BBias asymmetric subgap states mimicking Majorana signatures
J. C. Estrada Saldaña , ∗ A. Vekris , , L. Pavešič , , P.Krogstrup , , R. Žitko , , K. Grove-Rasmussen , and J. Nygård Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, 2100 Copenhagen, Denmark Sino-Danish Center for Education and Research (SDC) SDC Building,Yanqihu Campus, University of Chinese Academy of Sciences,380 Huaibeizhuang, Huairou District, 101408 Beijing, China Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia and Microsoft Quantum Materials Lab Copenhagen, Niels Bohr Institute,University of Copenhagen, 2100 Copenhagen, Denmark
Superconducting islands (SIs) on hybrid nanowires are routinely used to search for signatures ofMajorana zero modes (MZMs). Whereas the coupling of a quantum dot (QD) to a large groundedsuperconducting reservoir produces discrete Yu-Shiba-Rusinov subgap states with a perfect positive-negative bias symmetry, it has been recently predicted that the subgap spectrum resulting from thecoupling of a QD to a small electrically isolated SI is in general not symmetric. It is furthermoreaffected by the even-odd parity alternation in the occupation of the SI enforced by Coulomb blockadedue to large charging energy. By performing differential conductance measurements on a QD-SI device and with support of theoretical modelling, we show that the subgap states indeed lacksymmetry of positive and negative bias positions and that, in particular instances, they even lackcontinuity. For odd parity of the SI in the ground state and weak QD-SI hybridisation, we observea bias-polarity dependent current blockade in the stability diagram. At finite magnetic field andfor strong QD-SI hybridisation, zero-bias crossings of the states can mimic signatures of electronteleportation via two well-separated MZMs at the ends of the SI.
I. INTRODUCTION
An ambitious program in modern condensed matterphysics aims at coherent manipulation of excitations in-side the gap of a superconductor for quantum compu-tation and simulation . These states are typicallyelectron-hole (e-h) symmetric in energy, i.e., particle-addition and particle-removal processes have exactly thesame energy cost, a property which is used by contempo-rary transport theories . Majorana , Andreev and Yu-Shiba-Rusinov (YSR) excitations sharethis seemingly fundamental property, which comes as aresult of the e-h "symmetry" in superconductors . Intransport experiments, the e-h symmetry of the subgapexcitations is typically probed as a positive-negative sym-metry in bias position of the associated differential con-ductance peaks.In a concurrent publication , we predict that the pres-ence of a sizeable, realistic Coulomb interaction in the su-perconductor breaks this e-h symmetry. Specifically, thecalculations show that in a quantum dot (QD) coupled toa superconducting island (SI) the subgap excitation en-ergies turn strongly e-h asymmetric when the Coulombcharging energy of the SI, E c , becomes of the order of thesuperconducting gap, ∆ . This happens for any genericgate tuning of the QD and the SI (i.e., away from specialhighly symmetric parameters, see Fig. 1 and Ref. 41).This property sets these experimentally still unexplored Coulombic subgap states aside from their Majorana, An-dreev and YSR close relatives. The e-h asymmetry trans-lates into a positive-negative bias asymmetry if the spec- trum is probed by electron tunneling from one reservoirto the other through the QD-SI, i.e., through a processwhich changes the ground-state (GS) total particle num-ber as N GS → N GS ± → N GS .To satisfy the condition E c ∼ ∆ , an ultrasmall SI mustbe fabricated. To ensure charge quantisation, it mustbe measured at a sufficiently low temperature such that k B T (cid:28) E c , and the tunnel barriers of the SI to the reser-voirs must be of sufficient impedance, Γ source , Γ drain (cid:28) E c , where Γ ’s are the corresponding tunneling rates. Atthe same time a sufficient barrier transparency to theQD is required to produce subgap excitations. The tech-nical challenge therefore resides in the conjugation of thenearly depleted regime in which a QD can be formedwith the large density regime compatible with supercon-ductivity in the SI. In addition, local gate control of theQD and the SI are needed to probe the dispersion of thesubgap excitations.These requirements have been fulfilled by recentadvances in the technology of in-situ deposition ofultra-thin, low-capacitance superconducting Al films onsemiconductor nanowires combined with establishedlithographic techniques. Following this breakthrough,SIs coupled to semiconducting Rashba nanowires werebrought to prominence as possible platforms for topo-logical qubits based on Majorana zero-energy excita-tions . Early on, an ultrasmall hybrid device of thistype with E c ∼ ∆ was found to exhibit even-odd parityalternation attributed to a subgap state of unclear ori-gin . Subsequently, signatures of Majorana zero modes(MZMs) were observed in larger SIs ( E c < ∆ ) in the form a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n FIG. 1. Differences between usual Yu-Shiba-Rusinov (YSR)subgap states and the investigated
Coulombic subgap states .(a,b) Sketches of differential conductance, dI/dV sd , versussource-drain bias voltage, V sd , for (a) YSR and (b) Coulom-bic subgap states (blue peaks). E c is the charging energyof the superconductor. While the amplitudes of the con-ductance peaks associated to both types of states are gen-erally asymmetric in V sd , only the Coulombic subgap statesare also asymmetric in their positive-negative bias positions (away from special gate settings). The edge of the continuum,shaded in cyan, is also generally asymmetric in bias positionin the latter case, in contrast to the former. (c) Sketch ofthe model of the quantum dot (QD) - superconducting island(SI) device. A single QD level is coupled with hybridisationstrength Γ to N levels in the SI. The N levels are paired toeach other, producing the superconducting gap ∆ . N elec-trons (shown as dots) fill N/ levels of the SI. The QD andthe SI have charging energies U and E c , respectively. In theabsence of the QD, it costs an energy E c +∆ to excite a quasi-particle in the SI, which is lowered by the coupling to the QDto produce Coulombic subgap states. (d-f) Calculated chargeparabolas versus δn , the gate-induced electron number n in the SI shifted by N = 200 , for parameters indicated ineach plot and interdot charging energy V = 0 . Thick linescorrespond to Γ = 0 , thin lines to finite Γ . Dashed (continu-ous) arrows indicate excitations for finite Γ at symmetric hole(electron) δn positions. (d) Usual YSR screening at E c = 0 with no dispersion in the charge states. Finite Γ lowers thedoublet → singlet excitations below ∆ . (e) Finite E c resultsin δn -dependent excitations. However, at the electron-hole(e-h) symmetric filling point of the QD, ν = 1 , the electronand hole excitations are still symmetric with respect to thee-h symmetric filling point of the SI, δn = 0 , where ν is thegate-induced charge in the QD. (f) For the same parametersas (e), but ν = 0 . , the electron and hole excitations are nolonger symmetric around δn = 0 . Panels (d-f) do not includecontinuum parabolas. of equal sizes of even and odd parity domains of the SIat a sufficiently large magnetic field, B . In these cases,to probe the hybrid SI in N GS → N GS ± → N GS tun-neling transport, close-by sections of the nanowire were FIG. 2. Quantum dot (QD) - superconducting island (SI) de-vice. (a) False-colored scanning electron micrograph of thedevice, comprising a top-gated InAs nanowire with an AlSI (underneath top gate V s , and therefore not visible) con-tacted by normal leads. (b) Sketch of the electrostatics ofthe device. The capacitances of gates 1, 3 and 5 are notshown. (c) Schematic of tunneling through states of the SI-QD for asymmetric source-drain barriers and negative source-drain bias voltage, V sd . Tunneling events (1) and (2) changethe total particle number in the ground state (GS), N GS , as N GS → N GS + 1 → N GS . The dotted line represents theGS. Solid lines represent the energy-asymmetric SI-QD firstexcited states. Cyan bands indicate the continuum. tuned near charge depletion, which leaves open the pos-sibility of formation of unintentional QDs, as observed ineven apparently ballistic nanowires . YSR excitationsin directly grounded superconducting reservoirs ( E c =0 ) or in effectively grounded mesoscopic su-perconducting reservoirs ( E c (cid:28) k B T , Γ source ) coupled to QDs can produce the zero-bias spectral fea-ture characteristic of MZMs out of trivial YSR/Andreevbound state zero-bias crossings at finite B . Itremains to be seen whether confounding data can alsobe produced by subgap excitations when the supercon-ductor is an island, and even the basic zero-field gatedependence of the excitations is still a mystery .In this work, we use local gating to intentionally de-fine a QD next to an Al SI on an InAs nanowire andwe explore the resulting subgap transport when E c ∼ ∆ at different coupling (hybridisation) strengths Γ betweenthe QD and the SI. Motivated by the use of SIs in thesearch for MZMs, we also investigate the B dependenceof the subgap states and their zero-bias crossings. Us-ing the measured QD-SI parameters as starting input,the measured stability diagram and subgap spectra arefound to be in very good agreement at low B with the re-sults of our model of the system, where the SI is describedas an electron puddle with several hundreds of electrons.The corresponding Hamiltonian includes pairing betweentime-reversed states, constant Coulomb interaction, andcoupling to the QD, which is itself modelled as an An-derson impurity (Fig. 1c).The article is organized in sections. In Sec. II, we sum-marize our main results. In Sec. III, we describe thedevice, its relation to the model, and our experimentalmethods, with additional details provided in AppendixA. In Secs. IV, V and VI, we show our main resultsgrouped according to hybridisation strength, Γ (interme-diate, weak and strong Γ , respectively), with the mainphysical effect observed in each regime indicated in theheadline of each section. Finally, in Sec. VII, we presentour conclusions and the perspectives of the work. II. SUMMARY OF MAIN RESULTS
To facilitate reading, we first summarize the main re-sults of this work:
1. Broken positive-negative bias symmetry.
Thesubgap excitations break positive-negative bias symme-try except for special highly-symmetric tunings. They re-main inversion symmetric with respect to the half-fillingpoint of the QD for even integer filling of the SI. The sub-gap states disperse asymmetrically with respect to theplunger gate voltage of the SI.
2. Triplet Coulombic states.
In the presence ofthe magnetic field, B , the experimental data is repro-duced by the model only if the subgap triplet states arealso taken into consideration. Due to the large Zeemanenergy in this device, the triplet states can decrease inenergy below the singlet Coulombic subgap states beforethe superconducting state is suppressed. The observationof this unusual situation is facilitated by the tunabilityof the SI occupancy.
3. Majorana-mimicking signatures.
In all theregimes that we have explored, the application of B ofsufficient strength results in an approximate one-electron(1e) periodicity of the zero-bias subgap-state crossingsover an extended range of B . In the strong Γ regime,in particular, this 1e periodicity remains pinned in anunexpectedly wide B range.Additionally, the article contains the following inter-esting observations:
4. Current blockade.
For weak QD-SI coupling weobserve current blockade which depends on the polarityof the applied bias voltage. This occurs for odd ground-state occupancy of the SI.
