Abstract
Spin qubits involving individual spins in single quantum dots or coupled spins in double quantum dots have emerged as potential building blocks for quantum information processing applications^{1,2,3,4}. It has been suggested that triple quantum dots may provide additional tools and functionalities. These include encoding information either to obtain protection from decoherence or to permit allelectrical operation^{5}, efficient spin busing across a quantum circuit^{6}, and to enable quantum error correction using the threespin GreenbergerHornZeilinger quantum state. Towards these goals we demonstrate coherent manipulation of two interacting threespin states. We employ the Landau–Zener–Stückelberg^{7,8} approach for creating and manipulating coherent superpositions of quantum states^{9}. We confirm that we are able to maintain coherence when decreasing the exchange coupling of one spin with another while simultaneously increasing its coupling with the third. Such control of pairwise exchange is a requirement of most spin qubit architectures^{10}, but has not been previously demonstrated.
Main
Following the spin qubit proposal by Loss and DiVincenzo^{10} and the electrostatic isolation of single spins in quantum dots (QDs)^{11} and double quantum dots (DQDs)^{12}, coherent manipulation was demonstrated in twolevel systems based on singlespin up and down states^{2} as well as twospin singlet and triplet states^{1}. Here we demonstrate coherent manipulation of a twolevel system based on threespin states. We employ the triple quantum dot (TQD) device layout shown in Fig. 1a, consisting of multiple metallic gates on a GaAs/AlGaAs heterostructure. The gates are used to electrostatically define three QDs in series within a twodimensional electron gas 110 nm below the surface. The QDs are surrounded by two quantum point contact charge detectors (QPCs)^{13}. The QPC conductance identifies the number of electrons in each QD and its derivative with respect to a relevant gate voltage maps out the device configuration stability diagram. We tune the device to the qubit operating electronic configuration, (N_{L},N_{C},N_{R})=(1,1,1), between two spintocharge conversion regimes (1,0,2) and (2,0,1), where L, C and R refer to the left, centre and right QDs respectively. The detuning, ɛ, controls the energy difference between configurations (1,0,2), (1,1,1) and (2,0,1). The exchange coupling, J, depends on ɛ and the tunnel couplings.
In this paper we concentrate on two scenarios. In the first scenario, at each point in the stability diagram the exchange coupling to the centre spin from one or both of the edge spins is minimal (that is, one edge spin resembles a passive spectator). This configuration is used as a control to confirm that our device maps onto twospin results in this limit^{9}. In the second scenario a true threeinteractingspin regime is achieved. (Results from a third intermediate regime are shown in the Supplementary Information.)
The energy level spectrum of a TQD (ref. 14) consists of quadruplets Q with total spin S=3/2 separated by the Zeeman energy in a magnetic field and doublets Δ′ and Δ with S=1/2. The two states of our qubit consist of one of the quadruplets, Q_{3/2}, and one of the doublets, Δ′_{1/2}, where with , and where J_{LC} ( J_{RC}) is the exchange coupling between the left (right) and centre spins. (Other threespin states are described in more detail in the Supplementary Information.)
