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K. Hecker, L. Banszerus, A. Schäpers, S. Möller, A. Peters, E. Icking, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), C. Volk, C. Stampfer

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[Coherent charge oscillations in a bilayer graphene double quantum dot](https://mdr.nims.go.jp/datasets/996bd33d-6b4c-4b0a-9294-794bca3c678b)

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Coherent charge oscillations in a bilayer graphene double quantum dotArticle https://doi.org/10.1038/s41467-023-43541-3Coherent charge oscillations in a bilayergraphene double quantum dotK. Hecker1,2,5 , L. Banszerus 1,2,5, A. Schäpers1, S. Möller 1,2, A. Peters1,E. Icking1,2, K. Watanabe 3, T. Taniguchi 4, C. Volk 1,2 & C. Stampfer 1,2The coherent dynamics of a quantum mechanical two-level system passingthrough an anti-crossing of two energy levels can give rise to Landau-Zener-Stückelberg-Majorana (LZSM) interference. LZSM interference spectroscopyhas proven to be a fruitful tool to investigate charge noise and charge deco-herence in semiconductor quantum dots (QDs). Recently, bilayer graphenehas developed as a promising platform to host highly tunable QDs potentiallyuseful for hosting spin and valley qubits. So far, in this system no coherentoscillations have been observed and little is known about charge noise in thismaterial. Here, we report coherent charge oscillations and T *2 charge deco-herence times in a bilayer graphenedoubleQD. The charge decoherence timesare measured independently using LZSM interference and photon assistedtunneling. Both techniques yieldT *2 average values in the rangeof 400–500ps.The observation of charge coherence allows to study the origin and spectraldistribution of charge noise in future experiments.The concept of Landau-Zener-Stückelberg-Majorana (LZSM) inter-ference was first introduced to describe atomic collisions1–5 and hasfound renewed interest with the advent of artificially designed quan-tum mechanical two-level systems (TLSs) in a variety of solid-stateplatforms6–17. In particular, LZSM interferometry has been developedinto a major work horse to study quantum interference effects in sili-con nanowires6, nitrogen-vacancy centers in diamond7,8, super-conducting qubits9 and semiconductor quantum dots (QDs), where itallows to coherently control the electron spin10,11 or the spatial positionof an electron in a double quantum dot (DQD)12–16. Charge noise limitsthe charge decoherence time and, e.g., mediated via spin-orbit inter-action, the spin decoherence time. Semiconductor QDs have beenstudied in awide range ofmaterials, including silicon18–20, germanium21and GaAs22,23. More recently, 2D materials, such as bilayer graphene(BLG) and transition metal dichalcogenides have emerged as poten-tially interesting alternative materials with appealing properties forhighly controllable QDs, interesting for hosting spin and valleyqubits24. BLG offers a gate voltage-tunable band gap25–27, small spin-orbit interaction and weak hyperfine coupling24. Importantachievements in BLG QD research include the implementation ofchargedetection28,29, an understandingof spin-valley coupling30–32, andthe measurement of the spin relaxation rate33,34. In addition, QDs havealso been realized in WSe2 and MoS2 monolayers35,36, which are ofinterest due to their substantial intrinsic spin-orbit coupling andpotential for light-matter coupling. However, despite these recentexperimental advances, no coherent oscillations of either charge, spinor valley states have yet been reported in quantum devices based on2D materials. A priori, it is not obvious that charge coherence can beobserved in van der Waals heterostructures such as BLG encapsulatedbetween hexagonal boron nitride (hBN) crystals. In contrast to QDs insemiconductor heterostructures based on GaAs37,38 and SiGe20,39,which are buried tens of nmbelow the dielectric interface, the electronwave function of a BLG QD extends into the hBNmaking it susceptibleto charge noise present due to disorder at the BLG/hBN interface andto impurities in the hBN. So far, no light has been shed on the role ofcharge noise in graphene QDs.Here, we demonstrate coherent charge oscillations in a BLG DQD.In contrast to spin, the charge degree of freedom offers fasterReceived: 14 March 2023Accepted: 13 November 2023Check for updates1JARA-FIT and 2nd Institute of Physics, RWTHAachenUniversity, 52074Aachen,Germany. 2PeterGrünberg Institute (PGI-9), Forschungszentrum Jülich, 52425Jülich, Germany. 3Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4InternationalCenter for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5These authors contributed equally:K. Hecker, L. Banszerus. e-mail: Katrin.Hecker@rwth-aachen.deNature Communications |         (2023) 14:7911 11234567890():,;1234567890():,;http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43541-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43541-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43541-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43541-3&domain=pdfmailto:Katrin.Hecker@rwth-aachen.dedynamics that can be controlled all-electrically using gateelectrodes40,41. We operate the DQD in the few electron regime andtune its interdot tunnel coupling to the low GHz regime, which weverify by photon assisted tunneling (PAT) spectroscopy. In a pulsed-gate experiment, we observe LZSM interference oscillations of anexcess electron, a characteristic signature of quantum phase coher-ence. From the PAT experiments, we determine an average ensembledecoherence time T *2 of around (416 ± 110) ps, while from the analysisof the LZSM interference oscillations we extract an average deco-herence time of around (483 ± 24) ps. These time scales are en parwiththose reported for advanced GaAs QDs41,42 which is a first indicator forlow charge noise and an important quality measure for the van-der-Waals interfaces in the BLG-based heterostructure.ResultsThe device used to form a DQD consists of a BLG flake encapsulatedbetween two crystals of hBN placed on a global graphite back gate(BG), with two layers ofmetallic top gates (i.e., the split and finger gatelayer) separated by aluminium oxide. Figure 1a shows a false-colorscanning electron microscopy image of the gate structure of thedevice (see Methods for details)43. The BG and split gates (SGs) areused to form a p-type channel connecting source and drain reservoirs.The potential along the channel can be controlled using a set of fingergates (FGs). A DQD is formed by locally overcompensating thepotential set by the BG using two adjacent FGs, as schematicallydepicted in Fig. 1b (see also yellow FGs in Fig. 1a). Figure 1c shows acharge stability diagram of the DQD in the few electron regime. Whenincreasing the FG voltages, especially on the right FG, VR, the currentthrough the DQD increases and the co-tunneling lines become morepronounced. This indicates that the interdot tunnel coupling can besensitively tuned by the voltages applied to the FGs44.The first step towards studying quantum phase coherence in theDQD is to create an effective TLS and to characterize its energy dis-persion using PAT spectroscopy. We focus on a single pair of triplepoints, where a TLS is formed by a single excess electron that can belocated either in the left or the rightQD (see Fig. 2a) with the two statesRj i : = ðN,M + 1Þ and Lj i : = ðN + 1,MÞ, where N and M is the electronoccupation of the left (L) and the right (R) QD, respectively. For largedetuning energies, ε, compared to the interdot tunnel couplingenergy, Δ/2, the eigenenergies of the TLS are given by EL = ε/2 andER = − ε/2, where ε is the detuning energy between the two QDs (seewhite arrow in Fig. 2a and gray dashed lines in Fig. 2b). This approx-imation becomes invalid for ∣ε∣ ≲Δ, where the eigenenergies are givenby the more general form E ± = ± 12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2 +Δ2p(see solid lines in Fig. 2b).The state of such a TLS can be represented on the Bloch sphere shownin Fig. 2c, using the states Rj i and Lj i as a basis.PAT spectroscopy allows to map out the energy dispersion of theTLS and thereby to determine the tunnel coupling, which togetherwith the detuning fully characterizes the system. Thismethod relies onmicrowave excitation of electrons across the interdot tunnel barrier,which becomes possible whenever the microwave excitation is inresonance with the energy splitting of the QD states, i.e., hf = E+ − E−with the microwave frequency f and Planck’s constant h41,45–47. At asmall bias voltage, we detune the energy levels of the QDs such thattransport is blocked. An electron in the low energy state (see inset inFig. 2h) can then only transfer into the higher state if the systemabsorbs a resonant microwave photon. The excited electron then cantunnel to the reservoir, contributing to a current through the devicewhose direction depends on the sign of the detuning energy.Figure 2d shows a charge stability diagram of the triple point inFig. 