Bound states in superconductors and interfaces

Hybrid devices where Coulomb blockaded quantum dots are coupled to superconducting electrodes is an ideal experimental platform for exploring superconductors with impurities. Semiconductor nanowires and carbon nanotubes have been particularly important since these materials can readily be interfaced with different superconductors while allowing for a large degree of device tunability [1]. Their intrinsic properties can also be exploited, for example when strong spin-orbit interactions allow for topological phases [2]. I will firstly provide a survey of the progress in materials and illustrate how various device geometries can easily be realized, ranging from extended 1D hybrid wires to highly confined dots [3]. In the second part of the talk, I will discuss recent examples, including detailed transport spectroscopy of Yu-Shiba-Rusinov (YSR) bound states in quantum dots [4] and the behavior of double quantum dots [5,6]. The latter system opens up for correlated superconducting quantum impurity systems of higher spin. Quantum dot stability diagrams and the ground state phase diagrams extracted from experiments are found to match theoretical models based on NRG as well as simpler zero-bandwidth approaches [5,7]. The examples serve to demonstrate YSR as a useful framework for understanding hybrid dot systems. References [1] Hybrid superconductor–quantum dot devices, S. De Franceschi, L. Kouwenhoven, C. Schönenberger and W. Wernsdorfer, Nat. Nanotech. 5, 703 (2010). [2] Majorana zero modes in superconductor-semiconductor heterostructures, Lutchyn, R. M., Bakkers, E. P. A. M., Kouwenhoven, L. P., Krogstrup, P., Marcus, C. M., Oreg, Y., Nat. Rev. Mater. 3, 52 (2018). [3] Hybrid Superconducting Devices Based on Quantum Wires, K. Grove-Rasmussen, T. S. Jespersen, A. Jellinggaard, J. Nygård, in Oxford Handbook of Small Superconductors, Oxford University Press (2017). [4] Tuning Yu-Shiba-Rusinov states in a quantum dot, A. Jellinggaard, K. Grove-Rasmussen, M.H. Madsen, and J. Nygård, Phys. Rev. B 94, 064520 (2016). [5] Yu-Shiba-Rusinov screening of spins in double quantum dots, K. Grove-Rasmussen, G. Steffensen, A. Jellinggaard, M. H. Madsen, R. Žitko, J. Paaske, J. Nygård, Nat. Commun. 9, 2376 (2018). [6] Supercurrent in a double quantum dot, J. C. Estrada Saldaña, A. Vekris, G. Steffensen, R. Žitko, P. Krogstrup, J. Paaske, K. Grove-Rasmussen, J. Nygård, Phys. Rev. Lett. 121, 257701 (2018). [7] Two-Impurity Yu-Shiba-Rusinov States in Coupled Quantum Dots, J. C. Estrada Saldaña, A. Vekris, R. Žitko, G. Steffensen, P. Krogstrup, J. Paaske, K. Grove-Rasmussen, J. Nygård, arXiv:1812.09303

We coupled single multi-gated nanowires to superconductors to obtain either single or double quantum dot Josephson junctions. When half-filled, these single-level quantum dots behave as highly tunable artificial quantum impurities. In a first set of experiments, we investigated how a magnetic field affects the supercurrent through a single impurity defined in InAs nanowires coupled to tantalum leads (Estrada Saldaña et al, arXiv:1801.01855 (2018)). In a different experiment, we used an InAs nanowire with epitaxially grown aluminium leads to realize the superconducting analogy of the two-impurity Kondo model using double quantum dots in series. In this analog, the Kondo effect is prevented by the superconducting gap, but the supercurrent reveals that spin screening still occurs via the Yu-Shiba-Rusinov mechanism (Estrada Saldaña et al, ArXiv:1812.09303 (2018); see also Estrada Saldaña et al, Phys. Rev. Lett., 121, 257701 (2018)). Our work is a stepping stone in the use of hybrid nanowires for the realization of correlated superconducting quantum impurity systems of higher spin, which may bring a new light into standing Kondo problems.

In this presentation I will review our recent experiments on semiconducting nanowires (NWs) and carbon nanotubes (CNTs) in close proximity to one or two superconducting contacts. To perform well-controlled transport spectroscopy in such systems, we introduce two new device types, namely InAs NWs with atomically precise crystal phase engineered in-situ grown axial tunnel barriers, and CNTs fully encapsulated in hexagonal Boron Nitride, both contacted by bulk superconductors. In the latter devices we obtain very long and clean quantum dots, with multiple subgap states and a non-monotonic magnetic field dependence of the critical supercurrent. For the NW systems, we use the axial tunnel barriers to probe, for example, the formation of the superconducting proximity gap in a long NW segment, or the hybridization of Andreev-type bound states with quantum dot states. We believe that these device architectures will be crucial for investigating and characterizing the large variety of new topologically trivial and non-trivial subgap states expected to form in such systems.

