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Hadrien Duprez, Solenn Cances, Andraz Omahen, Michele Masseroni, Max J. Ruckriegel, Christoph Adam, Chuyao Tong, Rebekka Garreis, Jonas D. Gerber, Wister Huang, Lisa Gächter, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Thomas Ihn, Klaus Ensslin

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[Spin-valley locked excited states spectroscopy in a one-particle bilayer graphene quantum dot](https://mdr.nims.go.jp/datasets/7f6229ce-48ac-4f10-a033-4275d8c4e2a7)

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Spin-valley locked excited states spectroscopy in a one-particle bilayer graphene quantum dotArticle https://doi.org/10.1038/s41467-024-54121-4Spin-valley locked excited statesspectroscopy in a one-particle bilayergraphene quantum dotHadrien Duprez 1 , Solenn Cances1, Andraz Omahen 1,Michele Masseroni 1, Max J. Ruckriegel1, Christoph Adam1, Chuyao Tong 1,Rebekka Garreis 1, Jonas D. Gerber 1, Wister Huang 1, Lisa Gächter1,Kenji Watanabe 2, Takashi Taniguchi 3, Thomas Ihn 1 & Klaus Ensslin 1Current semiconductor qubits rely either on the spin or on the charge degreeof freedom to encode quantum information. By contrast, in bilayer graphenethe valley degree of freedom, stemming from the crystal lattice symmetry, is arobust quantumnumber that can therefore be harnessed for this purpose. Thesimplest implementation of a valley qubit would rely on two states withopposite valleys as in the case of a single-carrier bilayer graphene quantumdotimmersed in a small perpendicularmagnetic field (B⊥≲ 100mT). However, thesingle-carrier quantum dot excited states spectrum has not been resolved todate in the relevant magnetic field range. Here, we fill this gap, by measuringthe parallel and perpendicular magnetic field dependence of this spectrumwith an unprecedented resolution of 4 μeV. We use a time-resolved chargedetection technique that gives us access to individual tunnel events. Ourresults come as a direct verification of the predicted spectrum and establish anew upper-bound on inter-valley mixing, equal to our energy resolution. Ourcharge detection technique opens the door to measuring the relaxation timeof a valley qubit in a single-carrier bilayer graphene quantum dot.Quantum dots are an essential building block of electrical quantumcircuits due to their diverse and versatile nature, positioning them as aprimary physical platform for hosting qubits. Currently, semi-conductor qubits rely on charge or spin for encoding quantuminformation1,2. The valley degree of freedom, which stems from thecrystal lattice symmetry, stands as a possible alternative.Spin qubits are ultimately limited by phonons and charge noisethat cause relaxation or dephasing through one of the spin-orbit,exchange or hyperfine interactions2. On the other hand, relaxationand decoherence processes for the valley qubits remain to beexplored. Bernal-stacked bilayer graphene (BLG) is an ideal materialto study these effects, as the charge carriers’ angular momentum ischaracterized by both the spin and the valley that are each robustquantum numbers. The strong valley blockade3 leading to longrelaxation times of a valley singlet to a valley triplet state4 in a BLGdouble quantum dot showcases the valley degree of freedom ascomparatively less fragile than the spin. In a BLG single quantum dotpopulated with one charge carrier, the intrinsic spin–orbit interac-tion couples the spin- and orbital- angular momenta which conse-quently share the same, out-of-plane, quantization axis. As a result,the four-fold degeneracy is lifted and the spectrum is composed oftwo Kramers pairs, each consisting of states with both oppositevalley and spin, noted (∣K� "�, ∣K + #�) and (∣K� #�, ∣K + "�), as illu-strated in Fig. 1f5–7. These pairs exemplify the spin-valley lockingmechanism at play in 2D materials with spin-orbit interaction ofKane-Mele8 and Ising type9–12.Received: 19 February 2024Accepted: 31 October 2024Check for updates1Solid State Physics Laboratory, ETH Zurich, Zurich CH-8093 ZH, Switzerland. 2Research Center for Electronic and Optical Materials, National Institute forMaterials Science, Namiki 305-0044Tsukuba, Japan. 