Supporting online material SUPPLEMENTARY INFORMATION doi: 0.038/nPHYS8 A: Derivation of the measured initial degree of circular polarization. Under steady state conditions, prior to the emission of the XX 0 T±3 photon, the spin blockaded metastable biexciton may be described as a statistical mixture of T 0, T 3 and T -3 states. The emission initiated from the T 0 state is in different energies than that initiated from the T 3 and T -3 states, and can therefore be spectrally filtered out. The T 3(-3) biexciton state decays only by emitting a right- (left-) hand circularly polarized photon. The initial state of the system prior to the emission of the first photon is thus completely defined by the energy and polarization of the emitted photon. In the following, without loss of generality, we describe a cascade which starts from the initial biexciton state T 3. We use the following notation: b denotes the biexciton state, with energy E. u, d denote the intermediate excitonic states. In these states the spin projections are J=,- respectively, where J is the exciton spin projection on the growth-axis, and the heavyhole is in the second orbital state. We use the notation u, d for the dark exciton ground state with spin projections J=,- respectively. The relevant levels ant the transitions between them are schematically described in Fig. A. We describe the evolution of the density matrix of the system by the master equation, b i H, A A AA AA (A) The Hamiltonian of the system is given by nature physics www.nature.com/naturephysics
supplementary information b / / H E b b E u u d d d u u d d u u d (A) doi: 0.038/nphys8 We set the average energy of the ground state to be zero. Note that both the intermediate state and the ground states of the dark exciton are split. The splittings are denoted by and respectively. Though is not measured in this experiment, we expect it to be very similar to, since both splittings result from the isotropic e-h exchange interaction. This interaction is quite insensitive to the in-plane variations in the envelope functions of the confined carriers. The Lindblad operator which describes the decay from the biexciton state is given A u b, where b is the radiative decay rate of the spin blockaded, metastable by b biexciton. The non radiative decay is mediated by a phonon. The phonons couple only to the position degrees of freedom of the charge carriers. Therefore, without spin-orbit coupling, the operator describing the non-radiative decay does not distinguish between the two spin states of the exciton, and is given by A u u d d Here, is the rate for this decay process. We measured it directly to be about a factor of 50 faster than b. Spin-orbit effects, which may couple these degrees of freedom to the spin degrees of freedom can, in principle, cause spin flips and dephasing during the decay process. However, the rates f and d for such processes, respectively, are much slower than the radiative decay rate. We have verified experimentally that f b, using b polarization sensitive intensity correlation measurements. Complete anti-bunching was nature physics www.nature.com/naturephysics
doi: 0.038/nphys8 supplementary information measured in this work as well, between the spin T 3 transitions and the X 0 exciton (Fig.3). Similarly, the radiative cascades which initialize at the T 0 metastable biexciton and end in the bright X 0 exciton after phonon mediated relaxation of the heavy hole, also prove the same 3. The dephasing rates were not measured directly in this work. However, these are known to be at least an order of magnitude longer than the X 0 radiative lifetime for quantum dot s electrons 4. As argued in the main text, they are expected to be much longer for dark excitons. As we show below, the polarization of the dark exciton is unaffected by these processes. We consider operators of the form Ad d u u d d in the limit / 0 which describes dephasing, and Af f u d d u which describes spin-flips. The emission of a right hand circularly polarized photon at time s projects the state of the exciton to the state u. We solve for ( t s ) with ( s) u u. The solution is most conveniently expressed, in the eigenstates of the Hamiltonian: S u d, A u d and it is given by: SS ( t s ) ( t s ) - e ( ts ) tot AA and ( t s ) ( t s ) SA * AS tot i f i ts i ts ts d tot i tot e e e (A3) Where tot d f. nature physics www.nature.com/naturephysics 3
