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Rebekka Garreis, Jonas Daniel Gerber, Veronika Stará, Chuyao Tong, Carolin Gold, Marc Röösli, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Klaus Ensslin, Thomas Ihn, Annika Kurzmann

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[Counting statistics of single electron transport in bilayer graphene quantum dots](https://mdr.nims.go.jp/datasets/6f8a870a-f633-47c7-8da5-37cebb8bf1a3)

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Counting statistics of single electron transport in bilayer graphene quantum dotsPHYSICAL REVIEW RESEARCH 5, 013042 (2023)Counting statistics of single electron transport in bilayer graphene quantum dotsRebekka Garreis ,1,* Jonas Daniel Gerber ,1 Veronika Stará ,2 Chuyao Tong ,1 Carolin Gold ,1,3 Marc Röösli,1Kenji Watanabe ,4 Takashi Taniguchi ,4 Klaus Ensslin ,1 Thomas Ihn,1 and Annika Kurzmann1,51Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland2Central European Institute of Technology, Brno University of Technology, 612 00 Brno, Czech Republic3Department of Physics and Astronomy, Columbia University, New York NY 10027, USA4National Institute for Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan52nd Institute of Physics, RWTH Aachen University, Aachen 52074, Germany(Received 14 October 2022; revised 23 December 2022; accepted 4 January 2023; published 24 January 2023)We measure telegraph noise of current fluctuations in an electrostatically defined quantum dot in bilayergraphene by real-time detection of single electron tunneling with a capacitively coupled neighboring quantumdot. Suppression of the second and third cumulant (related to shot noise) in a tunable graphene quantum dotis demonstrated experimentally. With this method we demonstrate the ability to measure very low current andnoise levels. Furthermore, we use this method to investigate the first spin excited state, an essential prerequisiteto measure spin relaxation.DOI: 10.1103/PhysRevResearch.5.013042I. INTRODUCTIONThe physics of low-dimensional quantum systems hasbeen studied in various semiconductors, such as GaAs [1,2],InAs [3,4], and silicon [5–7]. Graphene provides a new andinteresting platform because of its natural two-dimensionalcharacter, its specific band structure, and the additional val-ley degeneracy [8–11]. With rapidly advancing developmentsin the fabrication process of clean and high-quality sam-ples [12,13], electrostatically defined quantum dots have beendemonstrated [14–20]. These works include the investigationof the excited state spectrum [21], the Kondo effect [22], thePauli spin and valley blockade [23], as well as the in situtuning from an electronlike to a holelike quantum dot at a con-stant bulk density [24,25]. Furthermore, capacitively couplingthe investigated quantum dot to a neighboring dot allows forcharge detection [26]. Most recently the spin relaxation timeT1 in such a system was measured using the Elzerman readouttechnique and performing single-shot measurements for spinto charge conversion [27]. Time-resolved charge detection[28] is essential for this technique.Here we demonstrate full control over an electrostaticallydefined quantum dot and its tunnel barriers, which allows forrecording a time trace of single-electron tunneling on and offthe dot, the so-called random telegraph noise enabling us toextract the full counting statistics. It allows us to study allpossible correlations and cumulants of charge transfer [29]as well as the properties reflecting profound aspects of the*garreisr@phys.ethz.chPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.underlying quantum dynamics and physical mechanisms [30].The measurement of such a signal requires high sensitiv-ity to single charging events, sufficient signal-to-noise ratioand a high-bandwidth detector with low back action on thequantum system. We use a capacitively coupled quantum dot(henceforth called the detector) to read out the charge state ofthe quantum dot (empty or occupied with a single electron).While the full counting statistics of transport through an elec-trostatically defined quantum dot in GaAs has been demon-strated before [31–33], here we present a measurement ofthe full counting statistics in a gate-defined bilayer graphenequantum dot. We show experimentally the suppression of thesecond and third cumulants of the distribution of current fluc-tuations when the quantum dot is symmetrically coupled tothe leads. Furthermore, we