Single Channel Josephson Effect in a High Transmission Atomic Contact

The Josephson effect in scanning tunneling microscopy (STM) is an excellent tool to probe the properties of the superconducting order parameter on a local scale through the Ambegaokar-Baratoff (AB) relation. Using single atomic contacts created by means of atom manipulation, we demonstrate that in the extreme case of a single transport channel through the atomic junction modifications of the current-phase relation lead to significant deviations from the linear AB formula relating the critical current to the involved gap parameters. Using the full current-phase relation for arbitrary channel transmission, we model the Josephson effect in the dynamical Coulomb blockade regime because the charging energy of the junction capacitance cannot be neglected. We find excellent agreement with the experimental data. Projecting the current-phase relation onto the charge transfer operator shows that at high transmission multiple Cooper pair tunneling may occur. These deviations become non-negligible in Josephson-STM, for example, when scanning across single adatoms.

e Josephson e ect in scanning tunneling microscopy (STM) is an excellent tool to probe the properties of the superconducting order parameter on a local scale through the Ambegaokar-Barato (AB) relation. Using single atomic contacts created by means of atom manipulation, we demonstrate that in the extreme case of a single transport channel through the atomic junction modi cations of the current-phase relation lead to signi cant deviations from the linear AB formula relating the critical current to the involved gap parameters. Using the full current-phase relation for arbitrary channel transmission, we model the Josephson e ect in the dynamical Coulomb blockade regime because the charging energy of the junction capacitance cannot be neglected. We nd excellent agreement with the experimental data. Projecting the current-phase relation onto the charge transfer operator shows that at high transmission multiple Cooper pair tunneling may occur. ese deviations become non-negligible in Josephson-STM, for example, when scanning across single adatoms.
Control of electronic properties in quantum-coherent nanostructures such as Josephson junctions is di cult to achieve as it requires deterministic structure design at the atomic scale. Without atomic scale design the conductance of identically prepared nanostructures exhibits uctuations of the order of the conductance quantum G 0 = 2e 2 /h. High level control has been achieved using atomic break junctions to realize few channels highly transparent Josephson atomic point contacts (JAPC) [1][2][3][4][5]. e highlight of JAPCs is that they can be tuned to the regime where electronic transport is dominated by a single transport channel with large, nearly re ectionless, transmission. As a result, the current-phase relation of the junction becomes non-sinusoidal and multiple Cooper pair processes may occur. At the same time, the excitation spectrum of Andreev levels carrying the Josephson current consists of a single Andreev bound state (ABS) that is well separated from other ABS and from the continuum of states above the gap. us, the maximum supercurrent carried by a JAPC, i. e. the critical current I C , does not only depend on the superconducting gap parameters in the two leads, but also on the details of the tunneling conductance [6,7], i. e. the number of transport channels and their transparency.
is scenario has been used to study experimentally the transition from coherent Josephson transport to the regime of multiple Andreev re ections (MARs) and also to reveal for the rst time coherent ABS dynamics [4]. In these and previous studies, the superconducting phase difference behaved as a classical variable, its quantum uctuations being negligible. Equivalently, charge quantization and charging e ects could be neglected such that the Josephson current was fully determined by the classical dynamics of the phase.
Design at the atomic scale can be perfectioned using a scanning tunneling microscope (STM) through direct atomic manipulation with more control and reproducibility than in break junctions. However, a downside in the STM is the limited design exibility concerning the in uence of the environment.
is implies non-negligible charging e ects and quantum uctuations of the phase [8][9][10]. Still, thermal uctuations can be reduced by operating in the low mK regime [11]. In this new scenario, the e ect of the electromagnetic environment seen by the junction leads to dynamical Coulomb blockade (DCB) type physics [12,13], which has remained largely unexplored until recently as it requires both signi cant charging e ects as well as a high transparency channel in order to be visible.
