Superconducting quantum interference at the atomic scale

A single spin in a Josephson junction can reverse the flow of the supercurrent by changing the sign of the superconducting phase difference across it. At mesoscopic length scales, these π-junctions are employed in various applications, such as finding the pairing symmetry of the underlying superconductor, as well as quantum computing. At the atomic scale, the counterpart of a single spin in a superconducting tunnel junction is known as a Yu–Shiba–Rusinov state. Observation of the supercurrent reversal in that setting has so far remained elusive. Here we demonstrate such a 0 to π transition of a Josephson junction through a Yu–Shiba–Rusinov state as we continuously change the impurity–superconductor coupling. We detect the sign change in the critical current by exploiting a second transport channel as reference in analogy to a superconducting quantum interference device, which provides our scanning tunnelling microscope with the required phase sensitivity. The measured change in the Josephson current is a signature of the quantum phase transition and allows its characterization with high resolution. Continuously changing the coupling between a magnetic impurity and a superconductor allows the observation of the reversal of supercurrent flow at the atomic scale.

Two superconductors that are connected by a weak link can sustain a supercurrent, which is carried by Cooper pairs -the well-known Josephson e ect [1].Inserting a single spin into the junction may completely change its behavior by reversing the direction of the supercurrent [2], which is the result of a -shi in the phase across the junction.Such junctions have been used in nding the pairing symmetry in unconventional superconductors [3][4][5][6][7] and they have been proposed as building blocks for energy e cient quantum computing or high-speed memory [8][9][10].At mesoscopic length scales (≈10 to 100 nm), -junctions may be realized by singly occupied quantum dots or ferromagnetic interlayers [11][12][13][14][15][16][17][18].At the atomic scale (≈0.1 nm), a single magnetic impurity, which is exchange coupled to a superconductor, induces a spin nondegenerate superconducting bound state, a Yu-Shiba-Rusinov (YSR) state [19][20][21].By tuning the magnetic exchange coupling, the YSR state can be driven through a quantum phase transition (QPT) with a concomitant -shi [22][23][24].e hallmark of this quantum phase transition (QPT) in YSR states is a discontinuous change in the total spin of the respective ground states: a previously free impurity spin turns into a screened spin, when the magnetic exchange coupling increases beyond a critical value.Consequently, a reversal in the ow of Cooper pairs through a YSR state has been predicted [25].Experimentally, the QPT can be identied by a zero energy crossing of the YSR state in di erential conductance spectra [26][27][28][29][30].However, the actual consequences for the fundamental Josephson e ect remain elusive in atomic scale junctions.e observation of a YSR state based -junction is experimentally challenging, because detecting such a phase shi between superconducting ground states requires a reference channel.At mesoscopic length scales, this is typically solved by employing a superconducting quantum interference device (SQUID) loop geometry [14][15][16][17].To reach similar conditions at the atomic scale, a scanning tunneling microscope (STM) requires a rudimentary phase sensitivity through an additional transport channel [15,18].
Here, we demonstrate a supercurrent reversal in an atomic scale Josephson junction through a YSR state as we move across the QPT.We produce a magnetic impurity at the apex of a superconducting vanadium tip (see Fig. 1(a)), which is approached to a superconducting V(100) sample.As we approach, the atomic forces pull on the impurity [26,[31][32][33], which reduces the impurity-superconductor coupling  along with the magnetic exchange coupling.is concomitantly allows the YSR state to pass from the strong sca ering (screened spin) to the weak sca ering (free spin) regime as outlined in Fig. 1(b).e two scenarios are schematically illustrated in Fig. 1(c), where the total spin in the free spin regime is  tot = 1 /2.In the screened spin regime, a Cooper pair is broken to screen the impurity spin changing the overall parity of the system (indicating whether the total number of particles is even or odd) as well as the total spin to  tot = 0.
To detect the supercurrent reversal, we exploit the parallel presence of a second transport channel featuring a conventional superconducting Bardeen-Cooper-Schrie er (BCS) gap without any YSR state as a reference channel (see Fig. 1(a)).e sign change in the supercurrent through the YSR state manifests itself as a step in the measured net Josephson current resulting from the changeover of a constructive to a destructive interference of the two transport channels across the QPT.e evolution of the YSR state as a function of the normal state conductance  N is shown in Fig. 1(d).