5. Single-dot behaviour.
For strong QD-SI cou-pling we observe a regime where the QD is almost mergedwith the superconductor and shows only vestigial effects.Subgap states remain present in this regime, however.
6. Discontinuous subgap states.
For strong QD-SI
TABLE I. Parameters of the QD-SI device and model pre-sented in Secs. IV, V, and VI. For each section, the top-rowparameters are estimates obtained from measurements (formethods, see Appendix A), while the bottom-row ones areobtained from best fits of the model output to the experi-mental data based on the measured parameters as the initialinput for subsequent fine tuning.Section Γ (meV) U (meV) E c (meV) ∆ (meV) V (meV)IV 0.05 0.8 -1.0 0.19 < . ≈ < . . < . -0.4 0.8 0.4 0.2 0.16 coupling we observe evidence of discontinuous changes inthe subgap state positions when the total electron occu-pancy in the device changes. III. METHODS
We first outline the device, introduce basic terminologyand list device parameters in three regimes of couplingbetween the QD and the SI.Figure 2a shows a false-colored scanning electron mi-crograph of the device. A 110-nm wide InAs nanowireis contacted by 5/200 nm Ti/Au (in yellow) source anddrain leads. Ti/Au top finger gates, of voltages V S and V N , act as respective plunger gates of a 7-nm thick, ≈ V S and V N are always swept from more positive to more negative val-ues. The SI was defined by chemically etching Al awayfrom the upper and lower sections of the nanowire beforecontacting. Top gates 1 and 3 control the tunnel cou-plings of the QD with the drain and the SI, respectively.Top gate 5 tunes the tunnel coupling of the SI with thesource. The top gates are separated from the nanowireand the leads by a 6-nm thick film of HfO . A Si/ SiO substrate backgate was kept at zero voltage throughoutthe experiment.A standard lock-in technique was used to measure thezero-bias and finite-bias differential conductances, G and dI/dV sd , of the sample by biasing the source with an ACexcitation of 5 µ V at a frequency of 223 Hz on top ofa DC source-drain bias voltage, V sd , and recording theresulting AC and DC currents on the grounded drainlead. The measurements were performed in an OxfordTriton dilution refrigerator at 30 mK, under an externalmagnetic field, B . The direction of B , when applied, isshown by an arrow in Fig. 2a.Table I shows device and model parameters for thethree Γ regimes explored. These regimes correspond tointermediate, weak and strong Γ , and are investigated in TABLE II. Effective g -factors of the SI and the QD, | g S | , | g N | ,extracted from the data in Figs. 9, Fig. 15 and Fig. 7, and bare g -factors g and g used as input in finite B calculations.Section | g S | | g N | | g | | g | IV 6.9 2.9 20 5V 7.5 - - -VI 5.6 - 20 5
Sections IV, V and VI, respectively. The meaning of themodel parameters is schematically given in Fig. 1c; formore rigorous definitions see Appendix F. The model wassupplemented by an interdot charging energy V , arisingfrom the interdot capacitance C m , as sketched in Fig. 2b.For ∆ = 0 , the stability diagram of the QD-SI system isidentical to that of the normal-state double QD, and V has the same definition as in that case in terms of thecapacitances of the device . The effect of the gate volt-ages V S and V N is equivalent to that of the gate-inducedcharges n and ν of the SI and the QD in the model.In the following, we will use integer numbers N S , N N tolabel the domains in charge stability diagrams, althoughthe only conserved quantum number is the total chargein both parts of the device, N = N GS (ground state occu-pancy). These labels should not be taken literally, theymainly serve as a guide to the reader. Inside each domainthe partition of N between the QD and the SI charge de-pends on the particular values of V S and V N , hence itvaries from point to point. This has important implica-tions for the definition of the effective g -factors for theQD and the SI, g N and g S , as discussed in later sections.In Tab. II, we collect for all three Γ regimes the experi-mentally measured effective g -factors as well as the bare g -factors that were used in the calculation in order toreproduce the increases in the domain sizes observed inthe experimental stability diagrams.The quantum numbers in our calculations (see Ap-pendix F for details) are the total number of electronsin the system, N , the z-component of the total spin, S z , and the index for states in a given ( N , S z ) sector, i = 0 , , . . . , sometimes known as seniority quantum num-ber. The Coulombic subgap state energies are given by E = E ( N = N GS ± , S z = S z ,GS ± / , i = ) − E ( N = N GS , S z = S z ,GS , i = 0) . The edges of the continuumexcitations are given by E edge = E ( N = N GS ± , S z = S z ,GS ± / , i = ) − E ( N = N GS , S z = S z ,GS , i = 0) .Due to the finite size of the SI, the continuum is in truthonly a quasi-continuum of states. The calculations do notprovide direct results for the differential conductance ofthe system, only information about the ground state andthe low-lying excitations. IV. BIAS-ASYMMETRIC SUBGAP STATES ATINTERMEDIATE QD-SI COUPLING
We first focus on the regime of intermediate Γ wherethe bias asymmetry of the subgap states is most clearlyrevealed because the discrete excitations are located deepinside the gap. Furthermore, in this Γ regime our min-imal model has the greatest predictive power since wefocus on low-energy excitations. Thus, due to universal-ity, the details at higher energy scales make little impact(unlike in the limits of small and large Γ where the sub-gap states are close to the continuum edge and detailsstart to matter). Stability diagram
To guide the discussion of the spectral data and itsrelation to the model, we first introduce the charge sta-bility diagram of the device. Figure 3a shows a colormapof G versus V S and V N , representing a portion of the sta-bility diagram. Conductance lines, which adopt a honey-comb pattern, represent N GS → N GS +1 transitions. Thehexagonal domains at the center of the plot repeat fivetimes within this range, a sign of absence of discerniblelevel structure in the SI ( ∆ E (cid:28) k B T , where ∆ E is thelevel spacing in the SI) . A small but resolvable 1,1singlet domain is seen between the 2,1 and 0,1 doubletdomains. In contrast, the lines to the sides of the cen-tral hexagonal domains, which separate the 0,2 and 2,2domains and the 0,0 and 2,0 domains, show no splittingat this resolution. The difference stems from finite Γ andfinite interdot charging energy (capacitive coupling), V ,which stabilize the singlet 1,1 but not the doublets 1,0and 1,2.Two dashed circles in Fig. 3a encircle a pair of pointsof stronger G which occur when five (big circle, quintuplepoint, FP) or three (small circle, triple point, TP) statesof different N S , N N converge into near degeneracy. Theseare reminiscent of the well-known TPs in double QDs inthe normal and superconducting states . Atthe FP, the states 0,2-1,2-2,2-2,1-1,1 converge, whereas atthe TP, only the states 0,2-0,1-1,1 do. Finite Γ and V are also responsible for increasing the distance betweenthe points of multiple degeneracy, for the acute anglebetween vertical and horizontal conductance lines and,in the case of Γ , for curving the conductance lines.Our model of the system produces a GS stability dia-gram in excellent agreement with the gate position of theconductance lines, as shown in Fig. 3b and in the trans-parent overlay of the calculation on the data in Fig. 3c. Tuning the quantum dot plunger gate
The ratio E c / ∆ = 0 . is important enough to trans-form the usual bias-symmetric position of YSR statesinto positive-negative bias asymmetric Coulombic subgap FIG. 3. Intermediate-coupling regime and broken positive-negative (electron-hole) symmetry of the subgap peak positions. (a)Zero-bias differential conductance, G , versus SI and QD plunger gates, V S and V N . Other gates are set at V G1 = − mV, V G3 = − mV, V G5 = − mV. An unwanted gate glitch, possibly arising from charging and discharging of a charge trapin the dielectric of the device under top gate V N , is indicated by an asterisk. (b) Ground state (GS) crossings (represented asred lines) versus charge induced in the QD, ν , and in the SI, n , calculated for N = 800 using the model parameters indicatedin Tab. I for Sec. IV. The graph is a collage of five identical plots with n ranging from . to . , which are stitchedcopies of the plot highlighted by the rectangle with thick borders. The top and bottom plots are cropped to account for thegate range of the data in (a). (c) Transparent overlay of (b) on (a) showing an excellent agreement between the experimentand the model. For ease of comparison, the color scale of the colormap in (b) was changed to a grey scale. (d) Finite-biasdifferential conductance, dI/dV sd , versus source-drain bias voltage, V sd , and V N , with V N swept along the horizontal blue arrowin (a). The grey and black line cuts are taken at the gate positions indicated by arrows of the same color, i.e. before and afterthe leftmost GS crossing, to highlight the corresponding change in peak amplitude. For ease of qualitative comparison, thetwo traces are set to the same conductance scale (not shown) and are shifted with respect to each other. (e) Calculated GSand first-excitation spectra using the same model and parameters as in (b). In the calculation, n = 800 , and ν is swept alongthe blue arrow in (b). The color of the dots at E/ ∆ = 0 indicates the total charge of the GS, N GS . Black dots correspond tothe lowest-energy excitations with N GS ± : E > to particle-addition transitions, E < to particle-removal transitions. Anestimate for the edge of the continuum is indicated by dashed red and blue lines. (f) Overlay of (e) on (d), after mirroring (e)with respect to E/ ∆ = 0 to account for the V sd drop across the device. In (a) and (d), the color scale has been saturated tohighlight faint conductance lines. states. Figure 3d shows a dI/dV sd colormap represent-ing the gate dispersion of the subgap states as a functionof the occupation in the QD. The subgap states, whichspan V sd ≈ − . → . mV, have broken symmetryin the polarity of V sd , in evident contrast to usual YSRstates . They are inversion symmetricwith respect to the e-h gate-symmetric filling point ofthe QD, which corresponds to the center of the 0,1 sec-tor (indicated by a dot in Fig. 3d), from where removingor adding an electron to the QD are equally energeticallyunfavorable.The usual YSR states in grounded superconductorsshow up as pairs of differential conductance peaks withstrictly symmetric positive-negative bias positions, buttypically with asymmetric amplitude of conductance suchthat the amplitudes (as well as positions) are inversion symmetric around the ν = 1 zero-bias point, forming ei-ther YSR loops or opposite-facing U features in energywith respect to this point . This property of theYSR amplitudes has been attributed to quasiparticle re-laxation from the YSR states to the continuum . Ad-ditionally, the spectral weight of the YSR states changesin jumps at zero energy crossings , reflecting the dis-continuous change in the GS degeneracy; these changesare expected to result in conductance jumps. In our de-vice, the conductance of the subgap states is not uni-form across the spectrum in Fig. 3d. We observe dI/dV sd jumps in magnitude at the , → , crossing (lower tohigher dI/dV sd ) and at the , → , crossing (higher tolower dI/dV sd ). Moreover, the amplitude of the conduc-tance is approximately inversion-symmetric with respectto the QD e-h symmetry point at ν = 1 (dot in Fig. 3d).These observations suggest that similar mechanisms forconductance asymmetries as the ones mentioned abovefor usual YSR states may be in play here, which could beinvestigated in more elaborate conductance calculationsthat are, however, outside of the scope of the presentmodel.Our model reproduces the main subgap features ob-served in the data. The theoretical spectrum (shown inFig. 