Enhanced spin coherence while displacing electron in a 2D array of quantum dots
Pierre-André Mortemousque, Baptiste Jadot, Emmanuel Chanrion, Vivien Thiney, Christopher Bäuerle, Arne Ludwig, Andreas D. Wieck, Matias Urdampilleta, Tristan Meunier
EEnhanced spin coherence while displacing electron in a 2D array of quantum dots
Pierre-Andr´e Mortemousque,
1, 2
Baptiste Jadot,
1, 2
Emmanuel Chanrion, Vivien Thiney, ChristopherB¨auerle, Arne Ludwig, Andreas D. Wieck, Matias Urdampilleta, and Tristan Meunier Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France Univ. Grenoble Alpes, CEA, Leti, F-38000 Grenoble, France Lehrstuhl f¨ur Angewandte Festk¨orperphysik, Ruhr-Universit¨at Bochum,Universit¨atsstraße 150, D-44780 Bochum, Germany
The ability to shuttle coherently individual electron spins in arrays of quantum dots is a keyprocedure for the development of scalable quantum information platforms. It allows the use ofsparsely populated electron spin arrays, envisioned to efficiently tackle the one- and two-qubit gatechallenges. When the electrons are displaced in an array, they are submitted to site-dependent envi-ronment interactions such as hyperfine coupling with substrate nuclear spins. Here, we demonstratethat the electron multi-directional displacement in a 3 × The control over the flow of electrons in semiconductorcircuits is core to the success of micro and nanoelectron-ics. In their quantum counterparts, similar processes areinvestigated at the single-particle level, to preserve thefragile quantum information stored in individual elec-tron spins [1]. Indeed, displacing these electron spinscoherently opens possibilities to convey on-chip quantuminformation and increase the connectivity of spin-basedsemiconductor quantum circuits [2–7]. Several strategieshave been recently demonstrated to preserve spin coher-ence while shuttling charged particles. A first exampleconsists of the confinement of electrons in moving quan-tum dots [8, 9]. Another protocol is based on repetitivecoherent spin tunnelling between adjacent dots. Such astrategy is particularly appealing for architecture basedon large 2D arrays of quantum dots [10, 11] and hasbeen demonstrated for linear [12–14] and circular [15] ar-rays. Therefore, understanding the mechanisms at playaffecting the fidelity of the displacement in 2D arrays isan important task to optimize this quantum informationconveyer procedure and increase as much as possible thedistance over which the spin transfer is coherent.For an electron spin in a quantum dot, the loss of coher-ence is driven by spin dephasing mechanisms that arisefrom intrinsic properties of the semiconducting nanos-tructures like 1/f noise coupled to spin-orbit interaction[16] or hyperfine interaction [17]. In both cases, the elec-tron spins are experiencing slow effective magnetic fieldfluctuations inducing uncertainty on the Larmor preces-sion of individual electron spins. Therefore, refocusingtechniques are extremely efficient on spin systems [18–21]. With this property in mind, displacing electron spinson fast timescales is expected to enhance the spin coher-ence times in virtue of the averaging of the magnetic field fluctuations, as demonstrated in NMR via a phenomenoncalled motional narrowing effect [22, 23]. The unprece-dented level of control recently demonstrated in a 2D ar-ray of quantum dots [24] permits to displace the electronthrough an important set of possible dot configurationson a timescale faster than the coherence time. This ar-ray turns out to be a relevant platform to investigate andreveal the consequences of electron displacement on spincoherence properties.Here we study the coherence of individual electronspins when they are displaced via tunnelling through a2D array of five quantum dots defined in an AlGaAs het-erostructure. In this displacement regime with an elec-tron speed below 100 m s − , the hyperfine interactionwith the nuclei of the semiconductor nanostructure is re-sponsible for the loss of coherence [22]. This hyperfine in-teraction results in an effective fluctuating magnetic field,called the Overhauser field, with a distribution imposedby the number of nuclei the electron spins are interactingwith. The larger the number of nuclei, the longer is thecoherence time, with a typical square-root dependence[25]. During the electron displacement, the system vis-its a number of dot configurations imposed by the gatevoltage sequence. We relate to each dot configuration aneffective local magnetic field. Two principal parametersare influencing the coherence time of the displaced elec-tron spins: (i) the number of effective local magnetic fieldconfigurations explored which can be varied by changingthe number of accessible quantum dots in the array, and(ii) the typical interaction time fixed by the time spentin each charge configuration. The demonstrated controlof the dot system enables us to vary precisely both pa-rameters and to study their impacts on the coherent spintransfer. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n V B i TL T TL RCL B V TL V L-TL a V T V L TRBL BR i TR i BR i BL V T-TL b V R S/2e S/T? c FIG. 1.