3ResearchCenter forMaterials Nanoarchitectonics, National Institute forMaterials Science, Namiki 305-0044 Tsukuba, Japan. e-mail: hadrien.duprez@polytechnique.eduNature Communications |         (2024) 15:9717 11234567890():,;1234567890():,;http://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0009-0001-2216-807Xhttp://orcid.org/0009-0001-2216-807Xhttp://orcid.org/0009-0001-2216-807Xhttp://orcid.org/0009-0001-2216-807Xhttp://orcid.org/0009-0001-2216-807Xhttp://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0002-1233-998Xhttp://orcid.org/0000-0002-1233-998Xhttp://orcid.org/0000-0002-1233-998Xhttp://orcid.org/0000-0002-1233-998Xhttp://orcid.org/0000-0002-1233-998Xhttp://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0001-9996-6371http://orcid.org/0000-0001-9996-6371http://orcid.org/0000-0001-9996-6371http://orcid.org/0000-0001-9996-6371http://orcid.org/0000-0001-9996-6371http://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/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54121-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54121-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54121-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54121-4&domain=pdfmailto:hadrien.duprez@polytechnique.eduwww.nature.com/naturecommunicationsIn BLG quantum dots, the N-particle ground state spectrum wasextensively studied13–17. However, the existing data on the excited statesspectrum of a single-particle quantum dot is more scarce despiteseveral attempts using different measurement approaches5,14,18,19. Inparticular, the excited states spectrum of a single-particle quantumdotwas never measured at low perpendicular magnetic fields(B⊥ ≲ 100mT), where the first crossing of excited states occurs7,20. Inaddition, only the spin- (and no valley-) excited state relaxation timewas measured18,21,22, due to insufficient energy resolution18,21 or too hightunnel rates19,22 in the appropriate magnetic field range. Here,we overcome both of these shortcomings and experimentally establishthe—so far simply assumed—single-charge carrier spectrum picture atzero and low magnetic fields. In addition, we determine a new upperbound on the inter-valleymixing energy scale, equivalent to our energyresolution of 4 μeV, a five-fold improvement on the previously repor-ted upper bound5. Moreover, the tunnel rates into and out of thequantum dot were tuned down to unprecedentedly low values of12.2 Hz at B = 0 T. We overcame the previous requirement of a finiteperpendicular magnetic field to reach sub-kHz tunnel rates4,22,23,likely thanks to a relatively large displacement field (D/ϵ0 ≈ 0.9 V/nm,where ϵ0 is the vacuum permittivity), which opens a bandgap largerthan the disorder potential. Finally, the time-resolved charge detec-tion scheme used here for the spectroscopy, in combination with ourlow effective electron temperature (46mK), opens the door tomeasuring the relaxation time of the different excited states of thefirst charge carrier spectrum via single-shot readout24 even at lowmagnetic fields.The rate atwhich a charge carrier tunnels into or out of a quantumdot is given by Fermi’s golden rule. This rate therefore depends on thespatial extent of the dot’s orbital wavefunction, the number of avail-able micro-states (ground and excited states), and selection rules.Notably, in the case of the transition from 0 to 1 carrier, there are noselection rules as the leads are neither valley- nor spin-polarized. Thesimplest model assumes that all of the micro-states of a given chargestate have an identical orbital wave function. In this case, the tunnelingin (out) rate Γin(out) is expected to increase in steps proportional to thenumber of available micro-states24–27:ΓoutðεÞ= Γ 1� f ðε� εgÞh ið1ÞΓinðεÞ= Γ f ðε� εgÞ+Xif ðε� εg � ΔεiÞ" #ð2Þwhere Γ is the tunnel rate intrinsic to the barrier configuration and theorbital’s spatial extent, f is the Fermi-Dirac distribution, εg is theground state energy, and Δεi = εi − εg is the energy difference betweenthe excited state i and the ground state.ecda0gΓ in/out (Hz)VP (V)ΔSO/α13.0813.130 1 2 30200I det (pA)V P (V)t (s)100 1-P in/out(t)10.010.10 0.2t (s)0.4502513.05 13.07 13.09 13.11 13.13ΔVPor200 nmVPVLVRVsTVsCVdPVdLVdRVsBVbiasIdet9.5kHzΓinΓoutΔSOgraphite backgatebottom hBNBernal BLGtop hBNAu split gateAlOxAu top gateedge contactfbEFig. 