2a, recorded at zero bias voltage while applying a microwaveexcitation of f = 9GHz to the left gate. Twoparallel current resonancesof opposite sign appear symmetrically around zero detuning. In orderto map the energy splitting as a function of detuning, the microwaveexcitation frequency is varied. In the PAT measurements, also excitedstates could play a role, if energetically accessible47. The absence ofthese excited states can be explained by the out-of-plane magneticfield of 1.8 T applied to the device, which polarizes the valley states(valley splitting of ≈ 1.5meV) and also partly the spin states (spinsplitting of ≈ 210μeV). The resonances split further with increasingfrequency, as shown for f = 15GHz and f = 25GHz in Fig. 2e, f, respec-tively (see Supplementary Fig. 2 for more data). For a quantitativeanalysis, we measure the splitting of the PAT resonances along the VLaxis, δVL, as a function of the appliedmicrowave frequency, see line cutin Fig. 2g. At δVL < 0.2mV, the PAT resonances begin to overlap, settinga lower bound to the frequency range that can be investigated. Therelation of δVL and f is determined by the resonance condition(hf = E+ − E−) and can be expressed asf ðδV LÞ=1hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðαδV LÞ2 +Δ2q, ð1Þwhere α is the lever arm that converts the VL axis to a detuning energy.We fit Eq. (1) to the data with the fit parameters α and Δ (see red line inFig. 2h). This yields a tunnel coupling of Δ/(2h) = 2.1 ± 0.2GHz andα = 194.5 ± 1.1μeV/mV, which is in good agreement with the lever armobtained from DC finite bias spectroscopy measurements (see Sup-plementary Fig. 1). Additionally, similar measurements were con-ducted for a set of two different FG voltages (VL = 3.475 V, VR = 3.38 V,and VL = 3.555 V,VR = 3.36 V), which yield Δ/2h ≈ 1.54GHz and Δ/2h ≈ 7.87GHz, respectively. For the following measurements theregime of intermediate tunnel coupling (2.1GHz) was chosen.Besides studying the dispersion of the TLS, PAT spectroscopymeasurements also probe its coherent properties. From the full widthat half maximum, γ, of the PAT resonance, the ensemble decoherencetime T *2,PAT = 2h=ðαγÞ of the charge degree of freedom can beestimated41,42,48 The fits in Fig. 2g, yield a T *2,PAT at the positive (red) andnegative (blue) PAT peak of 422 ps and 291 ps, respectively. We assignthe timescale extracted in this experiment to the ensembleabBias teeVLVAC3.50 3.54V  (V)L3.323.363.40V (V)RI (pA)c 110 210DQDS310DVL VR1/ft   pt   iVpVpSGSDSGVRFig. 1 | Bilayer graphene double quantum dot. a False-color scanning electronmicroscopy image of the DQD device. The scale bar measures 500 nm. The splitgates (SG) together with the back gate (not shown) are used to define a narrowconductive channel connecting the source (S) and drain (D) reservoirs (highlightedin blue). The channel is crossed by finger gates (see yellow structures) used tolocally tune the band edges to formQDs. Our so-called finger gate left is connectedto a bias tee which allow applying DC (VL) and AC voltages (VAC). The voltage VR isapplied to the right finger gate. b Schematic of the conduction and valence bandedge profile of the DQD highlighting the left and right finger gate, where VL and VRcan be applied. c Charge stability diagram of the DQD at low electron occupationmeasured at a source-drain bias voltage of VSD = 0.5mV.Article https://doi.org/10.1038/s41467-023-43541-3Nature Communications |         (2023) 14:7911 2decoherence time as it results from a current that is integrated overmany pulse cycles and can thus be viewed as a time-ensemble average.In order to obtain a better understanding of the dynamics of theTLS and to gain insight into the charge decoherence time, we per-form LZSM interferometry measurements, a powerful technique,where the TLS is driven non-adiabatically through the anti-crossingof the energy levels. To this end, a voltage pulse with a finite risetime, tr ≈ 140 ps, is applied to the left finger gate, see Fig. 3a. Weinitialize (i) the excess electron in the right QD by allowing it to relaxinto the ground state, Rj i, at a detuning εi < 0 for a time ti ≈ 3 ns. Toensure a sufficient initialization but still keep a high signal-to-noiseratio, the initialization time was chosen such that ti ≳ 1/Γcomb ≈ 1.6 ns,where Γcomb is the combined tunneling rate through the DQD, esti-mated from finite bias spectroscopy (see Supplementary Fig. 1). Theinitial state of the TLS is shown in the level scheme (Fig. 3b (1)) as wellas in the energy diagram and Bloch sphere representation (Fig. 3c(1)). After the initialization, the chemical potential of the left QD isshifted by a voltage pulse of nominal amplitude Vp, corresponding toan effective detuning pulse of amplitude Ap applied to the sample(seeMethods for details). The change in detuning is approximated tooccur at a constant rate v = ∣∂ε/∂t∣ ≈ 1.6 μeV/ps. When passing theanti-crossing at zero detuning, the