Superconductor - quantum dot - superconductor (SQS) devices in a transistor geometry allow for studying the competition of a tunable single magnetic impurity and the superconducting ground state of the leads. We perform transport measurements on an electromigrated junction in the normal state of the leads, in which we characterize a strongly coupled quantum dot as a function of level depth, temperature and magnetic field. In particular, we extract the gate-dependent Kondo temperature scale of the device. In the superconducting state of the leads, we observe gate-dispersing sub-gap resonances in the transport data, pointing to Yu-Shiba-Rusinov states. A zero-energy crossing of these bound states is seen as the oddly occupied level is lowered, indicating that the transition from a screened to an unscreened local magnetic ground state takes place within the odd-charge diamond, and is due to the competition with the Kondo effect. Our sub-gap spectroscopies are reproduced by calculations that include interaction effects. Finally, we discuss the temperature dependence of the transport processes through the sub-gap levels, such as thermal replica of Yu-Shiba-Rusinov states.

A quantum dot attached to a superconducting contact provides a tunable analogue of a magnetic impurity in a superconductor. Such a system is characterized by subgap states known as Andreev or Yu-Shiba-Rusinov states. In this work, we employ nanowire-based quantum dots coupled strongly to a superconductor and weakly to a normal metal tunnel probe to study Andreev states by tunneling spectroscopy. We first exploit the ability to tune the tunnel coupling between a quantum dot and a superconductor to study the scaling of Andreev states with respect to the relevant parameter space, thereby obtaining an experimental phase diagram of the possible ground states of the system: a spin singlet or a magnetic doublet. By applying external magnetic fields, we also investigate the spin texture of the Andreev states. Finally, we study how a mesoscopic tunnel probe can lead to numerous replicas in the resulting Andreev spectrum.

Magnetic impurities on superconductors induce via exchange scattering local bound states, so called Yu-Shiba-Rusinov states, in their vicinity. Depending on the coupling strength between the impurity and the substrate, this state is either occupied or not, corresponding to a screened- and free-spin ground state, respectively. Here, we use the flexibility of a Fe-porphin molecule on a Pb(111) surface to tune continuously and reversibly between these ground states, thereby resolving this quantum phase transition on a single impurity. By approaching the STM tip toward the molecule we modify on the one hand the bound state energy and on the other hand the junction transport properties so that we can resolve the YSR excitations by single-electron as well as by (multiple) Andreev reflections. [1] [1] Farinacci et al., PRL 121, 196803 (2018)

Exchange scattering at a magnetic impurity adsorbed on a superconducting substrate induces low-energy bound states, known as Yu-Shiba-Rusinov (YSR) states (1-3). These are low energy bound states inside the superconducting energy gap locally induced by magnetic exchange scattering. Here, we investigate YSR states of Fe atoms on the surface of 2H-NbSe$_2$ using low temperature scanning tunneling microscopy and spectroscopy. 2H-NbSe$_2$ belongs to the class of transition metal dichalcogenides and is a layered van der Waals material with strong 2D character. In this material, superconductivity coexists with a charge density wave (CDW) at low temperatures. As the CDW is incommensurate with the lattice there are different local charge modulations present on the surface. Hence, adatoms sitting in identical adsorption sites differ in their position relative to the CDW. We show that the local properties of the charge density wave affect the symmetry of the spatial distribution and the energy of the YSR resonances. (1) L. Yu, Acta Phys. Sin. 21, 75 (1965). (2) H. Shiba, Prog. Theor. Phys. 40, 435-451 (1968). (3) A.I. Rusinov, Sov. Phys. JETP 29, 11016 (1969).

Parafermions are interacting generalizations of Majorana fermions: interface bound states in topological superconductors that are associated with a degeneracy of the physical states. Going beyond Majorana, parafermions allow to encode larger degeneracies in pairs of bound states. We present a mapping between lattice Z4 parafermions and lattice spin-1/2 fermions which preserves the locality of operators with Z4 symmetry. Based on this mapping, we construct an exactly solvable, local, and interacting one-dimensional fermionic Hamiltonian which hosts zero-energy modes obeying parafermionic algebra. We numerically show that this parafermionic phase remains stable in a wide range of parameters, and discuss its signatures in the fermionic spectral function.