Here, we demonstrate in an STM single channel Josephson tunneling in the presence of DCB up to very high conductances > 0.9G 0 . We build a single atom contact by placing a single aluminium atom onto an Al(100) surface and approaching it with an atomically sharp tip made of polycrystalline aluminum. Operating at a base temperature of 15 mK, we ensure that both tip and sample are superconducting (T Al C = 1.2 K). We obtain a JAPC that features a single Josephson channel where the tip-sample positioning o ers unprecedented reproducibility of the channel transmission coe cient, from below 0.1 to above 0.95, with all other channels having lower transmission by at least one order of magnitude. e set of transport channels with their transmission, the so called mesoscopic pin code, is extracted from measurements of the current-voltage curves in the MAR regime, using a well established technique [3,14]. e interplay between quantum uctuations of the phase and the uctuations due to the electromagnetic environment is most prominent in the Josephson peak forming in the lower voltage portion of the current-voltage curve. At low transmission, the Joseph-arXiv:1810.10609v1 [cond-mat.supr-con] 24 Oct 2018 son e ect is well modeled using the Ambegaokar-Barato (AB) formula for the Josephson energy in the tunnel limit augmented by a description of the environmental interaction using P(E)-theory [8,15,16]. However, at transmissions exceeding 0.1 the results deviate signi cantly from the AB formula and a general theoretical model in this regime is currently lacking. Here, we provide a simple theoretical picture where Cooper pair tunneling occurs by incoherent transfer of single and possibly multiple Cooper pairs with rates calculated using P(E)-theory. is type of coherence loss between tunneling events is reminiscent of DCB physics, but is in contrast to the DCB regime of conventional Josephson junctions where tunneling occurs by incoherent single Cooper pairs. e theoretical model ts very well with the measured data for voltages below the threshold where MAR processes become relevant. us, our measurements provide an important step towards understanding DCB physics in the single channel Josephson regime and may inspire future theories on the transition between Josephson and MAR regimes in the presence of signi cant quantum uctuations of the phase.
e results also provide a be er understanding of the intricacies of the Josephson e ect as a local probe of the superconducting order parameter [15][16][17].
We build a single-atom junction by pulling an aluminum atom with the aluminum tip from the atomically at Al(100) surface (see Fig. 1(a)) and placing it on the surface again, which is shown in Fig. 1(b). e black depression at the lower le part of the image in (b) is the vacancy of the missing Al atom, which now appears as the rather large white protrusion (due to the image contrast) to the right of the center. is constitutes a reproducible way to create single-atom junctions with the STM as shown schematically in Fig. 1(c). As has been shown before, a single-atom contact does not necessarily constitute a single channel contact [2,3,18]. Most atoms have more than one valence orbital that is available for electron transport.
In order to demonstrate that the single aluminum atom contact realizes only a single dominant channel, we analyze the subgap current in the corresponding spectra [14]. Following previous ndings [2,3,18], we expect for the situtation of a superconducting contact made of an Al tip and an Al sample, that multiple Andreev re ections provide the most direct and most straightforward way of determining the the mesoscopic pin code. Experimental data for the current-voltage characteristics of the single-atom contact for di erent tipsample distances are shown in Fig. 2(a). e tip-sample distance decreases from the dark blue spectrum to the yellow spectrum as the normal state conductance increases. We distinguish between the low voltage regime with a peak-like structure (Josephson regime, blue shaded, below 70 µV) and the subgap MAR-regime with step-like structures. It is this la er regime that we explore rst in order to x the parameters that determine the physics of the low-voltage Josephson current.
We start with a di erential conductance spectrum at a small normal conductance setpoint su ciently away from FIG. 1: a) Topographic image of the Al(100) surface with an adsorbed foreign atom as a reference (white protrusion with black halo) before atomic manipulation. b) An Al atom has been pulled from the surface (black depression on the lower le ) and placed on to the surface again (white protrusion on the upper right). e contrast has been adjusted to display the details of the la ice corrugation, such that the adatom appears completely white. c) Schematic of the tunnel junction. e tip of the scanning tunneling microscope is directly over the Al adatom creating a contact between two single atoms. d) Fit of a quasiparticle di erential conductance spectrum at a conductance setpoint of 36 nS, where no Josephson e ect and no Andreev re ections are observed. the subgap domain, see Fig. 1

(d). By
ing the density of states to the experimental data with the Bardeen-Cooper-Schrie er (BCS) model of tip and sample, we nd the values of the gap parameters as ∆ tip = 180 µeV and ∆ sample = 180 µeV.
