SUPPLEMENTARY INFORMATION

Transcription

1 doi:1.138/nature177 In this supplemental information, we detail experimental methods and describe some of the physics of the Si/SiGe-based double quantum dots (QDs). Basic experimental methods are further described elsewhere in a GaAs-based context [1 5], but the use of Si/SiGe introduces a substantially different order of magnitude for hyperfine effects and additional physics due to the multivalley band structure of the material. We begin by detailing experimental methods, proceed to data analysis procedures, and finally describe the consequences of multivalley effects. I. EXPERIMENTAL METHODS A. Charge stability and tuning We determine the number of electrons in the QDs with a nearby quantum point contact (QPC), shown in Fig. 1b of the main text. Electron transitions into and out of single dots and transitions between dots are detected via QPC conductance changes. We vary the gate -.2 V (V) L (1,) (,) (1,1) (1,2) (,2) (,1) di QPC /dv (pa/mv) V (V) R Fig. SI-1: Charge stability diagram of the double dot system, measured via the QPC differential transconductance as gate voltages are swept. Bright lines indicate gate bias configurations where single-charge tunneling occurs and the double dot occupancy changes. The lack of transitions in the lower left corner indicates the double dots have been emptied of electrons. The background gradient is due to variations of QPC sensitivity with gate sweeping. 1

2 a) b) V L (V) (1,1) (,2) V R, V Q (V ) -.22 I Dots (pa) V L (V) (1,1) (,2) -.24` -.23 V R, V Q (V ) I Dots (pa) Fig. SI-2: Pauli blockade seen with transport. a) Transport triangles with 5 µv applied across dots. b) Transport triangles with -5 µv across dots indicating blockade by the absence of transport at the base of the triangles. The width of the blockade region corresponds to an energy splitting of 14 µev. voltages of the device through a series of charge transitions, which appear as bright lines in Fig. SI-1. At sufficiently negative bias on the gates, the (,) charge configuration is found where there are no further transitions at increasingly negative biases, which provides the evidence that the dots have been fully emptied of electrons. One may then count transitions to infer the regions in gate bias space corresponding to charge configurations with the desired number of electrons [1]. The honeycomb clearly shows the left dot has been emptied, but such a determination is less obvious for the right dot. The (,) (,1) transition disappears near the (,1) (1,) anticrossing because the tunneling rate for this transition with the right bath becomes too slow to detect given our pulse frequency (278 Hz). The transition line returns after an electron is loaded onto the left dot, which presumably pushes the right dot electron closer to its bath and quickens the tunneling rate. The absence of jumps in the (,) (1,) transition line, which extends beyond what is shown in the figure, confirms that the right dot has also been completely emptied. B. Realization of spin blockade For spin blockade, we tune the device in the vicinity of the (1, 1) (, 2) charge transition [6]. Finer tuning of the interdot and bath tunnel barriers then proceeds until spin blockade is observed via a transport measurement through the double dots [1, 3]. With a positive bias applied across the double dot, transport proceeds unimpeded from (,2) to (1,1) and full transport triangles are observed at the triple points (Fig. SI-2a). With a negative bias applied across the dots, transport from (1,1) to (,2) is spin-blocked when a triplet is loaded into (1,1), eliminating current at the base of the transport triangles (Fig. SI-2b). At higher bias, blockade is lifted by the presence of excited triplet states, which restore transport at the top of the transport triangle bias region. The width of the blockade region indicates an energy splitting of approximately 14 µev, an amount sufficient for state readout for the coherent electron manipulation experiments. This splitting is consistent with valley correction energies described in Sec. IV. C. Pulse sequence considerations Initialization in the (,2) ground singlet state for all experiments was achieved by pulsing into (,2) near the (,2)/(,1) charge state boundary for 3 µs, which is several times longer than the bath tunneling time. An electron from an excited (,2) state could then 2