3e) displays three distinct ground-state total occu-pations, N GS , which depend on ν , as red, yellow andblue dots, while N GS ± excited states are indicated byblack dots. The spectrum needs to be mirrored acrossthe horizontal axis to account for the convention in po-larity in the experiment. The drain lead is grounded,thus for V sd < the chemical potential of the source ispositive, µ source > , and electrons flow from source todrain (see schematics in Fig. 2c). If the voltage dropis entirely at the source contact (i.e., Γ source (cid:28) Γ drain ),then the device is in equilibrium with the drain contactand, for V sd < , the unoccupied states of the spectrumare probed, i.e. the E/ ∆ > part of the spectral functionis observed as a conductance line. This justifies the mir-roring of the spectral plot, which is then overlaid on thedata in Fig. 3f for comparison. From the comparison,we obtain E c + ∆ ≈ . meV. This validates our as-sumption of asymmetric tunnel barriers from the source-drain contacts to the QD-SI. Symmetric voltage drops atthe source and drain tunnel barriers would namely halvethis value, which would then be in disagreement with themeasured E c and the fitted E c / ∆ .In Fig. 3e, we show an estimate for the continuumedges as rugged dashed lines. In the overlay in Fig. 3f,the subgap states (black dots) are only visible as isolatedconductance lines when they are below the continuum;else they may be seen as resonances over a backgroundof conductance. Resonances of the continuum itself ordue to transport through the system can also be presentand may explain the broad intense conductance lines at | V sd | (cid:39) . ; however, they are not included in our model.Additional details on the B , Γ( V G3 ) and Γ source ( V G5 ) dependence of spectral features observed in Fig. 3d aregiven in Appendixes B and C. Tuning the superconducting island plunger gate
Having explored the effect of V N on the subgap states,we now turn our attention to the effect of V S . Figures 4a-eshow five colormaps of dI/dV sd taken with V S swept alongthe directions indicated by vertical arrows in Fig. 3a,i.e. for fixed V N values across a N N = 2 → transi-tion. For every zero-bias crossing of the subgap states inFigs. 4a,e (i.e., away from the N N = 2 → transition)there are two equidistant higher-excitation lines outsidethe range V sd ≈ ± . mV, which are consistent withsingle-electron SI filling above E c + ∆ . These lines arealso equidistant in Figs. 4b-d.Inside the range V sd ≈ ± . mV, from Fig. 4a to FIG. 4. Asymmetric V S dispersion in intermediate-couplingregime. (a-e) Finite-bias differential conductance, dI/dV sd ,versus source-drain bias voltage, V sd , and plunger gate voltageof the SI, V S , at various settings of the plunger gate voltageof the QD, V N , indicated by (a) red, (b) green, (c) purple,(d) yellow and (e) cyan vertical arrows in Fig. 3a. The tailsof the black arrows in panels (a-e) are attached to a subgapstate, while their heads point to the direction of the evolutionof said state with varying V N . The color scale is saturated tohighlight subgap states. Charge numbers of the QD and theSI, N N and N S , are indicated in (a) and (e). An unwantedgate glitch is indicated by an asterisk in (b). (f-j) Calculatedground state and first-excitation spectra using the same modelparameters as in Fig. 3b. In the calculation, ν is fixed to thevalues indicated on top of each plot, and n is swept along the(f) red, (g) green, (h) purple, (i) yellow and (j) cyan verticalarrows in Fig. 3b. An overlay of the calculated spectra in(f-j) has been added to the rightmost GS crossing of (a-e) forcomparison, after mirroring (f-j) with respect to E/ ∆ = 0 toaccount for the V sd drop across the device. Fig. 4e, a positive-bias subgap state, whose motion isshown by an arrow in each panel, detaches from the SIcontinuum in Fig. 4b and crosses in Fig. 4c the center ofthe preexisting 0,2 Coulomb diamond. It then continuesdownwards in Fig. 4d, and ends in Fig. 4e at negative biaswhile registering a parity change from the removal of anelectron in the QD. This strongly positive-negative biasasymmetric behavior comes as a consequence of the inver-sion symmetry in bias and gate voltages of the Coulom-bic subgap states in Fig. 3d. Additionally, we observethat, as the state attached to the arrow goes down inbias position, its conductance changes from high (low) atthe bottom (top) part of the state in Figs. 4a-c, to low(high) at the bottom (top) part of the state in Figs. 4d,e.At the tail of the arrow in Fig. 4c, the amplitude of theconductance of the state registers a jump at zero biassimilar to the jumps seen in Fig. 3d, which is also in thiscase related to a N GS → N GS + 1 crossing. The observedbehavior repeats itself three times in Figs. 4a-e as a con-sequence of the filling of the SI.The data in Figs. 4a-e exhibit some additional nega-tive dI/dV sd features. Their position in V sd is typicallynot positive-negative bias symmetric. Their starting andending V sd positions appear rather arbitrary, spanning awindow | V sd | = 0 . − . mV. Similar features have beeninterpreted earlier as the result of quasiparticle block-ing from different escape tunneling rates in subgap andcontinuum states in the SI , and they are commonin SIs in nanowires . Additional observations con-cerning this negative dI/dV sd feature are gathered fromits B dependence later in this section, and from measure-ments of the stability diagram at finite V sd in AppendixD.We calculate the subgap spectra which correspond tothe data in Figs. 4a-e using the same model parametersas in Fig. 3b (Tab. I), and show the result in Figs. 4f-j.As in Figs. 3d,e the measured and calculated spectra aremirrored across the horizontal axis. Mirrored overlays ofthe calculation added to the rightmost Coulomb diamondof Figs. 4a-e fit well the data. The model thus accountsfor the motion of the subgap state attached to the blackarrow and for the positive-slope, positive- dI/dV sd subgaplines. The main discrepancies are the presence of nega-tive dI/dV sd lines and negative-slope lines in the databelow | V sd | = 0 . , both of which are possibly relatedto transport through the system as mentioned above ,as well as features which occur due to transport throughthe SI continuum. This is beyond our model and couldbe addressed in future more complex modelling. Magnetic field dependence and effective g -factors To obtain information about the spin S z of the sub-gap states, we investigate the B dependence of the sta-bility diagram and of the subgap spectra. The critical B of the superconducting Al film was measured to be B c = 2 . T in nanowire devices made from the same batch of nanowires used in the fabrication of the presentdevice , which leaves ample room for B -resolvedmeasurements in the superconducting state.Figure 5a shows the B = 0 . counterpart of the B = 0 stability diagram of Fig. 3a. We observe thatthe 1,0 and 1,2 sectors open up vertically and becomeresolvable with increasing B . The 1,1 sector also growsin this direction. The FP from Fig. 3a (big dashed circle)splits in Fig. 5a into a QP (inside a dashed circle) and aTP (inside a continuous circle). Some of the other QPsare visibly split into two TPs, such as the one inside thegreen dashed circle.From these measurements, we evaluate the effective g -factors for the SI, g S , and for the QD, g N . To estimatethem, we assume that the splitting of the pair of hori-zontal conductance lines delimiting the vertical growthof the 1,2 sector and the vertical conductance lines de-limiting the horizontal growth of the 0,1 sector, respec-tively, are equal to the Zeeman splittings of the SI andthe QD. This would be exact if the QD and the SI occu-pancies increased in steps of exactly 1 electron across theGS transition lines; for non-zero Γ this is only approxi-mately true due to different (non-integer) charge distri-butions between the QD and the SI on either side of thetransition line, hence these effective g -factors are somenon-trivial function of the true (bare) g -factors which ap-pear in the Hamiltonian. In the following we show thatto reproduce the experimental results, the bare g -factorsneed to be significantly larger than the effective ones, in-dicating that the extraction of g -factors from the domaingrowth is not an accurate technique in the intermediate Γ regime. The true g factors can hence only be obtainedby comparison to the model. Nevertheless, the growth ofdomains directly indicates that both g factors are of thesame sign, and that the SI g -factor is larger than thatof the QD. As detailed in Appendix B, we find effective g -factors | g S | = 6 . and | g N | = 2 . . Given that the g -factor of Al is , the value of | g S | may seem rather largeat a first glance. We note, however, that the g -factor ofbulk InAs is -15. Thus, the effective g -factor in the SI isnot that of pure Al, but rather in between those of bulkInAs and Al, corroborating our picture of hybridisationbetween the InAs nanowire and the Al superconductor inthe SI . Furthermore, we can conclude that both signsneed to be negative. As shown in Table II, the | g S | valuesfor regimes of different Γ are comparable in magnitude.Our calculation of the stability diagram of N GS → N GS + 1 transitions of the system (Fig. 5b) is ableto reproduce the gate positions of conductance lines inthe experimental stability diagram if spin triplet states( S z = ± ) are also taken into consideration. These statesare often not addressed in the studies of subgap excita-tions induced by impurities in superconductors as theyare usually submerged deep inside the continuum. How-ever, they become relevant in the presence of sufficientlylarge magnetic fields due to the large Zeeman energy inour device . For N N = 1 , N S = odd, the lowest tripletexcitation, S z = 1 , can even become the GS. Through FIG. 5. Evidence for the presence of triplet
Coulombic subgap states. (a) Stability diagram from Fig. 3a at finite magneticfield, B = 0 . T. (b) Ground state (GS) crossings (represented as red lines) calculated for N = 200 using the model parametersindicated in Tabs. I and II for Sec. IV. The graph is a collage of five identical plots with n ranging from . to . (highlighted by the rectangle with thick borders) which are stitched together. The top and bottom plots are cropped toaccount for the gate range of the data in (a). (c) Transparent overlay of (b) on (a) showing good agreement between theexperiment and the model. For ease of comparison, the color scale of the colormap in (b) was changed to a grey scale. (d,e)Differential conductance subgap spectra similar to Fig. 4, measured at (d) B = 0 and (e) B = 0 . T with V S swept alongthe direction of the red arrow in (a). Dashed lines indicate the horizontal limits of the 1,0 charge domain. For comparison,calculated spectra shown in panel (f) are overlaid on the data after mirroring the excitations across the horizontal axis, withthe spin S z of visible excitations indicated. (f) Calculated GS and excitation spectra using the same model parameters as in(b) for ν = 0 and n swept along the red arrow in (b). The total number of particles in the GS, N GS , is given by the colorlegend above. S z of the ground and excited states are indicated in each spectrum. (g) Charge parabolas calculated at ν = 1 forZeeman energies of the QD and the SI indicated in each panel. Each parabola is tagged by its S z number. A red bar indicatesthe singlet-triplet exchange splitting. Black bars indicate doublet and triplet Zeeman splittings. The S z = 0 triplet state is notincluded in the calculation. The n range spanned by the calculation is indicated schematically by a bar in panel (a). the corresponding charge parabolas for ν = 1 at zero andfinite B , Fig. 5g shows how this comes to be. For zeroZeeman energies of the QD and the SI (i.e., B = 0 ), itis evident that the triplet states are very close in en-ergy to the singlet excited states; their splitting (redbar) is set by the exchange interaction, J . At finiteZeeman energies, it is instructive to consider the exci-tations at δn = 0 and δn = − . For δn = 0 , the S z = 1 / → S z = 1 excitation becomes the first availableexcitation, while for δn = − , the GS becomes S z = 1 and the S z = 1 → S z = 1 / becomes the first availableexcitation. The excellent agreement with the experimentis highlighted by the transparent overlay of the calcula-tion on the experimental data in Fig. 5c. From the com-parison, the bare g -factors are established as | g S | = 20 and | g N | = 5 , which are on par with experimental and theoretical expectations . Strong renormalization ofthe effective g -factors has also been predicted for E c = 0 ;i.e., in the usual YSR case .The evolution of the subgap spectra against B is con-sistent with our observations regarding the stability di-agram. To probe the triplet S z = 1 state as an excita-tion in the spectra, we turn our attention to spectroscopywhich keeps ν fixed at ν = 0 instead of ν = 1 . Figures5d,e show the measured data at B = 0 and B = 0 . T, at a V N setting which corresponds approximately to ν = 0 in our model ( N N = 0 ), where the effects of Γ and V are small. At B = 0 . T, the 1,0 domain, which wasbarely discernible in the data at B = 0 , is now fully visi-ble due to its growth by Zeeman splitting. Concomitantwith the opening of the 1,0 domain, the state pointedby a black arrow in Fig. 5e goes towards zero bias with B with an effective g -factor equal to g S . In turn, thenegative- dI/dV sd state pointed by a green arrow movestogether with the continuum towards larger bias. Theevolution of the spectra and of the stability diagram in awider range of B is presented in Appendix B.In Fig. 5f, we show calculations at B = 0 and B = 0 . T of the corresponding spectra for ν = 0 , for the sameparameters as the calculation in Fig. 5b. Only spectro-scopically accessible states are shown; i.e., those whichdiffer by ± / in their S z component from the S z com-ponent of the GS. For example, for GS of spin S z = 0 ,the excited state with spin S z = 1 is not shown. A finite B splits the doublet and triplet states, bringing downin energy the triplet S z = 1 excitation which is locatedwithin the SI continuum at B = 0 . A comparison of thecalculation and the experiment, as shown by the overlayin Figs. 5d,e, indicates that the S z = 1 excitation in the1,0 charge state corresponds to the feature identified bya black arrow. Both the S z = 1 and S z = 1 / excitationsin the 1,0 and 0,0 charge states appear as conductanceresonances at the edge of the SI continuum. The S z = 0 excitation in the 1,0 charge state is not visible in thedata. V. CURRENT BLOCKADE AT WEAK QD-SICOUPLING
In sufficiently weakly coupled double QDs, a finite biasvoltage leads to the formation of triangles in the cur-rent stability diagram, which arise as the two QD levelsare detuned from each other within the bias window .Similarly, in the QD-SI system, in the limit of weak Γ ,current triangles appear at finite bias due to the gate-dependent energy difference between the GS and the dis-crete and continuum excitations probed in transport. Inthis section, we report the observation of a bias-polaritydependent disappearance of the current triangles in thefinite-bias stability diagram. SI-QD parameters in thisregime are given in Table I (row labelled as Sec. V).At zero bias, the conductance stability diagram ofFig. 6a shows a considerable opening of the 1,2 sector,which reflects a larger E c / ∆ ratio than in Fig. 3a, pos-sibly due to a variation in the SI capacitance resultingfrom the significant change in V G5 . Our model repro-duces well the gate positions of conductance lines in thestability diagram, as shown in Fig. 6b. For full agree-ment, the theory diagram needs to be rotated by . degrees to account for a small capacitive cross-talk be-tween V N and V S . This was not needed in previous plotsas a fine-tuning of Γ could be done to compensate forthis rotation. To reproduce the presence of a quadruplepoint (QP) in the experimental stability diagram, we set E c ≈ V , which is also consistent with the experimentalvalues of these parameters (see Tab. I).The larger E c / ∆ ratio leads to the splitting of the FPin Fig. 3a into a QP and a TP in Fig. 6a. Three currenttriangles are in principle expected to emerge at finite bias FIG. 6. Parity-dependent current blockade in weak-couplingregime. (a) Zero-bias differential conductance, G , stabilitydiagram in the weak Γ regime. (b) GS crossings (representedas red lines) calculated for N = 200 . The graph is a collage ofthree identical plots with n ranging from to , whichare stitched copies of the plot highlighted by the rectanglewith thick borders. (c,e) Finite-bias current, I , stability dia-grams at two opposite bias polarities, indicated in each plot.Gate settings for (a,c,e): V G1 = − mV, V G3 = − mV,and V G5 = − mV. Note the +69 mV change in V G5 withrespect to Fig. 3a. An asterisk in (c) indicates an unwantedgate glitch. (d,f) Calculated stability diagrams depicting exci-tations within the (d) negative and (f) positive ranges of E/ ∆ indicated in each plot, to be contrasted with experimental di-agrams in (c) and (e). See text for details. Parameters ofthe experimental diagrams and calculations are indicated inTable I, row labeled as Sec. V. (g) Sketch of the odd → even transition in the occupation of the SI, N S , absent in the datain (c,e) for any parity of the occupation of the QD, N N . Thesuppression of this transition (by an unknown mechanism)leads to missing current bands and missing current triangles. out of the two TPs (solid circles) and the QP (open circle)now present . However, as seen in the finite-bias current( I ) diagrams in Figs. 6c,e, one of the three triangles isalways gone; i.e., only two triangles are visible per trioof two TPs and one QP. Occasionally, the base of themissing triangles remains. To guide the reader throughthe observed phenomena, we draw lines on top of the datain Figs. 6a,c,e. At V sd = 0 , in Fig. 6a, we indicate the GSboundaries as continuous lines, intersecting at two TPsflanking a QP. At finite bias, positive and negative, in0Figs. 6c,e, we use continuous lines of the same slope asthe zero-bias GS boundaries to identify the two visibletriangles, and thick dashed lines to identify the absenttriangle. As shown in Appendix D, this observation isrobust over several pairs of triangles, provided that Γ issufficiently weak for the triangles to be reasonably wellresolved.The origin of the missing triangles resides in a blockedtransition . We plot in Figs. 6d,f stability diagramsin which the blue bands depict all possible excitationswithin the given energy range, to be compared with theexperimental diagrams in Figs. 6c,e. As positive (nega-tive) bias corresponds to transport by emptying (filling)the QD-SI, negative (positive) ranges of energies are used.At V sd = ± . mV, the bias window covers ≈ ∓ . /e .It is apparent from the comparison that I ≈ for tran-sitions within red boxes in the calculated diagrams. Wegive an example of this observation in the 1,2-2,2 tran-sition, indicated in Figs. 6c,e by green and red arrows,respectively. At positive bias, the 2,2 → → odd transitions in-volving excitations to the SI continuum. In this case, aband of current is observed in Fig. 6c. However, at neg-ative bias, the 1,2 → → even transitions areinstead probed. The expected current band is absentin Fig. 6e. Transitions which change the parity of theQD but keep the parity of the SI constant are insteadallowed; e.g., the 0,1-0,2 transition produces a currentband at both positive and negative bias. Due to this,the current is not exactly zero inside the missing currenttriangles.In general, current bands are absent in Figs. 6c,e when-ever the GS corresponds to a state with odd occupationon the SI, and the respective discrete or continuum ex-citations involve a parity change of the SI from odd toeven (see Fig. 6g). In the absence of a QD, a singletransport path for elastic cotunneling is available in theodd-to-even parity transition of the SI, as opposed to themultiple paths available in the even-to-odd transition,which leads to a parity-dependent current suppression .To corroborate whether this also holds for the QD-SI sys-tem, future modelling of the current is required. VI. MAJORANA MIRAGE AT STRONG QD-SICOUPLING
Experiments using Majorana devices must avoid con-founding signatures of MZMs coming from unwantedQDs . QDs lurking in the nanowire next to the hybridSI may be easily detected in the weak and intermediate Γ regimes through the presence of an additional set ofconductance lines orthogonal to the lines coming fromfilling the SI. However, in cases where the QD and the SIare so strongly hybridized that they behave effectively asa single dot, signatures of the presence of a lurking QDmay not be as straightforward. In this section, we showthat, in this case, 1) The stability diagram of the QD-SI is indeed very similar to that expected from a SI withoutan adjacent QD. 2) The QD may still show its presencein the form of discontinuities in the subgap spectra, and3) one-electron (1e) periodicity (equal even and odd do-main size at a given B ), compatible with an erroneousinterpretation in terms of electron teleportation medi-ated by well-separated MZMs, can arise in the data .Strikingly, the 1e periodicity extends over a large rangeof B . 4) Though the low- B data is reproduced by ourmodel, the extended B range of 1e periodicity at large B is not, and its origin remains an open research question.Figure 7a shows the stability diagram in this large Γ regime. The strong hybridisation results in seeminglysingle-dot G lines with clear even-odd GS parity alterna-tion. In this diagram, line wiggling is the only indicationof the presence of the QD. Tuning of V G3 in order todecrease Γ unmistakably demonstrates the presence ofthe QD in this regime, as shown in Appendix E. TheZeeman splitting of the conductance lines (which repre-sent GS crossings) against B shown in Fig. 7b confirmsthe even-odd GS parity assigned in Fig. 7a . Fig. 7eshows the output of our model for the GS boundaries ofthe stability diagram. The calculated diagram matcheswell the conductance lines in the experiment.In Fig. 7c, we plot the extracted vertical sizes of theodd ( S o , continuous vertical line in Fig. 7b) and even( S e , dashed vertical line in Fig. 7b) domains as functionof B . From B = 0 to B = 0 . T (green region), the do-main sizes evolve towards equal energy with an effective g -factor | g S | ≈ . . At B ≈ . T, S o ≈ S e , indicating1e periodicity. From B = 0 . T to B = 1 . T (blue re-gion), S o ≈ S e , corresponding to 1e periodicity in this B range. The 0.7 T range of pinning of 1e periodicity cor-responds to an effective Zeeman energy of ≈ . meV.This energy is at least 7 times larger than the full widthat half maximum of the zero-bias crossings in bias volt-age, which is less than 0.04 meV, from which we concludethat the pinning cannot be explained by the finite widthof the zero-bias crossings of the subgap states, in con-trast to e.g. Ref. 28. In turn, an interpretation of thedata in terms of well-separated MZMs is incompatiblewith 1) the short physical length of the SI ( ≈
350 nm ),which has been experimentally shown to result in oscilla-tions around 1e periodicity , 2) the low- B origin of thestates responsible for the zero-bias conductance peaks,which resides on the QD-SI coupling, and 3) the pres-ence of slight oscillations around 1e periodicity in the S e , S o data when the B dependence is acquired after a small V N change in the scale of ν (see Fig. 17 in Appendix Eand the discussion below). The latter would instead becompatible with overlapping MZMs, which contradictsthe statement of well-separated MZMs . We note thata small V N change is not expected to significantly affectthe confinement of MZMs, and the ν scale of the changerather suggests a possible role of the QD in driving theoscillations.While our model fully describes the stability diagramand S o , S e convergence towards 1e-periodicity at low B ,1 -171 -16814 V N (mV) V S ( m V ) a b 0.0 0.4 0.8 1.2B (T)oddeven b