We present a Fermi liquid approach to superconducting Kondo problems applicable when the Kondo temperature is large compared to the superconducting gap. To illustrate the theory, we study the current-phase relation and the Andreev level spectrum for an Anderson impurity between two s-wave superconductors. In the particle-hole symmetric Kondo limit, we find a 4\pi periodic Andreev spectrum. The 4\pi periodicity persists under a small voltage bias which however causes an asymmetric distortion of Andreev levels. The latter distinguishes the present 4\pi effect from the one in topological Majorana junctions.

We implement the Abelian bosonization method to study a localized magnetic impurity coupled to a superconductor at zero temperature. Interestingly, within the bosonic representation, Shiba states become related to topological soliton solutions of the classical equations of motion. The topological “charge” and en- ergy of the solitons correspond, respectively, to the spin-polarization and energy of the ground-state. In addition to allowing a simple and direct rederivation of well-known results, the bosonic representation allows to go beyond the non-interacting limit and to study the effect of strong interactions on the low-energy properties of the system. Our results could have potential impact in the recent theoretical and experimental developments, where Shiba states have been observed, and in recent works where superconductivity is induced in semiconducting nanowires by the means of proximity effect.

We analyze the Andreev transport properties of a Kondo-correlated double quantum dot attached to one superconducting and two normal leads. In the absence of superconductor, depending on the quantum dots’ occupations, such system can exhibit the SU(2) and SU(4) Kondo effects. We show that finite superconducting pairing suppresses the SU(2) Kondo effect, which can however reemerge for relatively large values of couplings to the superconductor. In the SU(4) Kondo regime, a crossover from the SU(4) to the SU(2) Kondo state is found as the coupling to superconductor is enhanced. We also examine the case when one of the normal leads is decoupled. We show that such system can exhibit the two-stage Kondo screening caused solely by the proximity of the superconductor which induces nonlocal pairing by Andreev reflection processes. The analysis is performed by means of the density-matrix numerical renormalization group method.

Impurity engineered states are the key component of modern electronics. We expect impurity engineered states will continue to be an important ingredient in quantum technologies beyond Si. New platforms that take advantage of impurity engineered bands such as superconductors and Dirac Materials are emerging. In this talk I will review the status and future perspectives of the field of impurity bound states in superconductors and will discuss striking similarities and important differences between defect states in superconductors and Dirac materials. I will highlight the recent progress made in our understanding of Yu-Shiba-Rusinov states in superconductors and topological states and illustrate the proposed induction of novel odd frequency Berezinskii pairing by YSR impurities. I will also discuss the universal nature of impurity resonances in unconventional SC and Dirac Materials and the implications these resonances have on functional properties of doped Dirac materials. Work supported by ERC, KAW and Villum Center for Dirac Materials.

Magnetic impurities on the surface of spin-orbit coupled but otherwise conventional superconductors offer the possibility to create topological phases with Majorana bound states (MBSs) without having to apply an external magnetic field. In this talk I will present some of our resent results in modeling both magnetic wires and islands on the surface of spin-orbit coupled superconductors, including an accurate treatment of the superconducting state through a self-consistent approach. In particular, I will show how wire end point MBSs very strongly hybridize with the in-gap Yu-Shiba-Rusinov (YSR) states, causing large oscillations in the MBSs energies. Still, by treating the MBSs as topological boundary modes dependent only on the effective mass gap, we arrive at a fully parameter-free fitting of the Majorana localization length, which stays very short. I will also show how the wire end point MBSs are very robust against disorder, despite individual YSR states being extremely sensitive to disorder.

