of the Figure S3: The width of the Gaussian function used to account for gate voltage fluctuations, according to Eq. (S3). applied gate voltage fluctuations that leads toσ(ε) plotted in Fig. 3 is 10µV. These fluctuations predominantly alter the red tail of the absorption resonance. 4 Excitonic Anderson Model (EAM) The coupled QD-FR system is described by an extension of the Single Impurity Anderson Model, which we call Excitonic Anderson Model (EAM) [7]: The Hamiltonian is given by H = H eh +H c +H t, where H eh = σ (ε eσ n eσ +ε hσ n hσ )+U ee n e n e σσ U eh n eσ n hσ (S4) describes the dot, with electron number n eσ = e σe σ and hole number n hσ = h σh σ. The Coulomb repulsion U ee, excitonic attraction U eh and hole energy ε hσ are taken to be positive. The energiesε a eσ andε h σ both shift linearly with gate voltagev g, with a slope of opposite sign, but same magnitude, ε a eσ = e α g V g, ε h σ = e α g V g, where e is the unit of charge and for the current experiment, the lever arm is given byα g = 7. 5 (S5)

17 H c = kσ ε kσc kσ c kσ represents the FR, which consists of non-interacting electrons with energies D < ε kσ < +D, with reservoir bandwidth 2D = 1/ρ and constant density of states per spin ρ. H t = Γ/πρ σ (e σc σ + h.c.) describes the tunnel-coupling between local level and reservoir, determined by the level-width Γ. When an incident photon is absorbed by the semiconductor quantum dot, it creates a particlehole pair. The interaction between photon and dot is given byh L (e σh σe iω Lt +h.c.), where e σ and h σ create a QD electron and a hole, with well defined spins σ and σ = σ. For the experimental situation of present interest, the average valence band hole occupations of the initial configuration are given by n i hσ = ni h σ = 0, and for the final configuration by nf hσ = 0 and n f h σ = 1. It is therefore convenient to define two different Hamiltonians,Hi/f = H i/f e +H c +H t, describing the system before and after absorption, where H a e = σ ε a eσn eσ +U ee n e n e +δ af ε h σ (a = i,f). (S6) These Hamiltonians differ (i) in the position of their e-levels (ε i eσ and ε f eσ = ε i eσ U eh ), where the e-level of the final Hamiltonian is pulled down by the excitonic Coulomb attraction and (ii) in the term δ af ε h σ which accounts for the energy of the hole. The absorption spectrum can then be calculated according to Fermi s golden rule A σ (ν) = 2π mm ρ i m f m e σ m i 2 δ(ω L E f m +Ei m) (S7) where m a and Em a are the eigenstates and -energies of H a. The detuning ν = ω L ω th is defined relative to the threshold frequency ω th EG f Ei G below which at T = 0 no photons can be absorbed, which is given by the difference of the ground state energies EG a of Ha. The positive hole is taken as a static spectator interacting with the dot through Coulomb interaction U eh only. 6

18 5 Numerical Renormalization Group approach for spectral functions The quantities m a and Em a occuring in Eq. S7 can be calculated using the numerical renormalization group[3] (NRG). This is an iterative method for numerically diagonalizing quantum impurity models, which also applies to the EAM Hamiltonians H a (a = i,f) specified around Eq. (S6). The spectrum of states of the Fermi reservoir is coarse-grained using a logarithmic discretization scheme governed by a parameter Λ > 1 (we typically use Λ = 1.8), followed by an exact mapping of the discretized model onto a semi-infinite chain, the so-called Wilson chain, whose hopping amplitudes decay exponentially along the chain, as t k Λ k/2. This produces a separation of energy scales and makes it possible to diagonalize the Hamiltonian iteratively: knowing the eigenstates of a chain of lengthk 1, one adds sitek and calculates the shell of eigenenergies of the Hamiltonian for the chain of length k. The high-lying eigenstates of that shell are discarded, while the low-lying states are kept and used for the next iteration. The spectrum of eigenenergies so obtained typically flows past one or more non-stable fixed-points and finally converges towards a stable fixed point, whereupon the iterative procedure can be stopped. In practice one thus deals with a finite Wilson chain, whose length is set by the smallest energy scale in the system (e.g. the Kondo temperature, temperature, or magnetic field). By combining NRG data from all iterations, it is possible to construct a complete set[4] of approximate many-body eigenstates of the full Hamiltonian. These can be used to evaluate equilibrium spectral functions via their Lehmann-representations; at finite temperatures, this can be done using the full density matrix (FDM)-NRG[5]. Since Eq. (S7) expresses the Fermi golden rule absorption rate via a Lehmann representation, it, too, can be evaluated using NRG [6], systematically so at finite temperatures by using 7