11 14 06 −0.4 V s d ( m V ) d d I/ d V s d ( . / h ) V S (mV) d even0.3 1.20.60.0 B (T)0.0 S o , S e ( m e V ) S o S e c F i g . S o S e (cid:31) n Experiment
210 210 E z / ∆ S o S e S o , S e ( e ) S z = / S z = S z = / S z = gTheorye even0E c + Δ -E c - Δ E n
200 201
TheoryExperiment Experiment Experimenth N GS = 201 202200E z / ∆ f Theory o S e QD SI (N/2 level) S z =1/2 S z =1/2 S z =0 S z =1 iComparison even Theory
010 013 G ( . / h ) FIG. 7. Majorana mirage in the strong-coupling regime and seeming single-dot behaviour. (a) Zero-bias differential conductance, G , versus SI and QD plunger gates, V S and V N , taken with other gates set at V G1 = − mV, V G3 = − mV, V G5 = − mV. Note that V G3 has been changed by -13 mV with respect to Fig. 4a. For reference, a horizontal green bar whose length is ≈ U , the charging energy of the QD, has been placed on the top of the graphic. U was obtained by changing V G3 to obtain aless hybridized QD-SI regime (see Appendix E). (b) G vs. magnetic field, B . (c) B dependence of the spacing between paritycrossings delimiting even ( S e ) and odd ( S o ) parity domains, showing convergence to equal energy at large B and sticking toequal energy over at least a 0.7 T range. The data was extracted from the topmost trio of peaks in the colormap in (b).Error bars correspond to the sum of the full widths at half maximum of the pair of conductance peaks delimiting each of thesedomains. To convert from gate voltage to energy, we use the lever-arm parameter of the SI, α S = 0 . mV/mV obtained inthe intermediate Γ regime, assuming that it stays the same in this regime (for details of its extraction, see Appendix A). (d)Colormap of finite-bias differential conductance, dI/dV sd , versus V sd taken with V S swept along the dashed line in (a). Arrowspoint to subgap-state discontinuities. To avoid cluttering, in (d) the odd-parity GS which occurs between two even GSs is notindicated. The color scale is saturated to highlight subgap states. The calculation in (h) is overlaid on the data after mirroringthe spectra with respect to its horizontal axis. (e) Calculation of the GS stability diagram. To match the experimental stabilitydiagram in (a), the calculated diagram has to be rotated by 3.4 degrees to account for V N , V S capacitive cross-coupling (notshown). (f) Zeeman splittings of GS crossings in (e) for ν = 1 . The Zeeman energy given on the horizontal axis corresponds tothe bare Zeeman energy of the SI, E zS . The Zeeman energy of the QD, E zN , is kept at the ratio of E zN /E zS = 1 / as in Fig. 6,for simplicity, as the g -factor of the QD could not be measured independently in the strong Γ case. (g) Extracted S e and S o from the calculation in (f) for ν = 1 (blue arrow in (e)). (h) Calculation of GS (colored dots) and first excitation spectra (blackdots) for ν = 0 (red arrow in (e)). Parameters of the experiment and calculation are given in Tab. I for Sec. VI. (i) Spin statesformed by an electron occupying the QD and/or the N/ level of the SI. Electron spin is shown as up/down arrows. at large B it predicts a single crossing point of equal S o , S e rather than an extended range of B where S o , S e remain equal, as shown in Fig. 7g for ν = 1 . The transi-tion comes with a change of GS spin for the even statesfrom singlet to triplet. After the initial 1e-periodic B crossing point governed by the modelled physics of thediscrete states, subsequent 1e-periodic B points can oc-cur as the field drives continuum excitations of the SItowards zero bias, as well as higher-spin states not in-cluded in the calculation. The crossing of a continuumexcitation with zero bias may well be responsible for theenhancement of the conductance observed in Fig. 7b inthe B = 0 . − . T range. In principle, it could also pro-duce conductance asymmetries between adjacent peaksas expected for MZMs , as our model predicts thatgenerically the edge of the continuum should reach zeroenergy at different B for adjacent GS crossings, thoughthis was not observed in this device. The finite widthof the conductance peaks may smooth out these various1e-periodic crossings. The observation of slight oscilla-tions with B around 1e periodicity of the S o and S e spac-ings for slightly different V N (shown in Appendix E) sug- gests that ν -tuning can vary the visibility of these variouscrossings in the data. The hypothesis that higher dis-crete/continuum excitations are responsible for extended1e periodicity and conductance enhancement in B in theQD-SI system deserves further investigation in futuremodelling and experimental work, as it can be crucialin discerning these from topological excitations. We alsonote that zero-energy pinning of non-topological subgapstates over extended B ranges has been theoretically ex-plained before by including the spin-orbit interaction ina QD which was coupled to a grounded superconduc-tor . Spin-orbit coupling, known to be present in InAsnanowires, has not been included in the present versionof the model .In Fig. 7d, we show the QD-SI subgap spectra mea-sured with V S swept along the respective dashed line inFig. 7a. The subgap states show discontinuities (indi-cated by arrows) when a new charge is removed from theQD-SI, as evidenced by the alignment of the discontinu-ities to the edge of the Coulomb diamonds correspond-ing to the SI continuum at V sd ≈ . mV. To highlightthis alignment, a vertical dashed line has been added at2 V S = 13 . mV. At the discontinuities, the subgap statesjump by ≈ . mV. This is a rather ubiquituous phe-nomenon, also present in similar linecuts through thestability diagram of Fig. 7a, as shown in Appendix E. Ifthis is indeed a universal feature in the strong Γ regime,it may be used as a tool to discern lurking QDs whichare strongly hybridized to the SI and which may not beapparent in the stability diagram.The corresponding model spectrum is shown in Fig. 7hfor ν = 0 . In order to compare it with the experimentaldata in Fig. 7d, we mirror and overlay it on the measure-ment, as done before. It is found to match reasonablywell the gate positions of the discrete subgap states; how-ever, it fails to reproduce the observed discontinuities. Asthe model is not a transport model and does not accountfor the continuum, it also does not reproduce negative dI/dV sd nor most of the features above | V sd | ≈ . mV.Due to the relation of the discontinuities with the removalof a particle from the QD-SI, the observed discontinuitiesare possibly related to transport through the system (forexample, to the effect of V on transport through the SIcontinuum), which highlights once more the need of morecomplex modelling. VII. CONCLUSIONS
We established the existence of novel Coulombic sub-gap states induced by a quantum dot through its ex-change coupling to a small superconducting island with acharging energy comparable to the superconducting gap, E c ∼ ∆ . These states lack the characteristic positive-negative symmetry in V sd of their Yu-Shiba-Rusinov, An-dreev and Majorana relatives. The subgap states areasymmetric for generic gate tuning. Their only remain-ing symmetry is the inversion symmetry with respect tothe half-filling point of the quantum dot for even integerfilling of the superconductor.When the QD and the SI are strongly hybridized andbehave as a single dot, it is not possible to discern thepresence of the lurking QD from the stability diagramalone. Instead, it is necessary to analyze the subgapstates which exhibit discontinuities against V S . Inter-estingly, the B dependence of the zero-bias subgap statecrossings of the strongly hybridized QD-SI system showsthat they evolve towards 1e periodicity. Moreover, theconductance resonances stay pinned at this periodicityover a Zeeman energy range much larger than the widthof the zero-bias conductance resonances. As ways to ruleout this possible source of confounding results, the signa-tures of the coupling of the QD to the SI shown through-out this article may be useful in experiments aiming atthe observation of MZMs in Rashba nanowires throughthe B dependence of Coulomb peaks of the SI. In ad-dition, the coupling of a QD to a Majorana island asproposed for parity-projection measurement schemes canproduce the undesired effect that excitations from theQD-SI coupling can coexist near zero energy with Majo- rana zero-energy excitations for any Γ strength .Our work opens a number of interesting research ques-tions: 1) What microscopic mechanism is responsiblefor the observed instances of pinning at 1e periodicity(Fig. 7c), recurring 1e periodicity (Fig. 15b), and oscil-lations around 1e periodicity (Fig. 17b) of the zero-biascrossings of the QD-SI subgap states against large B ? 2)Would these effects depend on the size of the SI through achange in the E c / ∆ ratio ? 3) Are the negative dI/dV sd and the negative-slope subgap conductance features, nei-ther of which can be reproduced in our equilibrium cal-culations for the minimal model, simply non-equilibriumtransport properties of the present model or do they re-quire more complex modelling of the system? 4) Thenumerical technique used in this work is most appropri-ate for the regime of intermediate Γ , where its use iskey to correctly determine the positions of subgap statesdeep in the gap, as well as their properties. At verylow Γ the subgap states instead lie close to gap edgesand could be determined perturbatively. At very large Γ , our numerical method becomes computationally ex-pensive due to large bond dimensions required to obtainfull convergence and a change of basis might be appro-priate. 5) What transport mechanisms are possible indevices of this class and how do their contributions tothe total current depend on the relative magnitudes of E c and ∆ ? Is Andreev reflection possible? How is it af-fected by the charging energy and gate-voltage-inducedasymmetries? 6) What is the role of the high-spin exci-tations in the regime of large magnetic field B ? Is theuse of the reduced Hamiltonian for the superconductor(which includes only the pairing interaction between thetime-reversal-conjugate pairs of states) still appropriate,or should additional interaction terms be retained?The existence of a finite E c in the SI may allow its useas an additional (over)screening channel in future experi-ments on S-QD-SI devices, constituting a YSR-analog tothe normal-state two-channel Kondo effect . Whenproperly tuned, this two-channel YSR realization couldprovide a window to unexplored non-Fermi liquid physicsin the superconducting state , perhaps shedding lighton the physics of unconventional superconductors nearquantum critical points . In addition, as a result ofthe two superconducting channels, this system should ex-hibit a finite Josephson energy, E J , and could potentiallyconstitute a fully-tunable two-grain, one-spin toy modelfor granular superconductors of different grain packingand grain size (i.e., different E J and E c ), having magneticimpurities and/or localized spins at grain boundaries,which may provide insights into the superconductor-insulator transition . Though it may prove hard totune the impurity away from the e-h symmetric fillingpoint, the observed states could also be looked for inscanning tunneling microscopy of ultra-small islands oninsulating substrates (e.g., Pb islands on SiOx), using amagnetic impurity attached on the scanning tip to ex-plore regimes of larger U and E c .3 ACKNOWLEDGMENTS
The project received funding from the EuropeanUnion’s Horizon 2020 research and innovation pro-gram under the Marie Sklodowska-Curie grant agreementNo. 832645. We additionally acknowledge financial sup-port from the Carlsberg Foundation, the IndependentResearch Fund Denmark, QuantERA ’SuperTop’ (NN127900), the Danish National Research Foundation, Vil-lum Foundation project No. 25310, and the Sino-DanishCenter. P. K. acknowledges support from Microsoft andthe ERC starting Grant No. 716655 under the Horizon2020 program. L. P. and R. Ž. acknowledge the sup-port from the Slovenian Research Agency (ARRS) underGrant No. P1-0044.
