Bound states in superconductors and interfaces

Boundary surfaces of nodal gap superconductors can host Andreev bound states (ABS), which develop a paramagnetic response under external RF field in contrast to the diamagnetic response of the bulk superconductor. At low temperature, this surface paramagnetic response dominates and enhances the nonlinear RF response of the sample. With a recently developed photoresponse imaging technique [Phys. Rev. Lett. 110, 087002 (2013)], the anisotropy of this “paramagnetic” nonlinear Meissner response, and its current direction (angular), RF power, and DC magnetic field dependence has been systematically studied in a $d_{x^2-y^2}$ superconductor. A theoretical model describing the current flow in the surface paramagnetic Andreev bound state, the bulk diamagnetic Meissner state, and their response to optical illumination is proposed and it shows good agreement with the experimental results. We also discuss recent experimental results on the surface impedance and nonlinear microwave spectroscopy of a proximity coupled superconductor / topological insulator bilayer system. Thin films of the Kondo topological insulator $SmB_6$ are prepared in proximity contact with the conventional superconductor $YB_6$, and studied in both zero and non-zero DC magnetic field.

We study the dynamics of a charge current in the double quantum dot system, coupled between the superconducting and normal reservoirs in the row configuration. The quantum dot's energy levels are separately periodically driven by an oscillating gates potential. Within the Keldysh formalism for the Green's function extended by the Floquet dimension we study emergence of the higher order harmonics and investigate their contribution to the total current. We analyze an ongoing transfer of the spectral weights between the subgap bound tates and discuss feasible experimental methods to observe it.

We analyze the mutual relations between local pairing, correlations and ,,leaking'' Majorana zero mode on a quantum dot placed in proximity of Rashba wire. If nanoobjects like quantum dots are coupled to superconductors, the proximity induced SC correlations lead to local pairing. On the other hand, strong Coulomb interactions disfavour any double occupancy. This competition leads to ground state phase transition between SC singlet and spinfull doublet configuration. If quantum dot is additionally coupled to Rashba chain hosting Majorana fermion, the Majorana zero mode ''leak'' into QD region. We inspect the qualitative rearangement of induced zero modes conditional on singlet - doublet crossover.

In the present work, we develop an efficient approach to study mesoscopic superconducting systems with given disorder in the form of a small amount of point impurities. In essence, we solve the Gor’kov equation by expressing the Green functions of a system with defects in terms of the Green functions of the clean system. We use our result to study the influence of a point impurity on the local density of states in three inhomogeneous superconducting systems. First, we analyze a short 3D SNS junction with an impurity inside the normal layer [1]. The impurity (either nonmagnetic or magnetic) induces two quasibound subgap states, which are somewhat similar to Shiba states in bulk superconductors. If the impurity is located close to a flat boundary of the junction, the energies of impurity states oscillate as functions of the position of the impurity. We also find that two nonmagnetic impurities induce four quasibound states, their energies depending in a complicated manner on the relative position of the defects. Finally, the influence of the defects on the Josephson current is calculated. Second, we study a normal bubble (ball) inside a bulk superconductor. The spectrum of this system has a global minigap, and within the quasiclassical approximation, there are local minigaps at higher energies. A point defect generally induces two quasiparticle states inside the global minigap, even when the defect is nonmagnetic and is located inside the superconductor. In addition, impurity states appear in local minigaps. Third, we consider a 2D Abrikosov vortex in the Kramer-Pesch limit – essentially, we take a vanishing vortex core size. The spectrum of the clean system is quite peculiar: the anomalous branch has an infinite slope at zero angular momentum, and higher branches emerge from the gap edge when the momentum exceeds some critical value. In this system, a point defect induces up to four impurity states. [1] A. A. Bespalov, Phys. Rev. B 97, 134504 (2018).