Sample and spin loading and readout se-quence. a,
Electron micrograph of a sample similar to theone used in this work. The nine circles indicate the 3 × red gate voltages, andits outer edges by the green , light blue and blue gate voltages.The couplings of the array to electron reservoirs are achievedthrough the corner QDs TL, BL, BR, and TR and by con-trolling the light blue and blue gates. The purple gates areused to define four quantum point contacts operated as localelectrometers, whose conductances set the measured currents i TL , BL , BR , TR . Scale bar (white) is 200 nm. b,c,
Schematicsof the sequence used to load a singlet spin in C ( b ), and toread out the final spin state after manipulation ( c ). ELECTRON SHUTTLING
The sample [24], shown in Fig. 1a, is an array of 3 × V L , B , R , T voltagesbefore rising the potential of the TL dot (Fig. 1b). First,we prove that all the 15 expected charge states for twoelectrons in five dots ( (cid:0) (cid:1) = 5 where both electrons are inthe same dot, (cid:0) (cid:1) = 10 where electrons are separated indifferent dots) are accessible by tuning the 4 linear com-binations δV ± X , Y of the V L , B , R , T voltages (Methods). Fig-ure 2a-c shows charge stability diagrams of two electronsrecorded as functions of δV − X , Y , which act like in-planeelectric dipole gates, for different values of δV +X , Y . By ad-justing simultaneously δV +X , Y , it is possible to change therelative potential of the dot C so that the central regionof the diagrams either contains two electrons (Fig. 2a),or no electron revealing the regions where the electronscan be separated in (T, B) (Fig. 2b) and (L, R) (Fig. 2c).Figure 2d-f shows simulations (Methods) of the stability diagrams shown in Fig. 2a-c, respectively, in which theidentification of the charge distributions is confirmed. Inour precedent report on the coherent control of electronspin in this structure [24], we have shown that all chargestates exhibited in the stability diagrams of Fig. 2 can beaccessed while setting sufficiently high tunnel couplingsfor the coherent spin transfers between dots. Up to 10charge configurations where the electrons are separatedin different dots are explored, and multi-directional andcomplex one- and two-electron displacements are per-formed. COHERENT ELECTRON SPIN SHUTTLING INA 2D ARRAY
We investigate the two-electron spin coherence whilethe electrons are displaced within the (L, B, R, T, C) sub-set of quantum dots. Before all coherent spin shuttling,two electrons in a singlet state (Methods) are loaded inTL before being transferred to L then C (Fig. 1b). Next,a sequence of voltage pulses δV L , B , R , T is used to separateand displace the electrons coherently in the array. Byshuttling one or two electrons within the dot array, theseparated electrons will experience decoherence, result-ing in the spin mixing with the triplet states. Finally,the two electrons are recombined in C and transferredto L and TL for spin readout (Fig. 1c). To consider thethree vectorial components of the hyperfine field equally,the experiments are performed under zero external mag-netic field [15, 25].We first investigate the quasi-static case where theelectrons are separated in different dots, one electronremaining in one dot, the other visiting a single timean increasing number of dots. For a first electron fixedin dot T, it results in N config = 1 to 4 distinct possi-ble configurations (Fig. 3a-d, respectively). The corre-sponding voltage pulse sequences are depicted in Supp.Fig. S1. For each experiment, the time spent in eachconfiguration, referred to as the resting time τ R , is var-ied. The resulting singlet probabilities are plotted (filledcircles) in Fig. 3e as a function of the total separationtime τ S = N config × τ R . The coherence time is directlyextracted by fitting a Gaussian decay e − ( τ S /T coh ) (solidlines) and increases with (cid:112) N config . Indeed, the sepa-rated electrons explore a large ensemble of nuclear spinsduring the displacement, averaging the hyperfine inter-action [25–27]. We have repeated this set of experimentsfor different permutations of QDs to extract a statisticaldependence of T coh on N config and average out the in-fluence of the non-regular coupling between the differentdots [24]. For each size of visited subsets, the averageand the standard deviation of T coh is plotted in Fig. 3fas a function of N config . They quantitatively fit with theexpected square root law (solid line). It confirms the im-portance of the hyperfine interaction and the number of ca bd e f FIG. 2.