3 tunnel to the right dot s bath and be replaced by one forming the ground singlet state. To guarantee we pulsed adiabatically with respect to tunnel coupling, we instituted a 5 ns ramp when traversing the (,2) (1,1) charge state boundary. Given an estimated tunnel coupling of 3 µev obtained via a spin funnel, gate capacitive lever arms obtained from transport triangle dimensions, and pulse amplitude for the (,2) (1,1) traversal, and using the Landau-Zener transition formalism [7], we estimate the probability of adiabatically transferring the singlet state to be greater than 99.99%. Given a tunnel coupling of 3 µev, a 3 mt applied magnetic field, and an electron g factor of 2, the Zeeman splitting is expected to bring the singlet and (1, 1)T crossing to the (,2) side of the singlet (,2)/(1,1) boundary. To avoid transferring into the (1, 1)T state, we need to pulse non-adiabatically with respect to the hyperfine-mediated mixing time when traversing the S/T anticrossing. This traversal occurs during the previously mentioned 5 ns ramp across the (,2)/(1,1) boundary. Again employing the Landau- Zener transition formalism with an estimate of the hyperfine field we evaluate the transition probability into T to be much less than.1%. To observe Rabi oscillations, we need to adiabatically transfer into the low-j electron ground state and thus need to pulse sufficiently slow and deep into (1,1). Given our pulse parameters, we experimentally observed the transition between adiabatic and nonadiabatic transfer to occur around 2.5 µs pulse-ramp duration. Numerical simulation conducted for a two-level system with J t 2 c/ ε confirm the transition should occur around 2 µs. For the Rabi oscillation data presented in the main text we used a 5 µs pulse duration for the adiabatic transfer into the low-j electron ground state. D. Data acquisition, scaling, normalization, and averaging Four transformations are utilized to convert raw QPC differential transconductance charge-state sensing data into singlet/triplet state probabilities. 1. All raw data is initially scaled to take into account the known fraction of the time spent in the measurement phase of the pulse cycle. Given that the left gate twiddle is only applied during the measurement pulse sequence phase, the non-measurement portions do not contribute to the lock-in signal, requiring this pulse-sequence-dependent scaling. Even though our transconductance technique does not require that the measurement phase dominate the pulse sequence in principle, we still conservatively used pulse sequences with measurement phases generally consisting of > 9% of the sequences to minimize any potential unexpected problems associated with this scale factor. For the Rabi and T2 experiments, we used a measurement phase time of 2 µs. 2. We normalize by the QPC response to charge transfer between (,2) and (1,1) charge configurations, which converts the blockade data into singlet/triplet state probabilities. The differential transconductance signal associated with traversing the (,2) (1,1) charge state boundary provides this normalization factor (the QPC s conductance changed approximately 1% between (,2) and (1,1) charge state configurations). Additionally, to complete the conversion of blockade data to state probabilities we also subtracted off the background differential transconductance signal associated with the direct coupling between the left gate twiddle and the QPC. To maximize the accuracy of the background reference signal, the reference signal was acquired inside the (,2) charge state configuration blockade measurement region. For the T2 and Rabi oscillation data, the reference signal was determined while applying the corresponding pulse sequences at zero separation time and zero exchange pulse duration, respectively. For these pulse conditions one would expect 1% singlet probability and no blockade signature. This expectation was confirmed with no discernable signal difference inside and outside of the blockade measurement region. At this point, the processed data is in terms of singlet/triplet state probabilities, but two more transformations are implemented to improve the accuracy of the state probability estimation. 3