characterize the spin excited stateof the first electron in the quantum dot. The measurementsdemonstrate the high level of understanding, and control overour graphene devices and pave the way for future experimentsin bilayer graphene such as the measurement of degeneracy ofcharge states via the tunneling rates [34] or entropy [35].II. TELEGRAPH SIGNALAs shown in Fig. 1(a), we define a quantum dot (QD),referred to as the signal dot, between two tunnel barriers (TLand TR) and use a plunger gate (PG) to tune the quantum dotto contain only the first electron. A second quantum dot in theCoulomb blockade regime, referred to as the detector, formsbelow a single finger gate (FG) in the neighboring channel andis employed as a charge detector. The same device has beenused for the spin-relaxation time measurements presented inRef. [27] and its geometry is described in detail in the Sup-plemental Material S1 [36]. The detector is biased with aconstant current of 10 pA, the voltage signal Vdet is measuredwith a detector bandwidth of about 1 kHz and sampled witha rate of 10 kHz. The sample is mounted in a dilution re-2643-1564/2023/5(1)/013042(7) 013042-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-1233-998Xhttps://orcid.org/0000-0002-4164-8765https://orcid.org/0000-0003-4818-4366https://orcid.org/0000-0003-4947-6002https://orcid.org/0000-0003-1700-3462https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0001-7007-6949http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.5.013042&domain=pdf&date_stamp=2023-01-24https://doi.org/10.1103/PhysRevResearch.5.013042https://creativecommons.org/licenses/by/4.0/REBEKKA GARREIS et al. PHYSICAL REVIEW RESEARCH 5, 013042 (2023)FIG. 1. (a) False-color micrograph of the device. The quantumdot (QD) is defined between two tunnel barriers (TL and TR), andits chemical potential is tuned by the plunger gate voltage VPG. Asecond capacitively coupled dot formed underneath the finger gate(FG) is used as a charge detector. (b) Time traces of the voltagedrop in the detector corresponding to charge fluctuations betweenan empty dot and one electron in the dot for three different VPGsand corresponding level schematics. Top: The chemical potentialof the quantum dot is above the chemical potential of source anddrain. Middle: The chemical potentials of the dot, source, and drainare aligned. Bottom: The chemical potential of the quantum dot isbelow the chemical potential of source and drain. (c) Probabilitydensity of the times τin and τout obtained from the time trace shownin the top panel of (b) with a time bin size of 0.013 s. (d) Evolutionof the tunneling rates versus plunger gate detuning from the centerof the resonance �VPG. Fitting a Fermi-Dirac distribution yields anelectron temperature of 52(3) mK.frigerator with a nominal base temperature of 10 mK. Unlessstated otherwise, all data presented in this paper are taken at aperpendicular magnetic field B⊥ = 3.1 T. Here, we expect thesingle electron ground state to be spin and valley polarized.Due to the finite bandwidth of the detector, a finite magneticfield is necessary to achieve low enough tunneling rates fortime resolved charge detection.A charge carrier tunneling in and out of the signal dot actscapacitively on the detector, hence shifting its conductanceresonances [26]. If an operating point on the rising (falling)edge of a detector resonance is chosen, a step up (down) inthe time-dependent detection signal is observed whenever anelectron tunnels off (on) the signal dot. Examples of such timetraces are shown in Fig. 1(b). In any of these time traces, wecan determine the waiting times τin (τout) during which thedot is empty (occupied) between two consecutive tunnelingevents, i.e., the signal is above (below) a predefined threshold(black dashed line). In this device the two levels are wellseparated and we achieve a signal to noise ratio above 6 (seeSupplemental Material S2 [36]).For statistically independent tunneling events, the wait-ing times are exponentially distributed with pin(out)(t ) dt =�in(out) exp(−�in(out)t ) dt , where pin(out) is the probability den-sity that an electron enters (leaves) the QD at time t aftera complementary event, and �in(out) = 1/〈τin(out)〉 is the tun-neling rate of the electron hopping on (off) the quantum dot[37]. Here, 〈τin(out)〉 denotes the statistical mean of the set ofexperimentally determined waiting times.Figure 1(c) shows the measured waiting time distributionsfor