Cooper Pair Spectroscopy for Odd-Frequency Superconductors

Abstract

In conventional spin-singlet superconductors with even parity, the anomalous pairing amplitude is even in frequency. Odd-frequency superconductivity is an unconventional superconducting state in which the pair amplitude is odd under exchange of time coordinates. Since superconducting pairing is encoded in two-particle correlations, double photoelectron spectroscopy provides a natural framework for probing such states through the simultaneous detection of two photoemitted electrons. When applied to superconducting samples, this approach is commonly referred to as Cooper pair spectroscopy. In this thesis, a Green’s-function formulation of Cooper pair spectroscopy for odd-frequency superconductors is developed by extending the theoretical structure of angle-resolved photoemission spectroscopy to the doublephotoemission case. Starting from an expansion of the S-matrix, the double-photoelectron counting rate is derived to second order, which is the lowest order giving a non-vanishing contribution. The result separates into an anomalous contribution, expressed through greater anomalous Green’s functions, and a normal contribution, expressed through lesser normal Green’s functions. The anomalous part describes correlated emission associated with a single Cooper pair, while the normal part describes contributions from ordinary single-particle correlations. The main result is that odd-frequency pairing leads to a different momentum structure in the anomalous contribution than conventional even-frequency pairing. In particular, the odd-frequency kernel contains a difference of screened Coulomb potentials, whereas the even-frequency case contains their sum. Consequently, for a symmetric screened Coulomb potential, the momentum dependence of the electron– photon coupling becomes essential for obtaining a finite odd-frequency anomalous contribution. The derived expression therefore identifies a key formal distinction between even- and odd-frequency superconductors in double photoelectron spectroscopy and provides a basis for future numerical and experimental investigations.