4 3. We normalize by the QPC drift. The third scaling is designed to compensate for QPC drift over time. The overall device and QPC were found to be very stable over long periods of time. For instance, in a typical day of data acquisition the QPC sensitivity fluctuated less than 2%. Over a period of approximately a month the sensitivity drifted approximately 1%, with no QPC retuning efforts. Even though the QPC drift was relatively modest, we sought to cancel its effect. In between QPC sensitivity calibration scans, we used the background (,2) differential transconductance signal previously discussed as a proxy for the QPC (,2) vs. (1,1) charge state sensitivity. For a given scan, we started with the QPC sensitivity as determined from the last calibration scan and scaled it proportionally with changes in the background differential transconductance signal. 4. We compensate for signal loss due to blockade relaxation during the 2 µs measurement phase used in the pulse sequences. To estimate the signal loss, we recorded the blockade signal as a function of measurement time from 5 µs 21 µs and extrapolated the signal back to zero measurement time. By following this method we estimated the signal loss due to blockade relaxation to be 15%. The dominant source of uncertainty in the measured singlet/triplet state probabilities is attributed to the uncertainty in this relaxation scale factor. These four transformations combine into a linear scaling of data from QPC differential transconductance into singlet probability. No averaging on top of that performed by the lock-in amplifier was used for the spin funnel and Rabi oscillation data. For the T2 data, two additional averaging steps were employed to obtain data of sufficient quality to yield a tight bound on the σ hf measurement. During acquisition we fixed the separation pulse detuning and varied separation time to generate a single trace, and then repeated this sequence at the different detunings. For averaging, first we acquired and averaged approximately 2 measurements at each separation pulse time before stepping to the next separation time. Although this initial round of averaging improved our signal to noise, we discovered this technique was not effective for averaging over a sufficient distribution of hf necessary to resolve the oscillations seen in Fig. 4c of the main text. The observed time scale for significant fluctuations of hf was slow compared with the rate of data acquisition, which yielded strongly correlated data points for adjacent separation times. To eliminate this correlation, we acquired 16 independent traces at each separation pulse detuning, which were then averaged. This proved to be an efficient method for averaging over a distribution of hf. In total, the twelve T2 traces took approximately eight full days to acquire with the limiting factor being the inherent timescale of the 29 Si nuclear fluctuations. (Only six of the twelve measured and fit traces are illustrated in Fig. 4c of the main text, for clarity.) II. HYPERFINE AND EXCHANGE PHYSICS The spin physics of our silicon-based singlet-triplet double-dot system is very similar to the GaAs case [4, 5], as discussed in Sec. IV. Perhaps the strongest evidence that the blockade phenomena and coherence/decoherence phenomena we observe is indeed related to spin singlets and triplets rather than charge-related states (e.g. low-lying valley states) is the remarkable agreement of our observations with models based on interactions with the 29 Si nuclear spins in the silicon material. This is also the area where our results contrast most sharply with the GaAs case, as these interactions, although qualitatively the same, are far weaker in silicon, as we now detail. The Hamiltonian describing the subspace of two separated electron spins, each with vector spin operator S j, and interacting with an ensemble of nuclear spins with vector spin operators I n, is H = JS 1 S 2 + gµ B B(S1 z + S2)+ z A jn S j I n. (1) jn The first term is the effective exchange energy, controlled by the voltage bias across the left and right gates of the device. The second term is the Zeeman energy due to the 4