Individual local magnetic moments act destructively on Cooper pairs, leading to discrete spin-polarized states inside the superconducting energy gap as predicted by Yu, Shiba and Rusinov (YSR) [1,2,3]. Rusinov suggested that around magnetic atoms the decaying YSR wavefunction should have a spatially oscillating structure. We experimentally evidenced YSR bound states with spatially oscillating particle-hole asymmetry extending tens of nanometers from individual iron atoms embedded in a 2H-NbSe2 crystal and in Pb/Si(111) monolayer [4]. We theoretically elucidate how reduced dimensionality enhances the spatial extent of these bound states and describe their energy and spatial structure. The large spatial extent of the YSR states in 2D superconductors can be use with profit for stabilizing topological phases in arrays of magnetic impurities. Zero bias anomalies localized at the edge of a Fe chain of Pb(110) were recently interpreted as fingerprints of the emergence of topological superconductivity [5]. In contrast to the 1D case, two-dimensional superconductors have a one-dimensional boundary which naturally leads to chiral Majorana edge states characterized by a Dirac-like dispersion. We have recently observed some hint of dispersive Majorana edge states in a single atomic layer Pb superconductor. This material has strong triplet correlations but is not topological by itself [6]. We will show that by applying a Zeeman field with the help of a buried Co-Si nano-magnet one can provoke a transition to a topological state [7]. In addition to their dispersive edge states, 2D topological superconductors are also supposed to support localized Majorana bound states in topological defects. We will show that some recent measurements seem to support this theoretical prediction [8]. References [1] L. Yu, Bound state in superconductors with paramagnetic impurities, Acta Phys. Sin. 21, 75 (1965). [2] H. Shiba, Classical Spins in Superconductors, Prog. Theor. Phys. 40, 435 (1968). [3] A.I. Rusinov, On the theory of gapless superconductivity in alloys containing paramagnetic impurities, JETP Lett. 9, 85 (1969). [4] G. Ménard et al., Coherent long-range magnetic bound states in a superconductor, Nature Physics 11, 1013 (2015) [5] S. Nadj-Perge et al., Science 346, 602 (2014) [6] C. Brun et al., “Remarkable effects of disorder on superconductivity of single atomic layers of lead on silicon” Nature Phys. 444, 10 (2014). [7] Gerbold C. Ménard, et al., Two-dimensional topological superconductivity in Pb/Co/Si (111), Nature communications 8, 2040 (2017) [8] G. C. Ménard, et al., Isolated pairs of Majorana zero modes in a disordered superconducting lead monolayer, arXiv:1810.09541 (2018)

Chains of magnetic atoms on high-Z s-wave superconductors can exhibit topological superconductivity and may host Majorana bound states at their ends [1-8]. Most previous experimental work towards the realization of such systems focused on the self-assembly of chains [9,10] where a full control of the atomic composition in the chain is difficult to achieve. In this study, we use superconducting Ta and Re substrates which enable scanning tunneling microscope tip induced assembly of chains with a control on the chemical composition and position of each individual atom in the chain [11,12]. We investigate the evolution of the Yu-Shiba-Rusinov states [13] of the building blocks of the chains, the single atoms [14], into bands in the two extreme regimes, i.e. the dilute chain and wire limits. For particular combinations of substrate and adatom species, we find indications for topological superconductivity in the wire limit [12]. We acknowledge funding by the ERC via the Advanced Grant ADMIRE (No. 786020). [1] T.-P.- Choy et al. PRB 84, 195442 (2011); [2] I. Martin et al., PRB 85, 144505 (2012); [3] Klinovaja et al., PRL 111, 186805 (2013); [4] F. Pientka et al., PRB 88, 155420 (2013); [5] B. Braunecker et al., PRL 111, 147202 (2013); [6] M. M. Vazifeh et al., PRL 111, 206802 (2013); [7] J. Li et al. PRB 90, 235433 (2014); [8] M. Schecter et al., PRB 93, 140503 (2016); [9] S. Nadj-Perge et al., Science 346, 6209 (2014); [10] M. Ruby et al., Nano Letters 17, 4473, (2017); [11] A. Kamlapure et al., Nature Comm. 9, 3253 (2018); [12] H. Kim et al., Science Advances 4, eaar5251 (2018); [13] A. V. Balatsky et al., Rev. Mod. Phys. 78, 373 (2006); [14] L. Cornils et al., PRL 119, 197002 (2017).

We review Weyl singularities of the Andreev bound state spectrum in multi-terminal supeconducting nanostructures and discuss the recent developments in the field: Weyl disks, coupled Weyl points, spin-weyl double qubits.

We consider a new model system supporting Majorana zero modes based on semiconductor nanowires with a full superconducting shell. We demonstrate that, in the presence of spin-orbit coupling in the semiconductor induced by a radial electric field, the winding of the superconducting order parameter leads to a topological phase supporting Majorana zero modes. The topological phase persists over a large range of chemical potentials and can be induced by a predictable and weak magnetic field piercing the cylinder. The system can be readily realized in semiconductor nanowires covered by a full superconducting shell, opening a pathway for realizing topological quantum computing proposals.