Motivated by recent experimental reports of significant spin-orbit coupling (SOC) and a sign-changing order-parameter in the Li$_{1-x}$Fe$_x$(OHFe)$_{1-y}$Zn$_y$Se superconductor with only electron Fermi surface present, we study the possible Cooper-pairing symmetries and their quasiparticle interference (QPI) signatures. We find that each of the resulting states - $s$-wave, $d$-wave and helical $p$-wave - can have a fully gapped density of states (DOS) consistent with angle-resolved photoemission experiments (ARPES) experiments and, due to spin-orbit coupling, are a mixture of spin singlet and triplet components leading to intra- and inter-band features in the QPI signal. Analyzing predicted QPI patterns we find that only the $s$- and $d$-wave pairing states with a dominant even parity triplet component can fit the experimental data with two dominant peaks in the DOS at energies roughly corresponding to the gap sizes at each pocket. Moreover, we show that pairing states with dominant triplet component may exist and can be identified using spin-resolved STM.

Most of the proposed entanglement detection methods of spatially separated electrons of Cooper pairs require the use of ferromagnetic electrodes with high spin polarization and difficult experimental techniques. Therefore, we show new detection models [1-3], based on ferromagnetic detectors attached to the Cooper pairs splitters [4-6], using entanglement witness approach. The presented models are based on electric current cross correlation measurements and, simpler to perform, dc current measurements. An advantage of the analyzed detection models over those proposed so far is their easier experimental implementation due to the use of dc current measurements and the reduced system requirements, e.g., in terms of detector efficiency. By proper optimization we demonstrate that under certain experimental conditions entanglement detection is even possible with magnetic detectors of any efficiency greater than zero, and we determine the minimal number of current cross correlation measurements necessary for correct detection of quantum entanglement of a Cooper pair in the singlet state. The proposed methods can be used for experimental detection of quantum entanglement of spatially separated electrons of Cooper pairs, and will allow for further progress of the quantum communication electronic technology. 1. W. Kłobus, A. Grudka, A. Baumgartner, D. Tomaszewski, C. Schönenberger, J. Martinek, Phys. Rev. B 89, 125404 (2014). 2. P. Busz, D. Tomaszewski, J. Martinek, Phys. Rev. B 96, 064520 (2017). 3. P. Busz, D. Tomaszewski, J. Martinek, to be published (2019). 4. L. Hofstetter, S. Csonka, J. Nygård, and C. Schönenberger, Nature 461, 960 (2009). 5. J. Wei and V. Chandrasekhar, Nature Physics 6, 494 (2010). 6. Z. B. Tan, D. Cox, T. Nieminen, P. Lähteenmäki, D. Golubev, G. B. Lesovik, and P. J. Hakonen, Phys. Rev. Lett. 114, 096602 (2015).

We investigate a hybrid heterostructure with magnetic skyrmions (Sk) inside a chiral ferromagnet interfaced by a thin superconducting film via an insulating barrier. The barrier prevents the electronic transport between the superconductor and the chiral magnet, such that the coupling can only occur through the magnetic fields generated by these materials. We find that Pearl vortices (PV) are generated spontaneously in the superconductor within the skyrmion radius, while anti-Pearl vortices (\(\overline{\mathrm{PV}}\)) compensating the magnetic moment of the Pearl vortices are generated outside of the Sk radius, forming an energetically stable topological hybrid structure. Finally, we analyze the interplay of skyrmion and vortex lattices and their mutual feedback on each other. In particular, we argue that the size of the skyrmions will be greatly affected by the presence of the vortices offering another prospect of manipulating the skyrmionic size by the proximity to a superconductor.

Samuel Häusler (1), Dominik Husmann (1), Shun Uchino (2), Martin Lebrat (1), Sebastian Krinner (1), Philipp Fabritius (1), Jeffery Mohan (1), Laura Corman (1), Jean-Philippe Brantut (3), Thierry Giamarchi (4), Tilman Esslinger (1)

(1) ETH Zürich, Switzerland, (2) Waseda University, Japan, (3) EPFL Lausanne, Switzerland, (4) University of Geneva, Switzerland

Recent advances with ultracold gases permit to study transport with neutral atoms through mesoscopic structures engineered with light potentials. Here we focus on the direct current through a quantum point contact in the strongly interacting unitary regime. At low temperatures we observe a non-linear convex current-bias characteristic, reminescent of superconducting weak links. The curves are quantitatively reproduced using a Keldysh Green function technique applied to a point contact model with superfluid reservoirs. The model explains the microscopic origin of the current that flows even though the bias is insufficient to bridge the superfluid gap. It is carried by multiple Andreev reflections between the reservoirs where several pairs are coherently transferred to bring a quasi-particle across the gap. This process is very efficient due to the high transparency of the junction. We investigate the behaviour at various densities and temperatures revealing special properties of the unitary Fermi gas. Especially, the large conductance at high temperatures where the gas is in the normal phase.

D. Husmann et al., Science 350, 1498-1501 (2015)

Superconducting quantum dot modeled by a single-level impurity with Coulomb repulsion attached to superconducting leads displays an impurity quantum phase transition from a spin singlet state in the weak coupling to a spin doublet in the strong coupling. The supercurrent through the impurity due to the phase difference of the left and right superconducting leads changes discontinuously its value and direction at this transition. This (0- $\pi$) transition is signaled by a continuous crossing of the discrete gap states and is mostly related to the emergence of the Kondo scale from the normal quantum dot. We showed earlier that the 0- $\pi$ transition at zero temperature can be reached in weak coupling without referring to the Kondo effect [1-3]. To go beyond the transition and to non-zero temperatures we must, however, lift the degeneracy in the spin space and allow for a spin-polarized equilibrium state. We study the superconducting quantum dot in an external magnetic field and analyze the behavior of the gap states across the 0- $\pi$ transition. We solve explicitly the impurity model without Coulomb repulsion and in the (generalized) atomic limit to show that the crossing is continuous and that the low-magnetic state of the 0-phase comes over to a high-magnetic state of the $\pi$-phase. We pinpoint the flaw in the mean-field solution and outline the way towards consistent approximations within the many-body perturbation expansion. [1] M. Žonda, V. Pokorný, V. Janiš, and T. Novotný, Sci. Rep. 5, 8821 (2015). [2] M. Žonda, V. Pokorný, V. Janiš, and T. Novotný, Phys. Rev. B 93 024523 (2016). [3] V. Janiš, V. Pokorný, and M. Žonda, Eur. Phys. J. B 89, 197 (2016).

We study the Anderson single-level quantum dot attached to two BCS superconducting leads with the same gap size. We reveal that a system with asymmetric tunnel coupling to the leads ($\Gamma_L=\Gamma_R$) can be related to the symmetric system with the same net coupling strength ($\Gamma=\Gamma_L+\Gamma_R$). Surprisingly, it is the symmetric case which is the most general, meaning that all physical quantities in case of asymmetric coupling are fully determined by the symmetric ones. We give ready-to-use conversion formulas for the 0 − π phase boundary, on-dot quantities, and the Josephson current, and illustrate them on the NRG results of Oguri, Tanaka and Bauer [Phys. Rev. B 87, 075432 (2013)] for the three-terminal setup. For the Kondo regime, we newly present a formula describing the position of the phase boundary in parameter space, taking the asymmetry of the junction into account.

A junction between two boundaries of a topological superconductor, mediated by localized edge modes of Majorana fermions, is investigated. The tunneling of fermions across the junction depends on the magnetic flux. It breaks the time-reversal symmetry at the boundary of the sample. The persistent current is determined by the emergence of Majorana edge modes. The structure of the edge modes depends on the magnitude of the tunneling amplitude across the junction. It is shown that there are two different regimes, which correspond to strong and weak tunneling of Majorana fermions, distinctive in the persistent current behavior. In a strong tunneling regime, the fermion parity of edge modes is not conserved and the persistent current is a $2\pi$-periodic function of the magnetic flux. When the tunneling is weak the chiral Majorana states, which are propagating along the edges have the same fermion parity. They form a $4\pi$-phase periodic persistent current along the boundaries. The regions in the space of parameters, which correspond to the emergence of $2\pi$- and of $4\pi$-harmonics, are numerically determined. The peculiarities in the persistent current behavior are studied. I.N.Karnaukhov, Scientific Reports 7, 7124 (2017) Form of Presentation talk or poster

Interplay between superconductivity, spin-orbit coupling and magnetic field can lead to realisation of the topologically non--trivial states. In finite 1D Rasha nanowires those states are manifested by emergence of a pair of zero-energy Majorana bound states. On the other hand, in 2D systems spin currents contributed by the edge states might appear. Creation of junction between those two systems opens new possibilities for probing the behaviour of Majorana Bound States. Therefore, we investigate properties of such bound states in multi-dimensional hybrid structures consisting of 1D Rashba nanowire and bounded 2D surfaces.