system undergoes a Landau-Zener(LZ) transition from the ground to the excited state with a transitionprobability given by49PLZ = exp �π2Δ2hv !, ð2Þand picks up a relative Stokes phaseφS, marked in blue on the equatorof the Bloch sphere in Fig. 3c (2). The LZ transition results in a super-position state with weights 1 − PLZ and PLZ in the ground and excitedstate, respectively (Fig. 3b (2), c (2)). The ratio of the weights isexperimentally accessible via v. In the adiabatic limit (hv≪Δ2), thesystem remains in the ground state, while in the non-adiabatic limit(hv≫Δ2), the entire wave function transitions to the excited state.After the first LZ transition, the system is allowed to time-evolve freelyat the point ofmaximumdetuning εi +Ap for a time tp (Fig. 3b (3), c (3)).It accumulates another phase contribution given by49φev =12hZ tLZ2tLZ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiεðtÞ2 +Δ2qdt, ð3Þwhere tLZ1ð2Þis the time of the first (second) LZ transition (see grayshaded areas in Fig. 3a, c (3)). While the system is then returned to εi at“rate" v, it undergoes a second LZ transition at zero detuning (Fig. 3b(4), c (4)), adding another phase contribution of φS. Both parts of thewave function interfere coherently, such that the final excitationprobability is a function of the total relative phase φ =φev + 2φS. Con-structive (destructive) interference into the excited (ground) stateoccurs for φ = 2nπ (φ = (2n + 1)π), n 2 N013. At the end of the pulsecycle, a projective readout of the final state is performed. An electronin the excited state Lj i can tunnel out of the DQD and contribute to acurrent that is averaged over many pulse cycles, while an electron inthe ground state Rj i cannot leave the DQD (Fig. 3b(5)). In the Blochsphere representation, the readout corresponds to a projection on thez axis (Fig. 3c (5)).d3.4623.3653.462 3.4643.3663.3703.370-40 400 -20I (pA)I (pA)9 GHzeEnergy�0b3.458 3.4663.374aV (V)R0 20I (pA)V  (V)L3.364V (V)R V (V)RV  (V)L V  (V)LRLLR2�VLh3.4640 2015 GHz3.462 3.4643.3660 103.370-10I (pA)25 GHzfhfcV (V)RV  (V)LL R2L R2L R2L R2 LRLR05-510I (pA)V  (V)L3.4625 3.46302�VLg1020300.2 0.4 0.6�V  (mV)f (GHz)00Lhf�   �   Fig. 2 | Photon-assisted tunneling. aCharge stability diagram of a pair of triplepoints at VSD = 50 μV. The charge ground states Rj i and Lj i, corresponding tothe position of an excess electron in the DQD (left or right) and the axis ofthe detuning energy, ε, are indicated. b Energy diagram of the TLS. Theenergies of the uncoupled charge states Rj i and Lj i are shown by dashed graylines. For non-zero tunnel coupling, Δ ≠ 0, a pair of hybridized eigenstatesemerges with eigenenergies represented by the solid black lines, showing asplitting of Δ at zero detuning. The resonance condition required for PAT isgiven by hf =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2 +Δ2p. c Bloch sphere representation of the TLS with thecharge states Rj i and Lj i on the poles. d Charge stability diagram of the triplepoint shown in (a) while applying a microwave excitation of f = 9 GHz and-36 dBm to the left FG. e, f Measurements as in (d) for f = 15 GHz and f = 25GHz, respectively. More data sets for different frequencies are presented inSupplementary Fig. 2. The separation of the peaks, 2δVL, is measured toinvestigate the energy splitting of the TLS. g Cut through panel e along VL.Lorentzians are fitted to the negative (blue) and positive (red) PAT peak. Thearrow indicates where 2δVL is extracted (see Supplementary Fig. 3 for moredata). h Resonant excitation frequency as a function of δVL (see e). Thedashed red curve shows a fit according to hf =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðαδV LÞ2 +Δ2qwhile the shadedbands mark the ± 1σ confidence interval. The dashed black line is a straightline through the origin with the slope of the lever arm α = 194.5 μeV/mV. Thegray data points in the background are the results for two distinct sets of FGvoltages (VL = 3.475 V, VR = 3.38 V, Δ/2h ≈ 1.54 GHz andVL = 3.555 V, VR = 3.36 V, Δ/2h ≈ 7.87 GHz). Inset: Schematic representation ofthe electrochemical potentials in a DQD illustrating the process of PAT.Article https://doi.org/10.1038/s41467-023-43541-3Nature Communications |         (2023) 14:7911 3Figure 4a shows a charge stability diagram of the transitionbetween Lj i= ðN + 1,MÞ and Rj i= ðN,M + 1Þ while applying a voltagepulse to the left FG (VL) as described above. Parallel to the zerodetuning line of the triple point, a series of additional lines of increasedcurrent can be observed, that are constrained by the co-tunneling linesand the pulse amplitude. This is a clear signature of coherent