The Anderson impurity model is a paradigmatic example in the study of strongly correlated quantum systems and describes an interacting quantum dot coupled to electronic leads. Here we investigate its dynamics following a quantum quench based on matrix product state simulations. We examine the behavior of its impurity magnetization. Its relaxation allows us to extract the predicted scaling of the Kondo temperature as a function of the impurity-lead hybridization and quantum dot repulsion. Additionally, our simulations provide estimates of the currents in the nonequilibrium quasisteady state appearing after the quench. Through their values, we examine the dependence of the conductance on the voltage bias
Vb and on the impurity chemical potential
Vg, which displays a zero-bias Kondo peak. Our results are relevant for transport measurements in Coulomb blockaded devices, and, in particular, in quantum dots induced in nanowires.
As the complexity of mesoscopic quantum devices increases, simulations are becoming an invaluable tool for understanding their behavior. This is especially true for the superconductor-semiconductor heterostructures used to build Majorana-based topological qubits, where quantitatively understanding the interplay of topological superconductivity, disorder, semiconductor quantum dots, Coulomb blockade and noise has been essential for progress on device design and interpretation of measurements. In this paper, we describe a general framework to simulate the low-energy quantum dynamics of such complex systems. We illustrate our approach by computing the dispersive gate sensing (DGS) response of quantum dots coupled to topological superconductors. We start by formulating the DGS response as an open-system quantum dynamics problem, which allows a consistent treatment of drive backaction as well as quantum and classical noise. For microscopic quantum problems subject to Coulomb-blockade, where a direct solution in the exponentially large many-body Hilbert space would be prohibitive, we introduce a series of controlled approximations that incorporate ideas from tensor network theory and quantum chemistry to reduce this Hilbert space to a few low-energy degrees of freedom that accurately capture the low-energy quantum dynamics. We demonstrate the methods introduced in this paper on the example of a single quantum dot coupled to a topological superconductor and a microscopic realization of the fermion parity readout setup of Aghaee et al. arXiv:2401.09549 (2024).
Superconductivity in its various manifestations has been stimulating both experimental and theoretical progress in condensed-matter physics for more than a hundred years. The remarkable property of electrons to pair up and form quasi-particles gives rise to a plethora of phenomena featuring important practical applications not only in science, but, for instance, also in medicine and metrology. Recently, new directions in investigating this fascinating subject emerged, such as superconductivity out-of equilibrium and topological superconductors. Providing experimental evidence for enhanced superconducting correlations in optically pumped copper oxides at temperatures far above the equilibrium transition temperature, the first issue caused considerable excitement. On the other hand, topological superconductors are believed to provide realizations of highly fault-tolerant qubits by means of hosting non-Abelian quasi-particles, which can be the building blocks of scalable quantum computers. Experimentally verifying the emergence of these Majorana edge modes, exotic quasi-particles in heterostructures consisting of a conventional superconductor and semiconductors or topological insulators, is one of the most urgent questions to be answered right now. Both subjects cannot be accounted for with analytically solvable approximations only, and also provide very challenging numerical problems. We implemented a matrix-product state (MPS) based toolkit exploiting
U(1) symmetries, providing a flexible and efficient platform to study these complex systems. In order to efficiently simulate out-of equilibrium setups we studied, compared, and developed time-evolution algorithms for MPS enabling us to choose the most suitable method for a given task. We also developed a new framework to represent operators in an enlarged Hilbert space so that benefits from conserving
U(1) symmetries can also be exploited in systems that originally break such symmetries (projected purification). Using this method we could efficiently model mesoscopic phenomena such as a charging energy controlled by a gate electrode without further approximations. Equipped with this techniques we studied out-of equilibrium spectral functions to explore how to identify superconducting correlations more reliably on ultra-short timescales. We found conclusive evidence that in particular two-particle spectral functions yield excellent probes for the formation of a (quasi-)condensate out-of equilibrium. Furthermore, we also investigated the question whether in a particular model system there is the possibility of true long-range order out-of equilibrium by studying correlation matrices and the scaling of their eigenvalues. Here, we observe a change in the algebraic decay of the correlations, even though the extrapolated order parameter is still zero within the error bounds. Furthermore, we also investigated the effects of coupling a superconductor-semiconductor heterostructure, which is subject to an in-plane magnetic field and a charging energy controlled by a gate voltage, to normal leads. In the context of experimentally verifying the existence of Majorana edge modes, such systems are believed to be the most promising and recent studies seem to underline this expectation. However, in order to consistently analyze the experimental data, the effects of quantum fluctuations caused by hybridization of the heterostructure with the leads have to be understood. Here, only perturbative limits are available so far, i.e., the weak and strong tunneling limit, while the experimentally relevant regime is expected to be somewhere inbetween. We aimed to fill this gap using the projected purification method to calculate the ground state phase diagram over a wide parameter regime. Our results indicate that the experimental situation is much more involved than what is predicted from perturbative analysis.
A general theoretical formulation for the effect of a strong on-site Coulomb interaction on the time-dependent electron transport through a quantum dot under the influence of arbitrary time-varying bias voltages and/or external fields is presented, based on slave bosons and the Keldysh nonequilibrium Green's function (GF) techniques. To avoid the difficulties of computing double-time GFs, we generalize the propagation scheme recently developed by Croy and Saalmann to combine the auxiliary-mode expansion with the celebrated Lacroix's decoupling approximation in dealing with the second-order correlated GFs and then establish a closed set of coupled equations of motion, called second-order quantum rate equations (SOQREs), for an exact description of transient dynamics of electron correlated tunneling. We verify that the stationary solution of our SOQREs is able to correctly describe the Kondo effect on a qualitative level. Moreover, a comparison with other methods, such as the second-order von Neumann approach and Hubbard-I approximation, is performed. As illustrations, we investigate the transient current behaviors in response to a step voltage pulse and a harmonic driving voltage, and linear admittance as well, in the cotunneling regime.
Dynamical processes induced by the external time-dependent fields can provide valuable insight into the characteristic energy scales of a given physical system. We investigate them here in a nanoscopic heterostructure, consisting of the double quantum dot coupled in series to the superconducting and the metallic reservoirs, analyzing its response to (i) abrupt bias voltage applied across the junction, (ii) sudden change of the energy levels, and imposed by (iii) their periodic driving. We explore subgap properties of this setup which are strictly related to the in-gap quasiparticles and discuss their signatures manifested in the time-dependent charge currents. The characteristic multi-mode oscillations, their beating patters and photon-assisted harmonics reveal a rich spectrum of dynamical features that might be important for designing the superconducting qubits.
Non-local conductance experiments using tunnel junctions can provide valuable spectroscopic information on both the transport and relaxation of quasiparticles in superconductors, as these techniques directly probe the quasiparticle charge and energy imbalance even at mK temperatures. In this work, we employ mesoscopic three terminal Cu and Al NIS devices to study non-local quasiparticle transport over length-scales on the order of the superconducting coherence length in this regime. Via a dual-bias scheme, which utilizes detector biases both above and below the superconducting gap, we are able to extract the effect of quasiparticle energy imbalance via its impact on the self consistent pair potential by symmetry considerations. We observe non-local conductance features due to pair-breaking which are anti-symmetric with respect to the polarity of the voltage bias, with a sharp onset during single electron tunneling at energies around
3Δ. We compare these findings with quasiclassical simulations including inelastic effects to obtain estimates of the energy dependent inelastic scattering time. In addition, we demonstrate kinetic effects due to a large applied supercurrent which can also be captured in this formalism and decomposed with respect to the particle-hole symmetry and supercurrent direction, and discuss further opportunities for the advancement of this method.