1 | Time-resolved charge detection and tunneling spectroscopy in a bilayergraphene quantum dot. a 3D schematic of the stack b False-colored atomic forcemicroscopy image of the circuit and schematic of the measurement apparatus: aglobal backgate p-dopes the BLG where metallic top gates (in red) are patternedand polarized to form two conductive channels (dark areas). A second layer of topgates (in blue) enables the formation of a quantum dot in each channel, theirlocation is indicated by white circles. The top left one is the system-dot, while thebottom right one is used as a detector. c The plunger gate voltage VP(t) drive andd the simultaneously acquired detector current IdetðtÞ [marked in (b)]. The red(blue) shaded areas correspond to individual tunneling-in(-out) times. e Plot of theprobability of a single hole to have already tunneled into (out) of the dot as afunction of the waiting time, as red (blue) data points. The continuous linesrepresent the exponential probability expected from the average waiting times.f Schematic of the spin-valley locked states constituting the Kramers pairs, sepa-rated in energy by the spin-orbit coupling. The sketches depict the valley and spinmagneticmomenta.gAverage tunneling-in(-out) rates in red (blue) as a function ofthe plunger gate voltage VP. The two vertical black lines delimit the extent of theΓin = 2Γ plateau. The two vertical dashed gray lines indicate the two values of VP atwhich the data of (c–e) were acquired. The insets are schematics representing thedifferent tunnelingpossibilities in andout of a dotwith two levels that are each two-fold degenerate.Article https://doi.org/10.1038/s41467-024-54121-4Nature Communications |         (2024) 15:9717 2www.nature.com/naturecommunicationsIt was previously shown that BLG quantum dots can be tunedsuch that a single orbital is populated at a time16,17. Here, we takeadvantage of this tunability to determine the number of under-lying spin and valley micro-states, by measuring the tunnel ratesof the first charge carrier into and out of a single-carrier BLGquantum dot using a charge detector. Resolving the energydependence of the rates then gives access to the spectrum and itsmagnetic field dependence.ResultsDeviceThe device is based on a stack of two-dimensional materials as illu-strated on Fig. 1a,whichwas assembled using the standarddry transfertechnique28–30. It consists of a 35 nm thick top hBN layer, the BernalBLG sheet, a 28 nm thick bottom hBN, and a graphite backgate layer.The BLG was subsequently contacted with 1D edge contacts28. Threemetallic top gates (3 nm Cr, 20 nm Au) were evaporated on top of thestack to define two conducting channels. Most notable is the centralsplit gate (labeled sC in Fig. 1b), which has a nominalwidth of 140 nmatthe narrowest point. An additional set of top gates (3 nmCr, 20 nmAu)was also deposited, on top of a 26nm thick insulating layer of alumi-num oxide deposited by atomic layer deposition. For this experiment,the backgate is polarized at −7.25 V and the split gates (labeled sT, sC,and sB in Fig. 1b) at around 8.4 V, yielding a displacement field of 0.9V/nm (assuming a relative dielectric constant of 3.5 for the hBN). At suchdisplacement fields, the bandgap in BLG was previously measured tobe ~100meV31,32. This arrangement of patterned gates enables us toform two capacitively coupled quantum dots, one of which acts as acharge detector for the other33–35.Time-resolved measurement of the excited statesThe tunnel rates were measured by driving the dot’s occupancybetween the two charge states: 0 and 1 hole. The nearby chargedetector (represented by the bottom right white circle in Fig. 1b)enables us to acquire statistics on the random time takenby the chargecarrier to tunnel into (out of) the dot (top left white circle in Fig. 1b).The bias across the detector-dot Vbias = 14 μV was kept small on pur-pose to avoid photo-assisted tunneling36 and heatingwhile the system-dot remained unbiased and its barriers