chargeoscillations of one excess electron, where the fringes indicate con-structive interference. The first and strongest fringe corresponds tothe situationwhere the pulse just reaches zero detuning, i.e., εi +Ap = 0during the time span tp, and the electron can first tunnel over into theother QD. At every following line, the relative phase has increased byanother 2π. The region of negative current in Fig. 4a can be attributedto charge pumping occuring outside of the gate configurations wherethe measurement scheme presented in Fig. 3a operates. We find thedata in agreement with the adiabatic-impulse approximation49 as nointerference signature is visible before reaching the first fringe.The loss of charge coherence over time can be explored by per-forming LZSM interferometry in the time domain, i.e., as a function ofthe pulse duration tp and detuning energy during initialization εi, asshown in Fig. 4b. An effective pulse amplitude of Ap ≈ 228μeV can be1Initialization LZ transition Phase acc. LZ transition ReadoutaPLZ1-PLZ PL0.5φev1 2 3 4 5cxyz0b1 2 4+ 3 5LRRL0φev01 2 3 4 5tt  i +t  pt  i+ApL R�i�i�Fig. 3 | Pulse scheme and state manipulation in a LZSM interference experi-ment. a Time-dependent detuning driven by a voltage pulse applied to the left FGcharacterized by pulse amplitude, Ap, and durations tp and ti. b Schematics of theenergy levels in the DQD while the pulse scheme in (a) is applied. c Time evolutionof the TLS under the influence of the pulse scheme, shown in the energy diagramand on the Bloch sphere. (1) First, the system is initialized at εi for the duration of ti(see a). An electron tunnels to the right QD (see b) and the system is in the groundstate Rj i, as shown by the red dot and the red arrow. (2) The system is detuned toεi +Ap. The process is adiabatic except at the point of ε =0 (see a, b), where a LZtransition into the excited state is possible. This creates a superposition state,depicted by two dots in the energy diagram, and a rotation of the Bloch vectorabout the x axis. In this process, a relative Stokes phaseφS is pickedup, representedby the blue shaded area in the equator plane of the Bloch sphere. (3) During tp, bothcomponents of the wave function evolve separately in time at a detuning εi +Ap,accumulating a relative phaseφev represented by the area shaded in gray. (4)Whencrossing ε =0 again (see also panels a,b), a second LZ transition takes place and theinitially separated wave functions interfere. (5) In the readout configuration, thedetuning is set to εi again (see a). An electron in the excited state Lj i can tunnel tothe drain and contribute to the measured current, while an electron in the groundstate would be trapped due to Coulomb blockade (see b). By this method theoccupation probability of the excited state, PL, is determined.3.464I (pA) 0 10 200�2�4�3.4623.3643.372aV  (V)LV (V)RI II IIIEtIEtIIEtIII2010020100I (pA)-200 -100 0I (pA)35025015050c d I e-200 -100 0b 0 10 20I (pA)25050300150200100t  (ps)p   (μeV)t  (ps)pi����   (μeV)i�Fig. 4 | Coherent charge oscillations in the time domain. a Charge stabilitydiagram of the triple point as in Fig. 2a, while a pulse of duration tp = 165 ps andamplitude Ap = 228μeV is applied to the DQD by the left FG. The black dashed lineshighlight the co-tunneling lines. Fringes of increased current indicate coherentcharge oscillations. Complementary data sets for different tp and Ap are presentedin Supplementary Fig. 4. bOscillations in the time domain, obtained along the linecut indicatedby the black arrow in (a). Complementary data sets for different pulseAp are presented in Supplementary Fig. 5. cCut along the detuning axis as indicatedby the horizontal black and red arrows in (b). Up to six interference maxima withdecreasing intensity can be distinguished.dCut along the tp axis as indicatedby thevertical black and red arrows in (b). e Schematics illustrating the configurations atthemaxima labeled (I)–(III) in (c,d) indicating the influenceof theparameters εi andtp on the relative phase φev.Article https://doi.org/10.1038/s41467-023-43541-3Nature Communications |         (2023) 14:7911 4deduced from the position of the first interference fringe attp ≈ 200ps, which yields a rate of v ≈ 1.6μeV/ps. Oscillations appearalong the detuning and time axes. This observation can be explainedby consideringhowbothparameters influence the accumulatedphase.Figure 4c plots line cuts along the detuning axis in the time domain(see horizontal dashed lines in Fig. 4b). Up to six interference maximacan be identified before the signal is overlaid by the broad feature atεi = 0, which originates from the tunneling of charge carriers throughthe DQD when the electrochemical potential in the left QD is alignedwith the bias during the initialization step. In Fig. 4c, two oscillationmaxima are highlighted by (I) and (II) and the acquired phase is indi-cated by the gray area in the corresponding schematics shown inFig. 