Understanding the transport properties of cuprate superconductors is one of the central challenges in the physics of strongly correlated electrons. The most common approach is to define and solve a low-energy lattice model, but it is still unclear what the minimal model is to capture all relevant mechanisms and provide quantitative predictions. The main uncertainty concerns the choice of the orbital degrees of freedom to be included in the model, as well as the definition of the effective coupling. In this paper, we study the two most commonly considered models, namely the single-orbital Hubbard model and the three-orbital Emery model. We investigate and compare their spectral and transport properties, and find that the two models present a similar, but not the same, physical picture. We identify several strong quantitative differences which might allow one to discriminate between the two models by comparing theory with experiments. We compare our results for several physical quantities with 7 different experiments on 3 different La_2CuO_4-based cuprates, and in general find excellent agreement. The dc resistivity and the effective mass results suggest that the coupling constant in the effective Hubbard model is larger than expected. We find several more properties that are sensitive to the precise value of the coupling constant, including the critical doping for the Lifshitz transition, and the local spectral weight in the vicinity of the Fermi level; the latter provides a promising way to estimate the effective coupling constant in future photoemission experiments.

We use the Schrieffer-Wolff transformation (SWT) to analyze Josephson junctions between superconducting leads described by the charge-conserving BCS theory. Starting from the single-electron tunneling terms, we directly recover the conventional effective Hamiltonian, -EJ cos phi, with an operator-valued phase bias phi. The SWT approach has the advantage that it can be systematically extended to more complex scenarios. We show that if a Bogoliubov quasiparticle is present its motion couples to that of Cooper pairs, introducing correlated dynamics that reshape the energy spectrum of the junction. Furthermore, higher-order terms in the SWT naturally describe Josephson harmonics, whose amplitudes are directly related to the microscopic properties of the superconducting leads and the junction. We derive expressions that could facilitate tuning the ratio between the different harmonics in a controlled way.

We study the proximity effect in the Hubbard model coupled to BCS superconductors describing a single-layer strongly correlated electron system in a phase-biased Josephson junction. We find two distinct gapped solutions, one Mott-like insulating (M-phase) and one proximitized superconducting phase (S-phase), separated by first-order transition with hysteresis. In the M-phase the large correlation charge gap strongly suppresses the critical current, while the S-phase behaves as a 0-junction, with a proximitized gap that closes for ϕ=π to yield a correlated metal. Phase bias and junction transparency can thus serve as tuning knobs to switch between conducting and insulating regimes. Working within the dynamical mean field theory using the numerical renormalization group as the impurity solver, we associate M- and S-phase solutions with the doublet and singlet fixed points of the underlying superconducting Anderson impurity problem. We obtain detailed insight into the spectral structure on all energy scales. In the M-phase, the self-energy has sub-gap resonances symmetrically located around the Fermi level resulting from the splitting of the ”mid-gap pole” found in Mott insulators; this structure accounts for phase insensitivity.

We analytically and numerically investigate an Aharonov-Bohm interferometer with two superconducting terminals and a strongly correlated quantum dot in one arm. Through a rigorous derivation, we prove that this double-path interferometer is spectrally equivalent to a simpler system: an interacting quantum dot coupled to a non-interacting side-coupled proximitized mode and a semiconductor lead. This equivalence reveals a simple interpretation of the interferometer's behavior through the competition of a geometric factor χ, a key parameter characterizing the anomalous part of the hybridization function, with the properties of the side-coupled mode. We identify the conditions for the formation of doublet chimney in the phase diagrams in more general setting. Moreover, we show how the obtained Andreev bound state spectra clearly indicate the presence of Josephson diode effect generated by interferometric phenomena.

We investigate the superconducting Anderson impurity model for interacting quantum dot Josephson junctions with spin-orbit coupling and a term accounting for tunnelling through higher-energy orbitals. These elements establish the conditions required for spin polarization in the absence of external magnetic field at finite superconducting phase bias. This Hamiltonian has been previously used to model the Andreev spin qubit, where quantum information is encoded in spinful odd-parity subgap states. Here we instead analyse the even-parity sector, i.e., the Andreev pair qubit based on Andreev bound states (ABS). The model is solved using the zero-bandwidth approximation and the numerical renormalization group, with further insight from variational calculations. Electron-electron interaction admixes single-occupancy Yu-Shiba-Rusinov (YSR) components into the ABS states, thereby strongly enhancing spin transitions in the presence of spin-orbit coupling. The ABS states can thus become sensitive to local magnetic field fluctuations, which has implications for decoherence in Andreev pair qubits. For strong interaction U, especially in the cross-over region between the ABS and YSR regimes for U similar to 2 triangle, charge, spin, and inductive transitions can all become strong, offering avenues for spin control and quantum transduction.