were set tominimize the tunnelrates. A periodic square drive with amplitude ΔVP = 52mV and fre-quency 1 Hz,was applied to the system-dot’s plunger gate (labeledwithVP in Fig. 1b), such that the dot’s most stable state alternated betweensingle hole occupancy and empty. The detector current was channeledthrough an analog low-passfilterwith a 9.5 kHzbandwidth to avoid anyaliasing effect at our recording sampling frequency of 20 kHz. Thedetector current was then digitally filtered with a notch filter centeredat 698Hz to reduce the triboelectric noise originating from vibrationsof the dilution refrigerator’s pulse tube. In addition, a moving medianfilter spanning 15points (corresponding to a0.75ms timewindow)wasutilized to enhance our signal-to-noise ratio, while keeping a sharpresponse of the detector current. An example of such a processeddetector current trace is plotted in Fig. 1d, concurrently with theplunger gate drive in Fig. 1c. Figure 1d illustrates that the detectorcurrent jumps between two levels, centered around 60pA and 180pA,corresponding to the first hole being out of and in the dot, respec-tively. By comparing the gatedrive (Fig. 1c) and thedetector’s response(Fig. 1d), it is clear that the tunnel events are not synchronizedwith thegate drive but rather occur with some delay. The delays for eachindividual event were measured and represented by the blue (red)shaded areas in Fig. 1d. For each configuration, we acquired 82 times6 s long traces, adding up to ~ 450 events, enabling us to obtain areliable averagewaiting time for a charge carrier to tunnel into and outof the dot. It is possible to deduce the tunnel rates from the inverse ofthese average waiting times (see Supplementary Information for howwe account for the finite time measurement window on the averagerates). Importantly, as the sequential tunneling of a charge to or from adot is a Poisson process, the distributions of waiting times are expec-ted to be exponential. More specifically, the probability for the chargecarrier to have tunneled in (out) after time t should followPinðoutÞðtÞ= 1� exp½ΓinðoutÞt�. Therefore, we collected the waiting timesfor each type of event, tunneling in and out. In Fig. 1e, we plot thequantity 1 − nin(out)(t)/Nin(out) which corresponds to 1 − Pin(out)(t), wherenin(out)(t) is the number of in (out) tunneling events that occurredafter time t, andNin(out) is the total number of tunneling in (out) events.The solid lines in Fig. 1e depict the exponential laws obtained from theinverse average waiting times, showing excellent agreement withthe expected exponential decay of 1 − Pin(out).We now turn to the dependence of the measured rates on theplunger gate voltage VP. We applied the previously described schemefor measuring the tunnel rates in addition to varying the plunger gatevoltage, keeping the amplitude ΔVP of the square drive constant. Theobtained average rates are plotted in Fig. 1g, where it can be seen thatthe tunneling out rate (in blue) is constant far away from the transitionas expected from equation (1). This constitutes a calibration of theintrinsic tunnel rate Γ = 12.9Hz. By contrast, there is a step in Γin,separating two plateaus at 2.12Γ = 27.4Hz and 4.02Γ = 51.7 Hz,respectively consistent with two and four accessible micro-stateswithin the corresponding energy windows. The extent of the plateauΓin ≈ 2Γ corresponds to the spin–orbit splitting ΔSO, which lifts thedegeneracy between the two spin-valley locked states of each of theKramers pairs5,6. By multiplying the extent of this plateau by theindependently characterized plunger gate lever arm α = 0.003, weobtain the value of the spin–orbit splitting ΔSO ≈ 61 ± 4μeV. Ouruncertainty results from thermal broadening (seeMethods). This valueis compatible with previously reported values ranging between 40 μeVand 80 μeV5,6,37. In contrast to these standard bias spectroscopy tech-niques, that only gives information on the excited states energy, ourmethod additionally gives a direct measurement of each of the states’degeneracy.Magnetic field dependenceIn the following, we focus on the magnetic field dependence of theexcited state spectrum. The predicted energy separations between theground state ∣K� "� and each of the three excited states ∣K + #�,∣K� #�, and ∣K + "� are given by7Δε1 = ðgv + gsÞμBB?, ð3ÞΔε2 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ΔSO + gsμBB?