# Time-dependent Andreev bound states of a quantum dot coupled to two superconducting leads

## TL;DR

This paper theoretically investigates the time-dependent behavior of Andreev bound states and transport properties in a quantum dot coupled to superconducting and normal leads, revealing oscillations, damping, and time delays in conductance development.

## Contribution

It provides analytical expressions and insights into the transient dynamics and time delays in Andreev state formation in a quantum dot system with superconducting leads.

## Key findings

- Oscillations in currents governed by QD-superconducting coupling
- Damping due to relaxation from the normal lead
- Finite time delay for Andreev state development

## Abstract

Sub-gap transport properties of a quantum dot (QD) coupled to two superconducting and one metallic leads are studied theoretically, solving the time-dependent equation of motion by the Laplace transform technique. We focus on time-dependent response of the system induced by a sudden switching on the QD-leads couplings, studying the influence of initial conditions on the transient currents and the differential conductance. We derive analytical expressions for measurable quantities and find that they oscillate in time with the frequency governed by the QD-superconducting lead coupling and acquire damping, due to relaxation driven by the normal lead. Period of these oscillations increases with the superconducting phase difference $\phi$. In particular, for $\phi=\pi$ the QD occupancy and the normal current evolve monotonically (without any oscillations) to their stationary values. In such case the induced electron pairing vanishes and the superconducting current is completely blocked. We also analyze time-dependent development of the Andreev bound states. We show, that the measurable conductance peaks do not appear immediately after sudden switching of the QD coupling to external leads but it takes some finite time-interval for the system needs create these Andreev states. Such time-delay is mainly controlled by the QD-normal lead coupling.