Figure 1b illustrates the threespin energy spectrum as a function of detuning (zero detuning is defined as the centre of the (1,1,1) regime as shown). Experimentally we can tune the (1,1,1) region size by using gate C primarily^{15}. The eigenvalues of the four lowest states relevant for our experiments are:
The hyperfine interaction^{16} couples the state Δ′_{1/2} to the state Q_{3/2}(Q_{1/2}) at their anticrossing (asymptotic approach), see Fig. 1c. (Q_{1/2} and Δ_{1/2} are also hyperfine coupled.) Figure 1c also illustrates the two types of experiment we describe in this paper. With the single anticrossing (SA) pulse, based on the methodology in ref. 9, the system starts in the Δ′_{1/2} state in the (2,0,1) (or (1,0,2)) regime and then a pulse is applied to reach the (1,1,1) regime. The pulse rise time (see Supplementary Information) ensures that Landau–Zener (LZ) tunnelling creates a coherent superposition of Q_{3/2} and Δ′_{1/2} on passage through the anticrossing. After a state evolution time, τ, the pulse steps down, completing the spin interferometer on the return passage through the anticrossing. The probability of the Δ′_{1/2} state occupation, ${P}_{\text{\Delta}{\prime}_{1/2}}$, is directly obtained by this projection back into the (2,0,1) (or (1,0,2)) regime, where the required spintocharge information conversion is achieved by the Pauli blockade^{17} of the Q_{3/2} state. An experiment with a double anticrossing (DA) pulse is also illustrated in Fig. 1c. The sequence is similar, with the important distinction that a larger pulse enables LZ tunnelling processes through both anticrossings before again projecting back in the (2,0,1) regime having passed through both anticrossings twice. Important calibration information is obtained if the pulse time is longer than the coherence time (that is, τ>T_{2}^{*}) where the mixing at the Δ′_{1/2}−Q_{3/2} anticrossing is detected independently of coherence effects. Figure 1d plots this against magnetic field for a 9mVwide (1,1,1) regime midway between the narrow and wide (1,1,1) regimes. The two anticrossings form a ‘spin arch’ which is used to extract the coupling parameters for the model.
The distinction between our two regimes is now clear. In the case of a wide (1,1,1) region, close to zero detuning, both J_{LC} and J_{RC}∼0, so ${E}_{\text{\Delta}{\prime}_{1/2}}$≈${E}_{{\Delta}_{1/2}}$≈${E}_{{\text{Q}}_{\text{1/2}}}$ . Away from zero detuning only two of the spins are coupled: right–centre (left–centre) at negative (positive) detuning. Experiments using DA pulses in this regime involve coupling to not only Q_{3/2} but also to Q_{1/2}. Thus this regime is not suitable for a twolevel system involving three interacting spins. As a control experiment, however, in Fig. 2 we plot the coherent Landau–Zener–Stückelberg (LZS) oscillations obtained in this regime for both positive and negative detuning with a SA pulse. These compare to the first LZS experimental results with DQDs from ref. 9, later described theoretically in refs 18, 19. The degree of LZ tunnelling, that is, the relative size of A and B in the coherent AΔ′_{1/2}〉+Be^{iϕ(t)}Q_{3/2}〉state, depends on the speed, v, through the anticrossing: P_{LZ}=${\text{e}}^{\text{(2}\pi {\Delta}^{\text{2}}/\hslash v)}$ , where 2Δ is the energy splitting at the anticrossing. The visibility of the oscillations is a balance between this speed and T_{2}^{*}. For an infinite T_{2}^{*}, a rise time ∼0.2 μs would produce a 50/50 superposition (see also ref. 9). Experimentally it is found that a 6.6 ns pulse rise time (or 3.3 ns Gaussian time constant) leads to oscillations with the highest visibility. The value of T_{2}^{*}, obtained from a single parameter fit to the data, ranges from 5 to 18 ns, consistent with previous DQD experiments where T_{2}^{*} was limited by fluctuations in the nuclear field environment^{1}.
In Figs 3 and 4 we show results for experiments with DA pulses in a narrow (1,1,1) regime, where J_{LC} and J_{RC} are finite throughout and two welldefined qubit states exist between the two anticrossings (that is, simulations based on experimentally extracted parameters confirm that Δ′_{1/2} has moved far enough below the Q_{1/2} state such that no experimental features are related to interactions with the Q_{1/2} state). The energy level diagrams for this regime are shown in Fig. 4a. The stability diagram, measured in the presence of a fixed amplitude DA pulse at 25 mT, is shown in Fig. 3a. The results reveal LZS resonances parallel to both charge transfer lines, consistent with theoretical simulations (Fig. 3b) and confirming that coherence is maintained as the Δ′_{1/2} state is transformed from one dominated by coupling between left and centre spins to one dominated by right and centre spins, effectively demonstrating coherent pairwise exchange control.