�2 + ðgsμBBkÞ2q, ð4ÞΔε3 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ΔSO + gvμBB?�2 + ðgsμBBkÞ2q, ð5Þwhere gv is the valley Landé factor, gs = 2 the spin Landé factor, μB theBohr magneton, and B⊥(∥) the magnitude of the perpendicular (paral-lel) magnetic field.To further establish the obtained value of ΔSO at zero magneticfields, we study the parallel magnetic field dependence of the excitedstate spectrum, keeping B⊥ = 0. The full (open) circles in Fig. 2a arethe measured Γin(out) for several values of parallel magnetic fields,where the extent of the plateau Γin = 2Γ broadens as the magneticfield increases. The value of the intrinsic tunnel rate Γ ≈ 12.2 Hz wasfirst characterized by fitting each measured Γin(out)(ε) to equation (1),and averaging the obtained values for Γ. Then, the position of theground state εg was established at each magnetic field (see Supple-mentary information for details on the ground state determination).We then identified the point Δε2 = Δε3 where Γout = 3Γ (see Supple-mentary information) for each magnetic field value and emphasizedthese points on Fig. 2a with red crosses. The energy differencesArticle https://doi.org/10.1038/s41467-024-54121-4Nature Communications |         (2024) 15:9717 3www.nature.com/naturecommunicationsΔε2 = Δε3 are plotted as the blue circles in Fig. 2b. The dashed line is afit to the expected splitting as given by equations (4) and (5), whichsimplify to Δϵ2 =Δϵ3 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2SO + ðgsμBBkÞ2q, in absence of a perpendi-cular magnetic field. Here, the only free parameter is ΔSO ≈ 59 μeV,compatible with the value measured at zero magnetic field. As aconfirmation of the previously determined values of Γ, εg, and ΔSO,we plot the black continuous lines in Fig. 2a that follow equations (1)and (2). The effective electronic temperature T ≈ 46mK entering inthe Fermi-Dirac distribution is separately characterized from theaverage current values, depends on the actual electronic tempera-ture and the dot’s energy level fluctuation (see SupplementaryInformation), and determines the energy resolution of this method.These black lines are not fits to the individual sets of points, but aredirectly the theoretical curves of equations (1) and (2), with allparameters independently determined.Finally, we look at the perpendicular magnetic field depen-dence of the spectrum, for which it is expected that all degen-eracies are lifted. The tunneling rates Γin(out) were measured insteps of 20mT up to 120mT, and are shown as full (open) circlesin Fig. 3a. We used the same procedure as before to determineΔεi, by first characterizing εg and then identifying the pointswhere Γin = {1.5, 2.5, 3.5}Γ, indicated by the red crosses in Fig. 3a.The resulting values of Δεi are plotted as a function of the per-pendicular magnetic field in Fig. 3b. The points are then sepa-rated into three categories, differentiated by shapes and colors inFig. 3b. Each set of points follows a linear dependence on B⊥. Thepoints located at the crossings are attributed to both possiblesets. We subsequently plot the theoretical line for equation (4)that requires no fitting parameter, as a continuous gray line, usingthe previously determined ΔSO ≈ 59 μeV, showing excellentagreement with the gray downward triangles. The two remainingsets correspond to the first and third excited states (equations (3)and (5)), and require only one additional parameter gv, which isadjusted simultaneously for both sets. The red and blue dashedlines are the results of the parameter optimization for equations(3) and (5), giving gv ≈ 12.7.This value of the valley Landé factor is lower than previouslyreported values, which range from 18 to 9013–15,17,38. This discrepancycould be explained by the gv dependence on the displacement fieldstrength15,17,38 as we here operated at a large value of D ≈ 0.9V/nm, orby an unusually elongated shape of the quantum dot15. We then used0 mT170 mT340 