5 small magnetic field of 3 mt used in the experiment. This field is applied in the in-plane direction, which we notate as z above, but it is orthogonal to the semiconductor growth direction (1). Due to the magnetic field, the flip-flop terms of the third, contact Fermi hyperfine term (e.g. S + j I n ) play a negligible role at timescales shorter than the electron s T 1 time, since they do not conserve magnetic energy. In this approximation, the subspace of the spin singlet and zero-projecting triplet states shows a common energy shift due to all terms of Eq. (1) that are symmetric in S z 1 and S z 2. The remaining terms may be recast as operators on a singlet-triplet qubit by defining new effective Pauli operators σ x = S z 1 S z 2, σ y = 2(S x 1 S y 2 Sy 1 Sx 2 ), σ z = 2(S x 1 S x 2 + S y 1 Sy 2 ). The dynamic terms of H in this subspace may be rewritten where H = J 2 σz + hf 2 σx, (2) hf = n (A 1n A 2n )I z n. (3) Projections of qubit states onto these Pauli operators provide the Bloch sphere schematics shown in Figs. 3d and 4d of the main text, with the north pole corresponding to the singlet ground state, or the 1 eigenstate of σ z. The coupling constants of the Fermi contact hyperfine interaction are A jn = 8π 3 g µ B µ n ψ j (R n ) 2, (4) where g µ B /2 is the magnetic moment of a bare electron, and µ n I is the magnetic moment of nucleus n, in our case a randomly located 29 Si nucleus with I =1/2. The strength of each A jn term is proportional to the microscopic electron density of the jth electron, ψ j (r) 2, at each nuclear position R n. In the effective mass approximation commonly applied to semiconductors, this microscopic density is given by the product of the cell-averaged density ρ(r) defined by the local value of the multi-valley envelope function, and a factor η describing the relative concentration of the microscopic Bloch density at a nuclear site. A reliable estimate of this concentration factor is available from both magnetic resonance experiments in bulk silicon and ab-initio calculations [8]. The cell-averaged density depends on the QD volume, which varies only slightly when changing detuning. The nuclear spin distribution is a thermal average, and at the magnetic field and temperature employed in our experiment, each nuclear spin state is almost completely random. Therefore, hf follows the normal distribution resulting from a large ensemble of independent coin-flips, with variance σ 2 hf = I(I + 1) 3 (A 1n A 2n ) 2 n I(I + 1) 3 [ n dot L A 2 1n + n dot R Assuming a spatially uniform and reasonably high density n n of 29 Si nuclei, A 2 2n ]. (5) σhf 2 16π2 9 g2 µ 2 Bµ 2 nη 2 n n (W 1 L + W 1 R ) 3 2. (6) Here the final factor of 3/2 is due to valley beating, which introduces a fast oscillation of microscopic electron density of the ±Z valley mixed states, and ( 1 W L,R = d 3 rρl,r(r)) 2 (7) 5

6 is the span of electron states in the left (L) and right (R) dots. This span has dimensions of volume and rigorously accounts for non-uniformity, in particular the tails, of the electron density. An effective number n n W L,R of 29 Si spins interacts with electron spins in the left and right dots. These equations allow a rough estimate of σ hf without detailed calculation. Assuming isotopically natural silicon and a typical electron state size of nm 3 in our structures, each electron interacts with approximately 1 4 individual 29 Si nuclei, indicating a root-mean-square hyperfine coupling σ hf on the order of a nev. A more accurate calculation is subject to details of ρ L,R (r) controlled by the microscopic device geometry and tuning configuration. For this we employ self-consistent 3D simulations of field-gated depletion-mode devices that faithfully reproduce global (device-scale) potentials and charge distributions, concurrent with a full-configuration-interaction treatment of ground and excited few-electron states in the active area of the device. The virtual model incorporates our best knowledge of the heterostructure/dielectric/metal stacks. The SEM image of the measured device (Fig. 1b of the main text) was digitized to provide the gate layout. The simulated device is then tuned, following the general blueprint for tuning real devices, to the appropriate regime with specific electron density (of cm 2 ) in the leads, a well-formed DQD, and a (1,1) charge configuration. The simulated electron density ρ L,R for the L and R dots is overlayed on Fig. 1b of the main text. A sample simulation yields W L 54 nm 3 and W R 46 nm 3 for left and right dots with insignificant variation with gate bias voltages in the relevant range. From this simulation we obtain a σ hf of 1.9 nev. However, the effects of insufficient knowledge of device geometry and potential disorder on W L,R require further analysis. To compare the hyperfine