Two-electron stability diagrams. a-c,
Charge stability diagrams of two electrons in the five dots L, B, R, T,C ( a ), or in the four dots L, B, R, T ( b, c ). The charge state of the array of dots is controlled by sweeping δV − X , Y . Thesignal is recorded as linear combinations of the quantum point contact current derivatives: ( a, c ) ∂ V − Y i TL + i BL , and ( b ) ∂ V − Y i TL + i BL − i BR . The dashed white lines in ( a ) are guide for the eye for weak intensity degeneracy lines. The stabilitydiagrams are obtained for different gate voltage tunings. d-f, Simulations of the stability diagrams (Methods). The labelsindicate the position of the isolated electron in the QD array. nuclei the electrons interact with. It also demonstratesthe successful coherent displacement within the 2D arrayof dots.
ENHANCEMENT OF THE SPIN COHERENCETIME VIA ELECTRON SHUTTLING
We now analyse the impact of the electron tunnellingon their spin coherence. In comparison with the previ-ous case where the electrons explore a single time dis-tinct charge configurations, we study the situation wherethe charge configurations are periodically explored. Foreach experiment, the resting time spent in each chargeconfiguration τ R is set to a constant value. We use twoperiodic pulse sequences corresponding to two-electrondisplacement patterns: (i) single- (ii) two-electron peri-odic displacement. In the single-electron periodic dis-placement case, the first electron is maintained in L, andthe second is displaced among B, C, T, and R, so thatfour isochronous dot configurations are periodically vis-ited (Fig. 4a and Supp. Fig. S2a). In the two-electronperiodic displacement case, an electron is first moved inB and the second in R. From this point, each electron is sequentially displaced along a two-step trajectory, eithervertically, or horizontally. Eight isochronous QD configu-rations are periodically visited with C the only dot visitedby both electrons (Fig. 4b and Supp. Fig. S2b). The ex-perimental singlet probabilities (black points) recordedfor the single-electron ( τ R = 1 . τ R = 2 . ± ± . ± . . ± . µ m (assuming a conservative interdot dis-tance of 100 nm), respectively. These decay times areplotted as green squares in Fig. 3f, and their projection onthe square root curve as open blue squares ( N config equiv-alent to 11 and 21, respectively). A significant improve-ment is observed compared to the quasi-static case: theyare 1.6 times higher than the expected coherence timefor four and eight static dot configurations, respectively.Moreover, the decoherence law changes from Gaussian-like (Fig. 3e) to exponential (Fig. 4c,d) decay when vis-iting the charge configuration multiple times, which con-firms the significant impact of the electron dynamics ontheir spin coherence.By varying the resting time τ R , we can explore different a fe N con fi g =2 N con fi g =1 N con fi g =3 N con fi g =4 b c d FIG. 3.