mT500 mT680 mT850 mTε - εg  (μeV)150001050051- 001- 05--150-100-50050Γ in/out (Hz)aB∥ (mT)Δε2=Δε3 (μeV)002 0040 600 8006080100bcΔε2=Δε3|K-↓ , |K+↑|K-↑ , |K+↓0ΔSOB∥EFig. 2 | Tunneling rate spectroscopy in presence of an in-plane magnetic field.a Measured tunneling-in(-out) rates for different parallel magnetic fields. Each setof curves is plotted in adistinct color andwith an incremental offset of 3Γ for clarity.The tunneling-in(-out) rates correspond to the full (open) symbols. b Qualitativepicture of the expected single-carrier excited spectrum upon applying an in-planemagnetic field. c Parallel magnetic field dependence of the single-carrier spectrum.The data points are directly obtained from the red crosses of panel (a) whereΔε2 = Δε3, with uncertainty in energy given by the diameter of the circles. Thedashed blue line is the result of a least-square optimization carried out on the datapoints, taking equation (4) as a model and with ΔSO as a unique fit parameter.150001050051- 001- 05-Γ (Hz)-100-500500 mT20 mT40 mT60 mT80 mT100 mT120 mT001060 02 04050100B⊥ (mT)08 021ε - εg  (μeV)Δεi  (μeV)bcΔε3 |K-↓|K+↑|K+↓|K-↑0ΔSOB⊥EΔε2Δε1aFig. 3 | Tunneling rate spectroscopy in presence of an out-of-planemagnetic field. a Measured tunneling-in(-out) rates are plotted in full (open)symbols, for differentmagnetic field values each plotted in a distinct color andwithan incremental offset of 2Γ for clarity. b Qualitative picture of the expected single-carrier excited spectrum upon applying a perpendicular magnetic field.cPerpendicularmagneticfielddependenceof the single-carrier spectrum. The datapoints are directly obtained from the red crosses of panel (a). They aresubsequently separated into three categories, each corresponding to an excitedstate: the blue upward triangles correspond to excited state 1, the gray downwardtriangles to excited state 2, and the red circles to excited state 3. The uncertainty inenergy is given by the size of the symbols. The continuous gray line corresponds toequation (4), with B∥ = 0T, ΔSO = 59μeV and gs = 2. The dashed blue and red linescorrespond to the result of a least square optimization on gv simultaneously per-formed on (3) and (5), to the data set with matching colors.Article https://doi.org/10.1038/s41467-024-54121-4Nature Communications |         (2024) 15:9717 4www.nature.com/naturecommunicationsthis value of gv as well as the previously determined parameters inequations (1) and (2) to plot the continuous black lines on Fig. 3a. Theslight deviations occurring at higher energies could be due to anenergy-dependence of the barrier, which in turn would be magnetic-field dependent, as observed in ref. 27. Noticeably, there is no valleymixing of the states up to our experimental resolution. This attests ofthe absence of crystalline defects in the bilayer graphene, over thespatial extent of the quantum dot’s wavefunction. We thus establish anew upper bound on the states valley-mixing of 4 μeV (limited by theelectronic temperature in our device), five times lower than the pre-viously reported value5.ConclusionIn summary, we measured the spectrum of a single-carrier bilayergraphene quantum dot with an energy resolution of 4 μeV. We used atime-resolved charge detection technique that provides direct accessto the number of states in the system, and therefore to their degen-eracy. This enabled us to resolve all four (ground and excited) states ofa single-carrier quantum dot and their behavior as a function of bothperpendicular and parallel magnetic fields. On the technological side,we demonstrated that the tunneling rates into and out of a bilayergraphene quantum dot could be made as low as ~12Hz even in theabsence of a magnetic field. Finally, our measurement scheme opensthe door to measuring the lifetimes of the various excited states ofdifferent spin and/or valley nature in bilayer graphene quantum dots,close to zero magnetic fields39.MethodsEffective electronic temperatureThe electronic temperature is obtained from the value of the detectorcurrent averaged over the last 0.3(0.05) s of each pulse, for a gatefrequency of fVP = 1(2) Hz, to ensure the dot is as close as possible toequilibrium. Each