In this thesis, sub-gap states in bottom-gated InAs N–dot–S, N–double
dot–S, and N–dot–S–dot–N devices are investigated, and several different
theories are developed to model these states. Experimental results
include tracking single levels of the dot in an N–dot–S device as the
tunnel couplings are tuned electrostatically. This includes tuning the
odd occupation of the dot through a quantum phase transition, where
it forms a singlet with excitations in the superconductor. We detail the
fabrication of these bottom gated devices, which additionally feature
ancillary sensor dots connected with floating gates.
A numerical technique is developed, which predicts the position of
Yu-Shiba-Rusinov sub-gap states in the proximitized Anderson model
as well as properties of these states. This theory is valid for all occupations
of the dot and for weak to intermediate coupling. We compare it
to the Numerical Renormalization Group (NRG) process.
The thesis also details an implementation of the NRG process,
which was written for this project and includes an original method
for mapping the discretized hybridization hamiltonian to a chain. We
take significant steps towards a justification of the NRG process, which
is based on the general properties of Krylov subspaces alone, and is
thus not tied to a specific physical system.