e YSR state moves across the QPT when the YSR energies are closest to each other.Because both tip and sample are superconduct- ing, in the spectrum the tip YSR states appear at voltages  shi ed by the sample gap  s , i.e.  =  +  s with the YSR state energy  varying with the normal state conductance  N .Interestingly, there are no distinct coherence peaks visible at the sum of the tip gap and the sample gap ±( t +  s ), which indicates that a second transport channel through an empty gap (i.e.without any YSR state and hence with coherence peaks) has a much weaker, but still nite transmission compared to the YSR state.
is can be more directly seen in a single spectrum near the QPT, which is shown in Fig. 1(e).
e coherence peaks at  =  t +  s are greatly reduced and the YSR peaks are prominently enhanced by almost a factor of 100.
To con rm that the impurity-superconductor coupling (in our case the impurity-tip coupling) decreases with increasing conductance, we have measured the Kondo e ect in the same junction by quenching the superconductivity in a magnetic eld of 1.5 T. e Kondo spectra are shown in the inset of Fig. 1(f), from which we extract the half width at half maximum  K .As  K is directly related to the Kondo temperature and the magnetic exchange coupling, we conclude that the impurity-superconductor coupling decreases with decreasing tip-sample distance (i.e.increasing junction conductance).Physically, the impurity is pulled away from the tip by the a ractive atomic forces of the approaching sample substrate [33].e supercurrent, which is carried by tunneling Cooper pairs (Josephson e ect), is visible throughout the range of conductance values (see red arrow in Fig. 1(e)).In the dynamical Coulomb blockade (DCB) regime, in which the STM operates [34], the typical voltage-biased measurement shows a negative current peak followed by a positive current peak of equal size near zero bias voltage.e evolution of the Josephson e ect as function of conductance is shown in Fig. 2(a).Each spectrum is shown in a bias voltage range of ±60 eV and o set horizontally.Assuming a harmonic current-phase relation in the DCB (i.e. () =  C sin , where  C is the critical current), the Josephson current is predicted to scale with the square of the critical current, i.e.  ( ) ∝  2 C ∝  2 N [35][36][37][38].It can be directly seen in the data that this square dependence is not ful lled in the data set in Fig. 2(a).In particular, the region indicated by the horizontal bracket shows signi cant deviations, even a slight decrease in the Josephson current with increasing conductance.e conductance at which the QPT occurs is indicated by a vertical dashed line, which falls directly into the region of the horizontal bracket.
For a more quantitative analysis, we plot the current maxima  S (switching current) for each conductance as a blue line in a double logarithmic plot in Fig. 2(b).e expected square dependence on the conductance ( S ∝  2 C ∝  2 N ) can be clearly seen for very small and very large conductances.In the transition region (indicated by the horizontal bracket), the behavior of the switching current  S strongly changes.For comparison, we plot the experimentally extracted energies of the YSR state (red line), which has a minimum at the QPT (vertical dashed line) [39]. is indicates a drastic change in the behavior of the Josephson e ect across the QPT.
To put the evolution of the switching current in reference to other Josephson junctions, we calculate √  S  N , which is shown in Fig. 2(c) ( N is the normal state tunneling resistance).is quantity is proportional to the product  C  N for a harmonic energy-phase relation.In this way, the overall conductance dependence is eliminated such that the measurement appears like a step in Fig. 2(c) with a sizeable reduction in height almost by a factor of two across the QPT.We will show below that this is due to a supercurrent reversal in the YSR channel which leads to a crossover from a constructive to a destructive interference between the two transport channels.
In order to compare the experimental data to the theory, we have to renormalize the normal state resistance  N for the YSR spectra due to the enhanced density of states from Kondo correlations (for details see the Supplementary Information [40]).e reference spectra (orange line in Fig. 2(c)) are measured for a Josephson junction without any YSR states, where the √  S  N is constant as expected from the Ambegaokar-Barato formula [41,42].
To understand the behavior of the Josephson e ect in Fig. 2, we rst have a look at the energy-phase relations far away from the QPT at high and at low conductance.In Fig. 3(a), the energy-phase relations for the BCS channel and the YSR channel, which is calculated from a mean eld Anderson impurity model, are shown in red and blue, respectively (for details see the Supplementary Information [40]).To calculate the energy-phase relation, we apply a constant phase difference  across the tunnel junction, but no bias voltage.A Fourier expansion of the energy-phase relation reveals that the most relevant contribution to the Josephson e ect is the harmonic term proportional to cos().Zooming in to both channels (cf.Fig. 3(b)), we estimate that the ratio of the chan-nel transmissions is about 4:1 (YSR:BCS): is results in a signi cantly smaller amplitude for the energy-phase relation of the BCS channel (red) than in the YSR channel (blue) (Individual Channels in Fig. 3(b)).e coherent superposition of these two channels (Channel Sum in Fig. 3(b)) leads to an overall sign change as well as di erent amplitudes, when the channels are in phase (upper row in Fig. 3(b)) or out of phase (lower row in Fig. 3(b)).In the measurement, we are only sensitive to the change in amplitude  ( ) ∝ ( YSR +  BCS ) 2 though, which results in the obvious step in Fig. 2(c).We attribute the width of the step to the nite temperature in our experiment.