We study theoretically the relaxation of electron spins in graphene in proximity to an s-wave superconductor in the presence of resonant magnetic and spin-orbit impurities [1]. Off resonance, the relaxation behaves as predicted from superconducting coherence: with lower temperatures the spin relaxation increases when electrons scatter off magnetic impurities (Hebel-Slichter effect), and decreases when the scatterers induce spin-orbit coupling. This distinct temperature dependence, not available in the normal state, can uniquely discriminate between the two scattering mechanisms. But the Hebel-Slichter picture breaks down at resonances. The emergence of Yu-Shiba-Rusinov bound states within the superconducting gap shifts the spectral weight of the magnetic resonances and leads to a significant decrease of the spin relaxation rate at lower temperatures. Our findings should be valid for generic s-wave superconductors that host resonant magnetic impurities. [1] Denis Kochan, and Jaroslav Fabian; arXiv: 1902.05474

Starting from the Bogolubov -- de Gennes theory we show that a finite curvature of the semiconducting nanowires strongly affects the p-wave superconducting order parameter induced in these systems in the presence of strong Zeeman and spin-orbit interactions. Depending on the magnetic field direction the change of the nanowire shape can cause the inhomogeneity of both the amplitude and phase of the induced order parameter. This inhomogeneity reveals itself either through the spontaneous supercurrents or via the appearance of the localized quasiparticle modes at the Fermi level. Thus, the superconducting ground state and quasiparticle modes in Majorana nanowires can be engineered by a change of the wire shape and geometry.

InAs nanowires are strong candidates for realizing Majorana bound states (MBSs). Quantum dots (QDs) formed between the superconductor and the normal contact have been utilized as a spectroscopic tool for probing the midgap states in proximitized nanowires. Recently, the nonlocality of MBSs was assessed by evaluating the hybridization of the QD states with the MBSs [1]. The QDs that have previously been employed are typically obtained by relying on electrostatic gating and/or spontaneous formation. In a recent study, we have probed the midgap states in the proximitized nanowire lead between the QD and the superconducting contact using a crystal-phase defined QD which provide more robust and controllable transport characteristics [2]. Here, the QDs are defined during nanowire growth by inserting two segments of wurtzite crystal phase in otherwise zinc blende InAs nanowire [3,4]. The hard-wall potential barriers provided by the conduction-band edge offset between the two crystal phases allow for probing the evolution of the proximity gap in a wide gate range. As a result of the gate-induced carrier-density modulation, the system transforms form being in the short junction limit where a constant proximity gap is detected, to the long junction limit where the gap closes. As a next step, we use epitaxial markers in the form of a GaSb shell selectively grown on the zinc blend segments [5] to controllably align the superconducting contact with the crystal-phase QD. Thus, allowing for probing the proximitized region directly underneath the superconductor. [1] M.-T. Deng, S. Vaitiekėnas, E. Prada, P. San-Jose, J. Nygård, P. Krogstrup, R. Aguado, and C. M. Marcus, Phys. Rev. B 98, 085125, (2018). [2] C. Jünger, A. Baumgartner, R. Delagrange, D. Chevallier, S. Lehmann, M. Nilsson, K. A. Dick, C. Thelander, and C. Schönenberger, arXiv:1812.06850, (2018). [3] M. Nilsson, L. Namazi, S. Lehmann, M. Leijnse, K. A. Dick, and C. Thelander Phys. Rev. B 93, 195422, (2016). [4] M. Nilsson, F. Viñas Boström, S. Lehmann, K. A. Dick, M. Leijnse, and C. Thelander Phys. Rev. Lett. 121, 156802, (2018). [5] L. Namazi, M. Nilsson, S. Lehmann, C. Thelander, and K. A. Dick, Nanoscale, 7, 10472-10481, (2015).