average current curve is then fitted to a slantedFermi distributionaIe1 + exp αeðVP � V0Þ=kBTe� � + Ioff ð6Þwith a the slope (corresponding to the cross-talk of VP on thedetector), Ie the step height, Te the electronic temperature, V0 thecenter of the transition and Ioff an offset contribution, which are allfitting parameters.Ahistogramof all obtainedTe for all our gathereddata sets used inthe main text are plotted in Fig. 4a. The median value is 27.4mK. Wecan clearly see that the bin close to Te = 0 is the most prominent. Weunderstand this as resulting from sudden jumps of the transition thathave an amplitude on the order of kBTe, over the course of our plungergate sweep. As these artificially reduce themedian, we exclude the firstbin to obtain amore reliable value of themedianofTe ≈ 31.2mK, whichwe then use.To take into account the fast jumps as well as the slow drifts of thetransition in the effective temperature, we also plot a histogram of thetransition centers variation δV0 = V0 − 〈V0(B)〉, with V0 the transitioncenter of an individual sweep and 〈V0(B)〉 the average of the transitioncenters at eachmagnetic field, in Fig. 4b.We can see that the transitionfluctuates according to a Gaussian probability distribution with astandard deviation of σV0 = 0.95mV (red line in Fig. 4b). These fluc-tuations can be attributed to charge noise in the vicinity of the dotwhich affects the dot’s electrochemical potential.To estimate an effective temperature, we consider the slowcharge noise to be independent of the temperature extracted from theFermi fits and we obtain T �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2e + ðeασV0=kBÞ2q� 46mK, which weuse in the main text.Digitization procedureTo quantify the lifetimes, we first digitize the measured detectorcurrent on two levels. In the first step, we fit the histograms of thecurrent values with two Gaussian distributions to obtain the averagecurrent value of each level 〈Iup〉 and hIdni. Second, we define twothresholds Ith-up and Ith�dn such thatIth�up = hIupi � γðhIupi � hIdniÞ=2 ð7ÞIth�dn = hIdni+ γðhIupi � hIdniÞ=2 ð8Þwith γ a parameter that we have to fix. We found that the digitizationwas almost identical whenever 0 ≤ γ ≤0.4 for all of our data. Wetherefore used the value γ = 0.2 for digitizing all of the traces acquiredfor this work. Third, we apply an algorithm that scans each pointsequentially and attributes each point to a category as follows. Anypoint above Ith-up is digitized as a 1, any point below Ith�dn is digitized asa 0, and any point located in between the two thresholds is attributedthe same value as the previous point. (For the edge case where the firstfew points of a trace lie in between the two thresholds, those points areignored). This double threshold method enables us to reliably digitizeour traces that have a signal-to-noise ratio of SNR � ðhIupi �hIdniÞ=½ðσup + σdnÞ=2� � 5:3 (for the data displayed in Fig. 1c of the maintext, which is representative of all our data). Note that this SNR couldbe made larger by using a larger detector bias, but is voluntarily keptrelatively small to avoid photo-assisted tunneling in our system dot.Delay on the gate pulseThe plunger gate voltage VP is controlled by a voltage source that has acharacteristic time of ~ 10−6 s, and through a line that is low-passfiltered with a simple RC circuit, which has a characteristic time of10−5 s. The gate response time is therefore negligible in comparison toour measured lifetimes. It is in fact possible to observe this directly inthe data: in Fig. 1c of the main text, there are very sharp peaks in thedetector current, that are synchronizedwith the gate pulses. Those arethe signature of the high-pass filter behaviour of the system-dot’splunger gate on the detector current, and it can be observed that thistime scale is much smaller than that of the lifetimes.0δV0 (mV)Te (mV)50 100 150-15 -5 0 01 515-10050100abcounts050100150countsFig. 