the tunneling-in (-out) events for the time trace, a partof which is shown in the top panel of Fig. 1(b). In thismeasurement, zero source-drain bias voltage was applied tothe signal dot, and therefore, the thermodynamic equilibriumstate of the quantum dot was probed. The distribution wasobtained by binning the set of waiting times τin(out) determinedfrom the time trace. It is evident that the experimental datais well described by the exponential distribution. Since themeasurement setup and detection circuit have a finite band-width, all tunnel rates presented in this paper are correctedusing the finite-bandwidth correction introduced by Naamanand Aumentado [38]. In our case, the bandwidth is limited bythe detector resistance and cable and filter capacitances. Sincethe detector resistance depends significantly on the occupationof the dot, the detection bandwidth �det,up(down) is also dif-ferent for tunneling in and out (for details see SupplementalMaterial S3 [36]). The finite bandwidth correction relatesthe true tunnel rates �in(out) with the measured rates �∗in(out)according to�in(out) = �∗in(out)�det,down�det,up ± �∗in�∗out(�det,up−�det,down)�det,down�det,up − �∗in�det,up − �∗out�det,down.(1)For a first characterization of the quantum dot and itstunneling rates, we change the voltage applied on the plungergate (VPG) and thereby tune the occupation probability of thedot. Figure 1(b) shows three exemplary time traces. In thetop panel the dot resides in the unoccupied state most ofthe time, in the middle panel the electrons tunnel in and outwith similar rates, and in the bottom panel the quantum dot isoccupied with one electron most of the time. The schematicsdepict the corresponding level schemes with arrows indicatingthe possible tunneling paths. Dashed lines indicate tunnelingprocesses enabled by thermal activation. Here, we only trackthe charge occupation of the quantum dot, but not the directionof the electron movements. Since tunneling to (from) bothleads is possible, the measured tunneling rates represent thesum of the tunneling rates between the dot and the left and theright lead.Collecting time traces of length T = 90 s, which is muchlonger than the waiting times, for several plunger gate volt-ages, we can map the temperature broadened Fermi-Diracdistribution of the leads. To this end, Fig. 1(d) shows theevolution of the tunneling rates �in(out) versus plunger gatedetuning. While �in is seen to increase with increasing plungergate voltage, �out decreases. The qualitative reason of thisbehavior is the following: If the electrochemical potential ofthe quantum dot, μQD, is energetically above the source anddrain electrochemical potentials μS/D at the most negativeplunger gate voltages, an electron can tunnel out very quickly013042-2COUNTING STATISTICS OF SINGLE ELECTRON … PHYSICAL REVIEW RESEARCH 5, 013042 (2023)due to many available unoccupied states in the leads. At thesame time, �in is much smaller than �out due to the smallnumber of occupied states in the leads above μS/D. In analogy,if μQD is resonant with μS/D, the tunneling rates �in and�out are equal, given that the one-electron state in the dotis nondegenerate. When μQD < μS/D (most positive plungergate voltage), �in > �out.The gate voltage axis can be converted into an energy axis�E = eα�VPG using the lever arm α = 0.113 determinedfrom finite bias measurements similar to the one shown inFig. 4(a). Note that we use e > 0 and therefore �E = μ − E .Fitting �in and �out with temperature broadened Fermi-Diracdistributions �(0)out f (−�E/kBTe ) and �(0)in [1 − f (−�E/kBTe )]yields an electron temperature 52(3) mK, and the tunneling-inand -out rates �(0)in = 0.092(15) kHz and �(0)out = 0.10(1) kHz.We compare the electron temperature obtained from the tun-neling rates with the temperature broadened width of thedetection step in dc measurements and find that they are ingood agreement. We therefore conclude that the detector hasa negligible back action on the quantum dot.III. FULL COUNTING STATISTICSIn order to investigate the full counting statistics of thecurrent through the quantum dot, one needs to be able todistinguish electrons tunneling into (out of) the left or theright lead. This can be achieved by tuning the sample intoa nonequilibrium regime, where a finite bias voltage VSDmuch larger than the thermal broadening is applied to thesignal dot. In this configuration, detecting an electron tun-neling out of the dot can be directly associated with acontribution to the current. We therefore apply a source-drain voltage VSD = 75 µV to the signal dot. Figure 2(a)shows the resulting tunneling-in and -out rates versus theenergy detuning �E of μQD to the midpoint between thesource and drain electrochemical potentials. The tunnel-ing rates follow �in = �L f ((μD − E )/kBTe) + �R f ((μS −E )/kBTe) and �out = �L[1 − f ((μD − E )/kBTe)] + �R[1 −f ((μS − E )/kBTe)], where μS(D) is the chemical potential ofsource (drain) and �L(R) the tunneling rate across the left(right) barrier. The shape of the double Fermi distribution re-flects the bias window of 75 µV, where the central plateau cor-responds to μQD being energetically in between the source anddrain electrochemical potentials. Each step left and right of thecenter corresponds to μQD being aligned with either the elec-trochemical potential of source or drain. In the sequential tun-neling regime, �in(out) is purely dominated by the right (left)tunnel barrier, hence the difference of the two tunneling ratesbetween the Fermi steps, marked with a black double arrow, isrelated to the asymmetry a = (�R − �L)/(�R + �L) betweenthe two barriers. Here �R(L) denotes the mean of the tunnelingrates in the sequential tunneling regime labeled in Fig. 2.To investigate the statistical properties of sequential elec-tron transport through the quantum dot, we analyze time tracesfor energies �E well within the bias window. As shown inthe inset of Fig. 2(a), we divide the time trace into subtracesof length t0 and count the number of events n, i.e., stepsup (marked with arrows) per time period. Evaluating T/t0subtraces yields the histogram shown in Fig. 2(b), i.e., the fullcounting statistics of the current, where we plot n on the hori-FIG. 2. (a) Evolution of the tunneling rates versus energy de-tuning �E with a finite bias applied to the dot. The center plateaucorresponds to the finite bias window. Inset: Time trace correspond-ing to an energy detuning around zero. The time trace is dividedinto subtraces of length t0 and the number of events n, i.e., stepsup (marked with arrows) per time period is counted. (b) Statisticaldistribution of the number n of electrons leaving the quantum dotduring a given time period t0. We extract the first three cumulantsof this distribution. (c) Second and third normalized cumulants ofthe distribution of n as a function of the asymmetry of the tunnelingrates. To improve the statistics, each point at a certain asymmetry isthe mean of all time traces obtained within the bias window for giventunnel barrier voltages.zontal axis and the counts of subtraces with n tunneling eventson the vertical axis. Using this distribution, we determine thefirst three cumulants: C1, the mean 〈n〉, and Ci = 〈(n − 〈n〉)i〉for i = 2, 3. These cumulants are related to the mean currentflowing through the dot (first cumulant), and the shot noise SI(second and higher cumulants), which specifies the statisticalcurrent fluctuations in the quantum dot [32].We repeat the measurement of the full counting statisticsof the tunneling current for different tunneling asymmetries a.At each value of a, we measure the full counting statisticsseveral times. From each of them the statistical cumulantsare determined for a specific value of a. Then their meanvalues and the uncertainty of the mean are computed. Fig-ure 2(c) shows the resulting Fano factors C2C1, as well as theratio C3C1versus the asymmetry a. Compared to a purely Pois-sonian process, the cumulants are expected to be reducedfor a Coulomb-blockaded system, since an electron can onlyenter the dot if the previous one has left, which leads tocorrelations in electron tunneling. It is theoretically expectedthat the normalized cumulants follow C2/C1 = 0.5(1 + a2)and C3/C1 = 0.25(1 + 3a4) [29]. These theoretical expres-sions [dashed lines in Fig. 2(c)] describe the experimentallyobtained data very well, with no fitting parameters involved.Note that most data points lie slightly below the predictedcurve: due to the finite measurement bandwidth, events veryclose to the border between two time periods t0 are associatedwith the wrong time bin.013042-3REBEKKA GARREIS et al. PHYSICAL REVIEW RESEARCH 5, 013042 (2023)FIG. 3. (a) Time trace for a two-level pulse applied to the plungergate. The signal is color coded by the parts where the electrochemicalpotential of the quantum dot is above (olive) and below (green) theelectrochemical potential of source and drain. The pulse sequenceis plotted in blue. (b) Same time trace as shown in (a), but sepa-rated into the two regimes of total �E to be analyzed individually.(c) Tunneling rates versus energy detuning for an unpulsed mea-surement. (d) Tunneling rates for the same gate configuration as in(c), but with an additional square pulse applied to the plunger gate.The resolution for energies where the Fermi-Dirac distribution is atthe higher plateau is increased considerably. The green and olivelines mark the pulsed energy configuration of the trace shown in(a) and (b).IV. EXCITED STATE SPECTROSCOPYIn Fig. 3(c), for detunings above 15 µeV the uncertain-ties of the tunneling-in rates drastically increase. Here, thetunneling-in and -out rates are very asymmetric, hence thenumber of events for filling a mostly filled level becomes verysmall. The investigation of excited states needs a precise mea-surement with high statistics of the tunneling rates over a largedetuning range, hence also for very asymmetric tunneling-inand -out rates. Therefore, we add a two-level pulse to theplunger gate voltage with a frequency νpulse = 2 Hz. At thisfrequency, the waiting time for tunneling in at �E > 0 ismuch smaller than half the duty cycle, whereas the waitingtime for tunneling out is much larger. A similar argumentapplies to the case where �E < 0. The pulse amplitude of22.6 µeV is much larger than the step sizes in dc plunger gatevoltage used for the sweep in Fig. 3(d). In case of μQD belowthe electrochemical potential of source and drain, this shortensthe long waiting time after a tunneling-in event by pulsing thelevel so much up in energy that it is quickly filled. The levelis then pulsed down again to the desired energy. This inducestwo tunnel events for each pulse at two different total plungergate voltages and improves the number of events detectedwithin a given measurement time [39]. This method measuresfast tunneling rates more efficiently, while very slow tunnelingrates can not be measured accurately.In our experiment, the square pulse is combined with the dcoffset VPG via two resistors at room temperature as depictedin Fig. 1(a). Figure 3(a) shows an exemplary resulting timetrace (olive and green) and the applied pulse sequence (blue)switching between two different total plunger gate voltages. Inthe green gate configuration the dot is statistically more likelyto be occupied, vice versa in the olive gate configuration it islikely to be empty. Immediately after the pulse step up (down),the dot occupation stays constant for a finite amount of timeuntil a tunneling-out (-in) event occurs. Similar to the non-pulsed case, this wait time can be converted into a tunnelingrate. Based on the pulse frequency, the trace is rearranged andsplit into two time traces corresponding to two different ab-solute gate voltages, and are evaluated separately [Fig. 3(b)].The advantage of this measurement technique is demonstratedby comparing Figs. 3(c) and 3(d). Since the tunneling events’statistic are improved, the measured �in(out) has a lower uncer-tainty. Furthermore, reliable tunneling rates can be measuredfor a larger range of �E , because the events are no longerlimited by thermal broadening of the leads, but by the pulseamplitude. Note that the lower tunneling rate in the pulsing-induced tunneling regime [e.g., �out around the green line inFig. 3(d)] does not go to zero entirely, but reaches a constantvalue of � = 2νpulse. Additionally, for |�E | < 20 µeV, eachtunneling rate is evaluated from two different pulsed tracesmeasured at two different absolute plunger gate voltages.Charge fluctuations in the sample, resulting in small shifts in�E , yield a small offset between the two data sets. These twoeffects have no impact on the following discussions, since weare mostly interested in the higher tunneling rates.The pulsing technique also allows for dVdet/dVPG measure-ments with a pulse frequency of 205 Hz, which is in the sameorder of magnitude as the tunneling rates. The pulse amplitudeis 290 µV. In this domain, the differential change of the quan-tum dot occupation is measured, if Vdet is demodulated at thepulse frequency. Figure 4(a) shows a finite bias measurementof the first electron in the quantum dot at a perpendicularmagnetic field B⊥ = 2.1 T. For large enough bias, the firstspin excited state appears in the bias window (at the energycorresponding to the sum of zero-field spin-orbit splitting andZeeman splitting [27]), which