Since the temperature in our experiment (10 mK) is still non-zero, we expect uctuations due to thermal excitations close to the QPT.e probability for the system to be in the ground state (blue) or the excited state (orange) is indicated in Fig. 3(c) using an e ective temperature of 75 mK. is will broaden the expected sharp features associated with the quantum phase transition.Taking the excitation probability due to the nite temperature into account, we can calculate the expected Josephson current in the DCB regime (see Supplementary Information [40]).e t is shown in Fig. 3(d) with excellent agreement to the data.e only free parameters are the e ective temperature  e = 75 mK, which is determined by the width of the transition and the ratio of the two channel transmissions, which is determined by the step height.For a best t, we nd that the YSR channel contributes 78.4% and the BCS reference channel contributes 21.6% to the total conductance relevant to the Josephson e ect, which is consistent with the prominent YSR states and the strongly reduced coherence peaks in the quasiparticle spectra (see Fig. 1(e)).All other parameters are given by the experimentally extracted values.In this way, we demonstrate that the supercurrent through an atomic scale YSR state reverses upon  e tunneling process in the free spin and in the screened spin regime is shown from the initial to the nal state via an intermediate (virtual) state in the excitation picture.
e order is set by the numbered red arrows.e total spin  tot describes the total spin of the YSR system including the spin of the impurity.(a) In the free spin regime, the order of the spins is exchanged compared to the intial state, which results in the supercurrent reversal.(b) In the screened spin regime, the order of the spins is retained, such that there is no sign change in the supercurrent.
crossing the QPT, which can be detected in the STM by means of a BCS reference channel in analogy to a SQUID geometry (see Supplementary Information for more details [40]).
To be er understand the origin of this supercurrent reversal and to illustrate the crucial role of the impurity spin, we discuss the Cooper pair tunneling process in Fig. 4 using the excitation picture.Zero energy denotes the ground state,  is the energy of the excited YSR state, and  marks the beginning of the quasiparticle continuum.
e order is given by the numbered red arrows.Figure 4(a) describes the free spin regime, where the total spin is  tot = 1 /2 [22,25]. is indicates that the total parity (superconductor + impurity) must be odd.
e Cooper pair transfer process involves a swap between two fermions, the one associated with the impurity and one associated with the Cooper pair, as depicted by the arrows 3 and 4 in Fig. 4(a).Formally, this appears as an exchange of fermion operators inducing a negative sign (shi ) [15,43].By contrast, Fig. 4(b) shows the screened spin regime, which has a ground state with total spin  tot = 0 and correspondingly even parity.Here, the Cooper pair transfers conventionally as in an empty BCS gap except that the YSR state is used as intermediate (virtual) state instead of the continuum.It is this switching between transport regimes when crossing the QPT, which we observe experimentally.Recalling Fig. 1(f) that the impurity-substrate coupling reduces upon increasing the conductance, we move from the screened spin regime across the QPT to the free spin regime as the tip approaches the sample.is is consistent with the evolution of the Josephson current from an in-phase superposition (zero junction) to an out-of-phase superposition ( junction) as the conductance increases.
At the QPT, a system typically becomes very sensitive to external parameters, such as temperature.Here, we note that the width of the QPT step in Fig. 3(d) depends only on temperature, but experiences no broadening from voltage noise.
is is in contrast to conventional scanning tunneling spectroscopy, where temperature broadening is typically obscured by voltage noise as well as interactions with the environment [34].Hence, YSR-tip functionalization may open new developments for low temperature thermomentry with high spatial resolution where measuring the slope of the QPT step accesses the temperature.
In summary, the experimental results directly reveal the consequences of the discrete parity change across the QPT in YSR states as well as the role of the impurity spin, which manifests itself in the supercurrent reversal.Our results establish an important connection to mesoscopic -junctions providing the perspective to transfer some of their concepts, for example as sensing tools, to the atomic scale.Having direct tunable access to the QPT could be exploited to enhance the sensitivity in quantum sensing applications, such as a local temperature measurement.Also, demonstrating the coherent superposition of di erent transport channels in the DCB regime introduces a rudimentary phase sensitivity in STM measurements that can be exploited in other scenarios as well.
Supplementary Material for Superconducting antum Interference at the Atomic Scale