19 complete basis sets using a FDM-NRG approach [7]. However, it contains matrix elements between initial and final states that are eigenstates of different Hamiltonians, H i and H f. Hence, two separate NRG runs are required to calculate these (similar in spirit to what is done for time-dependent NRG[4]). The strategy is then as follows: NRG run #1 generates a complete set of approximate eigenstates m i and eigenenergies E i m for the initial Hamiltonian H i (without exciton). NRG run #2 generates a complete set of approximate eigenstates n f and eigenenergies E f m for the final Hamiltonian H f (with exciton). The double sum in Eq. (S7), over all initial and final eigenstates, is performed in two steps. First we perform a backward run, with site indexk running from the end to the beginning of the Wilson chain[4], and calculate for each shell k the contribution ρ i k towards the initial density matrix from that shell (obtained using data from NRG run #1). This is followed by the usual forward run, in which the matrix elements f n e σ m i 2 between shell-k eigenstates from NRG runs #2 and NRG #1 are calculated, combined withρ i k, and binned (see below) according to the corresponding frequency difference En f Em. i The T = 0 threshold frequency for the onset of absorption is given by the difference of ground state energies of NRG runs #2 and #1,ω th EG f Ei G. The absorption spectrum is expected to have divergences at the threshold ω th, hence all frequency data are shifted by the overall threshold energy ω th prior to binning. (For finite temperature, the sharp onset is broadened and divergences are cut off.) The discrete eigenenergies of shell k are spread over an energy range comparable to the characteristic energy Λ k/2 scale of that iteration, which decreases exponentially with k. Thus, the bins used for collecting the discrete data are likewise chosen to have widths de- 8

20 creasing exponentially with decreasing energy. The discrete, binned data are subsequently broadened using a log-gaussian broadening scheme, characterized by a broadening parameterα[5], here taken as α = Fitting model parameters To determine the values of Γ, U ee and D, we fit the numerical predictions for the gate-voltage dependence of the transition energy E transition to the experimental data (Fig. 2A, blue symbols, which give the frequency where the absorption spectrum reaches its maximum). Within our model the transition energy E transition = ω th +E Stark is given by the sum of the threshold frequency ω th = E f G Ei G and a linear Stark shift, E Stark ε. The threshold frequency ω th is obtained numerically by simply calculating the ground state energies E i G and Ef G of the two Anderson Hamiltonians Hi and H f specified around Eq. (S6), taking care to incorporate the gate-voltage dependence of the electron and hole levels, according to Eq. (S5). When comparing experimental data with NRG-results,E Stark is treated as fit parameter, together withγ,d andu ee. Since only relative changes in gate voltage have physical relevance, the horizontal offset of the experimental data is chosen such that the value ofv g for which the absorption contrast is strongest corresponds to the level-position ε ε i eσ = U/2. Since the band gap, which contributes toε h σ and hence toeg f, is not precisely known, the experimental and theoretical data are both shifted along the values of there minima are E transition = 0. 7 Other quantum dot transitions In the main text we focus exclusively on the X transition; the exchange coupling of a QD electron to the FR is strongest in this gate voltage regime, due to the gate voltage dependent tunneling rate. However other QD transitions also show the signatures of the exchange interaction: for examplex 0 andx + (Fig. 7a)transition energies as a function of gate voltage deviated 9

21 from the expected linear DC Stark shift as well. In contrast tox, the transition energy is lowered at the plateau edges. This is due to the fact, that both,x 0 andx + contain a single electron in the excited state; the energy of this state is lowered due to virtual tunnel events. (Fig.7b&c). Figure S4: a)photoluminescence as a function of V g. b) and c) Schematic of the X 0 and X + transition repectively. In contrast tox the final state energy is renormalized. 10

22 References and Notes [1] Alen, B. et al. Stark-shift modulation absorption spectroscopy of single quantum dots Appl. Phys. Lett. 83, 2235, (2003) [2] Karrai, K., Warburton, R. Optical transmission and reflection spectroscopy of single quantum dots, Superlattice Microst. 311, 33, (2003) [3] Krishna-murthy, H. R., Wilkins J. W. & Wilson, K. G. Renormalization-group approach to the Anderson model of dilute magnetic alloys. II. static properties for the asymmetric case. Phys. Rev. B 21, (1980) [4] Anders, F. B. & Schiller, A. Real-time dynamics in quantum-impurity systems: a timedependent numerical renormalization-group approach. Phys. Rev. Lett. 95, (2005). [5] Weichselbaum, A. & von Delft, J. Sum-rule conserving spectral functions from the numerical renormalization group. Phys. Rev. Lett. 99, (2007). [6] Helmes, R. W., Sindel, M., Borda, L. & von Delft, J. Absorption and emission in quantum dots: Fermi surface effects of Anderson excitoins. Phys. Rev. B 72, (2005) [7] Türeci, H. E., et al. Shedding light on non-equilibrium dynamics of a spin coupled to fermionic reservoir. arxiv: