Automated tuning of inter-dot tunnel couplings in quantum dot arrays
C. J. van Diepen, P. T. Eendebak, B. T. Buijtendorp, U. Mukhopadhyay, T. Fujita, C. Reichl, W. Wegscheider, L. M. K. Vandersypen
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Automated tuning of inter-dot tunnel couplings in quantum dot arrays
C. J. van Diepen,
1, 2, 3
P. T. Eendebak,
1, 3
B. T. Buijtendorp,
1, 2, 3
U. Mukhopadhyay,
1, 2
T. Fujita,
1, 2
C. Reichl, W. Wegscheider, and L. M. K. Vandersypen
1, 2 QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft,The Netherlands Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 155, 2600 AD Delft,The Netherlands Solid State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland (Dated: 29 March 2018)
Semiconductor quantum dot arrays defined electrostatically in a 2D electron gas provide a scalable platformfor quantum information processing and quantum simulations. For the operation of quantum dot arrays,appropriate voltages need to be applied to the gate electrodes that define the quantum dot potential landscape.Tuning the gate voltages has proven to be a time-consuming task, because of initial electrostatic disorder andcapacitive cross-talk effects. Here, we report on the automated tuning of the inter-dot tunnel coupling ina linear array of gate-defined semiconductor quantum dots. The automation of the tuning of the inter-dottunnel coupling is the next step forward in scalable and efficient control of larger quantum dot arrays. Thiswork greatly reduces the effort of tuning semiconductor quantum dots for quantum information processingand quantum simulation.Electrostatically defined semiconductor quantum dotsare actively studied as a platform for quantumcomputation and quantum simulation.
Control overthe inter-dot tunnel coupling is a key ingredient for bothapplications. Via control over the tunnel coupling wehave control over the exchange coupling, which is vital forrealizing the various proposals for spin-based qubits.
Based on the natural description of semiconductor quan-tum dots in terms of the Fermi-Hubbard model, controlover the tunnel coupling allows for analog simulations toexplore the physics of interacting electrons on a lattice.
An obstacle for the efficient use of semiconductor quan-tum dots are the background charged impurities and vari-ations in the gate patterns, which lead to a disorderedpotential landscape. Initial disorder can be compensatedfor by applying individually adjusted gate voltages. Ad-ditionally, even though gates are designed to specificallycontrol a chemical potential or a tunnel coupling, in prac-tice capacitive coupling induces cross-talk from all gatesto dot chemical potentials and tunnel couplings. Thedisorder and cross-talk increase the complexity of tuningup ever larger dot arrays. The effort of tuning can be re-duced by automation based on image processing. Earlierwork on automation of tuning for semiconductor quan-tum dots has shown that it is possible to automaticallyform double quantum dots with a sensing dot, and tofind the single electron regime in the double dot. Morerecently, these automated tuning routines were used todetermine the initialization, read-out and manipulationpoints for a singlet-triplet qubit. Automated controlover the inter-dot tunnel coupling is an important nextstep forward in control for scaling up the number of spinqubits in semiconductor quantum dots.In this Letter, we present and implement a computer-automated algorithm for the tuning of the inter-dot tun-nel coupling in semiconductor quantum dot arrays. Thealgorithm consists of two parts. Part I determines a vir- tual barrier gate, which corresponds to a linear combi-nation of voltages to apply on multiple gates in order toadjust the tunnel barrier without influencing the chem-ical potentials in the dots. To determine such a virtualbarrier gate we model and fit the capacitive anti-crossingsmeasured in charge stability diagrams. Part II tunes thetunnel coupling using a feed-back loop, which consistsof stepping the virtual barrier gate value and measuringthe tunnel coupling, until the tunnel coupling convergesto the target value. To measure the tunnel coupling weuse two methods. The first method is based on photon-assisted tunneling (PAT), while the second method isbased on the broadening of the inter-dot transition line. We describe the algorithm and demonstrate its powerby automatically tuning the tunnel coupling to a targetvalue for two double dots. We show results for tuningboth to higher and lower tunnel couplings for several dif-ferent initial values, both for a single electron and for twoelectrons on the double dot.The platform used for the demonstration of the algo-rithm is a linear triple quantum dot device. A scanningelectron microscopy image of a device similar to the oneused in our experiment is shown in Fig. 1(a). By ap-plying voltages on gate electrodes on the surface of aGaAs/AlGaAs heterostructure, we shape the potentiallandscape in the two-dimensional electron gas 85 nm be-low the surface. Gates LS and RS are designed to controlthe tunnel couplings to the left and right reservoir, re-spectively. Additionally, plunger gates, P i , are designedto control the chemical potential of dot i , and barriergates, D i , are designed to control the inter-dot tunnelcoupling between dot i and dot i + 1 . The device allowsfor the formation of three quantum dots in a linear con-figuration, which are indicated with three white dashedcircles in the bottom part of Fig. 1(a) and one additionaldot, indicated with the larger white dashed circle in theupper part. We refer to this additional dot as the sensing II measuretunnel coupling a cc e p t predict & adjustvoltagescalculatevirtual gates targettunnel coupling I measure & (cid:1) tanti-crossings n o y e s y e s
500 nm V
RF,SD D (a) (b) RSLS D P P P FIG. 1. (a) A scanning electron microscopy image of a device nominally identical to the one used for the measurements. Thethree smaller dashed circles indicate the positions of the dots in the array. The larger dashed circle indicates the location ofthe sensing dot. Squares indicate Fermi reservoirs, which are connected to ohmic contacts. (b) A flowchart of the automatedtunnel coupling tuning algorithm. The dashed boxes indicate the two parts of the algorithm. dot (SD), because it is operated as a charge sensor, uti-lizing its capacitive coupling to the three other quantumdots. One of the SD contacts is connected via a bias-tee to a resonator circuit, permitting fast read-out of thecharge configuration in the bottom dots, by measuringthe SD conductance with radio-frequency reflectometry.To optimize the sensitivity of the charge sensor, we op-erate the SD half-way on the flank of a Coulomb peak.Automation on the tuning of the sensing dot for read-outwas already shown in Ref. 10. One of the bottom gates, P , is connected to a microwave source, used for PATmeasurements.As starting point for our algorithm, we assume thatthe device is tuned near an inter-dot charge transition.Such a starting point can be obtained from a computer-automated tuning algorithm . We also require a roughestimate of the electron temperature for the modellingof charge transition line widths. For the PAT measure-ments, we calibrated the microwave power such that weonly observe single-photon lines. Part I of the algorithm, see Fig. 1(b), determinesthe virtual plunger and barrier gates by measuring thecross-capacitance matrix (see supplementary materialII), which describes the capacitive couplings from gatesto dot chemical potentials. To determine this matrix wemeasure charge stability diagrams with charge sensingand fit the avoided crossing with a classical model (sup-plementary Fig. 1). The fitting of the anti-crossings isbased on finding the minimum of the sum over all pix-els of the difference between the processed data and atwo-dimensional classical model of the avoided crossing(see supplementary material III). From the fit of the anti-crossing, we obtain the slopes of all five transition lines:four addition lines, where an electron moves between areservoir and a dot, and the inter-dot transition line,where a charge moves from one dot to the other. We fitthe anti-crossing to charge stability diagrams measuredfor any combination of P i , P i +1 and D i over a range of
40 mV around the starting point, to fill in the entries ofthe cross-capacitance matrix. From the inverse of thismatrix we obtain both the virtual barrier, e D i , and thevirtual plungers, e P i and e P i +1 . The effectiveness of thisbasis transformation in voltage-space becomes clear fromthe right angles between addition lines in the charge sta-bility diagram in the 2D-scan of e P i and e P i +1 in Fig. 2(a). The anti-crossing fit also provides the voltages at thecenter position on the inter-dot transition line, indicatedwith the white dot. The white dotted line indicates thedetuning axis, which will be used as a scanning axis inthe second half of the algorithm.Before describing part II of the algorithm let us firstexplain the two methods we use to measure the tunnelcoupling. The first method is PAT, see Fig. 2(b) and (e),which is based on the re-population of states induced bya microwave field. We can observe the re-population us-ing the sensing dot, when the different states correspondto different charge configurations. While varying the fre-quency of the microwave source, we observe resonancepeaks when the frequency is equal to the energy differencebetween two states. By scanning over the detuning axisand finding the resonance peaks we perform microwavespectroscopy to map out (part) of the energy level dia-gram, from which we determine the tunnel coupling. Weobtain the tunnel coupling by using a fitting procedurethat consists of three steps. First we process the dataper microwave frequency, mainly subtracting a smoothedbackground signal taken when the microwave source isoff. Second we find the extrema in this processed signalper microwave frequency and last we fit the curve(s) thatconnects the extrema using a model of the energy leveldiagram. For the PAT measurement with a single elec-tron as shown in Fig. 2(b), we model the system in termsof two levels with energies as shown in Fig. 2(c). Theresonance curve is then described by hf = √ ε + 4 t ,where h is Planck’s constant, f the applied microwavefrequency, t the inter-dot tunnel coupling and ε the de-tuning, which is given by α ( δ e P i − δ e P i +1 ) , with α thelever arm, a conversion factor between voltage and en-ergy scales. If two electrons occupy the two dots at zeromagnetic field, there are three relevant energy levels, twocorresponding to singlet states and the other to threefolddegenerate triplet states, see Fig. 2(f). This level struc-ture results in three possible transitions, with energiesdescribed by hf = ε ± √ ε + 8 t and hf = √ ε + 8 t .In the measurement shown in Fig. 2(e) we only observetwo out of the three transitions. This we explain by ob-serving that the thermal occupation of the lowest excitedstate is negligible. We note that some PAT transitionsinvolve a spin-flip, which is mediated by spin-orbit inter-action and a difference in the Overhauser fields between FIG. 2. In all subfigures, ( N , N ) indicates charge oc-cupation of the left and middle dot, with no dot formedon the right. (a) A double quantum dot charge stabil-ity diagram, showing the processed sensing dot signal asa function of virtual plunger gate voltages. The fitted anti-crossing model is indicated with dashed lines. The detun-ing axis is indicated with the white dotted line and thecenter point on the inter-dot transition line with a whitedot. (b) Photon-assisted tunneling measurement showingthe charge detector signal (background subtracted) as afunction of frequency and inter-dot detuning at the (0,1)to (1,0) transition. The red dashed line is a fit of the form hf = √ ε + 4 t . The detuning lever arm is extracted fromthe slope of the hyperbola in the large detuning limit. (c)The energy level diagram for one-electron occupation. Theeigenenergies are ± √ ε + 4 t . A microwave photon (redwiggly arrow) can induce a transition (and potentially tun-nelling between the dots) when the difference between theenergy levels corresponds to the photon energy (PAT). (d)Excess charge extracted from a fit to the sensing dot sig-nal as a function of ε for different t , measured by scanningover the detuning axis for the single-electron occupation.The model used to fit to the SD signal is V ( ε ) = V + δV Q ( ε ) + h δVδε (cid:12)(cid:12) Q =0 + (cid:16) δVδε (cid:12)(cid:12) Q =1 − δVδε (cid:12)(cid:12) Q =0 (cid:17) Q ( ε ) i ε . Here V is the background signal, δV is a measure of the charge sen-sitivity, Q the excess charge as a fraction of the electroncharge and δVδε the gate-sensor coupling when ε is varied. (e) Photon-assisted tunneling measurement similar to (b)but for the inter-dot transition from (2,0) to (1,1). Coloureddashed lines are fits to the measured data. (f) The energylevel diagram for the two electron transition. Coloured wig-gly arrows indicate microwave photon excitations. The en-ergy levels are given by ε ± √ ε + 8 t for the singlets andare for the degenerate triplets. the two dots. The variation in intensity for differenthorizontal lines in Fig. 2(b) and (e) is caused by thefrequency dependence of the transmission of the high-frequency wiring. One could compensate for this by ad-justing the output power of the microwave source per fre-quency. The blue tails in Fig. 2(e) are caused by sweep-ing gate voltages at a rate which is of the same orderof magnitude as the triplet-singlet relaxation rate. Thiswas confirmed by inverting the sweep direction and ob-serving that the blue tails appear on the other side of thetransition line.The second method to measure the tunnel coupling isbased on the broadening of the inter-dot transition line ,see Fig. 2(d). The broadening reflects a smoothly vary-ing charge distribution when scanning along the detuningaxis, caused by the tunnel coupling via the hybridisa-tion of the relevant states and the temperature throughthe thermal occupation of excited states. For the single-electron case, the average excess charge on the left (right)dot is given by Q = 1 Z X n ( c n e − E n /k B T e ) , (1)with Z the partition function, c n = ∓ ε/E n the prob-ability of finding the excess charge on the left (right) dot for the eigenstate with energy E n and the ther-mal energy k B T e ≈ . µ eV , with T e the effective elec-tron temperature. An analogous expression applies tothe two-electron case, with c n = 0 for the triplets and c n = (1 ± ε/ √ ε + 8 t ) for the hybridized singlets. Thelever arm used for measuring the tunnel coupling fromthe broadening of the inter-dot transition line is obtainedfrom PAT, but could also be measured with Coulomb di-amonds or bias triangles. Based on Eqn. 1 we obtain themodel for the charge sensor response when scanning overthe detuning axis, see the caption of Fig. 2. Here we compare the two methods for extracting thetunnel coupling. An advantage of the method based onthe broadening of the inter-dot transition line is that itis about two orders of magnitude faster than PAT (seeTable I in the supplementary material), because it is ef-fectively a single scan over the detuning axis while PATis a series of scans over the detuning axis for different mi-crowave frequencies. Another difference is in the rangeof tunnel couplings over which the two methods workwell. For PAT the upper limit depends on the maximumfrequency that the microwave source can produce. Weexpect that the lower limit for PAT is determined bycharge noise, resulting in broadening of the PAT peaks.With PAT, we were able to automatically measure tun- t ( μ e V μ one electron (aμ PATinter-dot broad.target t t ( μ e V μ t o electrons (bμ PATinter-dot broad.target t
FIG. 3. Results of the algorithmfor a transition involving (a) single-electron states and (b) two-electronstates. Each panel shows results fordifferent runs of the algorithm in-dicated by different colors. Trian-gles indicate tunnel coupling valuesmeasured with PAT and squares in-dicate measurements based on inter-dot transition line broadening. Solidlines are added as a guide to the eye.The black dashed lines indicate thehigh and low target tunnel couplingvalues. nel coupling values as low as µ eV . The lower limit forthe inter-dot transition broadening method is set by theeffective electron temperature, k B T e , here ≈ . µ eV . The upper limit for this method is that for very largetunnel couplings, the broadening of the inter-dot tran-sition line extends to the boundaries of the charge sta-bility region. In the measurements shown here, we didnot come close to this upper limit, but tunnel couplingsup to
75 GHz ≈ µ eV have been measured with theinter-dot transition line broadening method. . We ob-serve that the two methods are in good correspondencewith one another, i.e. the difference between the two issmaller than 10% of their average value (see supplemen-tary material V). Measurement errors are usually smallerthan the accuracy in target tunnel coupling we are inter-ested in, while potential outliers will typically be causedby unpredictable charge jumps.Now, let us describe part II of the algorithm, seeFig. 1(b), which performs a feedback loop. For each it-eration the virtual barrier gate value is adjusted and thetunnel coupling is measured. Before the first step of thealgorithm we measure the tunnel coupling with PAT. Ifwe are not yet within µ eV , of the target tunnel couplingvalue, we step the virtual barrier gate value with step sizeequal to the maximal step size in the positive direction ifthe tunnel coupling is too low and vice versa. We limitthe barrier gate step size to
20 mV such that the positionof the anti-crossing can again be located automaticallyby fitting the anti-crossing model. For larger step sizes,the position of the anti-crossing becomes harder to pre-dict due to non-linearities. After stepping the virtualbarrier gate we measure the tunnel coupling again usingPAT. Then we have measured the tunnel coupling for twosettings and we determine the next step for the virtualbarrier, by predicting the voltage required to reach thetarget value from an exponential fit to the measuredtunnel couplings and their respective virtual barrier val-ues (we thereby force the exponential to go to zero forvery negative barrier voltages). After the tunnel cou-pling has been measured five times with PAT we also havefive measured lever arm values for different gate voltages.The small differences in lever arm we interpret as causedby small shifts in the dot positions with the gate volt- ages. We predict the lever arm for other voltages usinga linear approximation (see supplementary material V).Using this knowledge of the lever arm, the algorithm canbe sped up for the subsequent iterations by measuring thetunnel coupling from inter-dot transition broadening.Following the procedure described above the algorithmautomatically tunes the inter-dot tunnel coupling to atarget value, within the range of the measurable tunnelcoupling values and the achievable values with our gatedesign and electron occupations. Fig. 3 shows the re-sults of the tuning algorithm for various initial and targettunnel coupling values, indicated with different colours.The target tunnel coupling value are indicated with blackdashed lines. We clearly see that the algorithm finds thegate voltages that bring the tunnel coupling to the tar-get value, stepwise moving closer. In Fig. 3(a) resultsfor the left pair of dots with a single electron are shown,while Fig. 3(b) shows results for an occupation with twoelectrons. We have obtained similar results for the sec-ond pair of neighbouring dots in the triple dot (see sup-plementary material VII). The duration of a run of thealgorithm mainly depends on the difference between theinitial and the final tunnel coupling value, because welimit the maximum step size. The duration typically isin the order of 10 min (see supplementary material VIfor more details).In conclusion, we have shown automation of the tun-ing of the tunnel coupling between adjacent semiconduc-tor quantum dots. Key for this automation were im-age processing methods to automatically fit the shape ofan anti-crossing and to find the shape of the resonancecurve in a PAT measurement. The present methods formeasuring inter-dot tunnel couplings and the feedbackroutine can be extended to larger quantum dot arrays.This work demonstrates further automated control oversemiconductor quantum dots and is the next step for-ward in automated tuning of larger quantum dot arrays,necessary for scaling up the number of spin-based qubitsimplemented with semiconductor quantum dots.The authors acknowledge useful discussions withT. Hensgens, J. P. Dehollain and other members ofthe Vandersypen group, experimental assistance byC. A. Volk and A. M. J. Zwerver, and technical sup-port by M. Ammerlaan, J. Haanstra, S. Visser andR. Roeleveld. This work was supported by the Nether-lands Organization for Scientific Research (NWO Vici),and the Dutch Ministry of Economic Affairs through theallowance for Top Consortia for Knowledge and Innova-tion (TKI) and the Swiss National Science Foundation. D. Loss and D. P. Divincenzo, Phys. Rev. A , 120 (1998). R. Hanson, L. P. Kouwenhoven, J. R. 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I. SOFTWARE AND ALGORITHMS
The software was developed using Python , SciPy andthe QCoDeS framework. The image processing is per-formed in pixel coordinates. The parameters of algo-rithms are given in physical units such as mV. The corre-sponding parameter in pixel units is then determined bytranslating the value using the scan parameters. By spec-ifying the parameters in physical units the algorithmsremain valid also if scans are made with a different reso-lution. Of course making scans with a different resolutioncan lead to differences in rounding of numbers leading toslightly different results. II. VIRTUAL GATES
Due to the capacitive coupling from gates to dot chem-ical potentials and tunnel barriers, changing the voltageapplied on one of the gates influences not only one butall of the chemical potentials and tunnel barriers in thepotential landscape. To compensate for the cross-talkfrom gates to chemical potentials we make use of a cross-capacitance matrix. The entries of this matrix corre-spond to the coupling strengths. The columns in theinverse of the cross-capacitance matrix contain the co-efficients for the gate combinations defining the virtualgates. The virtual plungers, ˜ P i , which are linear combi-nations of plungers, P i , control the chemical potential inone dot while leaving the other chemical potentials un-affected. The virtual barrier, ˜ D i , changes the inter-dottunnel coupling without affecting the chemical potentials,hence contains compensation for the effect of the barrier, D , on the dot chemical potentials. An example of ameasured cross-capacitance matrix is δ e P δ e P δ e D = .
00 0 .
49 1 . .
55 0 .
88 1 . .
00 0 .
00 1 . δP δP δD . (2)The upper two rows are scaled such that the top-left entryis one. The left two entries of the bottom row describe theeffect of the plunger gates on the tunnel barrier. Theseentries are set to zero because the PAT and inter-dot linebroadening measurements are performed near an inter-dot transition, hence using these methods we could notindependently measure the effect of plungers on the tun-nel barrier. The last entry of the row for the couplingsto the barrier is set to one as we chose the effect of thephysical barrier on the virtual barrier to be one-to-one. III. AVOIDED CROSSING MODEL
Here, we describe the fitting routine of an avoidedcrossing in a charge stability diagram, see Fig. 4(a). For this fitting we developed a two-dimensional, classi-cal model of an avoided crossing which will be explainedbelow. First we describe the processing of the measureddata.The first step in the processing is calculating a deriva-tive of the image. This is done by applying a first orderGaussian filter. From the convoluted data we subtracta background signal. This background signal is a thirdorder polynomial fit to the data, which was convolutedwith the Gaussian filter. This background subtractionis done to remove the shape of the sensing dot Coulombpeak in the data. Next step is to straighten the measureddata into a square image. This straightening ensures thathorizontal and vertical directions are equally weighted inthe fitting with the 2D model. Then we normalize thesignal with its th percentile. proc . data = (conv . data pix . − bg fit ) p (conv . data pix . − bg fit ) . (3)We developed a classical model of an anti-crossing asobserved in charge stability diagrams. The model con-sists of a two-dimensional patch, see Fig. S4(b). The lineshapes in this model are based on a truncated cosine. Themodel has eight parameters, that need to be fit. Two pa-rameters describe the center of the avoided-crossing, asindicated with a white dot in Fig. 2(a) in the main text,five parameters describe the angles of the four additionlines and the inter-dot transition line, and one parametercorresponds to the length of the inter-dot transition line.Additional to these eight parameters, which are to be fit-ted, there are two more parameters, which we fix beforethe fitting. The first is the typical width of an additionline, which is based on the effective electron temperature.The second parameter is the length of the four line pieceswhich we fit on the addition lines. These are chosen suchthat they are significantly larger than the effective elec-tron temperature and smaller than the addition energy.The anti-crossing is fit by minimizing the following costfunction cost = X pixels [ | proc . data | − model] , (4)which is the sum over all pixel intensities of the processeddata minus the 2D patch of the model. This fitting pro-cedure results in a fit as shown in Fig. 4(c). IV. PHOTON ASSISTED TUNNELLING FITTING
Here we explain the fitting procedure for the photonassisted tunnelling measurements. The PAT fitting pro-cedure consists of three steps: processing the data, de-tecting the detuning values of the resonance peaks permicrowave frequency, see Fig. 5(a), and fitting the curvedescribing the energy difference to the detected peaks.The processing of the data is done per horizontal linein a PAT measurement, i.e. per applied microwave fre-quency. The first step is the subtraction of a background
FIG. 4. (a) A charge stability diagram with an avoided crossing, showing the unprocessed sensor signal. (b) The two-dimensionalpatch generated based on the classical model of the anti-crossing. (c) Processed sensor signal recorded during the charge stabilitymeasurement, from which the model fitted on the anti-crossing is subtracted. signal. We measure the background signal with a scanover the detuning axis while the microwave source is off,note that this is the same scan we would do when we wantto measure the tunnel coupling based on the broadeningof the inter-dot transition line. Before subtracting thebackground signal we smoothen both the signal and thebackground signal with a Gaussian filter with σ set to fivepixels. After the background subtraction we subtract theaverage of the signal and rescale it, resulting in the dataas shown in Fig. 5(b).We detect the resonance peaks as extrema in the pro-cessed signal. Just as for the processing of the data,the peak detection is done per horizontal line. First wefind the maximum and minimum per horizontal line. Weheuristically determined a threshold for the detected ex-trema based on the difference in signal for the two chargeconfigurations and the noise level. We filter the extremaby only accepting the detected peaks which have an ab-solute value higher then the threshold, note here that wealready normalized the processed signal.To the filtered extrema, indicated with red dots inFig. 5(c), we fit the energy transition model, which isdescribed in the main text. This fitting procedure re-sults in fits as shown in Fig. 2(b) and (e) in the maintext. V. COMPARE PAT AND INTER-DOT TRANSITION LINEBROADENING
We compare the tunnel coupling measurements basedon PAT and those based on the inter-dot transition linebroadening to check that they are in agreement with oneanother. We use both methods to measure the tunnelcoupling over a range of tunnel coupling values for whichboth methods are reliable. These measurements weredone in the single electron occupation regime. In Fig. 6we show measured tunnel couplings by both the PAT
Measurement Time
Anti-crossing 5 s / 1 minPAT 1 minPOL 2 sTuning alg. 10 min.TABLE I. The approximate time used per type of measure-ment. The fitting time is included in the shown durations. method and the method based on broadening of the inter-dot line. The lever arm we used for the inter-dot linebroadening measurement is taken from the PAT measure-ment at that virtual barrier gate voltage, see Fig. 6(b).
VI. TIME REQUIRED TO RUN THE ALGORITHM
In Table I an overview is given with the approximatetimes used for the different routines of the tuning al-gorithm and the tuning algorithm itself. The measure-ment time for the anti-crossing measurement is lower ifthe gates on which the voltages are swept are connectedto high-frequent lines, hence can be swept with an a.c.signal, or relatively longer if the voltages can only bechanged by stepping a d.c. voltage. For the triple dotdevice used for the demonstration of the algorithm onlythe plunger gates were connected to high-frequent lines.
VII. ADDITIONAL ALGORITHM RESULTS
In this section we present additional results of the com-puter automated tuning algorithm. Fig. 7 shows theresults of the tuning algorithm on the tunnel couplingbetween the right pair of dots. Again, different coloursindicate results of the algorithm for different initial andtarget tunnel coupling values. Fig. 7(a) shows results forthe pair of dots with a single electron, while Fig. 7(b) shows results for an occupation with two electrons. “Python – Python Programming Language” . “SciPy – Scientific computing tools for Python” . “QCoDeS - Python-based data acquisition framework” . FIG. 5. (a) Line cut of the sensing dot signal as a function of detuning with the microwave frequency, f , at