Quasi-static electron spin coherence. a-d,
Schematic pictures of the different electron spin displace-ment sequences inside the 2D array of five QDs employedto probe the electron spin coherence shown in ( e ). e, Sin-glet probability plotted as a function of the total electronseparation time τ S spent in one (T-C, blue , a ), two (T-Cand T-L, orange , b ), three (T-L, T-C, and T-B, green , c ),and four (T-C, T-L, T-B, and T-R, red , d ) different chargeconfigurations. The two electrons are initialised in the sin-glet state in the C-dot. Then, one electron is first trans-ferred to T and the second electron visits each other dot onlyonce over an equal time of τ R . As a consequence, the sec-ond electron spends for the different sequences a total sep-aration time τ S = 1 × τ R in C, τ S = 2 × τ R in C and L, τ S = 3 × τ R in L, C, and B, or τ S = 4 × τ R in C, L, B,and R, where τ R is in integer value of the AWG clock period.The data are fitted with Gaussian decays ( solid lines ) withcharacteristic times T (cid:63) equal to 7 . ± . . ± . . ± . . ± . f, T (cid:63) averaged over different possible charge configurations plottedas a function N config (black circles). The data are fitted with asquare-root function (red solid line) T (cid:63) ( N config = 1) (cid:112) N config ,with T (cid:63) ( N config = 1) = 9 . . T coh of 32 ± ± N config of 11 and21, respectively). shuttling regimes for the two-electron periodic displace-ment case. The resulting singlet probability is plotted inFig. 4e as a function of τ S = (8 N cycle + 1) τ R for differentvalues of τ R going from 1 ns to 13 ns (bottom to top). Nosinglet probability decay is observed for τ R longer thanthe static spin coherence time due to complete mixingbetween the singlet and triplet states. By progressivelyreducing τ R , we observe the emergence of a single ex-ponential decay of the singlet probability, demonstrat-ing preservation of the spin coherence during displace-ment. We extract the characteristic decay time T coh byfitting with exponential decay (Supp. Fig. S3c) whichwe plot against τ R (Fig. 4f). The points are fitted withan inverse function α/τ R , typical of a motional narrow- a bdc ef
11 ns9 ns7 ns5 ns3 ns2 ns1 ns
FIG. 4.
Enhanced spin coherence time for periodicelectron displacements. a-b,
Schematic pictures of thedifferent electron spin periodic displacement sequences insidethe 2D array of five QDs employed to probe the electron spincoherence shown in ( c-f ). c, d, Singlet probabilities plotted(black dots) as a functions of the total electron separationtime τ S , c for the single electron periodic displacement pattern( a ), and d for the two-electron periodic displacement pattern( b ). The data were acquired for configuration durations τ R of 1 . c ) and 2 . d ). The experimental data are fittedusing simulations of the spin dynamics (see text) as solid bluelines. The fit error functions (see text) are shown in insets. e, Singlet probability plotted (dots) as a function of the totalelectron separation time τ S , for increasing (bottom to top)values of τ R : 1, 2, 3, 5, 7, 9, 11, and 13 ns, for the two-electron periodic displacement case ( b ). The solid lines arethe simulated singlet probabilities (see text) computed for thecorresponding τ R times, for a common remanence value R =0. f, Characteristic coherence times T coh plotted as functionsof τ R and fitted with an inverse function T /τ R + T (cid:63) , with T = 130 ±
24 ns. ing phenomenon, with a scale factor α = 130 ±
24 ns .This value is in agreement with the expected squareof the dephasing time [23], estimated in Fig. 3f to be T (cid:63) ( N config = 1) = 9 . = 93(13) ns . b a dc FIG. 5.
Electron path randomisation consequences.a,b,
Simulated decoherence decay curves for one- ( a ) and two-( b ) displaced electrons for different values of the remanenceparameter R : 0 (blue), 0.5 (orange), 0.7 (green), 0.9 (red),and 1 (purple). c, Estimated remanence R plotted as a func-tion of the distance between the dot location of two consec-utive periods, for a charging energy of 1 (blue), 3 (orange),and 10 meV (green). d, Cross-sectional view of two-electrondensities computed using the fundamental Fock-Darwin state[28], and distant of 33 nm ( R = 0 . PATH RANDOMISATION
If only a definite set of dot configurations are cycli-cally explored, no difference in the spin dynamics shouldbe observed between the quasi-static and the periodicdisplacement protocols. Then, the only relevant param-eter is the time spent in each configuration, and a gaus-sian decay of the singlet probabilities would be expected.However, the difference observed in the experiment forcesus to reconsider the initial assumptions made about theelectron displacement: the electrons are not periodicallyexploring the same set of random magnetic field configu-rations. To better understand the impact of the electronshuttling on spin coherence time, we model the systemand study the spin dynamics (Supp. Fig. S4 and Meth-ods). We introduce the notion of randomization of theeffective hyperfine field during the coherent electron dis-placements. It takes the form of a remanent parameter0 ≤ R ≤
1, and it describes the change in effective hy-perfine field, between two consecutive displacement pe-riods i and i + 1, at equivalent dot j . It is expressed as (cid:126)B ji +1 = R (cid:126)B ji + (1 − R ) (cid:126)δB ji +1 , where (cid:126)δB ji +1 is a randomoccurence of the magnetic field distribution. For exam-ple, a full remanence R = 1 means that the effective hy-perfine interaction for an electron visiting multiple timesthe same dot is constant over the entire displacement se- quence of a single shot. Assuming (i) the evolution of theeffective hyperfine field much slower than the coherentspin displacement [29, 30], (ii) the nuclear spin immuneto the electrostatic manipulation, and (iii) not affected bythe electron motion, the remanence value is then inter-preted as a small electrostatic potential reorganization,leading to fluctuations in the dot positions. For R < . R , which is thefingerprint of a continuous motional narrowing effect. Onthe other side, for R > .