strength of isotopically natural silicon to GaAs, consider the maximum possible hyperfine energy shift (i.e. the case of fully polarized nuclei). This maximum energy shift is about 1,2 times larger in GaAs than in Si. Three major factors lead to this difference. First, the density of magnetically active nuclei in GaAs is 2 times higher than in isotopically natural Si. Second, the nuclear magnetic moments of all Ga and As nuclei are 3 to 5 times larger than that of 29 Si. Finally, there is a higher concentration of the microscopic electron density at nuclei in GaAs, in part due to more distinct s-character of Γ conduction electrons in GaAs in contrast to -electrons in Si. III. DATA ANALYSIS Both the T2 experiments and Rabi oscillation experiments described in the main text result in data that has excellent agreement with the hyperfine interactions discussed above. Although both data sets contain information about σ hf, this value is more reliably measured via the T2 experiment. The model used for this experiment is illustrated by the Bloch spheres in Fig. 4d of the main text. In these, the initial state of the system is nearly exactly on the north pole of the Bloch sphere, due to the rapid pulsing from a high-j (singlet) eigenstate. Rotation then follows for free evolution time t. During this time, the spins experience both random instances of the hyperfine field as well as an exchange interaction of energy J. Therefore they rotate about an axis with polar angle θ Ω = tan 1 ( hf /J), with angular frequency of rotation Ω = 2 hf + J 2 / h. Rotation axes sampled from the hyperfine distribution, with length proportional to Ω, are plotted in Fig. 4d. The ensemble-averaged trials of the T2 experiment then measure the probability of returning to the singlet state, P s (t). The averaged singlet probability function is ( ) Ωt P s (t) = 1 sin 2 θ Ω sin 2, (8) 2 where refers to an integral over hf with respect to its normal distribution. Asymptotic features of this integral are discussed in Ref. 9. Its qualitative features are seen by examining the individual trajectories, as in Fig. 4d of the main text. For smaller values of J/σ hf, the orbits make larger southerly excursions on the Bloch sphere, increasing visibility; for higher values of J/σ hf, trajectories stay more tightly confined to the north pole, reducing visibility. 6

7 For curve-fitting, we numerically integrate Eq. (8). An overall offset and a scale factor are included in the model, and these parameters fit within a few percent of the scale factors arrived at by the independent calibration discussed in Sec. I D. Unfortunately, these scale parameters and the global value of σ hf are highly correlated parameters. The small discrepancy between the best-fit scale factor and the experimentally determined scale factor is the primary source of experiment uncertainty in σ hf ; it is assumed to arise from systematic errors in the post-processing procedures used to compensate for blockade relaxation effects and drifts of the QPC sensitivity discussed in Sec. I D. We quantify this uncertainty via an informal bootstrapping process. Different fitting strategies (e.g. holding scale factors fixed at the best assessed values vs. including these parameters, or varying the time range of the fit) result in different values of σ hf. These differences are small, but still statistically significant with respect to measurement noise. A linearly time-dependent error model is then introduced as a fitting weight factor and in determining the parameter fit uncertainty for σ hf. The parameters of the error model are chosen to assure the uncertainty in σ hf encompasses values observed in these multiple, valid fitting strategies. The resulting confidence interval of this weighted fit is shown as the shaded region of Fig. 4c of the main text for six of the twelve traces measured. From this curve-fitting and error analysis, we measure σ hf =2.6 ±.2 nev. The nuclear dephasing timescale T2 is defined in our work as the time by which a spin coherence decays to 1/e of its initial value, assuming J =. This definition gives T2 = 2 h/σ hf, which for this result corresponds to T2 = 36 ± 3 ns. We also obtain the J values and their uncertainties at each bias for the twelve values chosen for this experiment, all of which are plotted in Fig. 5. These results are in turn used for the analysis of the Rabi experiment. This