Magnetic skyrmions are nanoscale particle-like spin configurations that are efficiently created and manipulated. They hold great promises for next-generation spintronics applications. In parallel to these developments, the interplay of magnetism, superconductivity and spin-orbit coupling has proved to be a versatile platform for engineering topological superconductivity predicted to host non-abelian excitations, Majorana zero modes (MZMs). In this talk, I will focus on the theoretical analysis of magnetic skyrmions proximitized by conventional superconductors. I will show that a topological superconducting phase can emerge in these systems and uncover a whole flat band of these modes on the edge of the magnetic texture, in contrast to a previously reported MZM in the core of the skyrmion [1]. I will discuss in details the origin of these MZMs by relating this problem to the the extensively-studied Rashba nanowire model. We have found that these modes are remarkably stable to electronic and geometric perturbations which we investigate by a combination of analytical arguments and numerical tight-binding calculations. Additionally, this analysis reveals that the number of MZMs on the edge scales linearly with its perimeter. I will then discuss possible experimental realizations and consequences of this phenomenon and argue that this system is suitable for the realization of the topological Kondo effect and of electron teleportation [2]. [1] G. Yang, P. Stano, J. Klinovaja & D. Loss, PRB 93, 224505 (2016). [2] M. Garnier, A. Mesaros, P. Simon, submitted.

Recent progress in experiments and modelling has improved our understanding of subgap states in superconductor-quantum dot systems. I will here present results on quantum states in hybrid superconductor-semiconductor (InAs) nanowire devices. When coupling a superconductor (S) to a quantum dot (QD) system, the confined electron spin may be screened by quasiparticles above the gap in the superconductor, similarly to the Kondo effect for QDs with normal electrodes. This screening results in so-called Yu-Shiba-Rusinov (YSR) sub-gap states. We explore such states in bottom-gated hybrid superconductor-nanowire single and double dot devices. The YSR states can be modelled by using a simple zero-band width approximation, which we compare to measurements on single [1] and double quantum dots (DQD) [2] coupled to a single superconductor, i.e. the N-QD-S and N-DQD-S configurations, respectively. In the latter case, the characteristic double dot honeycomb stability diagram is shown to be qualitatively different in the superconducting state. [1] A. Jellinggaard, K. Grove-Rasmussen, M.H. Madsen, J. Nygård, “Tuning Yu-Shiba-Rusinov states in a quantum dot”, Phys. Rev. B 94, 064520 (2016) [2] K. Grove-Rasmussen, G. Steffensen, A. Jellinggaard, M. H. Madsen, R. Žitko, J. Paaske & J. Nygård, “Yu-Shiba-Rusinov screening of spins in a double quantum dots”, Nat. Comm. 9:2376 (2018)

We probed the current-phase relation in a clean carbon nanotube quantum dot, close to orbital degeneracy, in a regime of strong competition between local electronic correlations and superconducting proximity effect. We show experimentally that the 0-pi transition can then be controlled by the superconducting phase [1] and that the nature of the transition depends crucially on the occupation and the width of the orbital levels [2]. When the transport of Cooper pairs takes place through only one of these levels, we find that the phase diagram of the phase-dependent 0-pi transition is a universal characteristic of a discontinuous level-crossing quantum transition at zero temperature. We have also measured the high frequency emission of a carbon nanotube junction due to the AC Josephson effect in the same regime of competition of the Kondo and superconducting proximity effect. We show that, strikingly, whereas the DC supercurrent is enhanced by the Kondo effect, the AC Josephson effect is not necessarily enhanced. This indicates possible signature of the dynamics of the Kondo screening. [1] R. Delagrange, D. J. Luitz, R. Weil, A. Kasumov, V. Meden, H. Bouchiat, R. Deblock, Phys. Rev. B 91, 241401(R) (2015). [2] R. Delagrange, R. Weil, A. Kasumov, M. Ferrier, H. Bouchiat and R. Deblock, Phys. Rev. B 93, 195437 (2016).

An overview on the theoretical understanding of the equilibrium Josephson current through a correlated quantum dot is presented. Results obtained by elaborate many-body methods are compared to experiments. The quantitative agreement indicates that the experimental systems can be understood in terms of the simplified model.