supplementary information doi: 0.038/nphys8 The non radiative decay time / was measured to be of the order of 0 ps, while the splitting gives precession time h / and h / of about 3 nsec. Therefore, in the limit /, /, / 0 and for long enough times such that f d t s / 0ps, the density matrix can be quite confidently approximated as i ( ts)/ e ( ts ) i ( ts)/ (A4) e This shows that the intermediate phonon emission, being fast and mostly spin conserving, does not affect the coherence of the DE's spin, and yields the same density matrix that would have been obtained for a pure optical process. Returning to the basis u, d the density matrix is therefore given by ( t ) cos[ ( ts ) / ] isin[ ( ts ) / ] s isin[ ( ts) / ] cos[ ( ts) / ] Specifically, the probability of finding the system in the state u at time t, given the emission of a right hand circularly polarized ( ) T 3 biexciton photon at time s is P t ( ) cos[ ( ) / ] uu ts ts (A5) u In order to calculate the temporal evolution of the degree of circular polarization (DCP) of the second photon in the experiment, one should consider the Poissonian nature of the charging, the spontaneous nature of the radiative recombination of the charged exciton and the finite resolution of the detectors used. We obtained an analytic expression which resembled well the measured data. Here, for simplicity we calculate the initial DCP, only. This is the DCP when the time difference between the detection of the first photon to the 4 nature physics www.nature.com/naturephysics
doi: 0.038/nphys8 supplementary information detection of the second photon vanishes. In this case the charging and the emission are immediate and one needs to consider the finite resolution of the detectors used, only. Let us consider positive charging. We denote by f(t) the normalized response function of the detectors. The first photon, right hand circularly polarized, emitted at time s was detected at time t. The probability to detect at time t the second photon, emitted at time s, left (right) hand circularly polarized, is given by convoluting over the response functions of the two detectors: t t P t t ds f( t s ) ds cos[ ( s s ) / ] f( t s ). s The integration limits insure causality (a photon is detected only after its emission, and the first photon precedes the second one). The value at zero time difference is then given by, t t P 0 ds f ( t s ) ds cos[ ( s s ) / ] f ( t s ) (A6) s For the normalized detectors response function we use the following analytical description: t RISE / t/ FALL RISE FALL e e, t 0 FALL f() t (A7) 0 t 0 In practice the detectors used in the experiment are best fitted by 80. RISE FALL ps (A8) The measured initial degree of circular polarization of the second photon is thus: DCP P (0) P (0 ) P (0) P (0) 0 Evaluating the integral (A6) using (A7) and (A8) we arrive at: nature physics www.nature.com/naturephysics 5
supplementary information doi: 0.038/nphys8 4 0 5 4 4 DCP T T. (A9) With the measured T h.9ns and 80 ps agreement with the measured value (Fig. 3b and 4a)., we obtain DCP(0) -0.67, in good A: Microscopic model for the cw background subtraction. The signal in cw intensity correlation measurements is composed of a contribution coming from directly related photon emissions the 'same cascade' events, and a 'nonsame-cascade' background resulting from the return of the system to steady state. If one is interested in the direct process only, a way to remove the non-same-cascade background is needed. The correlation functions between any two transitions in a system of well separated energy levels can be calculated using classical rate equations 5,6. These equations should include all relevant levels and the rates of all relevant processes transferring population from one level to another. The non-same-cascade part of the correlation function can then be obtained by artificially setting the rate of the process directly connecting the relevant levels to zero. Experimentally, however, it is very difficult to switch off a process. Therefore, in order to estimate the non-same-cascade part of the correlation between the XX 0 T3X 0 D and the X + h + transitions, we took a different path. We measured the correlation function between the XX 0 T3X 0 D transition and the X 0 B0 transition. This correlation is expected to have no same-cascade component. This is because the direct rate of the process X 0 D X 0 B, which does not involve charge addition, is vanishingly small. This is 6 nature physics www.nature.com/naturephysics