ese are then fed into a standard MAR model [14,[19][20][21] that extracts the mesoscopic pin code by assuming independent transmission channels to capture the experimental data at a given voltage (for details see the Supporting Information [22]). e ts are shown in Fig. 2(a) as thinner lines with darker color superimposed on the data. e ts accurately describe the MARs in the subgap regime (< 360 µV) with small discrepancies only appearing at the onset of MAR steps at larger tip-sample distances. We a ribute them to inelastic processes in the electron-hole tunneling which is not included in the modeling [10] and requires an extended description accounting for features known from dynamical Coulomb blockade [23].
In the mesoscopic pin code analysis, we use three transmission channels (cf. [2,24]) and nd that a single channel dominates the others by at least one order of magnitude for all tip-sample distances measured (see Fig. 2 us, we conclude that in our STM set-up a single-atom contact between Al tip and sample with a single transmission chan- nel is realized experimentally to very good approximation. With the transport parameters xed, we can turn to the lowvoltage regime (see Fig. 2(a)) to explore the Josephson e ect in a rather unconventional domain, where charge transfer through a single channel with tunable transmission meets dynamical Coulomb blockade physics. e Josephson peak arises due to inelastic Cooper pair tunneling, with broadening determined by the interaction with the environment. In our low-temperature STM experiment the granularity of charge transport determines the nature of these interactions and the observed peak is a manifestation of dynamical Coulomb-blockade. e interplay between (Josephson) tunneling and electromagnetic degrees of freedom of the environment can be described by P(E)-theory e calculated spectra have essentially no free t parameter, except for the rst spectrum at lowest transmission. A clear deviation between the Ambegaokar-Barato approach (AB) and the full Andreev bound state relation at arbitrary transmission (AT) can be seen at higher transmission.
For the transmissions in f) and g), non-adiabatic processes become signi cant at higher bias voltages, such that our model is only applicable within the blue shaded areas. Panel h) shows the χ 2 -values of the ts. e arbitrary transmission model yields low χ 2 -values throughout (indicating good agreement), except in the red shaded region, where non-adiabatic processes become signi cant. [8,12,25]. e current-voltage relation is where the Josephson energy E J is given by the AB formula and G N is the total normal state conductance. e P(E)function describes the probability density for exchanging energy E with the environment. Here, the energy E is given by the kinetic energy of the tunneling Cooper pair 2eV , where V is the applied junction bias voltage. is approach describes junctions with arbitrary many channels, as long as the transmission of each individual channel is small, τ i 1. e above theory for single-Cooper pair tunneling underestimates the Josephson peak at higher transmission (yellow line in Fig. 3), as compared to what is observed in our experimental data (blue dots in Fig. 3). It turns out that, at higher transmission, the nonlinear dependence of the energy on the transmission as well as to a smaller extent the tunneling of multiple Cooper pairs within a tunneling event are nonnegligible and need to be taken into account. ey can be traced back to the non-sinusoidal energy-phase relation expected for transparent single-channel contacts. In the following, we propose a simple extension of existing P(E)-theories that describes the dynamical Coulomb blockade regime of single and multiple Cooper pair transfer processes. e starting point is the ABS energy-phase relation for a single channel of arbitrary transmission [5,7] where the index ± labels the states between −∆ and +∆ below (−) and above (+) the Fermi level. Focussing on the lower (−) Andreev state (i. e., assuming low temperature and the adiabatic limit, see below), we can express the energy as with the coe cients E m given by In the spirit of P(E)-theory, we replace the phase ϕ in the energy-phase relation by an operator. e phase acquires signi cant uctuations in the dynamical Coulomb-blockade regime, where the number of transferred charges is a wellde ned quantity. In the resulting Hamiltonian, a perturbative treatment is applied to the operators e imϕ that induce the translation of m Cooper pairs. Instead of the single P(E)function in the tunnel limit describing inelastic single Cooper pair tunneling, each m-Cooper pair tunneling process involves a new P m (E)-function that gives the probability of energy exchange 2meV (see Supplemental Material for details [22]): and we nd a Josephson current for the single channel case, which depends on the P m (E) functions at the energies ±2meV . Note, that in the tunnel limit, τ 1, of Eq. (5) the coe cient |E 1 | = ∆τ /8 = E J /2 dominates and Eq. (7) reduces to Eq. (1).