Abstract Bose-Einstein condensates subjected to modulation of the interaction between atoms exhibit the emergence of density waves and matter-wave jets with a velocity proportional to the square root of the modulation frequency. Matter-wave jets have been studied in two- and one-dimensional systems showing that for sufficiently strong modulation additional higher harmonic matter-wave jets emerge. Here we report the experimental observation of incommensurable “golden” $$\frac{1+\sqrt{5}}{2}$$ 1 + 5 2 matter-wave jets in a Bose-Einstein condensate exposed to a single frequency interaction modulation. We study the formation of higher-order jets in quasi-one-dimensional geometry with numerical one dimensional (1D) Gross-Pitaevskii equation simulation. We explore the process of jet formation experimentally and theoretically for a wide range of modulation amplitudes and frequencies and establish a phase diagram delineating different regimes of jet formation. The observation of incommensurate jets provides a new route to an aperiodic density modulation of the condensate without employing an external potential.

We propose a set of variational wavefunctions for the sub-gap spin-doublet and spin-singlet eigenstates of the particle-hole symmetric superconducting Anderson impurity model. The wavefunctions include up to two Bogoliubov quasiparticles in the continuum which is necessary to correctly capture the weak-coupling asymptotics in all parameter regimes. The eigenvalue problems reduce to solving transcendental equations. We investigate how the lowest singlet state evolves with increasing charge repulsion U, transitioning from a proximitized state (a superposition of empty and doubly occupied impurity orbitals, corresponding to an Andreev bound state) to a local moment that is Kondo screened by Bogoliubov quasiparticles (Yu-Shiba-Rusinov state). This change occurs for U = 2O, where O is the BCS gap. At this point, the band-edge effects make the eigenenergy scale in a singular way as T-2/3, where T is the hybridization strength. Away from this special point, regular T-linear behavior is recovered, but only for T less than or similar to (U/2-O)(2)/O. The singular behavior thus extends over a broad range of parameters, including those relevant for some quantum devices in current use. The singular state is an equal-superposition state with maximal fluctuations between the local impurity charge configurations. Accurately capturing the band-edge singularity requires a continuum model, and it cannot be correctly described by discrete (truncated) models such as the zero-bandwidth approximation or the superconducting atomic limit. We determine the region of parameter space where the second spin-singlet state exists: in addition to the whole U < 2O ABS region, it also includes a small part of the U> 2O YSR region for finite values of T, as long as some ABS wavefunction component is admixed.

We consider an interacting quantum dot strongly coupled to two superconducting leads in a Josephson junction geometry. By defining symmetry-adapted superpositions of states from the leads, we formulate an effective Hamiltonian for the strong-hybridization regime with a single orbital directly coupled to the dot and three additional indirectly coupled orbitals. This minimal basis set allows one to account for the quasiparticles in the vicinity of the dot as well as those further away in the leads and to describe how their role evolves as a function of coupling strength and phase bias phi. This formulation also reveals the changing nature of the spin-doublet state for the experimentally relevant coupling strengths. The binding of a nearly decoupled quasiparticle in the vicinity of the quantum dot explains the "doublet chimney" in the phase diagram for phi similar to pi, in contrast to phi similar to 0, where the residual quasiparticle escapes to infinity and plays no active role.

We introduce a method based on auxiliary master equation for solving the problem of an impurity with local electron-electron and electron-phonon interaction embedded between two conduction leads with a finite bias voltage. The Anderson-Holstein Hamiltonian is transformed to a corresponding Lindblad equation with a reduced set of sites providing an optimal approximation of the hybridization function. The problem is solved in the superfermion representation using configuration interaction for fermions and bosons. The phonon basis is shifted and rotated with the intention of permitting a low phonon basis cutoff even in the strong-coupling regime. We benchmark this approach with the numerical renormalization group in equilibrium, finding excellent agreement. We observe, however, that the rotation brings no advantage beyond the bare shift. This is even more apparent out of equilibrium, where issues in convergence with respect to the size of the phononic Hilbert space occur only in the rotated basis. As an application of the method, we explore the evolution of the phononic peak in the differential conductance spectra with changing phonon frequency.