I will present an overview of relatively simple theoretical approaches developed in our group in past several years and applied to the problem of description of correlated quantum dots attached to the BCS superconducting leads. As a thorough Quantum Monte Carlo analysis [1] of the experimental data [2] showed realistic experimental setups can be even quantitatively captured by the Single Impurity Anderson Model with superconducting leads. Pioneering semi-analytical approaches have not matched the so far employed heavy numerical tools such as Numerical Renormalization Group and/or Quantum Monte Carlo in the ability of quantitatively predicting the properties of this model. However, we have shown recently that self-consistent perturbation expansion up to the second order in the interaction strength [3,4] yields at zero temperature and for a wide range of other parameters excellent results for the position of the 0 − π impurity quantum phase transition boundary and the Josephson current as well as the energy of Andreev bound states in the 0-phase. This method can be also extended to the three-terminal situation with an extra normal lead corresponding to the experimentally interesting STM setup [5], where it allows to study phase-dependent Kondo physics. Furthermore, we have discovered exact identities connecting symmetric and asymmetric coupling situations which significantly reduce computational requirements in experimentally generic asymmetric setups [6] and provided simple approximate analytical formulas for the fitting of the phase boundaries from finite-temperature experimental data [7]. [1] David J. Luitz, Fakher F. Assaad, Tomáš Novotný, Christoph Karrasch, and Volker Meden, Understanding the Josephson current through a Kondo-correlated quantum dot, Phys. Rev. Lett. 108, 227001 (2012) [2] H. Ingerslev Jørgensen, T. Novotný, K. Grove-Rasmussen, K. Flensberg, and P. E. Lindelof, Critical Current 0-π Transition in Designed Josephson Quantum Dot Junctions, Nano Lett. 7 (8), 2441 (2007) [3] M. Žonda, V. Pokorný, V. Janiš, and T. Novotný, Perturbation theory of a superconducting 0-π impurity quantum phase transition, Scientific Reports 5, 8821; DOI:10.1038/srep08821 (2015) [4] M. Žonda, V. Pokorný, V. Janiš, and T. Novotný, Perturbation theory for an Anderson quantum dot asymmetrically attached to two superconducting leads, Phys. Rev. B 93, 024523 (2016) [5] T. Domański, M. Žonda, V. Pokorný, G. Górski, V. Janiš, and T. Novotný, Josephson-phase-controlled interplay between correlation effects and electron pairing in a three-terminal nanostructure, Phys. Rev. B 95, 045104 (2017) [6] Alžběta Kadlecová, Martin Žonda, and Tomáš Novotný, Quantum dot attached to superconducting leads: Relation between symmetric and asymmetric coupling, Phys. Rev. B 95, 195114 (2017) [7] Alžběta Kadlecová, Martin Žonda, Vladislav Pokorný, and Tomáš Novotný, Practical Guide to Quantum Phase Transitions in Superconducting Quantum Dots, arXiv:1807.07017 (under review in Phys. Rev. Applied)

Using a circuit-QED setup, we measure the spectrum of excitations in an InAs nanowire connected to superconducting electrodes. Some transitions correspond to Andreev levels spin-split by spin-orbit interaction.

Topology is providing new insight into condensed matter physics problems. Concepts like Chern numbers and their relation to physical phenomena have become very familiar, but actually, key quantities like the quantum geometric tensor, which provides a much deeper information about quantum states, remain experimentally difficult to access. Recently it has been shown that multiterminal superconducting junctions constitute an ideal playground to mimic topological systems in a controlled manner. Here, we theoretically study the spectrum of Andreev bound states in topological Josephson matter and demonstrate that the full information of the quantum geometric tensor of the ground state manifold can be extracted with the help of microwave spectroscopy. In particular, we develop the concept of artificially polarized microwaves, which can be used to obtain both the quantum metric tensor and the Berry curvature. The quantized integrated absorption provides a direct evidence of topological quantum properties of the Andreev states.

We investigate the Josephson radiation of a voltage-biased diffusive normal metal bridge between two superconductors. The higher harmonics of the Josephson radiation strongly depend on the non-equilibrium population of Andreev bound states that is generated both by the dc and ac bias voltages. The combined effect of dc-bias and microwave excitation leads to strong Josephson emission even at higher temperatures, where the equilibrium Josephson current is suppressed by thermal excitation.

The formation of electron pairs is a prerequisite of superconductivity. The fermionic nature of electrons yields four classes of superconducting correlations with definite symmetry in spin, space and time. Here, we suggest double quantum dots coupled to conventional s-wave superconductors in the presence of inhomogeneous magnetic fields as a model system exhibiting unconventional pairing. Due to their small number of degrees of freedom, tunable by gate voltages, quantum-dot systems are ideal to gain fundamental insight in unconventional pairing. We propose detection schemes for superconducting triplet correlations, based on either Josephson or Andreev spectroscopy.

