Solution of master equations by fermionic-duality: Time-dependent charge and heat currents through an interacting quantum dot proximized by a superconductor

TL;DR

This paper presents an analytical solution for time-dependent charge and heat currents in an interacting quantum dot coupled to a superconductor, using fermionic duality to simplify the master equations and analyze non-equilibrium transport phenomena.

Contribution

It introduces a systematic approach exploiting fermionic duality from the start to solve master equations for complex quantum transport systems, including non-local observables and strong interactions.

Findings

Analytical time-dependent solution for quantum dot transport

Duality relates system evolution to a dual system with inverted parameters

Understanding of relaxation towards stationary state in complex systems

Abstract

We analyze the time-dependent solution of master equations by exploiting fermionic duality, a dissipative symmetry applicable to a large class of open systems describing quantum transport. Whereas previous studies mostly exploited duality relations after partially solving the evolution equations, we here systematically exploit the invariance under the fermionic duality mapping from the very beginning when setting up these equations. Moreover, we extend the resulting simplifications -- so far applied to the local state evolution- to non-local observables such as transport currents. We showcase the exploitation of fermionic duality for a quantum dot with strong interaction -- covering both the repulsive and attractive case -- proximized by contact with a large-gap superconductor which is weakly probed by charge and heat currents into a wide-band normal-metal electrode. We derive the complete time-dependent analytical solution of this problem involving non-equilibrium Cooper pair transport, Andreev bound states and strong interaction. Additionally exploiting detailed balance we show that even for this relatively complex problem the evolution towards the stationary state can be understood analytically in terms of the stationary state of the system itself via its relation to the stationary state of a dual system with inverted Coulomb interaction, superconducting pairing and applied voltages.