4 | Histograms of the results of the fits of the average detector current toequation (6), for all our gathered data, for two parameters. a Histogram of theTe parameter. The dashed red vertical line indicates the median value Te ≈ 31.2mK(ignoring the first bin close to Te = 0). bHistogram of the δV0 = V0 − 〈V0(B)〉, whereV0 is the fit parameter and 〈V0(B)〉 is the average for all plunger gate sweeps at agiven magnetic field. The continuous red line is a Gaussian distribution with astandard deviation of σV0 = 0.95mV.Article https://doi.org/10.1038/s41467-024-54121-4Nature Communications |         (2024) 15:9717 5www.nature.com/naturecommunicationsDetermination of the lifetimesWehere explain inmore detail howwe extract each individual lifetime.Each period of the plunger gate drive is constituted of twoconstant–voltage half-periods as shown in Fig. 1c of themain text. Eachof these half-periods is cut in 4 different time windows:• the begin edge time window (the first 1.1ms),• the checking time window (the subsequent 1ms)• the measurement time window (remaining of the time up to thelast window)• the end edge time window (the last 1.1ms)The two edge time windows, at the beginning and at the end ofevery half period, are cut out of the analysis to avoid false countingdue to the cross-talk of the plunger gate on the detector current (seeprevious Methods section). The checking time window consists inmonitoring the state of the system-dot and checking that it is in itsexpected initial state and remains so for this small amount of time. Ifit is not, or if it switches during this time, the events occurring duringthe whole half-period are not considered in our statistics. Finally, themeasurement time window starts at a time defined as t0: the begin-ning of our measurement of the lifetimes. The state of the dot ismonitored for this whole time andwe record the time tswitch when thestate of the system-dot switches for the first time during this timewindow. The lifetime is then defined as τ = tswitch − t0. Only the firstswitching event is considered, all other subsequent events occurringin the half-period are ignored. In this way, we circumvent the diffi-culty that arises in the analysis of telegraph noise, for which the stateof the system-dot is not well known at any time. 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Ultra-Long Relaxation of a Kramers Qubit Formedin a Bilayer Graphene Quantum Dot. https://doi.org/10.48550/arXiv.2403.08143 (2024)AcknowledgementsH.D. would like to thank Olivier Maillet, Everton Arrighi, Rebeca Ribeiro-Palau, and Artem Denisov for interesting discussions and suggestions.We also thank Lin Wang for pointing a mistake in the preprint. Weacknowledge financial support by the European Graphene FlagshipCore3 Project, H2020 European Research Council (ERC) Synergy Grantunder Grant Agreement 951541, the European Innovation Council undergrant agreement number 101046231/FantastiCOF, NCCR QSIT (SwissNational Science Foundation, grant number 51NF40-185902). K.W. andT.T. acknowledge support from the JSPS KAKENHI (Grant Numbers21H05233 and 23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.Author contributionsT.T., and K.W. grew the hBN. H.D. fabricated the device with inputs fromM.M. andM.R.. A.O. and H.D. characterized and tuned the dots. S.C. andH.D. acquired and analyzed the data with inputs from T.I. and K.E.. H.D.wrote the manuscript with inputs from S.C., A.O., M.M., M.R., C.A., C.T.,R.G., J.G., W.H., L.G., T.I., and K.E.Competing interestsThe authors have no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-54121-4.Correspondence and requests for materials should be addressed toHadrien Duprez.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-54121-4Nature Communications |         (2024) 15:9717 7https://doi.org/10.48550/arXiv.2403.08143https://doi.org/10.48550/arXiv.2403.08143https://doi.org/10.1038/s41467-024-54121-4http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/naturecommunications Spin-valley locked excited states spectroscopy in a one-particle bilayer graphene quantum dot Results Device Time-resolved measurement of the excited states Magnetic field dependence Conclusion Methods Effective electronic temperature Digitization procedure Delay on the gate pulse Determination of the lifetimes Data availability References Acknowledgements Author contributions Competing interests Additional information