e results of the extended DCB theory (red lines in Fig. 3) from Eq. (7) are compared to the experimental data and with the conventional DCB from Eq. (1). In Fig. 3(h), the χ 2 -values for the calculated curves are plo ed as function of total transmission. e lower χ 2 -values for the arbitrary transmission (AT) model indicate a much be er agreement compared to the AB model (details of the χ 2 calculation can be found in the Supporting Information [22]). Crucially, both theoretical calculations do not involve any free t parameters, but rely only on the mesoscopic pin code known from the MAR analysis, gap parameters for tip and sample obtained from the quasiparticle spectrum at low conductance, and on the tunnel junction parameters entering the P (m) (E)-function(s) (see Supporting Information [22]) determined by the Josephsonspectrum at lowest transmission [28]. Without introducing additional parameters or assumptions, the I-V given by Eq. (7) clearly improves upon the conventional DCB result. We nd good agreement with the experimental data over the whole voltage range of the Josephson peak up to τ ∼ 0.7. For larger τ this voltage range shrinks until no discernible agreement can be claimed at the highest transmission, 1 − τ 1. e level to which our extension of DCB-theory reproduces the STM measurements is fully consistent with expectations: (i) e improved agreement at high transmission can be a ributed to the dependence of coe cients E m on the channel transmission τ . ese coe cients E m of the extended theory play a similar role to E J in Eq. (1). e comparison between E m for (m = 1, 2, 3) and E J /2 is plo ed in Fig. 4(b) showing a signi cant increase of E 1 compared to E J /2 for transmissions τ 0.2. At higher transmission the probability for multiple Cooper pair transfers increases, such that a (small) part of the current is due to the energy exchange of multiple Cooper pairs with the environment. Distinguishing the contributions of processes with di erent m is not possible in our experiment. In the future, this could be achieved using a designed environment with su ciently narrow resonances, such that speci c m-Cooper pair process can be enhanced via the P m (E)-functions in Eq. (7). (ii) As a low-order perturbative approach any P(E)-type approach is bound to fail, if tunneling becomes too frequent for the environment to relax between consecutive tunneling events. A simple test (see Supporting Information [22]) suggests that the assumption of sequential, independent tunneling events inherent to the rate-picture of Eqs. (1) and (7) breaks down at about τ ∼ 0.9. (iii) Even without environmental e ects, however, the large transmission limit, 1 − τ 1, is challenging, since we can no longer separate a large-voltage regime of MAR from the low-voltage region of Josephson-tunneling, cf. Fig. 2(a), as high-order Andreev re ections scale with high powers of the transmission and are no longer suppressed. Within the equilibrium picture of ABS, E ± (ϕ) in Eq. (3), with ϕ being a classical variable, in the low-bias regime, the dissipative current can be understood in terms of Landau-Zener transitions between the ABSs, cf. Fig. 4(a) [4]. While well in the dynamical Coulomb blockade regime of large quantum uctuations of the phase, we can, nonetheless, use the classical phase picture and exploit the Landau-Zener transition probability, p = exp [−π (1 − τ )∆/eV ], to estimate a threshold voltage, V = (1 − τ )∆, where non-adiabatic transitions become non-negligible. is voltage, diminishing as the transmission approaches unity, ts with the observed range of va- FIG. 4: a) Andreev bound state relation for high transmissions. As the gap closes, the probability for transitions between branches (non-adiabatic processes) becomes more likely. b) Absolute value of the coupling coe cients |E m | at di erent transmissions in comparison to the coupling coe cient of the linear Ambegaokar-Barato model (E J /2). lidity in Fig. 3. For instance, for τ = 0.88 [ Fig. 3(g)], we expect and observe the adiabatic model to fail outside of the region −22µV ≤ V ≤ 22µV shaded in blue.