experiment is equivalent to the T2 experiment except that the initial polar angle of spins on the Bloch sphere, θ i, is increased. The spins are adiabatically prepared into an eigenstate of a Hamiltonian including a small residual exchange energy, J i, and a random instance of the hyperfine gradient energy hf. The value of J i is the same as the lowest value of J from the T2 analysis, which our analysis deduced as.6 ±.2 nev. The random initial polar angle of this eigenstate on the Bloch sphere is θ i = tan 1 ( hf /J i ). This initial condition for an assortment of values of θ i sampled from the hyperfine distribution are shown as black dots on the Bloch spheres of Fig. 3d of the main text. Following this preparation, the exchange energy is pulsed to a value J, corresponding to rotation about an axis with polar angle θ Ω. An ensemble of these rotation axes and the corresponding orbits are also shown on the spheres of Fig. 3d. Once again, for high J, the resulting rotation is nearly around the z-axis; for low J, the axis is more strongly tilted for most hyperfine values. The ensemble-averaged probability of returning to the initial eigenstate, P r (t), with exchange pulse duration t, is ( ) Ωt P R (t) = 1 sin 2 (θ Ω θ i ) sin 2. (9) 2 This integral may be decomposed into a visibility V (J) describing the initial amplitude of oscillations; an amplitude modulation A(J, t) satisfying A(J, ) = 1; and a phase modulation φ(j, t): P R (t) =1 V (J) The visibility may be analytically integrated to find [ E V (J) = sin 2 (θ Ω θ i ) = J J i J + J i 1 A(J, t) cos[jt/ h + φ(t)]. (1) 2 ( J 2σhf ) E ( Ji 2σhf )], (11) where E(x) = πx exp(x 2 )erfc(x). For the purpose of curve-fitting, the remaining terms are handled numerically. The phase modulation φ(j, t) is evident in the data, and including this effect in the curve-fit results in a more accurate determination of J than using a simpler sinusoidal function. The amplitude modulation in Eq. (1) due strictly to hyperfine effects is numerically predicted to resemble Gaussian damping at short times (with rough 7

8 ε (mv) Exchange Pulse Duration (µs) Singlet Probability Fig. SI-3: Full fit of Rabi data to Eq. (1), in grayscale. The dashed red line shows the location of the calculated minimum t min (J) of the first fringe, to be used in Fig. SI-4. The depth of the first fringe results from two factors in Eq. (1): the visibility V (J) at t =(indicated by the dark red line), where A(J, )=1, and the degree of damping which occurs at this first fringe, corresponding to A(J, t min (J)). timescale hj/σhf) 2 but to decay more weakly at longer times. We notate this damping function, calculated via Eq. (9), as A (J, t). Although many oscillations are observed in our experiment, the actual damping present at high J is faster than that predicted by A (J, t). Hence the data is modeled with an additional damping function, taken to be Gaussian, i.e. A(J, t) =A (J, t) exp{ [t/τ(j)] 2 }. The additional damping with timescale τ(j) is likely due to potential fluctuations, but further investigation is required. Figure SI-3 shows the results of fitting Eq. (1) to the Rabi data. This fit used a running Gaussian average in the detuning (vertical) and pulse duration (horizontal) directions to reduce noise prior to nonlinear least-squares minimization, with variable window size according to an initial estimate of fringe frequencies found by fast Fourier transform. Weighting errors were taken from a combination of the standard deviation of the running average and the covariance matrix resulting from the analysis of the T2 data. Details of this fit result, compared to the data prior to averaging, are shown in Fig. SI-4. The resulting parameter fit errors are used for the confidence interval shown in Fig. 5 of the main text. The estimated experimental visibility at high J cited in the text,.7 ±.1, is taken as the average and standard deviation of the last ten points of the singlet probability at the first fringe minimum shown in Fig. SI-4. It is found that within our error estimates, the data scaling discussed in Sec. I D results in a visibility V (J) which follows Eqs. (1) and (11) extremely well within the noise. Of course, from detuning to detuning the visibility approaches values both greater and smaller than the theoretical expectation, but this is the expected result of hyperfine fluctuations occurring on the timescale of collecting the data. The average of this large data set is in strong agreement with the hyperfine model, since the visibility and the fringe frequencies of the Rabi oscillations are tightly constrained by Eq. (1), leading to the conclusion that both the basic hyperfine model and the experimental calibration between QPC differential transconductance and singlet probability are correct on average, and that the J and σ hf (or T2 ) values inferred from the data are accurate within their stated uncertainties. The timescales for singlet-triplet coupling fluctuations are fully consistent with hyperfine 8