I will present the theory and experimental realization of tunneling between tip and substrate Yu-Shiba-Rusinov (YSR) states in the superconducting junction of a scanning tunneling microscope (STM) operating at 15 mK. The simple analytical results are in good agreement with conductance measurements exhibiting peaks in the tunnel current at a number of sub-gap bias voltages. [1] The voltages are identified as resonances of sub-gap discrete magnetic states, so called YSR states, that form inside a volume around the magnetic impurity of Fermi wave length size. [2] When a YSR state formed around an impurity in the STM tip is brought into proximity with a YSR state formed around an impurity in the substrate, new resonances arise at characteristic values of the applied bias voltage. The tunnel current at the new resonances is a result of the interplay between coherent transport processes and incoherent relaxation. [1] M. Ruby, F. Pientka, Y. Peng, F. von Oppen, B. W. Heinrich, and K. J. Franke, Phys. Rev. Lett. 115, 087001 (2015). [2] M. I. Salkola, A. V. Balatsky, and J. R. Schrieffer, Phys. Rev. B 55, 12648 (1997).

We theoretically study a scheme to distinguish the two ground states of a one-dimensional topological superconductor, which could serve as a basis for the readout of Majorana qubits. The scheme is based on parity-to-charge conversion, i.e., the ground-state parity of the superconductor is converted to the charge occupation on a tunnel-coupled auxiliary quantum dot. We describe how certain error mechanisms degrade the quality of the parity-to-charge conversion process. We consider (i) leakage due to a strong readout tunnel pulse, (ii) incomplete charge Rabi oscillations due to slow charge noise, and (iii) charge relaxation due to phonon emission and absorption. To describe these effects, we use simple model Hamiltonians based on the ideal Kitaev chain, and draw conclusions to generic one-dimensional topological superconductors wherever possible. In general, the effects of the error mechanisms can be minimized by choosing an optimal strength for the readout tunnel pulse. In a case study based on InAs heterostructure device parameters, we estimate that the parity-to-charge conversion error is mainly due to slow charge noise for weak tunnel pulses and leakage for strong tunnel pulses.

We theoretically investigate the transport properties of a quasi-one-dimensional ferromagnet-superconductor junction where the superconductor consists of mixed singlet and triplet pairings. We show that the relative orientation of the Stoner field (h̃) in the ferromagnetic lead and the d vector of the superconductor acts like a on-off switch for the zero bias conductance of the device. In the regime, where triplet pairing amplitude dominates over the singlet counterpart (topological phase), a pair of Majorana zero modes appear at each end of the superconducting part of the nanowire. When h̃ is parallel or antiparallel to the d vector, transport gets completely blocked due to blockage in pairing while, when h̃ and d are perpendicular to each other, the zero energy two terminal differential conductance spectra exhibits sharp transition from $4e^2/h$ to $2e^2/h$ as the magnetization strength in the lead becomes larger than the chemical potential indicating the spin-selective coupling of a pair of Majorana zero modes to the lead.

We study the temperature effects in a system consisting of a single-level quantum dot with local Coulomb interaction attached to two superconducting leads and optionally a third, metallic lead. The system is described by the single-impurity Anderson model coupled to BCS superconducting baths and solved using the continuous-time, hybridization-expansion (CT-HYB) quantum Monte Carlo as well as the numerical renormalization group (NRG). We study the behavior of the subgap (Andreev-Shiba) states, the Josephson current and the fate of the zero-pi (singlet-doublet) quantum phase transition. We also show the limits of usability of the stochastic optimization method for obtaining the spectral functions from the imaginary-time CT-HYB results.

We investigated the spectrum of Kondo impurities absorbed in a superconducting substrate. The avoidance of the double occupancy of the impurity due to large Coloumb repulsion is taken care of with Slave-boson formalism. The ground state properties of the impurity is studied within the mean-field theory. The Kondo resonance feature of a metallic system gives rise to a bound state for a superconducting substrate. The bound state energy starting from the Fermi energy increase with larger Kondo temperature. This bound state splits when two impurities interacting via an antiferromagnetic RKKY exchange interaction.