Further theoretical and experimental investigation of the large transmission regime will advance a more complete understanding, complementing the elaborate, self-consistent theory for the case of thermal phase uctuations [4], and can also clarify the impact of a renormalization of the charging energy at stronger tunneling [29].
We have demonstrated the single channel Josephson e ect in the STM from an atomic contact at arbitrary transmission up to the quantum of conductance and in presence of dynamical Coulomb blockade.
e Josephson current in this regime can be very well modeled by the energy-phase relation of the full Andreev bound state projected onto the charge transfer operators for single and multiple Cooper pair tunneling. We nd excellent agreement between theory and experiment with no free parameters as each parameter has been determined independently. Concerning few channel junctions, we believe, it is crucial to consider individual transport channels and their transmission separately instead of using the total conductance and the multi-channel approximation of the AB formula. Deviations can already be discerned at channel transmissions as low as τ = 0.1. ese ndings are an important step towards a detailed understanding of the dynamical Coulomb blockade regime in the Josephson e ect measured by the STM. is is urgently needed, as the Josephson e ect becomes important as a tool to extract local information about the superconducting properties of the sample. For instance, when scanning across a magnetic adatom that induces Yu-Shiba-Rusinov states and locally reduces the order parameter of the substrate [26,27], the inevitable changes in the number of channels and their transmission have to be considered.
We gratefully acknowledge fruitful discussions with Berthold Jäck, and Elke Scheer. Funding from the European Research Council for the Consolidator Grant ABSO-LUTESPIN, from the Spanish MINECO (Grant No. FIS2014-55486-P, FIS2017-84057-P, and FIS2017-84860-R), from the "María de Maeztu" Programme for Units of Excellence in R&D (MDM-2014-0377), from the Zeiss Foundation, from the DFG through AN336/11-1, and from the IQST is also gratefully acknowledged. valid as long as non-adiabatic processes are negligible, which are not included in the model here as we will discuss later [4].

Supplementary Information
FIG. S1: Linear plot of the experimental multiple Andreev re ection spectra along with their corresponding ts.

TIP AND SAMPLE PREPARATION
e experiments were carried out in a scanning tunneling microscope (STM) operating at a base temperature of 15 mK [1]. For the sample, we use an Al(100) single crystal. To obtain a clean Al(100) surface, the sample was cleaned by multiple cycles of Ar spu ering and subsequent annealing. e tip material was a polycrystalline Al wire, which was cut in air and prepared in ultrahigh vacuum by spu ering and eld emission. Single Al atoms were pulled from the surface by the tip and then placed on the surface to realize a single atomic contact. We demonstrate that the tip apex remains una ected by the manipulation through imaging of the reference impurity (sombrero shape in Fig. 1(a) and (b) of the main text).
Both images of the reference impurity look identical before and a er the manipulation.