An electron tunneling across a junction integrated into an electric circuit can generate an excitation in the photonic field (electromagnetic environment) and lose energy in the process. Such inelastic tunneling of particles is commonly described using the P ( E ) theory. In the conventional approach to this theory, the tunneling rate and the electric current through the junction are derived using Fermi's golden rule and by averaging over the environmental photonic degrees of freedom. In this work, we address the same problem of inelastic tunneling due to photonic environment in Lindbladian formalism and we present how the photonic degrees of freedom are traced out in the quantum master equation approach. The resulting quantum master equation is parametrized by the same P ( E ) function and enables us to obtain not only the electric current but various other quantities, for instance, the heat current, in a systematic and convenient way. We also demonstrate that the Lindbladian formalism provides a comprehensive description of Bogoliubov quasiparticle tunneling through superconducting junctions and that it properly accounts for the coherence factors. The coherence factors become important if the normal-state density of states is particle-hole asymmetric.

Spin noise based quantum random number generators first appeared in 2008 and have since then garnered little further interest, in part because their bit rate is limited by the transverse relaxation time T_2 which for coated alkali vapour cells is typically in the kbit/s range. Here we present two advances. The first is an improved bit generation protocol that allows generating bits at rates exceeding 1/T_2 with only a minor increase of serial correlations. The second is a significant reduction of the time T_2 itself by removing the coating, increasing the vapour temperature and introducing a magnetic-field gradient. In this way we managed to increase the bit generation rate to 1.04 Mbit/s. We analyse the quality of the generated random bits using entropy estimation and we discuss the extraction methods to obtain high-entropy bitstreams. We accurately predict the entropy output of the device backed with a stochastic model and numerical simulations.

Since the seminal works in the fifties it has been known that a bath of non-interacting electrons mediates an interaction between local moments (such as nuclear spins or magnetic impurities) coupled to distant sites of the lattice. Recent efforts in simultaneous control of multi-qubit arrays relies on defining quantum dots in environments which likewise contain electrons capable of mediating effective interactions. It is therefore interesting to analyze systems with more than two impurities in higher orders of perturbation theory. Here we derive and discuss the significance of the effective spin interactions between four impurities in fourth order of the coupling between the impurities and the substrate, mediated by conduction electrons of a two-dimensional square lattice. This effective interaction resembles the ring exchange - a coupling of four spins appearing in perturbative treatment of the Hubbard model. Despite it being a higher-order effect, it can dramatically influence the magnetic ordering. We show that as an effective interaction between impurity spins, the ring exchange can also compete with the more familiar two-spin interactions.

We solve the problem of an interacting quantum dot embedded in a Josephson junction between two superconductors with finite charging energy described by the transmon (Cooper pair box) Hamiltonian. The approach is based on the flat-band approximation of the Richardson model, which reduces the Hilbert space to the point where exact diagonalisation is possible while retaining all states that are necessary to describe the low energy phenomena. The presented method accounts for the physics of the quantum dot, the Josephson effect and the Coulomb repulsion (charging energy) at the same level. In particular, it captures the quantum fluctuations of the superconducting phase as well as the coupling between the superconducting phase and the quantum dot (spin) degrees of freedom. The method can be directly applied for modelling Andreev spin qubits embedded in transmon circuits in all parameter regimes, for describing time-dependent processes, and for the calculation of transition matrix elements for microwave-driven transmon, spin-flip and mixed transitions that involve coupling to charge or current degree of freedom.

Abstract Flicker (pink, 1/\(f\)) noise is ubiquitous in all electronic devices, including in oscillator circuits used in true random number generators (TRNGs) based on jitter. Flicker noise produces strong serial correlations with very slow decay in time. We present a stochastic model for counter-mode TRNGs that takes the effects of such time correlations into account. Key parameters in the model are the spectral strengths of the pink and white noise components, as well as the low-frequency cutoff for the flicker noise spectrum. The random bits are defined as the least significant bit of consecutive integer-valued count numbers. We present the dependence of autocorrelations and min-entropy of generated random bits on model parameters. The autocorrelation between the bits is suppressed by increasing the strength of either pink or white noise, but it remains long-ranged (power-law decay). The power-law exponent depends linearly on the strength of pink noise, while the prefactor depends exponentially on both strengths. We determine the min-entropy per bit from a careful analysis of long-time sequences. It approaches the value of 1 approximately as a stretched exponential function of the flicker noise strength: highly entropic random bit generators can thus be designed even in the presence of strong flicker noise. We also propose an effective and efficient online health test for generators of this type.

Controlled coupling between distant particles is a key requirement for the implementation of quantum information technologies. A promising platform are hybrid systems of semiconducting quantum dots coupled to superconducting islands, where the tunability of the dots is combined with the macroscopic coherence of the islands to produce states with non-local correlations, e.g. in Cooper pair splitters. Electrons in hybrid quantum dots are typically not amenable to long-distance spin alignment as they tend to be screened into a localized singlet state by bound superconducting quasiparticles. However, two quasiparticles coming from different superconductors can overscreen the quantum dot into a doublet state, leading to ferromagnetic correlations between the superconducting islands. We present experimental evidence of a stabilized overscreened state, implying correlated quasiparticles over a micrometer distance. We propose alternating chains of quantum dots and superconducting islands as a novel platform for controllable large-scale spin coupling.