Zero--energy Majorana quasiparticles can be induced at the edge of a low dimensional systems. Non--Abelian statistics of this state makes it a good candidate for the realization of quantum computing. From the practical point of view, it is crucial to obtain an intentional creation and manipulation of this type of bound states. We show such a possibility in a setup of several systems (e.g. optical trap or quantum dot-nanoring hybrid system) in which we specify a 'defect' (inhomogeneity) region via electrostatic means. States in such defect can lead to the emergence of Andreev and Majorana bound states in investigated system. We study the differences between those bound states and the possibility of their manipulation.

Superconductor-semiconductor structures can be driven into a topological regime by application of an external magnetic field, in which Majorana zero modes emerge. Our contribution extends previous work on Majorana modes in superconducting structures, by developing nanowire-based double-island devices where both the Josephson coupling as well as Majorana coupling between islands can be manipulated using gate electrodes, whereas the charge of islands can be measured with high-bandwidth proximal charge sensors. We experimentally study microwave induced transitions between two superconducting islands when the magnetic field along the nanowire axis is applied. At higher magnetic fields we observe a change in gate space periodicity of the microwave induced transitions. Results are compatible with single electron transitions between zero modes on both sides of the junction. From microwave spectroscopy extracted energy scales for Josephson and single electron coupling will be useful in the near future experiments.

Electron transport across Graphene-Superconductor junctions have recently attracted a lot of attention [1] after it was pointed out that the Andreev processes in such junction holds distinct features as compared to Andreev processes in Normal metal-Superconductor junction because of the ultra relativistic dispersion of the charge carriers in Graphene. In this work we report some interesting transport properties through a pair of such junctions due to Andreev and normal processes by exploiting the analogy with certain optical phenomena such as Goos-Hanchen shift [2] and discuss its experimentally verifiable features. References: [1] C. W. J. Beenakker, PRL 97, 067007, 2006 [2] C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, PRL 102, 146804, 2009

Tomasch oscillations occur in systems where there is a spatial variation in the electron-electron interactions which give rise to the superconducting pairing potential. For example they can occur in states bound within superconducting junctions or islands. Here we report on Tomasch-like oscillations which have been found to occur in Superconducting-Topological Insulator structures. In particular in superconducting islands deposited on top of the topological insulator Bi$_2$Se$_3$. We go on to consider their existence in junctions formed from Topological Insulators, or topological insulator surface states, and superconductors by calculating the local density fo states in these systems.

At low temperatures, non-equilibrium voltage fluctuations can be generated in current-biased superconducting nanowires and Josephson junctions due to the proliferation of quantum phase slips (QPS) or, equivalently, due to a quantum tunnelling of magnetic flux quanta. Examples include generation of a nonvanishing voltage drop across such systems under external current bias. Another QPS-related effect in superconducting nanowires and Josephson junctions is shot noise of the voltage. In our talk, we discuss the full counting statistics of voltage fluctuations generated by the QPS in superconducting bridges and Josephson junctions. With the aid of the Keldysh technique combined with duality arguments, we derive the generating function fully describing voltage fluctuations and explicitly evaluate this function in some limits. For example, at zero frequency or in the long-time limit voltage fluctuations are described by Poisson statistics, just as it could be expected intuitively. We also evaluate all voltage cumulants at nonzero frequencies and under external current bias.

I will describe my recent work in how supercurrent can be used as a probe in S-QD-S systems. My poster may include descriptions of the intricacies of utilizing supercurrent (phase-current) generated by thermal fluctuations of the josephson circuitry as a probe, or how the YSR physics can be qualitatively modeled by simple approximations without the need of powerful numerical machinery. What i will focus on is yet to be decided.