EXTRACTING THE MESOSCOPIC PIN CODE FROM THE SUBGAP CURRENT
To extract the number of channels and their transmission, we exploit the multiple Andreev re ections that are measured at di erent normal state conductances. For the ts, we use the model of multiple Andreev re ections outlined in Ref. [2] treating multiple channels as independent of each other and adding them for the total spectrum. As has been shown before [3], the aluminum spectra cannot be ed accurately at low temperatures without considering the spectral broadening due to the interaction with the environment (P(E)theory). However, the models considered in the context of multiple Andreev re ections do not include the environmental interactions to the degree needed here. In addition, the broadening cannot be modelled by an e ective Dynes parameter Γ either [4]. To remedy this shortcoming, we approximate the P(E)-function by a Gaussian P * (E), which includes an e ective capacitive voltage noise. In this way, the broadening is symmetric and we can include the in uence of the environment at least phenomenologically to lowest order by convolving one of the leads' density of states with this P * (E)function before calculating the Andreev re ections. (For the later analysis of the Josephson spectra, we employ the full P(E)-function.) To determine the ing parameters, we t a spectrum at low conductance that does not show any Andreev re ections nor any Josephson e ect. We calculate the di erential conductance dI /dV from the tunneling current with the tunneling probability from tip to sample e other tunneling direction ì Γ(V ) from sample to tip can be obtained by exchanging electrons and holes in Eq. (9). Here, R T is the tunneling resistance, f (E) = 1/(1 + exp(E/k B T )) is the Fermi function, and ρ t , ρ s are the densities of states of tip and sample, respectively. Here, we replace the full P(E)-function by a Gaussian modeling an e ective capacitive noise: where E C = Q 2 /2C J is the (e ective) charging energy for Cooper pairs (Q = 2e). We use the same e ective capacitance C J = 21.7 fF and e ective temperature T = 200 mK as for the modeling of the Josephson e ect, which does not introduce any new parameters (see below). e densities of states of the aluminum tip and the aluminum sample are given by the simple Bardeen-Cooper-Schrie er (BCS) expression: where the order parameters for both tip and sample are the same ∆ t,s = 180 µeV and the Dynes parameter Γ t,s = 0.01 µeV is only non-zero for numerical purposes.
In order to t the number of channels and their transmissions to the experimental data, we calculate a series of Andreev re ection spectra from the t parameters for di erent transmissions. Assuming independent channels, we t linear combinations of these spectra to the experimental data using a statistical search algorithm (Matlab).
We should point out, that the result of having single channel transmission is quite robust with respect to details of the t and the modeling, i. e. whether we use the symmetric P * (E)-function or a phenomenological broadening parameter.
is is because the number of channels and their conductance do not only modify the subgap structure of the spectrum, but they also have a sizeable e ect on the normal conducting part of the spectrum, i. e. at voltages |V | > ∆ t + ∆ s . e so-called excess current I exc [2] is a constant (voltage independent) current contribution to the normal conducting part of the spectrum. e excess current I exc increases monotonically with the channel transmission and is quite independent of the details of the modeling as well as of any broadening mechanisms. It is easy to see that the excess current I exc contribution becomes maximal in the case of a single channel contact. e excess current I exc is naturally included in the Andreev calculations and as such the whole spectrum inside and outside of the gap becomes an important indicator for single channel tunneling.

CALCULATING THE JOSEPHSON CURRENT WITHIN P(E)-THEORY AT ARBITRARY TRANSMISSION
e Josephson current in the STM at mK temperatures and low transmission, where only single Cooper-pair processes play a role, has previously successfully been modeled on the basis of the simple sinusoidal energy-phase relation and the standard P(E)-function for single Cooper pairs (cf. Eq. (1) in the main text). To calculate the I (V ) characteristics of the single-channel Josephson e ect at arbitary transmission, we start from the full energy-phase relation E ± (ϕ) of the Andreev bound states (cf. Eq. (3) in the main text), which includes multiple Cooper-pair processes. Since we only consider nonadiabatic processes in this model, we focus on the lower branch (− label) and omit the sign index in the following. We transform this energy-phase relation from phase space to charge space by a Fourier transform, E ϕ = ∞ m=−∞ E m e imϕ (cf. Eq. (4) and (5) in the main text), in order to include the interaction of multiple Cooper-pair transfers with the surrounding environment. Following the standard procedure of calculating tunneling currents