MATERIALS AND METHODS
Experiments were performed on Josephson nanojunctions built in a low temperature scanning tunneling microscope (STM) operated at 10 mK.Approaching a superconducting vanadium tip, tailored with a spin-1 ⁄ 2 impurity at its apex, to a crystalline V(100) substrate, we drove the impurity induced YSR states across the quantum phase transition (QPT) and detected a reversal of the supercurrent.e V(100) substrate was cleaned by repeated cycles of Ar ion spu ering, annealing to ∼ 925 K, and cooling to ambient temperature at a rate of 1-2 K/s.Oxygen di used from the bulk to the surface lead to typical surface reconstructions [1,2], which did not in uence the characteristics of superconducting vanadium.Surface defects mostly involve missing oxygen within the reconstruction, which appeared bright in STM topographs [3].Magnetic defects were found exhibiting YSR states at arbitrary energies within the gap as reported in Ref. [3].
e tip was spu ered in ultra-high vacuum and treated with eld emission as well as subsequent indentation into the vanadium substrate until the expected gap of bulk vanadium appeared in the conductance spectrum.YSR tips were designed following the method of random dipping explained in Ref. [3].We purposefully chose to use YSR tips for our experiment as it gave be er stability of the junction at higher conductance.Moreover, it o ered be er exibility in designing and de ning the junction over magnetic surface defects, which were mostly found to have a spatial extent of around 1 nm.