Appendix A: Parameter extraction methods
In this Appendix, we detail the methods which we usedto extract the QD and the SI parameters from the mea-surements. To obtain the charging energy (Hubbard pa-rameter) of the QD, U , we perform Coulomb diamondsspectroscopy for a fixed voltage of the SI plunger gate, V S ,such that the SI is placed in deep Coulomb blockade andonly acts as a cotunneling probe of the QD. The measure-ment, which is performed at finite magnetic field, B , topartially suppress superconductivity, is shown in Fig. 8a.From the length of the vertical arrow, which goes fromzero bias up to the apex of the central 0,1 Coulomb di-amond and which is equal to U/ for asymmetric sourceand drain tunnel barriers, we extract U = 0 . − . meV.From the ratio of U/ to the V N (the plunger gate volt-age of the QD) extension of the diamond, we find thelever-arm parameter of V N , α N = 0 . mV/mV.To obtain the charging energy of the SI, E c , we alsomade use of Coulomb diamonds spectroscopy. The mea-surement, performed at B = 0 , is shown in Fig. 8b. Inthis case, we fix the value of V N , such that the QD isplaced in deep Coulomb blockade and only acts as a co-tunneling probe of the SI, and we sweep V S . Black linesshow the prolongation of the equidistant (in V S ) con-ductance lines at high bias ( V sd outside of ≈ ± . mV)coming from higher excitations towards zero bias suchas to form a Coulomb diamond . The zero-bias cross-ings of this diamond correspond to N GS = N GS + 1 and N GS +2 = N GS +3 , and therefore to 2e spacing. Thus, theheight of the diamond is approximately E c (given by thevertical arrow), from which we find E c ≈ . meV. Thisapproximation ceases to be valid when E c is significantlylarger than ∆ . From the ratio of E c to the V S extensionof the diamond, we find the lever-arm parameter of V S , α S = 0 . mV/mV.The measurements of the hybridisation between theQD and the SI, Γ , and of the interdot charging energy, V , are based on the curvature and spacing of the GS tran-sition lines in the stability diagram, inspired by similarmeasurements in double QDs in the normal state . This -171 0.000.060.12-1680,1 0,20,0 V s d ( m V ) −0.4 d I/ d V s d ( e / h ) B=0.35 TU/210 d I/ d V s d ( e / h ) V s d ( m V ) −0.4 B=0 ab -169 -16812 c B=0 V S ( m V ) V N (mV) G ( e / h ) c -172 -16810 V N (mV)V S (mV)V N (mV) FPTP FIG. 8. Extraction of QD and SI parameters from experimen-tal data. The stability diagram from Fig. 3a has been copiedon the bottom left panel to illustrate with arrows and boxesthe gate-sweep directions/ranges of the data in (a-c). (a) Dif-ferential conductance, dI/dV sd , versus source-drain bias volt-age V sd at B = 0 . T taken with V N swept in the direction ofthe horizontal blue arrow in the stability diagram of the toppanel. (b) dI/dV sd versus V sd at B = 0 taken with V S swept inthe direction of the vertical red arrow in the stability diagramof the top panel. (c) Zoom on the region inside the green boxof the stability diagram on the bottom left panel. The mean-ing of the lines drawn on top of the plots is explained in thetext. extraction is performed in the portion of the stability di-agram magnified around the triple (TP) and quintuple(FP) points shown in Fig. 8c.To measure Γ , we first prolong with dashed lines theapproximately horizontal and vertical conductance linesin the diagram of Fig. 8c. Γ is then given by the length ofthe blue bars, which is the distance between the intersec-tion of the curved conductance lines with the line joiningthe TP and FP, and the intersection of the dashed lines.To convert this length, which is in units of gate voltage,to energy, we use the lever-arm parameters α N and α S .We find Γ ≈ . meV.Second, we assume that the system behaves approxi-mately as a double QD in the normal state in the two-electron charge sector and we equate the length of the redline to
2Γ + √ V . After using the measured lever-armparameters of the gates, we find V = 0 . meV.By using our model to interpret the spectral datain Fig. 3d we obtain ∆ = 0 . meV, which is slightly4smaller than ∆ = 0 . meV of the parent Al super-conductor in devices fabricated out of the same batch ofnanowires , and which may be attributed to weakerhybridisation between the InAs nanowire and the Al su-perconductor , as we reported before .The set of values is in reasonable agreement with themodel parameters which produce the best fit to the sta-bility diagram and the spectra in Figs. 3 and 4.To obtain the QD-SI parameters in the weak andstrong Γ regimes from Secs. V and VI, we assume thatthe lever-arm parameters of the gates found in the in-termediate Γ regime remain the same in these slightlydifferent gate settings. In both regimes, we find E c bymeasuring the distance in gate voltage V S between everyfirst and third peak in the stability diagrams of Figs. 14a(which is a wider-gate-range version of Fig. 6a) and 7a,and averaging it over the number of distances measured.We then convert the gate voltage distance, which is equalto E c /α S , to energy.In the weak Γ regime, U is measured as the horizon-tal V N distance between the midpoints of the verticalconductance lines delimiting the 0,1 charge sector. Themeasured distance, U/α N , is then converted to energy.To determine Γ and V , we employ on the stability dia-gram of Fig. 6a the same method as shown above for theintermediate Γ regime.In the strong Γ regime, determinations of U , Γ and V are difficult due to the nearly single-dot character ofthe stability diagram. To estimate U , we tune the devicequasi-continuously into a less hybridized regime. Thedetails of this method are shown in Appendix E. Appendix B: Additional magnetic field data forintermediate Γ and determination of effective g -factors In this Appendix, we present the method used to ex-tract the effective g -factors of the QD and the SI. Ad-ditionally, we show the stability diagram and spectra inFig. 3 at different B , the N N = 0 spectra in Figs. 5d,e atintermediate and larger B , and we complement the lat-ter with the spectra for N N = 1 at different B . From the B dependence of the stability diagram we extract the B dependence of S o and S e .Figure 9a-e shows the stability diagram of Fig. 3aat different B , complementing the data in Fig. 5a. At B = 0 . T and B = 0 . T, the spacings of the horizon-tal conductance lines which delimit the odd 1,2 and theeven 0,2 sectors, S o (continuous vertical line in Fig. 9d)and S e (dashed vertical line in Fig. 9d), become approx-imately equal, as plotted in Fig. 9f. The vertical size ofthe 1,1 and the 0,1 domains also becomes approximatelyequal, indicating that the 1e periodicity enforced by theZeeman energy occurs irrespective of the QD occupation.Additionally, at B = 0 . T, all GS multi-degeneracies areTPs (each encircled in black), which is consistent with 1echarging of the system.