4e. Maximum (I) corresponds to the configuration where a relativephase of 2π is acquired during one pulse cycle. Following the detuningaxis to the nextmaximum (II), the phase increases by another 2π as thesystem is taken further beyond zero detuning, leading to a highermaximum value of the integrand in Eq. (3) (compare gray areas inschematics (I) and (II)). The absence of oscillations on the first fringe(εi ≈ − 200μeV) in Fig. 4a can be attributed to the distortion of thepulse when transmitted through the setup, as has been studied inGaAs/AlGaAs DQDs16. Fig. 4d shows line cuts along the time axis (seevertical dashed lines in Fig. 4b), which show oscillations that aredamped due to the loss of quantum phase coherence. However, theresolution along the time axis is not sufficient for a quantitativedecoherence analysis. The expected oscillation period is given byT =h=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2 +Δ2p40. Going frommaximum (I) to (III) also adds a phase of2π, in this case by prolonging tp and thus expanding the integrationbounds in Eq. (3) (compare gray areas in schematics (I) and (III)in Fig. 4e).Next, we focus on the amplitude domain to quantitatively analyzethe loss of phase coherence9,13,50–52. The pulse amplitude Ap is anotherexperimental knob to tune the relative phase of the TLS superpositionstate. Figure 5a shows the current through the device in a pulsedmeasurement of constant tp = 200ps, as a function of εi and Ap. Thedata show several parallel interference fringes with decreasing inten-sity. The first and most prominent fringe, labeled (0), corresponds tothe situation sketched in the upper schematic in Fig. 5b, where thepulse takes the system to zero detuning during tp. At the second fringe,labeled (I), Ap has increased such that the pulse crosses zero detuningand a relative phase of 2π is accumulated, see lower schematic inFig. 5b. The intensity along a single fringe shows no periodic mod-ulation, which confirms that quantum coherence is only lost betweentwo pulses6,49. In the case of multiple coherent LZ transitions, theintensity along a given fringe would additionally be modulated due tothe interference of more than two consecutive LZ transitions, whereasonly a featureless current would be expected if decoherence occurredbetween all LZ transitions, i.e., faster than the pulse duration. Hence,the pulse duration (tp = 200ps) and the initialization time (here:k  (ps)�A(�eV)p0 4I (pA)5000-500k (ps)A0 5 10a c300200e600400800Et08-300 -200 -10053dln|I|FT-500Et100 (ps)T 2 ,FT*200 2400 5002000160 0 50Nf4t  (ps)p0�  (�eV)i6004008002000100 (ps)T 2 ,PA T*IIln|I |FTFT PATbFig. 5 | Coherent oscillations in the amplitude domain. a Current through thedevice as a function of the initialization detuning εi and pulse amplitude Ap, at aconstant duration of tp = 200ps. LZSM interference fringes of two consecutivepassages through zero detuning can be seen. Complementary data sets are shownin Supplementary Figs. 6 and 7. b Schematics illustrating the pulse and energydiagram on the points (0) and (I) in (a). The red dot illustrates the readout/initi-alization configuration. c Fourier transform (FT) of the data in (a). The crossing inthe center is attributed to the timedependenceof the phase50, while the decreasingbackground contains informationon thedecoherence indetuning-space.dLine cutalong the kε axis averaged over a small area in (c). The peak around kε =0 isattributed to low-frequency noise. Red lines are fits to Eq. (5). e Decoherence timeT *2,FT obtained from measurements as in panel a at different pulse widths tp,revealing an average of T *2,FT = ð483± 24Þps (dashed line and gray shaded area).f Histogram of T *2,PAT extracted from the FWHM, γ, of Lorentzian line shapes fit tothe PAT peaks as illustrated in Fig. 2g. T *2,PAT is calculated according toT *2,PAT = 2h=ðαγÞ41, 42, 48. The histogram contains data points from 18 different datasets measured in a frequency range from f = 9 to 30 GHz. For each frequency, PATpeaks at different VR have been evaluated. A total of 729 data points for T *2,PAT areshown in the histogram, where N denotes the number of points in one bin. Theaverage measures T *2,PAT = ð416 ± 110Þ ps, highlighted by the dashed black line andthe gray shaded area.Article https://doi.org/10.1038/s41467-023-43541-3Nature Communications |         (2023) 14:7911 5ti = 3 ns) give a lower and upper bound for the timescale ofdecoherence.In order to quantify the decoherence time, we consider theFourier transform (FT) of the current signal,IFTðkε,kAÞ=Z Zexpð�ikεε� ikAApÞIðε,ApÞdεdAp: ð4ÞThe relevant timescale can be obtained from the decay of theFourier amplitude13,50,52,lnjIFTðkε,kAÞj=a� kε=T*2,FT, ð5Þwherea is anunknownproportionality factor andT *2,FT is the ensembledecoherence time extracted by this experiment. Figure 5c shows theFourier transform of the symmetrized current signal (with respect toAp) fromFig. 