increases the overall currentthrough the dot, resulting in a peak in the dVdet/dVPG signal.As depicted in the schematics in Fig. 4(b), in the VSD − �VPGregion above this line, the excited state offers an additionalchannel to tunnel onto the quantum dot and the tunneling-inrate is expected to increase. However, due to Coulomb block-ade, only one electron can enter the dot at a time, hence thetunneling rate �out stays constant if the tunneling-out ratesof the ground and excited states are equal. Figures 4(c)–4(d)show the tunneling rates extracted from time traces collectedat the different line cuts marked in Fig. 4(a).In the following, we discuss the tunneling rates of theground (�G) and the excited state (�E) more quantitatively.Suppose only the ground state is within the bias window[the region enclosed by two dotted lines in Figs. 4(c)–4(f)],then only one state participates in tunneling. The tunnelingrates are given by �in = �Gin and �out = �Gout. Once the excitedstate enters the bias window, the tunneling-in rate increasesto �in = �Gin + �Ein. To calculate the tunneling-out rate, theprobabilities for three different alternative processes need tobe considered: the electron entered in the ground state andleaves the quantum dot with rate �Gout; the electron entered inthe excited state and leaves the quantum dot with �Eout; or, theelectron entered the excited state and relaxed into the groundstate with �1 and then leaves the quantum dot with �Gout. This013042-4COUNTING STATISTICS OF SINGLE ELECTRON … PHYSICAL REVIEW RESEARCH 5, 013042 (2023)FIG. 4. (a) Measured dVdet/dVPG at finite bias for the first chargecarrier transition. The spin excited state is marked with an arrow.(b) Schematic energy diagram of the quantum dot including thefirst spin excited state. Possible tunneling and relaxation paths arelabeled. (c)–(f) Tunneling rates versus energy detuning for differentsource drain biases as indicated in (a). As soon as the spin excitedstate enters the bias window, the additional tunneling path yields anincreased tunneling-in rate.yields an overall tunneling-out rate of:�out = 1〈τout〉 = �Gout(�Gin + �Ein)(�Eout + �1)�Ein(�Gout + �1) + �Gin(�Eout + �1) . (2)Hence, as soon as tunneling to the excited state is possible, astep in tunneling-out rate of��out = �Gout − �out = �Gout�Ein(�Gout − �Eout)�Ein(�Gout + �1) + �Gin(�Eout + �1)(3)is expected. The measurements show ��out = 0, hence as-suming �Gout�Ein �= 0 yields �Gout = �Eout. This means that inour experiment the spin-up and spin-down states of theone-electron quantum dot share identical tunneling-out rateswithin our measurement precision. In comparison, similarexperiments in GaAs show different tunneling in rates fordifferent excited states [40] as well as Pauli spin blockade indouble dots [41]. Here, we assume that the relaxation rate inEq. (3) is not much bigger than the tunneling rates. We vali-date this assumption by direct measurements of the relaxationrate �1 = 120 Hz using the Elzerman spin readout technique[42], which has been presented in Ref. [27] for this device andgate configuration.V. CONCLUSIONIn conclusion, we have demonstrated time-resolved mea-surements of an electron occupying and leaving a quantum dotin bilayer graphene. The high sample quality and sensitivityof the measurement technique allow us to detect random tele-graph noise, which we analyze by looking at the full countingstatistics. We also used a square pulse to improve the statisticsin the nonequilibrium gate configuration, which enables theinvestigation of the first spin excited state and is crucial forspin-relaxation measurements as demonstrated in Ref. [27].ACKNOWLEDGMENTSWe are grateful for the technical support by Peter Märkiand Thomas Bähler. We acknowledge financial support bythe European Graphene Flagship, the ERC Synergy GrantQuantropy, the European Union’s Horizon 2020 researchand innovation programme under Grant Agreement No.862660/QUANTUM E LEAPS and NCCR QSIT (Swiss Na-tional Science Foundation, Grant No. 51NF40-185902) andunder the Marie Sklodowska-Curie Grant Agreement No.766025. K.W. and T.T. acknowledge support from the Ele-mental Strategy Initiative conducted by the MEXT, Japan,Grant No. JPMXP0112101001, JSPS KAKENHI Grant No.JP20H00354 and the CREST(JPMJCR15F3), J.S.T. Fur-thermore, the extracted funding should be ERC SynergyQUANTROPY No 951541.[1] M. Ciorga, A. S. Sachrajda, P. Hawrylak, C. 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