FIG. 9. Magnetic-field evolution in intermediate-couplingregime. Stability diagram from Fig. 3a for a range of mag-netic fields, B : (a) B = 0 (duplicate of Fig. 3a), (b) B = 0 . T, (c) B = 0 . T (duplicate of Fig. 5a), (d) B = 0 . T, and(e) B = 0 . T. Asterisks denote unwanted glitches in V N . Thecolor scale has been saturated to highlight faint features. Thecolormaps in panels (b-e) share the same color scale. (f) B dependence of the spacing between parity crossings delimitingeven 0,2 ( S e ) and odd 1,2 ( S o ) parity domains, showing con-vergence to equal energy at large B . Error bars correspond tothe sum of the full widths at half maximum of the pair of con-ductance peaks delimiting each of these domains. To convertfrom gate voltage to energy, we use the lever-arm parameterof the superconducting island, α S = 0 . mV/mV. Equal S o and S e is compatible with the 1e period-icity of Coulomb peaks from MZMs in Majorana is-lands (in an otherwise topologically trivial QD-SI system). However, the 0.15 T range over which theequality is maintained corresponds to a Zeeman energyof ≈ . meV, which is comparable to the full width athalf maximum of the zero-bias crossings, 0.04 meV. Theshort B range over which the spacings are equal can thusbe explained by the width of the states, as previouslydone e.g. in Ref. 28. Well-separated MZMs are expectedto produce equal S o and S e over the entire B extensionof the topological phase .If we relate the vertical and horizontal growth (withincreasing B ) of the 0,1 and 1,2 domains to the Zeemansplittings of the SI and the QD, respectively, we can findeffective g -factors for the two. To extract g S , the effective g -factor of the SI, we subtract the V S extension betweenthe vertical black line at B = 0 . T (Fig. 9c) and B = 0 (Fig. 9a). Then, using α S to convert that splitting toZeeman energy, E ZS , we find | g S | = E ZS /µ B B = 6 . .This value is consistent with the slope of S o below B = FIG. 10. Magnetic-field dependence of V S dispersion inintermediate-coupling regime. (a-f) Differential conductance, dI/dV sd , vs. source-drain bias voltage, V sd , and plunger gatevoltage of the SI, V S , at two settings of the plunger gate volt-age of the QD, V N , and at different magnetic field, B . Thesesettings are indicated by (a-d) red and (e-h) light-blue arrowsin the stability diagram on the top panel, which is a duplicatefrom Fig. 3. The color scale has been saturated to highlightfaint features. . T in Fig. 5b. Through a similar procedure, usinginstead the horizontal black lines in Figs. 9c,a and α N ,we find | g N | = E ZN /µ B B = 2 . .In Fig. 10, we complement Figs. 5d,e for N N = 0 byshowing the B dependence in an extended range. Wefurther show the B dependence for N N = 1 . Dashed linesand green arrows have the same meaning as in Figs. 5d,e.A black arrow points to states which come down from thecontinuum in bias position with B . These states are the S z = 1 state in Figs. 10b-d and the S z = 1 / state inFigs. 10f-h. The data shows that both the 1,0 (Figs. 10a-d) and the 1,1 (Figs. 10e-h) domains grow with B . Inboth cases, there is a Coulomb blockaded quasiparticlein the SI. This corroborates the picture gained from thestability diagram, which is that, due to g S > g N , thelarge B behavior is dominated by the increasing distancebetween the GS transitions which come from charging ofthe SI due to a large SI Zeeman energy. At B = 0 . T,the spectra still shows some even-odd alternation, whichis consistent with Fig. 9f, i.e. the 1e periodicity is not yet
FIG. 11. Magnetic-field dependence of V N dispersion inintermediate-coupling regime. (a-d) Evolution in magneticfield, B , of the spectrum in Fig. 3d. Asterisks indicate un-wanted V N glitches. The color scale has been saturated tohighlight faint features. (e,g) Calculations of the spectra forthe same parameters as in Fig. 5b and (e) B = 0 , (g) B = 0 . T. (f,h) Overlay of the spectra on (e,g) on panels (a,b) forcomparison. achieved at such field.In Fig. 11, we show the B dependence of the spectra inFig. 3d. As B increases, two continuous changes occur.1) The onset of the broad conductance line at positivebias (pointed by a black arrow) decreases in bias as B in-creases. The extent of this decrease at B = 0 . T corre-sponds to a Zeeman splitting of . meV, which providesan effective g -factor of 6.9, which is in turn identical tothe independently extracted g S . Therefore, this decreasein bias position is attributed to the Zeeman splitting ofa higher excited state with the g -factor of the SI. Whenthis state enters into the bias window, it is responsiblefor the abrupt change in conductance at the broad peak.2) The apparent anticrossing between the subgap statesat negative bias and a higher excited state, encircled by adashed circle, closes with B . In the next section, we willshow that changing Γ through gating produces a similareffect.In Figs. 11e,g we show calculated spectra at B = 0 and B = 0 . T. In contrast to Fig. 3e, the calcula-tions include the triplet states S z = ± and are done for N = 200 . By overlaying the calculated spectra on theexperimental data in Figs. 11f,h, we find that at B = 0 the apparent anticrossing observed in the data at nega-6tive bias is not a true anticrossing. Instead, this featureis due to an increase in energy of the triplet states withrespect to the singlet state upon a N GS = N GS + 1 tran-sition. As B increases, the distance between the lowestof the triplet states, S z = 1 , and the singlet S z = 0 stateis reduced, closing the apparent anticrossing feature en-circled in the sequence of Figs. 11a-d. At B = 0 , thetriplet states at negative bias lie on top of a broad con-ductance resonance in the continuum. At positive bias,they are not visible below or inside the continuum at any B . However, at B = 0 . T the negative-bias triplet S z = 1 state detaches from this resonance and becomesvisible as a thin conductance line, which then overlapsthe singlet state at B = 0 . T. Appendix C: Gate-controlled changes inhybridisation and source barrier strength
In this Appendix, we explore the effects of changing thevoltages on the top gate between the QD and the SI, V G3 (Fig. 12), and on the top gate between the source contactand the SI, V G5 (Fig. 13). To highlight the changes, weuse copies of Figs. 3a,d in Figs. 12a,c and Figs. 13a,c, andcontrast them to their changed versions in Figs. 12b,dand Figs. 13b,d, respectively.The main effect of a more negative V G3 voltage is tostraighten the conductance lines in the stability diagram, FIG. 12. Tunability of QD-SI hybridisation Γ . (a,c) Copiesof the stability diagram and subgap spectra in Figs. 3a,d putnext to (b,d) similar measurements at a different value of V G3 .Asterisks indicate unwanted gate glitches in the gate voltage.The spectra in (c,d) was taken with the gate swept in thedirection as the blue arrow in (a,b). To indicate that thecharge states swept in (c) are the same as in (a), we namethe charge states in the same way, and assign negative chargevalues to states at more negative V S . The color scale hasbeen saturated to highlight faint conductance lines in (a,b)and faint subgap resonances in (c,d). FIG. 13. Tunability of SI-source lead hybridisation Γ source .(a,c) Copies of the stability diagram and subgap spectra inFigs. 3a,d of the main text put next to (b,d) similar mea-surements at different V G5 , indicated on each pair of plots.Asterisks indicate unwanted gate glitches in the gate voltage.The spectra in (c,d) was taken with the gate swept in thedirection of the blue arrow in (a,b). The color scale has beensaturated to highlight faint conductance lines in (a,b) andfaint subgap peaks in (c,d). a sign of weaker Γ (compare Figs. 12a,b). In addition,the change opens fully the 1,1, -1,0 and -1,2 domainsin the vertical direction, an indication of a larger ratio E c / ∆ . The opening splits the quintuple point (FP) intoone quadruple point (QP) and one triple (TP) point. Inthe corresponding subgap spectra, shown in Figs. 12c,dfor intermediate and weak Γ , respectively, the apparentavoided crossing between the subgap state and the higherexcitation inside of dashed circles is reduced in Fig. 12dwith respect to Fig. 12c. This effect was also observed inthe set of Figs. 11a-c due to the application of B .A change in V G5 , presented in Fig. 13, does not affectthe curvature of the conductance lines in the stabilitydiagram (compare Figs. 13a,b), but still opens up the1,0 and 1,2 domains of odd SI occupation visibly. Asin Fig. 6b, the opening of the 1,0 and 1,2 domains inFig. 13b indicates that the ratio E c / ∆ has effectively in-creased. Thus, small changes in both V G5 and V G3 seemto affect the capacitance of the hybrid SI, which is notunreasonable in view of their proximity to the SI. In ad-dition, the change in V G5 prompts a 10-fold decrease inthe conductance. Consequently, the subgap spectra inFig. 13d is much fainter than the one in 13c. Addition-ally, the subgap spectra exhibits negative dI/dV sd (bluecolor) at negative V sd . Such feature may arise due toblocking of a quasiparticle in the SI before it can en-ter into the QD. This depresses the current, which getsreestablished when the next excited state goes inside thebias window . Along the edge of the negative dI/dV sd region, positive dI/dV sd (red color) discrete states iden-7tified in the overlaid spectral calculation in Fig. 11f withtheir spin as S z = ± and S z = ± / are now barelydiscernible in Fig. 13d due to their contrast with thenegative dI/dV sd continuum background. The splittingbetween the S z = ± and the S z = 0 states, which wasnot visible in the data in Fig. 11f due to the overlap ofthe S z = ± states with the continuum, which renderedthem effectively invisible, is now barely distinguishablein Fig. 13d. Appendix D: Magnetic field and bias dependence ofthe stability diagram in the weak Γ regime In this Appendix, we explore the effect of an applied V sd and B on the differential conductance stability dia-gram of Fig. 6a, corresponding to the weak Γ regime (seeFigs. 14a-c and Fig. 15). In Figs. 14a,d,e, we also providewider gate ranges for the measurements in Figs. 6a-c, toshow that the reported current blockade is robust overseveral triangles, as long as Γ is weak. At V sd = 0 . mV,in Fig. 14c, as odd sectors in the SI occupation verticallyexpand and even sectors contract, the odd-even parity isreflected in the vertical alternation of the sign of dI/dV sd .Domains which are odd (even) in the SI occupation arefilled with blue (white) color, standing for negative (ap-proximately zero) dI/dV sd . At the leftmost TP of every FIG. 14. Weak-coupling stability diagrams in extended volt-age range. (a) Zero-bias, G , (b,c) finite-bias, dI/dV sd differ-ential conductance and (d,e) current, I , stability diagrams atvarious source-drain bias voltages, V sd , indicated in each plot.Details are given in the text. Asterisks indicate unwanted gateglitches. FIG. 15. Magnetic-field evolution in weak-coupling regime.