In this talk I will address the problem of supercurrent instability in highly transparent weak links. I will demonstrate that at subgap energies quantum behavior of superconducting weak links can be exactly accounted for by an effective Hamiltonian for a Josephson particle in a quantum dissipative environment formed by Andreev levels. I will identify several macroscopic quantum tunneling (MQT) regimes and predict a non-trivial non-monotonous dependence of the switching current distribution on temperature which can be observed in MQT experiments with transparent superconducting weak links. I will also argue that intrinsic dissipation in such weak links can be studied by means of qubit spectroscopy. Varying either the bias current or magnetic flux through a superconducting ring in the rf-SQUID setup, one can tune the level splitting value close to the bottom of the Josephson potential well. Monitoring the qubit energy relaxation time one can probe the spectral density of the effective dissipative bath and unambiguously identify the contribution emerging from Andreev levels.

Jean-Guy Caputo1, Romain Danneau2, Benoît Douçot3, Régis Mélin4 1 Laboratoire de Mathématiques, INSA de Rouen, Avenue de l’Université F-76801 Saint-Etienne du Rouvray, France 2Institute of Nanotechnology, Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany 3LPTHE, Sorbonne Université and CNRS, Paris, France 4Centre National de la Recherche Scientifique, Institut NEEL, F-38042 Grenoble Cedex 9, France and Université Grenoble-Alpes, Institut NEEL, F-38042 Grenoble Cedex 9, France Since the early 1960’s, the Josephson effect has attracted continuous interest and its development over the years has led to major applications in quantum information and technologies. It occurs when two superconductors connect a non-superconducting material and its physical mechanism can be described via the notion of Andreev bound states. Many physical properties of the Josephson junctions, such as the value of the Josephson current, depend on the energy-phase relation of the Andreev bound states at zero bias voltage and finite phase drop across the junction. In this talk, we will answer the following question: What is the fate of Andreev bound states in the presence of voltage biasing? We have recently provided theoretical insight on this issue for three-terminal superconductor-quantum dot-superconductor junctions. [1] An analogy between the Josephson effect and basic notions of band theory can be developed. Indeed, the nonequilibrium Andreev bound states resemble to the so-called “ladders of Wannier-Stark resonances” [2] in semiconducting superlattices which were observed by optical spectroscopy [3,4] in the 1980s. New numerical and theoretical results will be presented on the close connection between the finite frequency noise spectrum (a physical observable) and the Floquet spectrum (a mathematical object defined in terms of the resolvent). [5] The sign of the resonances in finite frequency noise as a function of inverse voltage depends on the overlap between semi-classical Floquet wave-functions, as well as on populations of the Floquet states. [6] These two recent works [5,6] support our strategy to realize finite frequency noise measurements on two-terminal superconductor-quantum dot-superconductor devices based on top-gated van der Waals bilayer graphene-boron nitride heterostructures, in order to demonstrate experimentally emergence of avoided crossings in the Floquet spectrum as a function of inverse bias voltage. [1] R. Mélin, J.-G. Caputo, K. Yang and B. Douçot, Simple Floquet-Wannier-Stark-Andreev viewpoint and emergence of low-energy scales in a voltage-biased three-terminal Josephson junction, Phys. Rev. B 95, 085415 (2017). [2] G.H. Wannier, Wave-functions and effective Hamiltonian for Bloch electrons in an electric field, Phys. Rev. 117, 432 (1960). [3] E.E. Mendez, F. Agullo-Rueda and J.M. Hong, Stark Localization in GaAs-GaAlAs Superlattices under an Electric Field, Phys. Rev. Lett. 60, 2426 (1988). [4] E.E. Mendez and G. Bastard, Wannier-Stark ladders and Bloch oscillations in superlattices, Phys. Today 46, 34 (1993). [5] R. Mélin, J.-G. Caputo, B. Douçot and R. Danneau, in preparation (2019). [6] B. Douçot, K. Yang and R. Mélin, in preparation (2019).