5, the motional narrowing pro-cess is limited by the number of visited sites, which leadsto Gaussian or Gaussian-like [25] coherence decays. Sucha transition from Gaussian to exponential dynamics is aclear manifestation of the motional narrowing effect thathas been observed in liquid NMR.The simulation well reproduces the experimental datain the single- and two-electron displacement with a rema-nence parameter
R ≈ (cid:46) R < .
5. One can note thatthe difference in decay times between the single- and thetwo-electron displacement cases is not significant regard-ing the fit uncertainties (Supp. Figure S3a,b). Moreover,this parameter also provides simulated coherence decaysvery similar to the experimental data of Fig. 4e, for thevarious values τ R of the displacement dynamics (solidlines), as well as for the data obtained in the quasi-staticexperiments (Supp. Fig. S5).Therefore, the cyclic nature of the evolution has to berandomized along the path of the electrons to reproducethe experimental data. Since the hyperfine interaction isa contact interaction, a displacement of the size of thedot will be sufficient to change the effective magneticfield completely. The relationship between R and theelectron centre of mass separation between two consec-utive periods d is given for various charging energies inFig. 5c. It is calculated as the overlap between the elec-tron density at two consecutive periods R = (cid:82) ψ i ψ i +1 dr .Therefore, between two consecutive displacement peri-ods, the electron centre of mass is expected to be distantof about d ≈
33 nm for a charging energy of 3 meV(Fig. 5d and Supp. Fig. S6). The schematic electroncentre of mass trajectories in the single- and the two-electron displacement experiments are superimposed onthe gate pattern and the simulated electrostatic potentialof Supp. Fig. S6a and b, respectively.Here we speculate that these fluctuations in dot po-sitions may occur because of the large voltage rangesemployed to displace the electron through the quantumdot array (as large as few hundreds of millivolts, Supp.Fig. S2). Such energy is expected to induce perturbationsof the semiconductor nanostructure resulting in a changein the disorder potential. Besides, the electrostatic po-tential simulations of Supp. Fig. S6a,b confirm that theconfinement potentials remain shallow during the elec-tron shuttling, making the dot positions very sensitive tofluctuations of charges in the substrate (e.g. donors inthe doping layer of the heterostructure, Supp. Fig. S6c).Moreover, the typical timescale of this alteration is com-parable to the one of the excitation potential and there-fore can be as fast as 1 nanosecond. It implies that theelectrons are displaced along a path that is fluctuating ona timescale comparable to the time needed to explore fewdot configurations. As a consequence, the effective mag-netic field configuration is not precisely cyclic anymoreand become randomized.A similar assumption on the electron path has beenreported recently by our group in a different displace-ment regime [9]. The electrons were then displaced inmoving quantum dots at a speed of 3000 m / s. Indeed, atthis speed, the motional narrowing process is extremelyefficient for hyperfine interaction, and the main mecha-nism for decoherence is mediated by spin-orbit interac-tion. Evidence of change in the disorder potential wasimprinted in the coherence of the transported electronsinitially prepared in a superposition of antiparallel spinstates. CONCLUSIONS
We have performed the one- and two-electron coher-ent shuttling in a two-dimensional array and exploredvarious displacement trajectories through different setsof quantum dots. We report an increase of the coher-ence time, which corresponds to a coherence length wellabove one micron for both one- and two-electron dis-placements. Furthermore, we specifically study the zeromagnetic field regime where all the three components ofthe hyperfine interaction contribute equally to the deco-herence of the two electron-spin singlet states. This hy-perfine interaction is expected to be averaged during theelectron displacement. A signature of a motional narrow-ing process is observed with a decoherence rate inverselyproportional to the time spent in the static phase. In-deed, reducing the time where the electrons are effectivelystatic and averaging faster than the spin dynamics overmany nuclear spin configurations increase the observedcoherence time [22]. An important consequence of theobservation is the signature that the path of the electronis slightly modified from one period of displacement tothe next. It results in the randomization of the effec-tive field distribution experienced by the electrons. Ex-perimental results are reproduced with a simple model.Therefore, only the single electron displacement duringthe gate movement from one dot to another is relevantto explain the increase in spin coherence time.