In recent years, there is substantial progress in creation of specially separated spin entangled electrons in solid state using the splitting of Cooper pairs, which is necessary ingredient of quantum communication and computing. It is also possible to generate a Josephson supercurrent form by split nonlocal Cooper pairs [1]. This new Josephson current required further studies especially its interference properties. While the behavior of single electrons under the influence of Aharonov-Bohm (AB) and Aharonov-Casher (AC) effects is well understood, it raises the question of the impact of these effects on nonlocal superconducting Cooper pairs that for s-wave superconductors are in spin singlet state. We analyze a normal ring, where a single electron interference is possible and two parallel nanowires connected to two superconducting electrodes, where a single-electron interference can be absent but a cross Andreev reflection is possible [2]. At low transmission, we can link the AB effect only to local Cooper pairs and the AC effect to nonlocal Cooper pair transport. We demonstrate that by using two quantum dots we can obtain different AC phases for the non-spin-flip and spin-flip transport processes that leads to a beating in the AC effect [3]. 1. R. S. Deacon, A. Oiwa, J. Sailer, S. Baba, Y. Kanai, K. Shibata, K. Hirakawa, and S. Tarucha, Nat. Commun. 6, 7446 (2015). 2. D. Tomaszewski, P. Busz, R. López, R. Žitko, M. Lee, and J. Martinek, Phys. Rev. B 97, 214506 (2018). 3. D. Tomaszewski, P. Busz, R. López, R. Žitko, M. Lee, and J. Martinek, Phys. Rev. B 98, 174504 (2018).

$\pi$ junctions are Josephson junctions (JJs) with a $\pi$ shifted current phase relationship. The flux qubit, a viable qubit candidate which includes a $\pi$ junction in a superconducting loop, was reported by Hilgenkamp et al. [1], crystallizing the technological importance of such junctions [1]. $\pi$ shifts as a result of a doublet ground state have been shown in a single-level spinful quantum dot JJs [2]. We demonstrate a double quantum dot (DQD) JJ, in which we show control of the ground state, and therefore of the phase, by changing with gate voltages the number of electrons in the system [3]. In contrast to the single QD case, the existence of molecular orbitals allows us to change the magnitude of Ic through level detuning, while keeping the junction in the $\pi$ phase. We compare our experimental results with good agreement to three theoretical methods: fourth order perturbation theory, the numerical renormalization group (NRG) method, and the zero-bandwidth model. Besides the possible applications of controlling the magnitude of the supercurrent in a $\pi$ junction, the DQD JJ is a stepping stone towards the implementation of an artificial Kitaev chain in nanowires [4]. [1] Hilgenkamp et al., Science, 312, 1495 (2006). [2] Van Dam et al., Nature, 442, 667 (2006). [3] J. C. Estrada Saldaña, A. Vekris, G. Steffensen, R. Žitko, P. Krogstrup, J. Paaske, K. Grove-Rasmussen, and J. Nygård Phys. Rev. Lett. 121, 257701 (2018) [4] Sau and Das Sarma, Nat. Commun., 3, 964 (2012).

We report on numerically exact iterative path-integral calculations (ISPI) for spin-dependent transport through small interacting quantum dots. Our ISPI method builds upon the truncation of exponentially vanishing real-time correlations at finite temperature and/or bias voltage. In particular, we study two distinct systems (i) a spin- valve, where ferromagnetic leads induce spin-dependent tunnel couplings between leads and dot and (ii) a hybrid structure, where the quantum dot is coupled to a normal and a superconducting lead. For the spin-valve setup, the observable of interest is the tunneling magnetoresistance through the quantum dot, which is investigated for various system parameters. We find that, especially at low tem- peratures, resonant tunneling effects are dominant and the sequential picture does not apply. The role of Coulomb interactions together with changing the temperature of the system is explored. Within the hybrid system, a finite gap parameter $\Delta$ induces anomalous self-energies in the Keldysh partition function. ISPI calculations are performed to deduce, e.g. the impact of Coulomb interactions on the Andreev bound state spectrum in a nonequilibrium situation.

We study the spin-resolved Andreev transport in a triple quantum dot system coupled to one superconducting and two ferromagnetic leads. The results are obtained by means of the real-time diagrammatic technique in the sequential tunneling regime with respect to the coupling to ferromagnetic leads, while the coupling to superconductor is arbitrary. We examine the behaviour of the Andreev current and differential conductance in the parallel and antiparallel alignments of ferromagnets’ magnetic moments, as well as the resulting tunnel magnetoresistance, in both the linear and nonlinear response regimes. Special emphasis is put on the analysis of the cross-correlations between the currents flowing through the left and right ferromagnetic junctions. Strong positive values of cross-correlations indicate transport regimes with high Cooper pair splitting efficiency of the device, while negative cross-correlations imply the presence of transport processes mutually suppressing the flow of the Andreev current. The mechanisms giving rise such transport behavior are thoroughly discussed.