THE JOSEPHSON CURRENT
Calculating the Josephson current is typically based on the energy-phase relation  () at zero applied bias voltage.In the DCB regime, the charging energy of the tunnel junction dominates, such that tunneling is charge dominated and sequential.
e measurement is typically voltage-biased and the Cooper pair tunneling relies on the exchange of energy with the environment.e corresponding Cooper pair transfer can be calculated by means of a Fourier transform of the energy-phase relation where   corresponds to the charge transfer operator and  is the number of Cooper pairs being transferred.e Fourier components   from Eq. S1 are used to calculate the Josephson current (S2) where   () is the probability to exchange energy  with the environment during the tunneling process, when  Cooper pairs are tunneling.It is de ned as a generalized  ()function [4] with the phase correlation function accounting for Cooper-pair-phase uctuations φ =  − 2 /ℏ around the mean value determined by the external voltage.For a more detailed discussion see Ref. [4].

DERIVATION OF THE ENERGY-PHASE RELATION
To describe the phase dependence of the energy of the YSR states we follow Ref. [5,6] and make use of a mean-eld Anderson model where a magnetic impurity coupled to superconducting leads is described by the following Hamiltonian  =  t +  s +  i +  hopping . (S5) Here,   , with  = t, s (t stands for tip and s for substrate), is the BCS Hamiltonian of the lead  given by where  †   and    are the creation and annihilation operators, respectively, of an electron of momentum , energy    , and spin  =↑, ↓ in lead ,   is the superconducting gap parameter, and   is the corresponding superconducting phase.On the other hand,  i is the Hamiltonian of the magnetic impurity, which is given by where   =  †    is the occupation number operator on the impurity,  U is the on-site energy, and  J is the exchange energy that breaks the spin degeneracy on the impurity.Finally, ity has to be taken into account when calculating the Josephson current as it can signi cantly broaden the region of the QPT even at mK temperatures.As the energy of the Andreev bound state in the BCS channel is  BCS >  √ 1 − , it is unlikely for this state to be thermally excited.We, therefore, only consider the thermal excitation of the YSR state.Here, the energy of the excited state is − YSR ().If we now de ne  as the probability for the YSR state to be thermally excited, we can write the lowest order harmonic coe cient as where  = 2/(1 + exp(| |/ B  )),  is the energy of the YSR state, and  is the temperature.We use the coe cient in Eq.S30 to calculate the Josephson current in Eq.S2, which is used to calculate the t in the main text.

THE 𝐼 C 𝑅 N PRODUCT FOR A YSR STATE
In the presence of a YSR state, the normal state resistance  N , which is relevant for modeling the Josephson current may be modi ed in the presence of Kondo correlations resulting from an enhanced density of states in the range of the superconducting gap.erefore, the normal state conductance  N that is typically measured outside the superconducting gap has to be renormalized as the assumption that the density of states is a constant no longer holds.We have used reference measurements through a tunnel junction without any YSR state [7], which has been shown previously to follow the expected Ambegaokar-Barato formula [8], to nd the renormalization coe cient for the normal state resistance  N in the presence of a YSR state.For the data set presented in the main text, we nd  N = 1/(2.05N ), where  N is the normal state conductance outside the superconducting gap.We note that this is a phenomenological result.e value of the renormalization factor only holds for this particular data set as the contributions from Kondo correlations to the local density of states may vary between impurities.

OTHER YSR TIPS
In Fig. S1, we show another data set with an STM tip that is functionalized with a YSR state.As in the main text, the YSR state energy changes as a function of conductance (i.e.tip-sample distance) as shown in Fig. S1(a).
e YSR state undergoes a QPT at 0.023 0 from a screened spin to a free spin regime (for comparison, the data in the main text has the QPT at 0.009 0 ).A spectrum near the QPT is shown in Fig. S1(b) to show the transmission ratio between the transport channel through the YSR state and the reference BCS gap.We can clearly see sizeable coherence peaks at ±( t +  s ) indicating that the reference BCS channel is much stronger than the YSR channel ( t =  s = 715 μeV).
e Josephson e ect can be seen throughout the range of conductance values shown in Fig. S1. Figure S2 shows the analysis of the Josephson e ect for the second data set in analogy to Fig. 2 of the main text.In Fig. S2(a), we see the evolution of the Josephson spectra as function of conductance.Each spectrum is shown in a range of ±60 μeV and horizontally o set by the corresponding conductance value.We can clearly see an increase in the signal, but which is again modulated in the region of the square bracket in Fig. S2(a).We plot the switching current  S (as indicated in Fig. S2(a)) as a function of conductance in Fig. S2(b) (blue line).e corresponding YSR energies are plo ed as a red line.We indicate the position of the QPT by a vertical dashed line, where the YSR energy has a minimum.e switching current also changes in the region of the QPT as in Fig. 2 of the main text.
In Fig. S2(c), the √  S  N product of the experimental data is plo ed (blue line) in relation to a reference junction (yellow line) and a t (orange line).For this data set, the renor-malization of the normal state resistance  YSR N = 1/(1.19N ) through the YSR state is not as pronoounced because the transmission through the YSR channel is much reduced compared to the data set in the main text.For the t, we use the same e ective temperature of  e = 75 mK as for the data in the main text.For the relative channel transmission, we nd 77.7% and 22.3% for the BCS and the YSR channel, respectively.e overall agreement between experimental data and t is excellent.