METHODS
The methods about the spin initialisation in the singletstate, the electrostatic potential, the stability diagramsimulations and the virtual gates employed in this studyare essentially similar to [24].
Acquisition protocol.
The voltage control apparatus isprogrammed to execute a list of single-shot experiments(about a few thousand different sequences). The total du-ration of each single-shot (comprising the electron spininitialisation, manipulation, readout and the instrumen-tal over-head) is about 50 ms. The singlet spin proba-bilities are computed by averaging 1000 single-shot (rep-etition of the programmed list of sequences). Therefore,the typical time between the repetition of two equivalentsequences is about a few minutes.
Materials and set-up.
Our device was fabricated us-ing a Si-doped AlGaAs/GaAs heterostructure grown bymolecular beam epitaxy, with a two-dimensional electrongas 100 nm below the crystal surface which has a carriermobility of 10 cm V − s − and an electron density of2 . × cm . The quantum dots are defined usingelectrostatic confinement generated using Ti/Au Schot-tky gates. It is anchored to the cold finger, which is inturn mechanically attached to the mixing chamber of ahomemade dilution refrigerator with a base temperatureof 60 mK. It is placed at the centre of a superconduct-ing solenoid generating the static out-of-plane magneticfield. Quantum dots are defined and controlled by apply-ing negative voltages on Ti/Au Schottky gates depositedon the surface of the crystal. Homemade electronics en-sure fast changes of both chemical potentials and tun-nel couplings with voltage pulse rise times approaching100 ns and refreshed every 16 µ s.A Tektronix 5014C with a typical channel voltage risetime (20% - 80%) of 0 . V L , V B , V R , and V T gate voltages. The charge con-figurations can be read out by four quantum point con-tacts, tuned to be sensitive local electrometers, and in-dependently biased with 300 µ V. The resulting currents i TL , i BL , i BR and i TR are measured using current-to-voltage converters with a typical bandwidth of 10 kHz.The single-shot repetition rate is about 20 Hz. Nuclear spin dynamics and experimentaltimescales
The experimental singlet probabilitiesare calculated as the average spin readout results ofeither 150 (Fig. 5a), or single 1000 shots (Figs. 3b and4c,d). Sequences of multiple displacement patternsare repeated to get a large enough number of shots.A sequence has a typical duration of 1 day, with asequence length of (cid:46) µ s [25, 31]. It allows the nuclear spin distributionto be randomised between each equivalent single-shotexperiment. Consequently, there is no preferentialquantification axis at zero magnetic fields, and thespin-orbit interactions are averaged out. Simulation of the electron spin dynamics.
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We would like to thank Maud Vinet, Xuedong Hu, andLieven M. K. Vandersypen for enlightening discussions.We acknowledge technical support from the Poles of theInstitut N´eel, and in particular, the NANOFAB teamwho helped with the sample realisation, as well as E.Eyraud, T. Crozes, P. Perrier, G. Pont, H. Rodenas, D.Lepoittevin, C. Hoarau and C. Guttin. M.U. acknowl-edges the support of project CODAQ (ANR-16-ACHN-0029). A.L. and A.D.W. acknowledge gratefully thesupport of DFG-TRR160, BMBF-Q.com-H 16KIS0109,and the DFH/UFA CDFA-05-06. T.M. acknowledges fi-nancial support from ERC QSPINMOTION, ERC QU-CUBE, ANR CMOSQSPIN (Grant No. ANR-17-CE24-0009), ANR SiQuBus and UGA IDEX (Grant No. ANR-15-IDEX-02).
AUTHOR CONTRIBUTIONS STATEMENT
P.-A. M. fabricated the sample and performed the ex-periments with the help of T.M. and C.B.. P.-A. M.and T.M. interpreted the data. P.-A. M. and T.M.wrote the manuscript with the input of all the other au-thors. H.F. contributed to the experimental setup. A.L.and A.D.W. performed the design and molecular-beam-epitaxy growth of the high mobility heterostructure. Allauthors discussed the results extensively, as well as themanuscript.