doi: 0.038/nphys8 supplementary information in contrast to the X 0 D X + process, which depends mostly on the rate by which positive charge is added. If the charging rate is artificially set to zero, and the emission from the X + and X 0 line is more or less equal, then the rate of return to steady state for both spectral lines is expected to be quite similar. In order to verify this simple, most intuitive idea, we first used a rate-equation model to calculate the correlation function between the XX 0 T3X 0 D and the X 0 B0 transitions. The use of classical rate equations here is justified since only the average populations of each degenerate or almost degenerate level are considered. We then use the same model, but without the direct charging process X 0 D X + to calculate the correlation between XX 0 T3X 0 D and X + h +. This gives the non-same-cascade background. Finally, we compare between the two calculated functions. The model we use here is similar to that of Ref. 6. It consists of levels. These include zero-, one- and two-exciton states, either in the presence of no additional charges, of one additional heavy hole, or of one additional electron. For the neutral states we discriminated between bright and dark excitons and between the ground and spinblockaded biexcitons. We considered the following dynamical processes: radiative recombination ( r ), optical e-h pair generation ( g ), and negative and positive charging rates, ( c e and c h ) respectively. The levels and rates are schematically described in Fig. A. The parameters of this model are the optical e-h pair generation rate ( g ) and the charging rate ( c ), which, due to our experimental conditions (equality of the X - and the X + line intensities, see Fig. in the main text), was assumed to be the same for both electrons and holes. For the radiative recombination rate ( r ) we used r=(0.8ns) -, independently measured for the neutral bright exciton. nature physics www.nature.com/naturephysics 7
supplementary information doi: 0.038/nphys8 Fig. A4 shows the calculated correlation functions, including the correlation between XX 0 T3X 0 D and X + h + calculated with the process X 0 D X +. The values g=0.3r and c=0.5r were found to best fit the measured correlation functions, also presented in Fig. A3. As can be seen in Fig. A3 for positive time differences, the measured (green circles) and the calculated (solid green line) correlation functions between the XX 0 T3X 0 D spectral line and the X 0 B0 line are very similar to the calculated non-same-cascade background (dashed red line). This provides therefore, a microscopic model which justifies our experimental way for estimating the cw background. References:. Ivchenko, E. L. & Pikus, G. E. Superlattices and Other Heterostructures: Symmetry and Optical Phenomena. (Springer, Berlin 997).. Poem, E. et al. Radiative cascades from charged quantum dots. Phys. Rev. B 8, 085306 (00). 3. Kodriano, Y. et al. Radiative cascade from quantum dot metastable spin-blockaded biexciton. arxiv:007.04v [cond-mat.mes-hall] (00). 4. Bracker, A. S. et al., Optical pumping of the electronic and nuclear spin of single charge-tunable quantum dots. Phys. Rev. Lett. 94, 04740 (005). 5. Regelman, D. V. et al., Semiconductor quantum dot: a quantum light source of multicolor photons with tunable statistics. Phys. Rev. Lett. 87, 5740 (00) 8 nature physics www.nature.com/naturephysics
doi: 0.038/nphys8 supplementary information 6. Baier, M. H. et al., Quantum-dot exciton dynamics probed by photon-correlation spectroscopy. Phys. Rev. B 73, 053 (006). Fig A. The states and energy scales in the master equation, Eq. (A). FIG. A. Schematic description of the model. Black horizontal lines represent the various levels. Arrows between levels represent transitions between them. Red (blue) arrows represent hole (electron) charging. Orange (green) arrows represent optical electron-hole pair recombination (generation). Thin arrow represents half the rate of a thick arrow (due to multiplicity of final states). The direct charging process connecting between the dark exciton and the positively charged exciton is highlighted. The same rate was used for electron and for heavy hole charging. The model thus has two adjustable parameters: g and c=c e =c h. FIG. A3. Verification of the background subtraction method. The green solid line (circles) represents the calculated (measured) correlation function between the XX 0 T3X 0 D and the X 0 B0 spectral lines. The blue solid line (triangles) represents the calculated (measured) correlation function between the XX 0 T3X 0 D and X + h + lines. The measured function are obtained with linear polarizations, thus no oscillations are seen. The red dashed line presents the calculated background deduced by calculating the correlation between the XX 0 T3X 0 D and X + h + lines, excluding the direct charging process X 0 D X +. nature physics www.nature.com/naturephysics 9
supplementary information doi: 0.038/nphys8 0 nature physics www.nature.com/naturephysics
doi: 0.038/nphys8 supplementary information XX - XX 0 T3 XX 0 XX + X - X 0 B X 0 D X + r g e - c e 0 c h h + nature physics www.nature.com/naturephysics
supplementary information doi: 0.038/nphys8.6.4 Normalized correlation. 0.8 0.6 0.4 0. 0 0 5 0 5 0 t t (ns) nature physics www.nature.com/naturephysics