9 Singlet Probability Exchange Pulse Duration (µs) J (nev) ε (mv) Singlet Probability at First Fringe Minimum τ (µs) Fig. SI-4: Details of the fit to Rabi data, and visibility versus effective exchange. The leftmost column of plots shows, as blue dots, cuts of the differential transconductance data shown in Fig. 3c of the main text. We normalized to singlet probability as discussed in Sec. I D, but with no further processing or averaging for either these traces or the image plots. The red lines show slices of the full Rabi fit (Fig. SI-3). The upper right panel shows the resulting values of J vs. ɛ from the fit as blue horizontal error bars, which show the very small fit parameter error. This data is the same as in Fig. 5 of the main text. Conversion from ɛ to J for the lower panels is accomplished by a smooth fit to this data, shown as the dotted red line and given by a fourth-order polynomial in ɛ divided by ( ɛ) 1.3. The middle right panel shows a sample of the Rabi data at the calculated location of the minimum of the first fringe, i.e. along the dotted red slice of Fig. SI-3. For example, the large green dots correspond to the large green dots on the corresponding slices shown to the left. The theoretical fringe minimum assuming no damping, 1 V (J), is shown as the dark red line. To account for the damping factor A(J, t), we use the fit value of the phenomenological damping parameter, τ(j), as shown with parameter fit error bars in the bottom plot, to show the expected value for this minimum at each J, as the dashed red line. (Note that for J less than about 1 nev, damping is dominated by hyperfine effects via A (J, t) and hence τ(j) becomes more uncertain.) We see that despite substantial noise, on average the fringe visibility follows the theoretical expectation for each value of J. estimates, but of course the data cannot exclude the possibility of contributions from other fluctuators, such as impurity electron spins. It is very unlikely, however, that multiple valley states affect the coherent oscillations we observe, based in part on the modeling of the data and in part on calculations relating to valley physics, which we now discuss. 9

10 (,2) (1,1 ) Energy (1,1) S T++ S + 3T ± ± S + 3 T ± ± S T ε S ++ S + 3T + S + (,2) Fig. SI-5: The energy of two-electron states in a double QD in the vicinity of the (1,1) (,2) charge transition as a function of the detuning. Only valleymixed states formed from the lowest orbitals in the left (L ± ) and right (R ± ) QDs are shown. Solid/red (dashed/blue) lines are singlets (triplets). Zeeman and hyperfine energies are too small on this scale to be visible. For similar diagrams in different regimes, see Ref. 1. IV. MULTI-VALLEY STATES IN SiGe DOUBLE QUANTUM DOTS A key difference between silicon-based QDs, as we discuss here, and similar systems in III-V semiconductors [2 5] is the presence of multiple equivalent minima (or valleys) in silicon s conduction band. The six-fold degeneracy of these valleys is partially lifted by strain and spatial confinement in (1) Si/SiGe heterostructures. Two equivalent valleys ±Z have the lowest energy, with the valley index acting as an additional discrete variable in the quantum-mechanical description of conduction electrons. While these valleys are independent in a perfect Si crystal, they are mixed by spatially sharp perturbations of the crystal potential, which complicates the structure of few-electron states. In particular, atoms of germanium in the random substitutional SiGe solid solution serve as these sharp perturbations and as a source of valley mixing. In thick SiGe layers, contributions of individual atoms interfere mostly destructively and self-average to a negligible residual value. In heterostructures, however, strong spatial