5a. Theprominent cross-like feature contains informationon the time between subsequent crossings of zero detuning and thephase accumulated in between50, but is of no further relevance for thedetermination of T *2,FT. To obtain T *2,FT, we follow refs. 50–52 andanalyze the decay of lnðIFTÞ as a function of kε. Figure 5d shows theaverage of 100 line cuts through the data in Fig. 5c along kε, that do notinclude the cross-like feature. Similar to other works, the central peakis likely causedby low-frequency noise and thereforenot considered inthe following13,51,52. The red solid lines represent the best fit to Eq. (5)resulting in T *2,FT = ð435± 66Þps. Figure 5e shows T *2,FT extracted frommeasurements as in Fig. 5a, c, d for different pulse widths tp. Theresults yield an average value of T *2,FT = ð483± 24Þps (see dashed lineand gray shaded area in Fig. 5e) with an additional systematicuncertainty of ≈ 70 ps that stems from the uncertainty in determiningthe lever arm.DiscussionFor comparison, Fig. 5f shows a histogram of T *2,PAT extracted from fitsto PAT peaks as shown exemplary in Fig. 2g, measured at differentexcitation frequencies in a range of f = 9 to 30GHz (see also Supple-mentary Figs. 2 and 3). Evaluating a total of 729 data points, we find adistribution around an average T *2,PAT = ð416± 110Þ ps (see dashed lineand gray shaded area in Fig. 5f), which is slightly smaller than theaverage T *2,FT obtained from LZSM interferometry. The significantvariation of T *2,PAT, evident from the large error and distribution inFig. 5f, is likely attributed to slight fluctuations in the power of theapplied pulse caused by the frequency-dependent transmissionthrough the setup. The discrepancy between T *2,PAT and T *2,FT can beexplained by charge fluctuations, which are slower than the coherencetimebut faster than the integration time of themeasurement, and thusmay result in a statistical broadening of the PAT resonances. Further-more, the appliedmicrowave excitation causes a power broadening ofthe PAT resonances. Both effects lead to an underestimation of T *2,PAT,which therefore only serves as a lower bound. Interestingly, our find-ings are comparable to the longest decoherence times determined byPAT, T *2,PAT, reported in GaAs (400 ps41, 250ps42) and are larger thanthose reported for carbon nanotubes (280ps46), whereas reportedT *2,FT determined by LZSM interference measurements ranges from100ps in a Si DQD52 to 4 ns in a GaAs DQD13. When comparing thesevalues with our results, it is important to note that the ensembledecoherence time T *2 is sensitive to noise contributions from a broadfrequency spectrum and depends on the exact measurement config-uration. In semiconductor materials, the dominant contributions areintrinsic noise originating from charge fluctuations in the material, aswell as charge noise coupling in via the gates40, where low-frequencynoise is assumed to be dominant, typically approximated with an 1/fspectral density42,53–56. Note that charge noise can affect both Δ and ε,whereas the latter effect is dominant due to the stronger dependenceof ε on the gate voltages. In first approximation, charge noise couplesin proportionally to the slope of the energy bands, ∣dE±/dε∣. This leadsto an increasing decoherence time when moving closer to the sweetspot ε =0 where ∣dE±/dε∣ vanishes40,42. In LZSM interferometry, such asin our experiment, the TLS evolves at finite detuning, ε ≠0, hence T *2,FTjust serves as a lower bound. In particular, the implementation of acharge sensor will allow the direct measurement of coherent chargeoscillations (e.g., in a Ramsey experiment) at the sweet spot where theinfluence of charge noise is further reduced to a minimum promisinglonger decoherence times56. Moreover, in further experiments,including charge echo measurements, the spectral distribution ofcharge noise in BLG DQDs could be investigated. These techniqueshave the potential to cancel out the effect of quasi-static noise andincrease the charge decoherence time57,58. Furthermore, temperature-dependent decoherence measurements allow to quantify electron-phonon