within the framework of P(E)theory (see, e.g., Refs. [5][6][7]), the tunneling rate associated with a forward tunneling process of m Cooper pairs through a single channel is given by (12) e corresponding backwards tunneling rates can be obtained by replacing m by −m. is is essentially a golden rule rate with E m e −imϕ as the perturbation. Here, we sum over all initial reservoir states R i with energies E R i weighted by the probability P β (R i ) to nd these states at the inverse temperature β = 1/(k B T ) and over all nal states R f with energies E R f . Performing the trace over the environmental degrees of freedom and exploiting the generalized Wick theorem, we nally arrive at Here, we have introduced the generalized P(E)-function with the phase correlation function accounting for Cooper-pair-phase uctuationsφ = ϕ − 2eV t/ around the mean value determined by the external voltage. For m = ±1, we recover again the standard P(E)-function used in the low-transmission limit for single Cooper-pair processes. Summing up all possible processes (forward and backward tunneling of one, two, up to in nitely many Cooper pairs), the current contribution of the lower Andreev bound state can be wri en as

VALIDITY OF THE P(E)-THEORY
To test for the applicability of P(E)-theory in this context, we calculate the product of the coupling constant with the P(E)-function [5]. In the tunneling regime, this condition of validity is typically wri en as: FIG. S2: a) Plots of the P(E)-function for m transferred Cooper pairs. For higher orders of m, the weight shi s to higher energies. b) Check for the applicability of P(E)-theory for single and multiple Cooper pair transfers. For all transmissions except the highest (τ > 0.9) the product of coupling constant E m with the maximum value of the P m (E)-function P max m is less than one, so that P(E)-theory is applicable. e blue shaded area shows the region, where non-adiabatic processes start dominating the spectrum. e P(E)-functions used in this analysis are plo ed in Fig.  S2(a). e parameters are given in the next section. e di erent curves are for multiple Cooper pair transfers up to m = 5. In order to simplify the analysis, we just use the maximum value of the P(E)-function P max , which in our scenario is typically near E = 0. Here, we consider sequential tunneling of single and multiple Cooper pairs, so we will calculate the product for each value of m separately. erefore, the coe cient E m takes the role of the coupling constant and the maximum of P m (E) is P max m , such that the condition reads as: e values for di erent Cooper pair transfers up to m = 3 are plo ed in Fig. S2(b). We can see that all values are less than one for all transmissions except the highest ones. Here, the validity condition is broken for single Cooper pair transfers at transmissions τ > 0.9. is is in agreement with our previous analysis of the validity of P(E)-theory in the DCB regime of the STM [8]. As we have shown in the main text that at these transmission values also non-adiabatic processes start to dominate, which are not directly captured in the theoretical model, breaking the applicability condition at these high transmission values is of minor relevance.
At the highest transmission of τ = 0.98, as shown in Fig.  3(h) in the main text, the Josephson e ect cannot be modeled adequately anymore by our models. is is due to the nonadiabatic processes becoming dominant as discussed in the main text as well as a breakdown of the applicability condition for P(E)-theory as shown in Fig. S2.

CALCULATING THE χ 2 -VALUES
We de ne the dimensionless χ 2 -values as where i indexes all data points in the data set, data i is a data point, and model i is the calculated value corresponding to the data point. e χ 2 -value is thus a dimensionless, normalized quantity indicating the deviation of the calculation from the data. e lower the χ 2 -value, the be er the agreement. e χ 2 -values have been calculated for a voltage interval |V | ≤ 30 µeV.

MODELING THE JOSEPHSON SPECTRA
e Josephson e ect is modeled with the full P(E)function. We cannot employ the simpli ed P * (E)-function used in the analysis of the Andreev spectra, because they are symmetric and would yield zero Josephson current. In order to extract the relevant parameters, we t a Josephson spectrum at very small conductances, where we can assume li le di erence between the Ambegaokar-Barato approach and the full Andreev bound state relation. e t function is based on the equation: where E J is the Josephson energy. e parameters are for the junction capacitance C J = 21 fF and for the e ective temperature T = 200 mK. For the environmental impedance Z (ω), we use the modi ed transmission line impedance introduced in Ref. [8], which describes the impedance of the surrounding vacuum as well as the resonances of the tip acting as a monopole antenna. e parameters for the modied transmission line impedance are the resonance energy ω R = 70 µeV, the e ective damping parameter α = 0.9, and the environmental dc resistance R env = 377 Ω [8].