(a) Evolution of the stability diagram in Fig. 6a with B .Dashed lines in the panel at B = 0 indicate GS boundariesnot visible due to low conductance. An asterisk indicates anunwanted gate glitch. (b) B dependence of extracted odd ( S o )and even ( S e ) vertical domain sizes given by vertical continu-ous and dashed lines in (a). Error bars correspond to the sumof the full widths at half maximum of the pair of conductancepeaks delimiting the odd and even domains. The data showsrecurrent 1e periodicity over B = 0 . to 0.8 T. trio of two TPs and one QP, the negative dI/dV sd regionhas an incursion into the dI/dV sd triangles (two examplesof which are pointed by black arrows), which indicatesthat the negative dI/dV sd signal is also dependent on theQD level position. This advocates for its origin being inthe QD-SI capacitive or tunnel couplings.In Figs. 14a,d,e, dashed boxes indicate the gate rangesfrom which the magnifications in Figs. 6a-c were ex-tracted. Black arrows in Fig. 14d point to examples ofcurrent triangles which nearly disappear when the polar-ity of the bias is reversed in Fig. 14e (pointed then by redarrows), and vice versa. In the sections of the stabilitydiagram within the boxes of solid contour, Γ is larger,as deduced from the finite curvature of the conductancelines in Fig. 14a. This results in washed-out current tri-angles and the bias-polarity dependent current blockadeis not as clearly seen.The B dependence of the stability diagram in the weak Γ regime, shown in Fig. 15a, is not significantly differentfrom that in the intermediate Γ regime (Fig. 9). As inFig. 9, the larger g -factor of the SI ( g S > g N ) results inan increase of the vertical size of all odd- N S domains.8Eventually, this leads to equal S o and S e spacings, asshown in Fig. 15b, leading to apparent 1e periodicity ofthe SI . Interestingly, the 1e periodicity is maintainedin three consecutive data points: B = 0 . T, B = 0 . T and B = 0 . T. The 0.3 T of range of B spanned bythese three points corresponds to a Zeeman energy of 0.13meV, which is significantly larger than the full width athalf maximum of the zero-bias conductance peaks (0.04meV), indicating that, in this particular case, the finitewidth of the peaks cannot explain this feature . In con-trast to the large-density scatter plot in Fig. 7c, the low-density of data points in Fig. 15b cannot provide conclu-sive evidence about the possible 1e-periodic pinning; thebehavior could well be similar to the oscillating patternin Fig. 17b (discussed in the following Appendix). How-ever, even within this low-density dataset, the discrep-ancy at large B with our model which excludes higherexcitations than the one with spin S z = 1 is confirmedas the model only predicts a single 1e-periodic point; theobservation of more than one 1e-periodic data point overa finite extension of B is at odds with the model. Appendix E: Additional data in the strong Γ regime In this Appendix we show gate tuning from the strongto an intermediate Γ regime, an additional example of the1e periodicity of the QD-SI at large B , and additionalexamples of subgap state discontinuities in the subgapspectra.In Fig. 7a, we showed an estimate of U by plottinga green bar. This was done as U could not be directlyfound from the stability diagram due to the approximatesingle-dot behavior of the strongly hybridized QD andSI. To obtain U , we reduced the QD-SI hybridisation.Figure 16 shows the evolution of the stability diagramfrom Fig. 7a (which has been copied in Fig. 16a with aless saturated color scale) with V G3 . As V G3 is decreased,the apparent hybridisation between the QD and the SIalso decreases, and an approximate honeycomb stabilitydiagram is recovered in the final diagram, Fig. 16c. Inthis final diagram, U corresponds approximately to thegreen bar which measures the distance between two ofthe conductance lines of less horizontal slope in Fig. 16c.Assuming that α N remains the same in this slightly dif-ferent gate setting, we obtain U = 1 . meV. FIG. 16. Revelation of lurking QD in strong-coupling regime.(a-c) Evolution of the stability diagram in Fig. 7a with V G3 ,from strong to intermediate Γ regimes. FIG. 17. Oscillations around S e ∼ S o in strong-couplingregime. (a) G vs. magnetic field, B , taken at the gate set-tings of Fig. 7a, for V N = − . mV, with V S swept throughthe corresponding vertical dashed line in Fig. 7a. (b) B de-pendence of the spacing between parity crossings delimitingeven ( S e ) and odd 1,2 ( S o ) parity domains, showing conver-gence to equal energy at large B and sticking to approximatelyequal energy over at least a 0.8 T range with small oscillationsaround 1e periodicity. Error bars correspond to the half widthat half maximum of the conductance peaks delimiting thesedomains. The data was taken from the lowest three conduc-tance peaks in (a). In Fig. 7b,c, we had shown that the zero-bias con-ductance peaks evolve towards 1e charging at B = 0 . T, sticking to approximately equal S o , S e spacings until B = 1 . T. In Fig. 17, we show an additional example ofthis phenomenon for a slightly different V N . Comparedto the evolution of the zero-bias crossings of the sub-gap states in Fig. 7b, Fig. 17a shows stronger wigglingof the conductance peaks versus B . The extracted S o , S e spacings, shown in Fig. 17b, converge faster than inFig. 7c towards equal energy, but also show slight oscilla-tions around equal energy, which are particularly visibleat B = 1 . − . T. These oscillations have been inter-preted before as coming from the partial overlapping ofMZMs at the ends of the SI . In the QD-SI system,these could come from Zeeman-driven changes of paritydue to higher excited states crossing zero energy, andmerit further investigation in future research.In Fig. 7d, we previously showed an example of subgap-state discontinuities when a charge was added to the QD-SI system. In Fig. 18, we present additional examples ofthis observation. These are consistent with the first ob-servation and indicate that, for these gate settings, thephenomena of subgap-state discontinuities is rather ubiq-uitous in the measured gate range. Appendix F: Model of the quantum dot -superconducting island system
The QD is described as a single non-degenerate im-purity level, as in the single-impurity Anderson model(SIAM). The SI is described as a set of equidistant en-ergy levels that represent time-reversal-conjugate pairsin the momentum/orbital space, coupled all-to-all by the9
FIG. 18. Subgap state discontinuities in strong-couplingregime. (a-f) Vertical subgap-spectra cuts through the sta-bility diagram in Fig. 7a, which has been copied on the top(same color scale as in Fig. 7a), with vertical lines superposedto indicate the V N values at which the cuts were made. Ar-rows in (a-f) point at examples of subgap-state discontinuities,which repeat along the maps for odd charging of the system.Additional changes in the spectra as V N is stepped are possi-bly due to gate modulation of the QD level. The color scalehas been saturated to highlight faint subgap features. pairing interaction. This step beyond the BCS mean-field approximation is required to accurately describe thestrong even-odd occupancy effects of the SI, which arisefrom its large charging energy E c . The QD is coupled toall levels of the SI via a hybridisation term. The Hamil- tonian used to model the system is H QD = ε QD ˆ n QD + U ˆ n QD , ↑ ˆ n QD , ↓ + E Z, QD ˆ S z, QD = U n QD − ν ) + E Z, QD ˆ S z, QD + const ., (F1) H SC = N (cid:88) i,σ ε σ,i c † i,σ c i,σ − αd (cid:88) i,j c † i, ↑ c † i, ↓ c j, ↓ c j, ↑ + E c (ˆ n SC − n ) , (F2) H hyb = v √ N (cid:88) i,σ (cid:0) c † i,σ d σ + h . c . (cid:1) + V (ˆ n QD − ν )(ˆ n SC − n ) , (F3) H = H QD + H SC + H hyb . (F4)Here d σ and c i,σ are annihilation operators of the QDand the bath, σ = ↑ , ↓ , ˆ n QD ,σ = d † σ d σ are impurity oc-cupancy operators, ˆ n QD = (cid:80) σ ˆ n QD ,σ , and ˆ S z, QD =(ˆ n QD , ↑ − ˆ n QD , ↑ ) / is the impurity spin operator in thedirection of the external magnetic field. ε QD is theenergy of the impurity level, U the electron repulsion, E Z, QD = g N µ B B is the impurity Zeeman energy, where g N is the corresponding bare g -ratio, µ B is the Bohrmagneton, and B the magnetic field, while ν = − ε QD U is the energy level in units of electron number.The superconductor energy levels ε i are spaced by d = 2 D/N , where the index i = 1 , . . . , N , D is thehalf-bandwidth and N is the number of levels. α is thestrength of the pairing interaction, E C is the chargingenergy and n is the gate voltage applied to the SI, ex-pressed in the units of electron number. The numberof electrons in the island is ˆ n SC = (cid:80) i,σ c † i,σ c i,σ . TheSI Zeeman energy is incorporated in the energy levelsas ε σ,i = ε i + n σ E Z, SI / where n ↑ = 1 and n ↓ = − , E Z, SI = g S µ B B .The hybridisation strength is Γ = πρv , where ρ =1 / D is the normal state bath density of states. The V term describes the capacitive coupling between the QDand the SI, which was found to be important to correctlyreproduce the charging diagrams of the device studied inthis work. We take the half-bandwidth D = 1 as the unitof energy.The hybridisation strengths of SI and QD with theneighboring lead, Γ source and Γ drain , respectively, do notenter the Hamiltonian directly. They are assumed weakand their small effect on the system can be accountedthrough a small renormalization of other parameters.For exploring the vast parameter space of the model,we found it useful to perform quick simulations with asignificantly smaller system size of N = 20 and a largervalue of α = 0 . , so that ∆ ≈ . D . To compensatefor the very large finite-size effects in this case, we cor-rected the resulting eigenvalues by subtracting the prod-uct of d/ and the absolute value of the excess charge in0the superconductor (for excess charge less than 2 in ab-solute value). This correction procedure works surpris-ingly well. Further improvement is possible by averagingeven- N and odd- N results since the N → ∞ limit is ap-proached from different sides , leading to a significantcancellation of finite-size effects.Highly converged simulation results that we used fordirect comparisons with the experimental measurements in the main text of this work were performed for a verylarge number of bath levels, N = 800 . This value is largeenough to minimize the finite-size effects even withoutfinite-size corrections. With finite-size corrections, equiv-alent results can be obtained for significantly smaller sys-tem size, N = 200 . We set α = 0 . , a magnitude appro-priate for Al grains , which determines the supercon-ducting gap in the absence of impurity, ∆ = 0 . D .The calculations were performed using the DMRG as described in Ref. 41 with three modifications of the matrix-product-operator (MPO) expression of the Hamiltonian: 1) incorporation of the QD-SI capacitive coupling V (ˆ n QD − ν )(ˆ n SC − n ) , 2) addition of the impurity Zeeman term E Z, QD , 3) addition of the bath Zeeman term E Z, SI . The fullexpression of the MPO is as follows (notation follows that of Ref. 41). Left-most site (impurity-site): W = (cid:16) I ( (cid:15) imp − V n )ˆ n imp + U ˆ n imp , ↑ ˆ n imp , ↓ + E Z, QD ˆ S z, QD − d ↑ F − d ↓ F + d †↑ F + d †↓ F V ˆ n imp (cid:17) . (F5)Here F = ( − n is the local fermionic-parity operator, which gives a phase of − if there is an odd number of electronson the site.Generic site (with g = αd ): W i = (cid:15) i − V ν + E c (1 − n )]ˆ n i + E Z, SI ˆ S z,i + ( g + 2 E c )ˆ n i ↑ ˆ n i ↓ gc i ↓ c i ↑ gc † i ↑ c † i ↓ E c ˆ n i I vc † i ↑ F i vc † i ↓ F i vc i ↑ F i vc i ↓ F i c † i ↑ c † i ↓ I c i ↓ c i ↑ I
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