confinement of electron states reduces the number of contributing Ge atoms, and abrupt changes in Ge content at heterointerfaces lead to constructive interference. These effects result in a substantial increase of valley-mixing. For relevant Si/SiGe QD structures, the largest intra-orbital matrix elements of the valley mixing operator Ûmix are estimated to be tens of µev or more, but still notably smaller than the QD inter-orbital spacing. The combinations of two valleys that diagonalize the valley mixing operator for a particular non-degenerate QD orbital O acquire additional energies of about ± Umix O (here U mix O = O Z Ûmix O +Z ), and the resulting valley-mixed states have very similar envelope wavefunctions. The effect of valley mixing on two-electron states in a double QD are shown in Fig. SI-5. For this sketch, the valley mixing terms U L,R mix for the lowest orbitals L and R in the two dots are chosen to be close in magnitude and phase, but not identical. (For the relevant strength of valley-mixing effects, differences of at least several µev in Umix O values are expected even between nominally identical dots due to randomness in the Ge positions in the SiGe alloy.) The lower states on the left of Fig. SI-5 are the sixteen (1, 1) states, which split into 4 sets of a singlet plus 3 triplet states (S +3T ). These differ in the mixed valley indices of the left and right electrons (and corresponding valley energy corrections ± Umix L ± U mix R ). The lowest six states on the right of Fig. SI-5 are all the (, 2) states allowed by the Pauli principle, with valley energy correction 2 Umix R for the ground singlet state S, for the next four S + +3T + states, and +2 Umix R for the last excited S ++ singlet. Only six of the lowest (1,1) states allow adiabatic charge transfer to the (,2) configuration while all others are Pauli blocked until, at larger detunings, transfer to higher right-dot orbitals becomes energetically feasible. The magnitude of the left and right valley mixing terms defines the order and energy separation of the states in the (1,1) manifold, while a phase difference is 1

11 responsible, in particular, for the relative strength of the lowest (singlet only, labeled S in Fig. SI-5) and second lowest (both singlet and triplet, labeled T) anticrossings. These two anticrossings border the domain of Pauli blockade for the lowest S +3T set of (1,1) states. When the magnitude of valley mixing terms substantially exceeds the tunnel coupling t c, as they do in the presented experiments (where t c 3 µev, as discussed in the main text), the lowest singlets (1,1)S, (,2)S, and three triplet states (1,1)T are the only relevant states in the vicinity of the singlet anticrossing. That is, the electrons in these states live within the nondegenerate valley ground state and the physics of their transitions near the anticrossing is similar to the case of single-valley III-V materials. Correspondingly, in this document and the main text, the subscripts are omitted for brevity. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressly or implied, of the United States Department of Defense or the U.S. Government. Approved for public release, distribution unlimited. [1] Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, (27). [2] Johnson, A. C. et al. Triplet-singlet spin relaxation via nuclei in a double quantum dot. Nature 435, (25). [3] Johnson, A. C., Petta, J. R., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Singlet-triplet spin blockade and charge sensing in a few-electron double quantum dot. Phys. Rev. B 72, (25). [4] Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 39, (25). [5] Laird, E. A. et al. Effect of exchange interaction on spin dephasing in a double quantum dot. Phys. Rev. Lett. 97, 5681 (26). [6] Borselli, M. G. et al. Pauli spin blockade in undoped Si/SiGe two-electron double quantum dots. Appl. Phys. Lett. 99, 6319 (211). [7] Shevchenko, S. N., Ashhab, S. & Nori, F. Landau-Zener-Stückelberg interferometry. Phys. Rep. 492, 1 3 (21). [8] Assali, L. V. C. et al. Hyperfine interactions in silicon quantum dots. Phys. Rev. B 83, (211). [9] Coish, W. A. & Loss, D. Singlet-triplet decoherence due to nuclear spins in a double quantum dot. Phys. Rev. B 72, (25). [1] Culcer, D., Cywiński, L., Li, Q., Hu, X. & Das Sarma, S. Realizing singlet-triplet qubits in multivalley Si quantum dots. Phys. Rev. B 8, 2532 (29). 11