Additional information
Correspondenceand requests for materials should be ad-dressed to [email protected] or [email protected]
Supplemental Materials: Enhanced spin coherence while displacing electron in a 2Darray of quantum dots c da b
FIG. S1.
Single-period electron displacement sequences. a-d,
Sequences of nanosecond voltage pulses used to performthe single period electron displacement of Fig. 3. a, The pulses are applied during τ S to displace the electrons in the TCconfiguration. b, The pulses are applied during τ S / c, The pulsesare applied during τ S / d, The pulses are applied during τ S / ab FIG. S2.
Periodic electron displacement sequences. a,b,
Sequences of nanosecond voltage pulses used to perform theperiodic electron displacements. a, One electron remains in the left quantum dot while the second electron is displaced in theB, C, T, R dots. b, One electron periodically visits L, C, R, and the second T, C, B. The x-axes labels denote the locationof the dot occupied by the electron. For example, LB stands for one electron in dot L, and one in dot B. The schematicrepresentations of the charge configurations are superimposed. Only one period of displacement is shown for each sequence.The voltage rising times are given as illustrations. a bc
FIG. S3.
Exponential spin coherence fits of the periodically displaced electron.
In order to extract the coherencetime T coh that are discussed in the manuscript, we fit the experimental data using a single exponential decay law. a, b
Samedataset as in Fig. 4c and d, respectively. The data are fitted with an exponential decay ( solid lines ) with characteristic timesof ( a ) 32 ±
7, and ( b ) 44 ± c Same dataset as in Fig. 4e. The data are fitted with an exponential decay ( solid lines )with the characteristic times: 91 . ±
10 (blue), 44 . ± . ± . ± . ±
15 (purple), and 8 . ± FIG. S4.
Simulated spin coherence time in a static double quantum dot.
The electron spin decoherence curvescan be simulated for different standard deviation σ ( B Z ) values of a Gaussian distribution in hyperfine fields (Methods). Thecoherence times extracted from Gaussian fits are plotted as functions of σ ( B Z ). The field corresponding to the spin coherencetime T (cid:63) ( N config = 1) = 9 . T (cid:63) ( (cid:112) N config ) fit of Fig. 3f) is pinpointed in dashed red line ( B Z = 3 . FIG. S5.
Simulated decays of the quasi-static electron spin coherence.
Same dataset as in Fig. 3e. The experimentaldata (dots) are compared with the simulations (solid lines) of the coherence spin in the quasi-static case (remanence parameteris irrelevant here as each configuration is explored only a single time). The overshoot in singlet probability described in [25] isobserved in the simulations only for the double quantum dot case (blue), and is not observed in the experimental data. Forthe two electrons in three (orange), four (green), or five (red) dots, the simulated decays are Gaussian-like. defbac
FIG. S6.
Electrostatic potential simulations. a, b,
Electrostatic potential simulations performed with a gate voltageconfigurations corresponding to a confinement of two electrons ( a ) in L-B, and ( b ) in T-R. Hypothetical electron trajectories(dots) are overlapped for the two displacement patterns considered. The solid lines are guides for the eye. a, A single electronis periodically displaced between dots (B, C, T, and R), the other electron remains in the L dot. Four dot configurationsare explored, but the dot positions evolve at each period (two periods are drawn, e.g. 1 → (cid:48) → b, Two electrons aredisplaced, one between (L, C, R) and the other between (T, C, B). Eight dot configurations are explored and here again, thedot positions evolve at each displacement period (a single period is drawn, e.g. 1 → (cid:48) ). The electrostatic potential simulationswere performed by solving the Laplace equation [34]. The orange circles have a radius of 20 nm corresponding to the Bohrradius of an electron with a charging energy E C of 3 meV. The dashed circles have a radius of 33 nm, value computed toreach a remanence R = 0 . c, Electrostatic potential simulation of a random donor distribution in the Si-doped layer of theheterostructure (Methods). The induced potential can localise the dots in different spatial positions than the one intended bythe gate geometry. d-f,
Schematic representations of the Bohr radii and corresponding dot displacements to reach a remanence R = 0 ..