R. Seoane Souto, A. Martín-Rodero and A. Levy Yeyati Departamento de Física Teórica de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) and Instituto Nicols Cabrera, Universidad Autónoma de Madrid E-28049 Madrid, Spain The shot noise and the higher order cumulants contain valuable information about interactions and correlations between electrons in nanodevices. While these studies have traditionally been restricted to the stationary case, recent technological advances in the direction of the single electron detection have attracted an increasing interest in the time resolved full counting statistics (FCS). On the other hand, superconducting nanodevices are of central interest as building blocks in quantum technologies. In this kind of devices, an unexplained excess of quasiparticles has been observed, which set the limit for possible applications. In superconducting atomic junctions these quasi-particles can decay to the Andreev bound states, giving rise to long lived odd parity states [1]. In this presentation I will focus in the dynamics of the charge transfer through nano-junctions coupled to superconducting electrodes. Under rather general conditions, the system gets trapped in a metastable state, characterized by a non-equilibrium population of the Andreev bound states. Although the trapped quasiparticles lead to a peculiar time dynamics, the quantum state of the system cannot be inferred only from the evolution of the single particle observables. Instead, the information provided by the full counting statistics is needed to fully characterize the state [2]. Finally, I will also demonstrate that the evolution of the roots of the FCS generating function, recently measured experimentally for the first time in a superconducting nano-junction [3], contains full information about the system transport properties. Their evolution provide a way for identifying dynamical quantum phase transitions, much in the same way as the Lee Yang zeros in the equilibrium statistical mechanics [4]. I will also discuss when a simplified description of the system dynamics based on the dominant zeros is possible and the kind of information that can be obtained from the lower current cumulants. [1] M. Zgirski, L. Bretheau, Q. Le Masne, H. Pothier, D. Esteve, and C. Urbina, Phys. Rev. Lett. 106, 257003 (2011). [2] R. S. Souto, A. Martín-Rodero, and A. L. Yeyati, Phys. Rev. Lett. 117, 267701 (2016); Phys. Rev. B 96, 165444 (2017). [3] K. Brandner, V. F. Maisi, J. P. Pekola, J. P. Garrahan and C. Flindt, Phys. Rev. Lett. 118, 180601 (2017). [4] C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952); Phys. Rev. 87, 410 (1952).

We consider a quantum dot coupled to both superconducting and ferromagnetic electrodes, and study the triad interplay of the Kondo effect, superconductivity, and ferromagnetism, any pair of which compete with and suppress each other. We find that the interplay leads to a mixed-valence quantum phase transition, which is usually a crossover rather than a true transition. At the transition, the system changes from the spin doublet to singlet state. The singlet phase is adiabatically connected (through crossovers) to the so-called ’charge Kondo state’ and to the superconducting state. We analyze in detail the physical characteristics of different states and propose that the measurement of the cross-current correlation and the charge relaxation resistance can clearly distinguish between them.

In this contribution we investigate how the transport properties of a superconductor (SC) is affected when an artificial impurity (quantum dot) is attached to the SC. A tunnel probe is connected to the SC at a distance from the impurity larger than the SC coherence length. We find a significant enhancement - called non-local signal - in the tunnel-probe current when the impurity is tuned to the charge degeneracy point. Direct transport measurement on the quantum dot reveals the presence of Yu-Shiba-Rusinov state. The magnitude of the non-local signal increases with up to a factor of 20 with applied external magnetic field. We compare our experimental results with NRG simulation, which reproduces the experimental findings. This work is supported by National Research Development and Innovation Office of Hungary (Project No. 2017-1.2.1-NKP-2017-00001).

Hybrid superconductor-semiconductor devices have drawn large attention both theoretically and experimentally in the recent years. In the strong coupling limit such devices can host exotic states, like Andreev bound states, Majorana bound states or parafermions. In the weak coupling limit when the semiconductor hosts two quantum dots in a three terminal geometry, the device can act as a source of entangled electron pairs, it splits the Cooper pairs via crossed Andreev reflection. Here we present the signatures of non-local couplings in the strongly coupled limit in the Cooper pair splitter geometry. The strong coupling to the superconductor gives rise to local Andreev bound states on the dots, while having non-local couplings between the dots leads to the hybridization of the Andreev bound states, forming a non-local state, the Andreev molecular state. We distinguish three different non-local couplings, the crossed Andreev reflection, the elastic cotunneling, where an electron tunnels through the superconductor and the direct interdot tunnel coupling. We show that one can tell the dominant coupling by measuring the stability diagram of the two quantum dots simultaneously. Further we discuss the triplet blockade, which is usually attributed to the crossed Andreev reflection, is not exclusively present in case of finite crossed Andreev coupling. Recent experiments on semiconducting nanowires showed that beside the here presented transport investigation of the Andreev molecular state another measurement tool can be the microwave spectroscopy, by coupling the Cooper pair splitter to resonator.

Classical magnetic moments induce Yu-Shiba-Rusinov states in superconducting hosts. A significantly large coupling between the impurity and Cooper pairs leads to a quantum phase transition, during which the polarization of the ground state jumps from $0$ to $1/2$. We examine the influence of the intrinsic, Kane-Mele type spin-orbit coupling on the phase transition, the topography of bound states, and the supercurrents induced around a single magnetic impurity.