An epitaxial semiconductor-superconductor nanowire is a superconducting system with a complex level structure originating from hybridization: in addition to a dense set of higher-energy states derived predominantly from the metallic superconducting shell above the bulk gap $\Delta$, there is a smaller number of lower-energy proximitized states from the semiconducting core that define the induced gap $\Delta^*$. Nanostructures built from such nanowires can furthermore incorporate quantum dots in order to obtain localized spins for storing and manipulating quantum information. We discuss the magnetic field dependence in three devices with different combinations of embedded quantum dots and superconducting islands. For strong fields, they show pinning of excitation energies to a uniform spacing, even if for weak fields they have non-universal properties with different behaviors for even and odd number of confined electrons. We propose a quantum impurity model for hybrid devices that incorporates all relevant physical effects. We show that the model accounts for the key observations and permits unambiguous interpretation in terms of many-particle states. In particular, we study the replicas of the Yu-Shiba-Rusinov states in the hybrid gap, their collapse and oscillation around zero bias with increasing field, and the strong smoothing effect of the spin-orbit coupling (SOC) on these oscillations. We propose that the SOC-induced mixing of many-body states is a generic mechanism and that magnetic pinning is likely to be a ubiquitous feature in hybrid nanowires.

We use a hybrid superconductor-semiconductor transmon device to perform spectroscopy of a quantum dot Josephson junction tuned to be in a spin-1/2 ground state with an unpaired quasiparticle. Because of spin-orbit coupling, we resolve two flux-sensitive branches in the transmon spectrum, depending on the spin of the quasiparticle. A finite magnetic field shifts the two branches in energy, favoring one spin state and resulting in the anomalous Josephson effect. We demonstrate the excitation of the direct spin-flip transition using all-electrical control. Manipulation and control of the spin-flip transition enable the future implementation of charging energy protected Andreev spin qubits.

We present a theoretical study of a system consisting of a superconducting island and two quantum dots, a possible platform for building qubits and Cooper pair splitters, in the regime where each dot hosts a single electron and, hence, carries a magnetic moment. We focus on the case where the dots are coupled to overlapping superconductor states and we study whether the spins are ferromagnetically or antiferromagnetically aligned. We show that if the total number of particles is even, the spins align antiferromagnetically in the flatband limit, i.e., when the bandwidth of the superconductor is negligibly small, but ferromagnetically if the bandwidth is finite and above some value. If the total number of particles is odd, the alignment is ferromagnetic independently from the bandwidth. This implies that the results of the flatband limit are applicable only within restricted parameter regime for realistic superconducting qubit systems and that some care is required in applying simplified models to devices such as Cooper pair splitters.

Spectroscopy of a Josephson junction device with an embedded quantum dot reveals the presence of a contribution to level splitting in external magnetic field that is proportional to $\cos \phi$, where $\phi$ is the gauge-invariant phase difference across the junction. To elucidate the origin of this unanticipated effect, we systematically study the Zeeman splitting of spinful subgap states in the superconducting Anderson impurity model. The magnitude of the splitting is renormalized by the exchange interaction between the local moment and the continuum of Bogoliubov quasiparticles in a variant of the Knight shift phenomenon. The leading term in the shift is linear in the hybridisation strength $\Gamma$ (quadratic in electron hopping), while the subleading term is quadratic in $\Gamma$ (quartic in electron hopping) and depends on $\phi$ due to spin-polarization-dependent corrections to the Josephson energy of the device. The amplitude of the $\phi$-dependent part is largest for experimentally relevant parameters beyond the perturbative regime where it is investigated using numerical renormalization group calculations. Such magnetic-field-tunable coupling between the quantum dot spin and the Josephson current could find wide use in superconducting spintronics.

Local quantum phase transitions driven by Kondo correlations have been theoretically proposed in several magnetic nanosystems; however, clear experimental signatures are scant. Modeling a nickelocene molecule on a Cu(100) substrate as a two-orbital Anderson impurity with single-ion anisotropy coupled to two conduction bands, we find that recent scanning tunneling spectra reveal the existence of a topological quantum phase transition from the usual local Fermi liquid with high zero-bias conductance to a non-Landau Fermi liquid, characterized by a non-trivial quantized Luttinger integral, with a small conductance. The effects of intermediate valence, finite temperature, and structural relaxation of the molecule position allow us to explain the different observed behaviors.