Figure 1 :
Figure 1: Atomic YSR state.(a) Schematic of the tunnel junction.e YSR impurity is at the tip with two transport channels (BCS and YSR) indicated as dashed lines.(b) Phase diagram of the YSR system as function of impurity-superconductor coupling and level (particle-hole) asymmetry.(c) Schematic of the free spin and the screened spin regime.In the screened regime, a Cooper pair is broken changing the overall parity of the system.(d) Di erential conductance spectra as function of bias voltage (-axis) and conductance (-axis).e prominent peaks are the YSR states, while the coherence peaks are only barely visible.(e) Horizontal line cut through (d) to show the prominent YSR peaks.(f) Half-width at half maximum  K of the Kondo peak in the same junction at a magnetic eld of 1.5 T, when superconductivity is quenched.e Kondo spectra are shown in the inset.

Figure 2 :
Figure 2: Josephson E ect.(a) Josephson spectra  ( ) in a range of ±60 μeV shi ed horizontally by the conductance at which they were measured.e horizontal bracket indicates the region, where the evolution deviates from the conventional Ambegaokar-Barato formula.e quantum phase transition (QPT) is indicated by the vertical dashed line.(b) e switching current  S , which is the current maximum indicated in (a) as function of the normal state conductance  N (blue line).e square dependence at low and high conductance is indicated by dashed lines (labeled by ∝  2 N ).e YSR energy as function of conductance is shown as a blue line.e minimum indicates the QPT (vertical dashed line).(c) is graph shows √  S  N (blue line) of the data in (b), which is proportional to  C  N for a harmonic energy-phase relation ( N : renormalized resistance (see text),  C : critical current).A reference junction without any YSR states is shown as a yellow line indicating the expected evolution according to the Ambegaokar-Barato formula.

Figure 3 :
Figure 3: Supercurrent Reversal.(a) Energy-phase relation for the BCS channel (red) and the YSR channel (blue).For the Josephson current only the oscillation is relevant, but not the energy o set.(b) shows a zoom-in to the oscillation for the two channels.e upper two panels in (b) show the in-phase oscillation in the screened spin regime.elower two panels in (b) show the out-of-phase oscillation in the free spin regime, which is indicative of the supercurrent reversal.e STM is not sensitive of the sign of the supercurrent, but the concomitant change in magnitude is clearly observable.(c) Probabilities for the system to be in the ground state (blue) or the excited state (red) as function of YSR state energy.(d) e √  S  N product comparing experimental data with a theoretical t. e only free parameters are the temperature, which de nes the width of the QPT and the relative channel transmission, which de nes the step height.

Figure 4 :
Figure 4: Cooper Pair Tunneling.etunneling process in the free spin and in the screened spin regime is shown from the initial to the nal state via an intermediate (virtual) state in the excitation picture.e order is set by the numbered red arrows.e total spin  tot describes the total spin of the YSR system including the spin of the impurity.(a) In the free spin regime, the order of the spins is exchanged compared to the intial state, which results in the supercurrent reversal.(b) In the screened spin regime, the order of the spins is retained, such that there is no sign change in the supercurrent.

Figure S2 :
Figure S2: Josephson E ect (Data Set #2) (a) Josephson spectra  ( ) in a range of ±60 μeV shi ed horizontally by the conductance at which they were measured.e horizontal bracket indicates the region, where the evolution deviates from the conventional Ambegaokar-Barato formula.e quantum phase transition (QPT) is indicated by the vertical dashed line.(b) e switching current  S , which is the current maximum indicated in (a) as function of the normal state conductance  N (blue line).e square dependence at low and high conductance is indicated by dashed lines (labeled by ∝  2 N ).e YSR energy as function of conductance is shown as a blue line.e minimum indicates the QPT (vertical dashed line).(c) is graph shows √  S  N (blue line) of the data in (b), which is proportional to  C  N for a harmonic energyphase relation ( N : renormalized resistance (see text),  C : critical current).A reference junction without any YSR states is shown as a yellow line indicating the expected evolution according to the Ambegaokar-Barato formula.