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Time Evolution in Superconducting Nanostructures

Lukáš Soták

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

This work focuses on the time evolution of a quantum dot coupled to superconduct- ing leads following sudden parameter changes (quenches). The system is modeled using a superconducting Anderson impurity framework reformulated through chain expansion. We simulate the system dynamics and identify three regimes: full relaxation, persistent oscillations, and oscillations centered around the expected equilibrium value. We show that relaxation strongly depends on the local density of states on the quantum dot corre- sponding to the post-quench system configuration. While relaxation is suppressed within the superconducting gap, it can reappear at specific energies due to Andreev bound states. 1

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This work focuses on the time evolution of a quantum dot coupled to superconduct- ing leads following sudden parameter changes (quenches). The system is modeled using a superconducting Anderson impurity framework reformulated through chain expansion. We simulate the system dynamics and identify three regimes: full relaxation, persistent oscillations, and oscillations centered around the expected equilibrium value. We show that relaxation strongly depends on the local density of states on the quantum dot corre- sponding to the post-quench system configuration. While relaxation is suppressed within the superconducting gap, it can reappear at specific energies due to Andreev bound states. 1
We introduce a versatile and scalable framework for constructing effective models of superconducting (SC) nanostructures described by the generalized SC Anderson impurity model with multiple quantum dots and leads. Our chain expansion (ChE) method maps each SC lead onto a finite tight-binding chain with parameters obtained from Pad\'e approximants of the tunneling self-energy. We provide an explicit algorithm for the general case as well as simple analytical expressions for the chain parameters in the wide-band and infinite-chain limits. This mapping preserves low-energy physics while enabling efficient simulations: short chains are tractable using exact diagonalization, and longer ones are handled with density matrix renormalization group methods. The approach remains reliable and computationally efficient across diverse geometries, both in and out of equilibrium. We use ChE to map the ground-state phase diagrams of double, triple, and quadruple quantum dots coupled to a single SC lead. While half-filled symmetric systems show similar overall diagrams, the particular phases differ substantially with the dot number. Here, large parameter regions are entirely missed by the widely used zero-bandwidth approximation but are captured by ChE. Away from half-filling, additional dots markedly increase diagram complexity, producing a rich variety of stable phases. These results demonstrate ChE as a fast, accurate, and systematically improvable tool for exploring complex SC nanostructures.
We study the quench dynamics of an Anderson impurity model using the configuration interaction (CI) method. In particular, we focus on the relaxation behavior of the impurity occupation. The system is found to behave very differently in the weak-coupling and strong-coupling regimes. In the weak-coupling regime, the impurity occupation relaxes to a time-independent constant quickly after only a few oscillations. In the strong-coupling regime, the impurity occupation develops a fast oscillation, with a much slower relaxation. We show that it is the multipeak structure in the many-body energy spectrum that separates these two regimes. The characteristic behavior, including the power-law decay and the period of oscillation, can also be related to certain features in the many-body energy spectrum. The respective advantages of several impurity solvers are discussed, and the convergence of different CI truncation schemes is provided.
We theoretically explore the non-equilibrium dynamics of a single quantum dot system coupled to both ferromagnetic and superconducting electrodes. To investigate its time evolution, we utilize the time-dependent numerical renormalization group technique, which captures the system's response to abrupt parameter changes in a fully non-perturbative manner. Our analysis focuses on dynamics following a sudden modification in the couplings to the leads or a shift of the orbital level. In particular, we calculate the time evolution of the induced local superconducting pairing correlations and magnetization. In this context, the relevant energy spectra are examined. Moreover, we study the behavior of the Loschmidt echo and the return function to shed light on the signatures of dynamical quantum phase transitions. The determined dependencies reveal non-trivial competition between relevant correlations, involving superconducting pairing and ferromagnetic-contacted induced exchange field, and deepen our understanding of nanoscale hybrid systems' dynamical behavior.
We study the time evolution and steady state of the charge current in a single-impurity Anderson model, using matrix product states techniques. A nonequilibrium situation is imposed by applying a bias voltage across one-dimensional tight-binding leads. Focusing on particle-hole symmetry, we extract current-voltage characteristics from universal low-bias up to high-bias regimes, where band effects start to play a dominant role. We discuss three quenches, which after strongly quench-dependent transients yield the same steady-state current. Among these quenches we identify those favorable for extracting steady-state observables. The period of short-time oscillations is shown to compare well to real-time renormalization group results for a simpler model of spinless fermions. We find indications that many-body effects play an important role at high-bias voltage and finite bandwidth of the metallic leads. The growth of entanglement entropy after a certain time scale $\ensuremath{\propto}{\ensuremath{\Delta}}^{\ensuremath{-}1}$ is the major limiting factor for calculating the time evolution. We show that the magnitude of the steady-state current positively correlates with entanglement entropy. The role of high-energy states for the steady-state current is explored by considering a damping term in the time evolution.
We study the relaxation dynamics of a quantum dot with local Coulomb correlations coupled to two noninteracting leads which are held in grand-canonical equilibrium. Only charge degrees of freedom are considered and the dot is described by a model which in the scaling limit becomes equivalent to the interacting resonant level model. We examine the time evolution of the current and the dot occupancy resulting from changes in the dot-lead coupling, the dots' on-site energy, or the charging energy. Abrupt and smooth parameter changes as well as setups with and without driving bias voltage are considered. For biased dots, we investigate the often studied response after turning on the dot-lead coupling but also the experimentally more relevant case in which the voltage is turned on. We identify and explain a variety of interesting many-body effects, and we clarify the role of initial correlations.
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK{T}_{K}TK​ (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt{T}_{K}tTK​t. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.