Tunneling conductance due to discrete spectrum of Andreev states

Abstract: We study tunneling spectroscopy of discrete subgap Andreev states in a superconducting system. If the tunneling coupling is weak, individual levels are resolved and the conductance $G(V)$ at small temperatures is composed of a set of resonant Lorentz peaks which cannot be described within perturbation theory over tunnelling strength. We establish a general formula for their widths and heights and show that the width of any peak scales as normal-state tunnel conductance, while its height is $\lesssim 2e^2/h$ and depends only on contact geometry and the spatial profile of the resonant Andreev level. We also establish an exact formula for the single-channel conductance that takes the whole Andreev spectrum into account. We use it to study the interference of Andreev reflection processes through different levels. The effect is most pronounced at low voltages, where an Andreev level at $E_j$ and its conjugate at $-E_j$ interfere destructively. This interference leads to the quantization of the zero-bias conductance: G(0) equals $2e^2/h$ (or 0) if there is (there is not) a Majorana fermion in the spectrum, in agreement with previous results from $S$-matrix theory. We also study $G(eV>0)$ for a system with a pair of almost decoupled Majorana fermions with splitting $E_0$ and show that at lowest $E_0$ there is a zero-bias Lorentz peak of width $W$ as expected for a single Majorana fermion (a topological NS-junction) with a narrow dip of width $E_0^2/W$ at zero bias, which ensures $G(0)=0$ (the NS-junction remains trivial on a very small energy scale). As the coupling $W$ gets stronger, the dip becomes narrower, which can be understood as enhanced decoupling of the remote Majorana fermion. Then the zero-bias dip requires extremely low temperatures $T\lesssim E_0^2/W$ to be observed.