To gain further insight, Fig. 4b and c show experimental and theoretical plots of the pulse duration dependence of LZS oscillations at different magnetic fields. Two boundaries, marked with horizontal white dashed lines, can be observed at fields above 25 mT. The region between the boundaries corresponds to the regime between the two anticrossings, while the resonances correspond to LZS oscillations. It can be seen (for example, curved dotted lines) that the resonances double back on themselves. This is a direct observation of tracking the resonance across the maximum in the Δ′_{1/2} versus detuning curve (see Figs 4a and 1b). We speculate that operating at this spot may provide more protection from charge noise, as the energy levels become locally flat versus detuning.
Although the frequency of coherent oscillations grows with field, owing to the increased spacing between the two qubit levels, it seems as if the experiment and theory differ by 20 mT for experimental data at 40 mT and by 15 mT for data at 25 mT. We attribute this to a dynamic nuclear polarization effect (DNP; ref. 20). To make this quantitative we extract horizontal slices in Fig. 4b at 40 mT (blue and white lines) and fit them to obtain T_{2}^{*}. The data are consistent with a 20 mT DNP effect. It is found experimentally that the values of T_{2}^{*} for the threespin qubit experiments in Fig. 3c,d (8–15 ns) are within error identical to the values from the twospin qubit experiments. This is consistent with T_{2}^{*} being dominated by local uncorrelated nuclear field fluctuations as both sets of qubit states differ by the same total spin^{21}. Finally, we note that we also observe a resonance beyond the second anticrossing, marked with a white spot in Fig. 4b. This is a nontrivial feature corresponding to a resonance condition of two interacting spin interferometers, one between the two anticrossings and a second, beyond the second anticrossing.
In conclusion, we have demonstrated coherent control of a qubit based on threeinteractingspin states. We have confirmed that there is no detectable change in the coherence time in the threespin experiments compared with the twospin experiments. We have realized the pairwise control of exchange for a threespin system by pulsing the detuning energy of a triple quantum dot. The same technique should carry over when more quantum dots are added in series to increase the number of qubits. Pairwise control of exchange, as demonstrated here, will then be useful for building complex quantum algorithms based on electron spin qubits in quantum dots.
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Acknowledgements
We thank D. G. Austing, W. Coish and E. Laird for discussions and O. Kodra for programming. A.S.S. and M.PL. acknowledge funding from NSERC. G.G., A.K, M.PL., and A.S.S. acknowledge funding from CIFAR. G.G. acknowledges funding from the NRCCNRS collaboration.
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Author notes
 L. Gaudreau
 & G. Granger
These authors contributed equally to this work
Affiliations
Institute for Microstructural Sciences, National Research Council Canada, Ottawa, Ontario K1A 0R6, Canada
 L. Gaudreau
 , G. Granger
 , A. Kam
 , G. C. Aers
 , S. A. Studenikin
 , P. Zawadzki
 , Z. R. Wasilewski
 & A. S. Sachrajda
Département de physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
 L. Gaudreau
 & M. PioroLadrière
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Contributions
Z.R.W. developed and grew the 2DEG heterostructure free of telegraphic noise; A.K. fabricated the triple quantum dot device capable of reaching the fewelectron regime; P.Z., L.G. and S.A.S. designed and built the highfrequency lines up to 50 GHz at millikelvin temperatures; P.Z., L.G. and G.G. ran the cryogenic equipment; L.G., G.G., S.A.S. and M.PL. developed the pulsing techniques; G.G., L.G. and S.A.S. performed the measurements; G.G., L.G. and G.C.A. analysed the data; G.C.A. performed theoretical simulations; G.G. and A.S.S. wrote the manuscript and Supplementary Information with input from all authors; G.G., L.G. and G.C.A made the figures; G.G., L.G., G.C.A., S.A.S., M.PL. and A.S.S. participated in discussions concerning the experimental and theoretical results; and A.S.S. supervised the project.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to A. S. Sachrajda.
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