Thesis presented December 15, 2022

Abstract:
This work aims to extend the comprehension of the out-of-equilibrium transport in Josephson junctions by both analytical and numerical methods. More specifically, it focuses on the Josephson radiation of a junction formed by a spin degenerate single level quantum dot connected to two superconducting leads. Such a junction hosts discrete subgap states whose energy depends periodically on the superconducting phase difference across the junction, they are the so-called Andreev bound states. Because of the energy dependence of the quantum dot transmission, these states are completely detached from the quasiparticle continuum and any finite detuning of the quantum dot from resonance conditions opens a second gap at the Fermi energy. The superconducting current flowing through the junction is proportional to the derivative of the junction energy with respect to the phase difference. Crucially, it depends on the Andreev bound states occupations. When a constant voltage is set across the junction, the phase difference oscillates at the Josephson frequency, which is proportional to the applied voltage.
Thus, a voltage bias can induce non-adiabatic changes of the Andreev bound states occupation. To investigate the consequences of this dynamics, we proposed a stochastic model of the Andreev bound states occupation that permits to analytically evaluate the current through the junction and its fluctuations. It predicted the existence of a parameter regime where the Josephson radiation is fractional. While those results provided analytical insights into the junction behaviour, strong assumptions were required. Thus, we turned to a microscopic description that models the system as a non-interacting quantum dot hosting a unique spin degenerate level which is tunnel coupled to two BCS superconducting leads. The current is then deduced from the full Green function, which is obtained by solving the Dyson equation. We developed a novel method to solve this equation in the time domain. Its complexity is O(N log(N)) in both operation and memory, where the time axis has been discretized into N time steps. By contrast, the usual time domain method requires O(N³) operations and O(N²) bytes of memory to solve the Dyson equation. This new method is not restrained in any way to the study of Josephson junctions, it can be used to solve any Dyson equations.

Keywords:
Kondo effect, Out of equilibrium, fractional josephson effect, josephson junction, Dyson equation

On-line thesis.