coupling, an intrinsic source of decoherence in QDs, asdemonstrated in GaAs QDs40. Charge coherence experiments can alsobe extended to DQDs in other 2Dmaterials such asWSe2 andMoSe235.The high sensitivity of this type of experiment can be exploited tocharacterize material-dependent properties such as defect-inducedand interfacial charge noise and electron-phonon coupling, which areexpected to be different from BLG.In summary, we have shown coherent charge oscillations in a BLGDQD, whichwe discuss in the framework of Landau-Zener-Stückelberginterference. Our findings constitute the first observation of phasecoherent oscillations in a graphene QD device and underline thismaterial’s potential in the field of quantum technology. We comparethe ensemble decoherence times, determined independently from aFourier analysis of coherent oscillations in the amplitude domain andfrom PAT spectroscopy. Both methods consistently yield averagedecoherence times T *2 in the range of 400–500ps.MethodsThe device was fabricated from a BLG flake encapsulated between twohBN crystals of approximately 25 nm (top) and 45 nm (bottom) thick-nesses using conventional van-der-Waals stacking techniques. A gra-phite flake is used as a back gate (BG). The source and drain are etchedthrough the tophBN to contact the BLGusing reactive ion etching. The30nm thick Cr/Au split gates (SGs) with a lateral separation of 80 nmare deposited on top of the heterostructure. Isolated from the SGs by15 nm thick atomic layer deposited (ALD) Al2O3, we fabricate 70 nmwide and 75 nm thick Cr/Au finger gates (FGs) with a pitch of 150nm.Figure 1a shows a false color scanning electronmicrograph of the gatepattern.In order to performhigh frequencygatemanipulation, the sampleis mounted on a custom-made printed circuit board (PCB). The DClines are low-pass-filtered (10 nF capacitors to ground). All FGs areconnected to on-board bias-tees, allowing for AC and DC control onthe same gate (see Fig. 1a). Microwaves are generated by an AgilentE8257D microwave source and pulse sequences are generated by aKeysightM8195Aarbitrarywaveformgeneratorwith a sampling rate of65GS/s. The AC signals are attenuated by −10 dB at room temperatureand further by −26 dB in the cryostat. The nominal pulse amplitude Vpoutput by the instruments is converted into an effective amplitude Apapplied to the sample bymeasuring a charge transition as a function ofε and Vp. This calibration converts the voltage Vp into an energy andtakes into account losses due to attenuators, cables and the PCB. For apulse width tp exceeding the finite rise time of tr ≈ 140ps, we deter-mine a ratio of Ap/Vp = 1.50 ±0.02μeV/mV which is significantlyreduced for tp < tr.Allmeasurements areperformed in a 3He/4Hedilution refrigeratorat a base temperature of around 15mK and at an electron temperatureof around 60mK using standard DC measurement techniques. Thecurrent through the device is amplified and converted into a voltagewith a home-built I–V converter at a gain of 108. A p-type channelbetween source and drain is defined by applying voltages of −3.5 V toArticle https://doi.org/10.1038/s41467-023-43541-3Nature Communications |         (2023) 14:7911 6the BG as well as 1.82V and 1.825 V to the outer and middle SGs,respectively. Positive voltages applied to the left and right FGs, VL andVR, formaDQD (see Fig. 1b). 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This project has received funding from the European Union’sHorizon 2020 research and innovation program under grant agreementNo. 881603 (Graphene Flagship) and from the European ResearchCouncil (ERC) under grant agreement No. 820254, the Deutsche For-schungsgemeinschaft (DFG, German Research Foundation) under Ger-many’s Excellence Strategy - Cluster of Excellence Matter and Light forQuantum Computing (ML4Q) EXC 2004/1 - 390534769, through DFG(STA 1146/11-1), and by the Helmholtz Nano Facility59. K.W. and T.T.acknowledge support from JSPS KAKENHI (Grant Numbers 19H05790,20H00354 and 21H05233).Author contributionsL.B., C.V. and C.S. designed the experiment. L.B., K.H., S.M. and E.I.fabricated the device. K.H., L.B., A.S. and A.P. performed the measure-ments and analyzed the data. K.W. and T.T. synthesized the hBNcrystals.C.V. and C.S. supervised the project. K.H., L.B., A.S., C.V. and C.S. wrotethe manuscript with contributions from all authors. K.H. and L.B. con-tributed equally to this work.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-43541-3.Correspondence and requests for materials should be addressed toK. Hecker.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. 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